Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013 Article ID 490673 9 pageshttpdxdoiorg1011552013490673

Research ArticleNonlocal Problems for Fractional Differential Equations viaResolvent Operators

Zhenbin Fan1 and Gisegravele Mophou2

1 Department of Mathematics Changshu Institute of Technology Suzhou Jiangsu 215500 China2 Laboratoire CEREGMIA Universite des Antilles et de la Guyane Campus FouilloleGuadeloupe (FWI) 97159 Pointe-a-Pitre France

Correspondence should be addressed to Gisele Mophou gmophouuniv-agfr

Received 7 April 2013 Accepted 15 May 2013

Academic Editor Sotiris Ntouyas

Copyright copy 2013 Z Fan and G Mophou This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchyoperator by means of the analytic resolvent method Based on this result we derive the existence of mild solutions for nonlocalfractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compactrespectively An example is also given to illustrate our theory

1 Introduction

In this paper we are concerned with the existence of mildsolutions for a fractional differential equation with nonlocalconditions of the form

119863120572

119906 (119905) = 119860119906 (119905) + 1198691minus120572

119905119891 (119905 119906 (119905)) 0 lt 119905 le 119887

119906 (0) = 1199060minus 119892 (119906)

(1)

where 119863120572 is the Caputo fractional derivative of order 120572 with

0 lt 120572 lt 1 119860 119863(119860) sub 119883 rarr 119883 is the infinitesimal generatorof a resolvent 119878

120572(119905) 119905 ge 0 119883 is a real Banach space endowed

with the norm sdot and 119891 and 119892 are appropriate continuousfunctions to be specified later

The theory of fractional differential equations hasreceived much attention over the past twenty years sincethey are important in describing the natural models such asdiffusion processes stochastic processes finance and hydrol-ogy Many notions associated with resolvent are developedsuch as integral resolvent solution operators 120572-resolventoperator functions (119886 119896)-regularized resolvent and 120572-orderfractional semigroups All of these notions play a central rolein the study of Volterra equations especially the fractionaldifferential equations Concerning the literature we refer the

reader to the books [1 2] the recent papers [3ndash20] and thereferences therein

On the other hand abstract differential equations withnonlocal conditions have also been studied extensively in theliterature since it is demonstrated that the nonlocal problemshave better effects in applications than the classical ones Itwas Byszewski and Lakshmikantham [21] who first studiedthe existence and uniqueness of mild solutions for nonlocaldifferential equations And the main difficulty in dealingwith the nonlocal problem is how to get the compactness ofsolution operator at zero especially when the nonlocal itemis only assumed to be Lipschitz continuous or continuousMany authors developed different techniques and methodsto solve this problem For more details on this topic we referto [10 11 22ndash33] and references therein

In this paper we combine the above two directionsand study the nonlocal fractional differential equation (1)governed by operator 119860 generating an analytic resolventA standard approach in deriving the mild solution of (1)is to define the solution operator 119876 Then conditions aregiven such that some fixed point theorems such as Browderrsquosand Schauderrsquos fixed point theorems can be applied to geta fixed point for solution operator 119876 which gives rise to amild solution of (1) The key step of using this approach is

2 International Journal of Differential Equations

to prove the compactness of Cauchy operator 119866 associatedwith solution operator 119876 When the operator 119860 generates acompact semigroup it is well known that theCauchy operator119866 is also compact However to the best of our knowledgeit is unknown when 119860 generates a compact resolvent Themain difficulty of this problem lies in the fact that there is noproperty of semigroups for resolvent

To this end wewill first discuss the continuity of resolventin the uniform operator topology in this paper In fact weprove that the compact analytic resolvent is continuous inthe uniform operator topology Based on this result we canprove the compactness of Cauchy operator As a consequencewe obtain the existence of mild solutions for (1) when thenonlocal item is Lipschitz continuous At the same time wealso derive the existence of mild solutions for (1) withoutthe Lipschitz or compact assumption on the nonlocal item119892 by using the techniques developed in [24 30] Actuallywe only assume that 119892 is continuous on 119862([0 119887] 119883) and 119892 iscompletely determined on [120575 119887] for some small 120575 gt 0 or 119892

is continuous on 119862([0 119887] 119883) with 1198711

([0 119887] 119883) topology (seeCorollaries 17ndash19)

This paper has four sections In Section 2 we recallsome definitions on Caputo fractional derivatives analyticresolvent and mild solutions to (1) In Section 3 we provethe compactness of Cauchy operator Finally in Section 4we establish the existence of mild solutions of (1) when thenonlocal item satisfies different conditions An example isalso given in this section

2 Preliminaries

Throughout this paper let 119887 gt 0 be fixed and let N Rand R

+be the set of positive integers real numbers and

nonnegative real numbers respectively We denote by 119883 theBanach space with the norm sdot 119862([0 119887] 119883) the space ofall 119883-valued continuous functions on [0 119887] with the norm119906 = sup119906(119905) 119905 isin [0 119887] and 119871

119901

([0 119887] 119883) the space of119883-valued Bochner integrable functions on [0 119887] with the norm119891119871119901 = (int

119887

0

119891(119905)119901d119905)1119901 where 1 le 119901 lt infin Also we denote

by L(119883) the space of bounded linear operators from 119883 into119883 endowed with the norm of operators

Now let us recall some basic definitions and results onfractional derivative and fractional differential equations

Definition 1 (see [1]) The fractional order integral of thefunction 119891 isin 119871

1

([0 119887] 119883) of order 120572 gt 0 is defined by

119869120572

119905119891 (119905) =

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) d119904 (2)

where Γ is the Gamma function

Definition 2 (see [1]) TheRiemann-Liouville fractional orderderivative of order 120572 of a function 119891 isin 119871

1

([0 119887] 119883) given onthe interval [0 119887] is defined by

119863120572

119871119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0

(119905 minus 119904)119899minus120572minus1

119891 (119904) d119904 (3)

where 120572 isin (119899 minus 1 119899] 119899 isin N

Definition 3 (see [1]) The Caputo fractional order derivativeof order 120572 of a function119891 isin 119871

1

([0 119887] 119883) given on the interval[0 119887] is defined by

119863120572

119891 (119905) =1

Γ (119899 minus 120572)int

119905

0

(119905 minus 119904)119899minus120572minus1

119891(119899)

(119904) d119904 (4)

where 120572 isin (119899 minus 1 119899] 119899 isin N

In the remainder of this paper we always suppose that 0 lt

120572 lt 1 and 119860 is a closed and densely defined linear operatoron 119883

Definition 4 A family 119878120572(119905)119905ge0

sube L(119883) of bounded linearoperators in 119883 is called a resolvent (or solution operator)generating by 119860 if the following conditions are satisfied

(S1) 119878120572(119905) is strong continuous on R

+and 119878120572(0) = 119868

(S2) 119878120572(119905)119863(119860) sube 119863(119860) and 119860119878

120572(119905)119909 = 119878

120572(119905)119860119909 for all 119909 isin

119863(119860) and 119905 ge 0(S3) the resolvent equation holds

119878120572(119905) 119909 = 119909 + int

119905

0

119892120572(119905 minus 119904) 119860119878

120572(119904) 119909d119904 forall119909 isin 119863 (119860) 119905 ge 0

(5)

For 120596 120579 isin R let

sum(120596 120579) = 120582 isin C 1003816100381610038161003816arg (120582 minus 120596)

1003816100381610038161003816 lt 120579 (6)

Definition 5 A resolvent 119878120572(119905) is called analytic if the

function 119878120572(sdot) R

+rarr L(119883) admits analytic extension

to a sector sum(0 1205790) for some 0 lt 120579

0le 1205872 An analytic

resolvent 119878120572(119905) is said to be of analyticity type (120596

0 1205790) if for

each 120579 lt 1205790and 120596 gt 120596

0there is 119872

1= 1198721(120596 120579) such that

119878(119911) le 1198721119890120596Re 119911 for 119911 isin sum(0 120579) where Re 119911 denotes the

real part of 119911

Definition 6 A resolvent 119878120572(119905) is called compact for 119905 gt 0 if

for every 119905 gt 0 119878120572(119905) is a compact operator

According to Proposition 12 in [2] we can give thefollowing definition of mild solutions for (1)

Definition 7 A function 119906 isin 119862([0 119887] 119883) is called a mildsolution of fractional evolution equation (1) if it satisfies

119906 (119905) = 119878120572(119905) [1199060minus 119892 (119906)]

+ int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119906 (119904)) d119904 0 le 119905 le 119887

(7)

for every 1199060isin 119883

Next we introduce the Hausdorff measure of noncom-pactness 120573(sdot) defined on each bounded subset Ω of Banachspace 119884 by

120573 (Ω) = inf 120576 gt 0 Ω has a finite 120576-net in 119884 (8)

Some basic properties of 120573(sdot) are given in the followinglemma

International Journal of Differential Equations 3

Lemma 8 (see [34]) Let 119884 be a real Banach space and let119861 119862 sube 119884 be bounded then the following properties aresatisfied

(1) 119861 is precompact if and only if 120573(119861) = 0

(2) 120573(119861) = 120573(119861) = 120573(conv 119861) where 119861 and conv 119861 meanthe closure and convex hull of 119861 respectively

(3) 120573(119861) le 120573(119862) when 119861 sube 119862

(4) 120573(119861 + 119862) le 120573(119861) + 120573(119862) where 119861 + 119862 = 119909 + 119910 119909 isin

119861 119910 isin 119862

(5) 120573(119861 cup 119862) le max120573(119861) 120573(119862)

(6) 120573(120582119861) = |120582|120573(119861) for any 120582 isin R

(7) if the map119876 119863(119876) sube 119884 rarr 119885 is Lipschitz continuouswith constant 119896 then 120573(119876119861) le 119896120573(119861) for any boundedsubset 119861 sube 119863(119876) where 119885 is a Banach space

The map 119876 119882 sube 119884 rarr 119884 is said to be a 120573-contractionif there exists a positive constant 119896 lt 1 such that 120573(119876119862) le

119896120573(119862) for any bounded closed subset 119862 sube 119882 where 119884 is aBanach space

Lemma 9 ([34] Darbo-Sadovskii) If 119882 sube 119884 is boundedclosed and convex the continuous map 119876 119882 rarr 119882 is a120573-contraction then the map 119876 has at least one fixed point in119882

3 Compactness of Cauchy Operators

Let Cauchy operator 119866 119862([0 119887] 119883) rarr 119862([0 119887] 119883) be def-ined by

(119866119891) (119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904 119905 isin [0 119887] (9)

If 119878120572(119905) is a compact 119862

0-semigroup it is well known that

119866 is compact However it is unknown in case of compactresolvent The main difficulty is that the resolvent does nothave the property of semigroups Thus it seems to be morecomplicated to prove the compactness of Cauchy operatorHere we will first discuss the continuity of resolvent in theuniform operator topology Then we can give the positiveanswer to the above problem

In the remainder of this paper we always assume that119872 = sup

119905isin[0119887]119878120572(119905) lt +infin

Lemma 10 Suppose 119878120572(119905) is a compact analytic resolvent of

analyticity type (1205960 1205790) Then the following are hold

(i) limℎrarr0

119878120572(119905 + ℎ) minus 119878

120572(119905) = 0 for 119905 gt 0

(ii) limℎrarr0

+119878120572(119905 + ℎ) minus 119878

120572(ℎ)119878120572(119905) = 0 for 119905 gt 0

(iii) limℎrarr0

+119878120572(119905) minus 119878

120572(ℎ)119878120572(119905 minus ℎ) = 0 for 119905 gt 0

Proof (i) Let 119878120572(119905) be an analytic resolvent of analyticity type

(1205960 1205790) and let 119905 gt 0 be given Then by means of Cauchy

integral formula we have

1198781015840

120572(119905) =

1

2120587119894int|119911minus119905|=119905 sin 120579

119878120572(119911)

(119911 minus 119905)2d119911 (10)

Thus for any 120596 gt 1205960 0 lt 120579 lt 120579

0 there exists a constant

1198721= 1198721(120596 120579) such that

100381710038171003817100381710038171198781015840

120572(119905)

10038171003817100381710038171003817le 1198721(2120587)minus1

int

120587

minus120587

119890120596(119905+119905 sin 120579 cos 120574)

(119905 sin 120579)minus1d120574

le 11987211198902120596119905

(119905 sin 120579)minus1

(11)

Now let 119909 le 1 and |ℎ| lt 119905 It follows from (11) that thereexists a constant 1198721015840 such that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(119905) 119909

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

int

119905+ℎ

119905

1198781015840

120572(119904) 119909 d119904

100381710038171003817100381710038171003817100381710038171003817

le 1198721015840

119909

100381610038161003816100381610038161003816100381610038161003816

int

119905+ℎ

119905

119904minus1 d119904

100381610038161003816100381610038161003816100381610038161003816

le 1198721015840

|ln (119905 + ℎ) minus ln 119905| 997888rarr 0 as ℎ 997888rarr 0

(12)

which implies that 119878120572(119905) is continuous in the uniform opera-

tor topology for 119905 gt 0 that is

limℎrarr0

1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878120572(119905)

1003817100381710038171003817 = 0 for 119905 gt 0 (13)

(ii) Let 119909 le 1 119905 gt 0 and 120576 gt 0 be given Since119878120572(119905) is compact the set 119882

119905= 119878

120572(119905)119909 119909 le 1 is

also compact Thus there exists a finite family 119878120572(119905)1199091

119878120572(119905)1199092 119878

120572(119905)119909119898 sub 119882

119905such that for any 119909 with 119909 le 1

there exists 119909119894(1 le 119894 le 119898) such that

1003817100381710038171003817119878120572 (119905) 119909 minus 119878120572(119905) 119909119894

1003817100381710038171003817 le120576

3 (119872 + 1) (14)

From the strong continuity of 119878120572(119905) 119905 ge 0 there exists 0 lt

ℎ1lt min119905 119887 such that

1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817 le120576

3 forall0 le ℎ le ℎ

1 1 le 119894 le 119898

(15)

On the other hand from (i) there exists 0 lt ℎ2

lt min119905 119887such that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(119905) 119909

1003817100381710038171003817 le120576

3 forall0 le ℎ le ℎ

2 119909 le 1

(16)

4 International Journal of Differential Equations

Thus for 0 le ℎ le minℎ1 ℎ2 and 119909 le 1 it follows from

(14)ndash(16) that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+ (119872 + 1)1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

le120576

3+

120576

3+

120576

3

le 120576

(17)

which implies that limℎrarr0

+119878120572(119905+ℎ)minus119878

120572(ℎ)119878120572(119905) = 0 for all

119905 gt 0(iii) Let 119905 gt 0 and 0 lt ℎ lt min119905 119887 Then there exists

119872 gt 0 such that

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) minus 119878

120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+ 1198721003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 minus ℎ)

1003817100381710038171003817

(18)

It follows from (i) and (ii) that

limℎrarr0

+

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817 = 0 for 119905 gt 0 (19)

This completes the proof

Lemma 11 Suppose 119878120572(119905) is a compact analytic resolvent of

analyticity type (1205960 1205790) Then Cauchy operator 119866 defined by

(9) is a compact operator

Proof We will show 119866 119862([0 119887] 119883) rarr 119862([0 119887] 119883) iscompact by using the Arzela-Ascoli theorem Let 119863

119897=

119891 isin 119862([0 119887] 119883) 119891 le 119897 be any bounded subset of119862([0 119887] 119883)

First we claim that the set 119866119863119897is equicontinuous on

119862([0 119887] 119883) In fact let 0 le 1199051

le 1199052

le 119887 and 119891 isin 119863119897 then

we have

119868 =1003817100381710038171003817(119866119891) (119905

2) minus (119866119891) (119905

1)1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199052

0

119878120572(1199052minus 119904) 119891 (119904) d119904 minus int

1199051

0

119878120572(1199051minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

1003817100381710038171003817119891 (119904)1003817100381710038171003817 d119904

+ int

1199052

1199051

1003817100381710038171003817119878120572 (1199052 minus 119904) 119891 (119904)1003817100381710038171003817 d119904

le 119897 int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

(20)

If 1199051= 0 it is easy to see that

lim1199052rarr0

119868 = 0 uniformly for 119891 isin 119863119897 (21)

If 0 lt 1199051lt 119887 for 0 lt 120575 lt 119905

1 we have

119868 le 119897 int

1199051minus120575

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

1199051

1199051minus120575

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

le 119887119897 sup119904isin[01199051minus120575]

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

+ 2119872119897120575 + 119872119897 (1199052minus 1199051)

(22)

Note that from Lemma 10(i) we know 119878120572(119905) is operator norm

continuous uniformly for 119905 isin [120575 119887] Combining this andthe arbitrariness of 120575 with the above estimation on 119868 we canconclude that

lim|1199051minus1199052|rarr 0

119868 = 0 uniformly for 119891 isin 119863119897 (23)

Thus 119866119863119897is equicontinuous on 119862([0 119887] 119883)

Next we will show that the set (119866119891)(119905) 119891 isin 119863119897 is

precompact in 119883 for every 119905 isin [0 119887] It is easy to see that theset (119866119891)(0) 119891 isin 119863

119897 is precompact in 119883 Now let 0 lt 119905 le 119887

be given and 0 lt 120576 lt 119905 Then

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 119891 isin 119863

119897 (24)

International Journal of Differential Equations 5

is precompact since 119878120572(120576) is compact Moreover for arbitrary

120576 lt 120575 lt 119905 we have10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le 119897 int

119905minus120576

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

119905minus120576

119905minus120575

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904 + 119897120575 (1198722

+ 119872)

(25)

From Lemma 10(iii) we know

119878120572(120576) 119878120572(119905 minus 119904 minus 120576) minus 119878

120572(119905 minus 119904) 997888rarr 0

as 120576 997888rarr 0 for 119904 isin [0 119905 minus 120575]

(26)

Then it follows from the Lebesgue dominated convergencetheorem and the arbitrariness of 120575 that

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(27)

On the other hand10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+ 119897120576119872

(28)

Thus

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(29)

which implies that (119866119891)(119905) 119891 isin 119863119897 is precompact in 119883 by

using the total boundedness and therefore 119866 is compact inview of Arzela-Ascoli theorem

4 Nonlocal Problems

In this section we always assume that 1199060

isin 119883 and that theoperator 119860 generates a compact analytic resolvent 119878

120572(119905) of

analyticity type (1205960 1205790) and we will prove the existence of

mild solutions of (1) when the nonlocal item 119892 is assumed tobe Lipschitz continuous and neither Lipschitz nor compactrespectively

Let 119903 be a fixed positive real number and

119882119903= 119906 isin 119862 ([0 119887] 119883) 119906 le 119903 (30)

Clearly119882119903is a bounded closed and convex set We make the

following assumptions

(H1) 119891 [0 119887] times 119883 rarr 119883 is continuous

(H2) 119892 119862([0 119887] 119883) rarr 119883 is Lipschitz continuous withLipschitz constant 119896 such that 119872119896 lt 1

Under these assumptions we can prove the first mainresult in this paper

Theorem 12 Assume that conditions (H1) and (H2) aresatisfied Then the nonlocal problem (1) has at least one mildsolution provided that

119872[10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817] le 119903

(31)

Proof We consider the solution operator 119876 119862([0 119887] 119883) rarr

119862([0 119887] 119883) defined by

(119876119906) (119905) = 119878120572(119905) [1199060minus 119892 (119906)] + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119906 (119904)) d119904

= (1198761119906) (119905) + (119876

2119906) (119905) 119905 isin [0 119887]

(32)

It is easy to see that the fixed point of 119876 is the mild solutionof nonlocal Cauchy problem (1) Subsequently we will provethat119876 has a fixed point by using Lemma 9 (Darbo-Sadovskiirsquosfixed point theorem)

Firstly we prove that the mapping 119876 is continuous on119862([0 119887] 119883) For this purpose let 119906

119899119899ge1

be a sequence in119862([0 119887] 119883) with lim

119899rarrinfin119906119899= 119906 in 119862([0 119887] 119883) Then

1003817100381710038171003817119876119906119899minus 119876119906

1003817100381710038171003817

le 119872[1003817100381710038171003817119892 (119906119899)minus119892 (119906)

1003817100381710038171003817 + 119887 sup119904isin[0119887]

1003817100381710038171003817119891 (119904 119906119899(119904))minus119891 (119904 119906 (119904))

1003817100381710038171003817]

(33)

By the continuity of 119891 and 119892 we deduce that119876 is continuouson 119862([0 119887] 119883)

6 International Journal of Differential Equations

Secondly we claim that 119876119882119903sube 119882119903 where 119882

119903is defined

by (30) In fact for any 119906 isin 119882119903 by (31) we have

119876119906 le 119872(10038171003817100381710038171199060

1003817100381710038171003817 +1003817100381710038171003817119892 (119906)

1003817100381710038171003817 + int

119887

0

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817 d119904)

le 119872(10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817)

le 119903

(34)

which implies that 119876 maps 119882119903into itself

Now according to Lemma 9 it remains to prove that119876 isa 120573-contraction in 119882

119903 From condition (H2) we get that 119876

1

119882119903

rarr 119862([0 119887] 119883) is Lipschitz continuous with constant119872119896 In fact for 119906 V isin 119882

119903 by (H2) we have

10038171003817100381710038171198761119906 minus 1198761V1003817100381710038171003817 le 119872

1003817100381710038171003817119892 (119906) minus 119892 (V)1003817100381710038171003817 le 119872119896 119906 minus V (35)

Thus it follows from Lemma 8-(30) that 120573(1198761119882119903) le

119872119896120573(119882119903)

For operator 1198762

119882119903

rarr 119862([0 119887] 119883) we have 1198762119906 =

119866119891119906 where 119891

119906(sdot) = 119891(sdot 119906(sdot)) is continuous on [0 119887] and

119866 is the Cauchy operator defined by (9) Thus in view ofLemma 11 we know that 119876

2is compact on 119862([0 119887] 119883) and

hence 120573(1198762119882119903) = 0 Consequently

120573 (119876119882119903) le 120573 (119876

1119882119903) + 120573 (119876

2119882119903) le 119872119896120573 (119882

119903) (36)

Since 119872119896 lt 1 the mapping 119876 is a 120573-contraction on 119882119903 By

Darbo-Sadovskiirsquos fixed point theorem the operator 119876 has afixed point in 119882

119903 which is just the mild solution of nonlocal

Cauchy problem (1)

Now we give the following technical condition on func-tion 119892

(H) 119892 119862([0 119887] 119883) rarr 119883 is continuous and the set119892(conv119876119882

119903) is precompact where conv119861 denotes the

convex closed hull of set 119861 sube 119862([0 119879] 119883) and 119876 isgiven by (32)

Remark 13 It is easy to see that condition (H) is weaker thanthe compactness and convexity of 119892 The same hypothesiscan be seen from [24 30] where the authors considered theexistence of mild solutions for semilinear nonlocal problemsof integer order when 119860 is a linear densely defined operatoron119883 which generates a 119862

0-semigroup After the proof of our

main results we will give some special types of nonlocal item119892 which is neither Lipschitz nor compact but satisfies thecondition (H) in the next Corollaries

Theorem 14 Assume that conditions (H1) and (H) are sat-isfied Then the nonlocal problem (1) has at least one mildsolution provided that (31) holds

Proof We will prove that 119876 has a fixed point by usingSchauderrsquos fixed point theorem According to the proof ofTheorem 12 we have proven that 119876 119882

119903rarr 119882

119903is

continuous Next we will prove that there exists a set119882 sube 119882119903

such that 119876 119882 rarr 119882 is compactFor this purpose let 0 lt 119905 le 119887 and 0 lt 120575 lt 119905 It is easy to

see that the set

119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)] 119906 isin 119882

119903 (37)

is precompact since 119878120572(120575) is compact On the other hand by

Lemma 10(iii) we obtain

lim120575rarr0

1003817100381710038171003817119878120572 (119905) [1199060 minus 119892 (119906)]

minus119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)]

1003817100381710038171003817 = 0 forall119906 isin 119882119903

(38)

which implies that 1198761119882119903(119905) is precompact in 119883 by using the

total boundedness Next we claim 1198761119882119903is equicontinuous

on [120578 119887] for any small positive number 120578 In fact for 119906 isin 119882119903

and 120578 le 1199051le 1199052le 119887 we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817

le1003817100381710038171003817119878120572 (1199052) minus 119878

120572(1199051)1003817100381710038171003817 (

100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817119892 (119906)1003817100381710038171003817)

(39)

By Lemma 10(i) 119878120572(119905) is operator norm continuous for 119905 gt

0 Thus 119878120572(119905) is operator norm continuous uniformly for 119905 isin

[120578 119887] and hence

lim|1199052minus1199051|rarr 0

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 = 0

uniformly for 119906 isin 119882119903

(40)

that is 1198761119882119903is equicontinuous on [120578 119887] Note that 119876

2is

compact by Lemma 11Therefore we have proven that119876119882119903(119905)

is precompact for every 119905 isin (0 119887] and119876119882119903is equicontinuous

on [120578 119887] for any small positive number 120578Now let 119882 = conv119876119882

119903 we get that 119882 is a bounded

closed and convex subset of 119862([0 119887] 119883) and 119876119882 sube 119882 It iseasy to see that119876119882(119905) is precompact in119883 for every 119905 isin (0 119887]

and 119876119882 is equicontinuous on [120578 119887] for any small positivenumber 120578 Moreover we have that 119892(119882) = 119892(conv119876119882

119903) is

precompact due to condition (H)Thus we can now claim that 119876 119882 rarr 119882 is compact In

fact it is easy to see that 1198761119882(0) = 119906

0minus 119892(119882) is precompact

since119892(119882) = 119892(conv119876119882119903) is precompact It remains to prove

that 1198761119882 is equicontinuous on [0 119887] To this end let 119906 isin 119882

and 0 le 1199051lt 1199052le 119887 then we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 le

1003817100381710038171003817119878120572 (1199052) minus 119878120572(1199051)1003817100381710038171003817

10038171003817100381710038171199060 minus 119892 (119906)1003817100381710038171003817

(41)

In view of the compactness of 119892(119882) and the strong continuityof 119878120572(119905) on [0 119887] we obtain the equicontinuous of 119876

1119882 on

[0 119887] Thus 1198761

119882 rarr 119862([0 119887] 119883) is compact by Arzela-Ascoli theorem and hence 119876 119882 rarr 119882 is also compactNow Schauderrsquos fixed point theorem implies that 119876 has afixed point on 119882 which gives rise to a mild solution ofnonlocal problem (1)

The following theorem is a direct consequence ofTheorem 14

International Journal of Differential Equations 7

Theorem 15 Assume that conditions (H1) and (H) are satis-fied for each 119903 gt 0 If

1003817100381710038171003817119892 (119906)1003817100381710038171003817

119906997888rarr 0 119906 997888rarr infin

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817

119909997888rarr 0 119909 997888rarr infin

(42)

uniformly for 119905 isin [0 119887] then the nonlocal problem (1) has atleast one mild solution

Remark 16 It is easy to see that if there exist constants 1198711

1198712gt 0 and 120574

1 1205742isin [0 1) such that1003817100381710038171003817119892 (119906)

1003817100381710038171003817 le 1198711(1 + 119906)

1205741

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817 le 1198712(1 + 119909)

1205742

(43)

for 119905 isin [0 119887] then conditions (42) are satisfied

Next we will give special types of nonlocal item 119892 whichis neither Lipschitz nor compact but satisfies condition (H)

We give the following assumptions

(H3) 119892 119862([0 119887] 119883) rarr 119883 is a continuous mapping whichmaps119882

119903into a bounded set and there is a 120575 = 120575(119903) isin

(0 119887) such that 119892(119906) = 119892(V) for any 119906 V isin 119882119903with

119906(119904) = V(119904) 119904 isin [120575 119887](H4) 119892 (119862([0 119887] 119883) sdot

1198711) rarr 119883 is continuous

Corollary 17 Assume that conditions (H1) and (H3) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(44)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) Moreover by condition (H3)119892(conv119876119882

119903) = 119892(conv(119876119882

119903)120575) is also precompact in

119862([0 119887] 119883) Thus all the hypotheses in Theorem 14 aresatisfied Therefore there is at least one mild solution ofnonlocal problem (1)

Corollary 18 Let condition (H1) be satisfied Suppose that119892(119906) = sum

119901

119895=1119888119895119906(119905119895) where 119888

119895are given positive constants and

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119901le 119887 Then the nonlocal problem (1) has

at least one mild solution on [0 119887] provided that

119872[

[

100381710038171003817100381711990601003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (45)

Proof It is easy to see that the mapping 119892 with 119892(119906) =

sum119901

119895=1119888119895119906(119905119895) satisfies condition (H3) And all the conditions

in Corollary 17 are satisfied So the conclusion holds

Corollary 19 Assume that conditions (H1) and (H4) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof According to Theorem 14 it is sufficient to prove thatthe hypothesis (H) is satisfied For arbitrary 120598 gt 0 there exists0 lt 120575 lt 119887 such that int120575

0

119906(119904)d119904 lt 120598 for all 119906 isin 119876119882119903 Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(46)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) which implies that (119876119882119903)120575is

precompact in 1198711

([0 119887] 119883) Thus 119876119882119903is precompact in

1198711

([0 119887] 119883) as it has an 120598-net (119876119882119903)120575 By condition 119892

(119862([0 119887] 119883) sdot 1198711) rarr 119883 is continuous and conv119876119882

119903sube

(119871)conv119876119882119903 it follows that condition (H) is satisfied where

(119871)conv119861 denotes the convex and closed hull of 119861 in1198711

([0 119887] 119883) Therefore the nonlocal problem (1) has at leastone mild solution on [0 119887]

Finally we give a simple example to illustrate our theory

Example 20 Consider the following fractional partial differ-ential heat equation in R

119863120572

119905119908 (119905 119909)

=1205972

119908 (119905 119909)

1205971199092

+ 1198691minus120572

119905119865 (119905 119908 (119905 119909)) 0 lt 119909 lt 1 0 lt 119905 lt 1

119908 (119905 0) = 119908 (119905 1) 1199081015840

119909(119905 0) = 119908

1015840

119909(119905 1) 0 le 119905 le 1

119908 (0 119909) = 1199080(119909) minus

119901

sum

119895=1

119888119895119908(119905119895 119909) 0 lt 119909 lt 1 0 lt 119905

119895le 1

(47)

where 0 lt 120572 lt 1 and 119863120572

119905denotes the Caputo fractional

derivativeWe introduce the abstract frame as follows Let 119883 = 119906 isin

119862([0 1]R) 119906(0) = 119906(1) Let 119860 be the linear operatorin 119883 defined by 119863(119860) = 119906 isin 119883 119906

1015840

11990610158401015840

isin 119883 andfor 119906 isin 119863(119860) 119860119906 = 119906

10158401015840 Then it is well known that 119860

generates a compact 1198620semigroup 119879(119905) for 119905 gt 0 on119883 From

the subordination principle [6 Theorems 31 and 33] 119860 alsogenerates a compact resolvent 119878

120572(119905) for 119905 gt 0

Assume that 119891 [0 1] times 119883 rarr 119883 is a continuousfunction defined by 119891(119905 119911)(119909) = 119865(119905 119911(119909)) 0 le 119909 le 1 and119892 119862([0 1] 119883) rarr 119883 is also a continuous function definedby 119892(119906)(119909) = 119908

0(119909) minus sum

119901

119895=1119888119895119906(119905119895)(119909) 0 le 119905 119909 le 1 where

119906(119905)(119909) = 119908(119905 119909)

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Differential Equations

to prove the compactness of Cauchy operator 119866 associatedwith solution operator 119876 When the operator 119860 generates acompact semigroup it is well known that theCauchy operator119866 is also compact However to the best of our knowledgeit is unknown when 119860 generates a compact resolvent Themain difficulty of this problem lies in the fact that there is noproperty of semigroups for resolvent

To this end wewill first discuss the continuity of resolventin the uniform operator topology in this paper In fact weprove that the compact analytic resolvent is continuous inthe uniform operator topology Based on this result we canprove the compactness of Cauchy operator As a consequencewe obtain the existence of mild solutions for (1) when thenonlocal item is Lipschitz continuous At the same time wealso derive the existence of mild solutions for (1) withoutthe Lipschitz or compact assumption on the nonlocal item119892 by using the techniques developed in [24 30] Actuallywe only assume that 119892 is continuous on 119862([0 119887] 119883) and 119892 iscompletely determined on [120575 119887] for some small 120575 gt 0 or 119892

is continuous on 119862([0 119887] 119883) with 1198711

([0 119887] 119883) topology (seeCorollaries 17ndash19)

This paper has four sections In Section 2 we recallsome definitions on Caputo fractional derivatives analyticresolvent and mild solutions to (1) In Section 3 we provethe compactness of Cauchy operator Finally in Section 4we establish the existence of mild solutions of (1) when thenonlocal item satisfies different conditions An example isalso given in this section

2 Preliminaries

Throughout this paper let 119887 gt 0 be fixed and let N Rand R

+be the set of positive integers real numbers and

nonnegative real numbers respectively We denote by 119883 theBanach space with the norm sdot 119862([0 119887] 119883) the space ofall 119883-valued continuous functions on [0 119887] with the norm119906 = sup119906(119905) 119905 isin [0 119887] and 119871

119901

([0 119887] 119883) the space of119883-valued Bochner integrable functions on [0 119887] with the norm119891119871119901 = (int

119887

0

119891(119905)119901d119905)1119901 where 1 le 119901 lt infin Also we denote

by L(119883) the space of bounded linear operators from 119883 into119883 endowed with the norm of operators

Now let us recall some basic definitions and results onfractional derivative and fractional differential equations

Definition 1 (see [1]) The fractional order integral of thefunction 119891 isin 119871

1

([0 119887] 119883) of order 120572 gt 0 is defined by

119869120572

119905119891 (119905) =

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) d119904 (2)

where Γ is the Gamma function

Definition 2 (see [1]) TheRiemann-Liouville fractional orderderivative of order 120572 of a function 119891 isin 119871

1

([0 119887] 119883) given onthe interval [0 119887] is defined by

119863120572

119871119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0

(119905 minus 119904)119899minus120572minus1

119891 (119904) d119904 (3)

where 120572 isin (119899 minus 1 119899] 119899 isin N

Definition 3 (see [1]) The Caputo fractional order derivativeof order 120572 of a function119891 isin 119871

1

([0 119887] 119883) given on the interval[0 119887] is defined by

119863120572

119891 (119905) =1

Γ (119899 minus 120572)int

119905

0

(119905 minus 119904)119899minus120572minus1

119891(119899)

(119904) d119904 (4)

where 120572 isin (119899 minus 1 119899] 119899 isin N

In the remainder of this paper we always suppose that 0 lt

120572 lt 1 and 119860 is a closed and densely defined linear operatoron 119883

Definition 4 A family 119878120572(119905)119905ge0

sube L(119883) of bounded linearoperators in 119883 is called a resolvent (or solution operator)generating by 119860 if the following conditions are satisfied

(S1) 119878120572(119905) is strong continuous on R

+and 119878120572(0) = 119868

(S2) 119878120572(119905)119863(119860) sube 119863(119860) and 119860119878

120572(119905)119909 = 119878

120572(119905)119860119909 for all 119909 isin

119863(119860) and 119905 ge 0(S3) the resolvent equation holds

119878120572(119905) 119909 = 119909 + int

119905

0

119892120572(119905 minus 119904) 119860119878

120572(119904) 119909d119904 forall119909 isin 119863 (119860) 119905 ge 0

(5)

For 120596 120579 isin R let

sum(120596 120579) = 120582 isin C 1003816100381610038161003816arg (120582 minus 120596)

1003816100381610038161003816 lt 120579 (6)

Definition 5 A resolvent 119878120572(119905) is called analytic if the

function 119878120572(sdot) R

+rarr L(119883) admits analytic extension

to a sector sum(0 1205790) for some 0 lt 120579

0le 1205872 An analytic

resolvent 119878120572(119905) is said to be of analyticity type (120596

0 1205790) if for

each 120579 lt 1205790and 120596 gt 120596

0there is 119872

1= 1198721(120596 120579) such that

119878(119911) le 1198721119890120596Re 119911 for 119911 isin sum(0 120579) where Re 119911 denotes the

real part of 119911

Definition 6 A resolvent 119878120572(119905) is called compact for 119905 gt 0 if

for every 119905 gt 0 119878120572(119905) is a compact operator

According to Proposition 12 in [2] we can give thefollowing definition of mild solutions for (1)

Definition 7 A function 119906 isin 119862([0 119887] 119883) is called a mildsolution of fractional evolution equation (1) if it satisfies

119906 (119905) = 119878120572(119905) [1199060minus 119892 (119906)]

+ int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119906 (119904)) d119904 0 le 119905 le 119887

(7)

for every 1199060isin 119883

Next we introduce the Hausdorff measure of noncom-pactness 120573(sdot) defined on each bounded subset Ω of Banachspace 119884 by

120573 (Ω) = inf 120576 gt 0 Ω has a finite 120576-net in 119884 (8)

Some basic properties of 120573(sdot) are given in the followinglemma

International Journal of Differential Equations 3

Lemma 8 (see [34]) Let 119884 be a real Banach space and let119861 119862 sube 119884 be bounded then the following properties aresatisfied

(1) 119861 is precompact if and only if 120573(119861) = 0

(2) 120573(119861) = 120573(119861) = 120573(conv 119861) where 119861 and conv 119861 meanthe closure and convex hull of 119861 respectively

(3) 120573(119861) le 120573(119862) when 119861 sube 119862

(4) 120573(119861 + 119862) le 120573(119861) + 120573(119862) where 119861 + 119862 = 119909 + 119910 119909 isin

119861 119910 isin 119862

(5) 120573(119861 cup 119862) le max120573(119861) 120573(119862)

(6) 120573(120582119861) = |120582|120573(119861) for any 120582 isin R

(7) if the map119876 119863(119876) sube 119884 rarr 119885 is Lipschitz continuouswith constant 119896 then 120573(119876119861) le 119896120573(119861) for any boundedsubset 119861 sube 119863(119876) where 119885 is a Banach space

The map 119876 119882 sube 119884 rarr 119884 is said to be a 120573-contractionif there exists a positive constant 119896 lt 1 such that 120573(119876119862) le

119896120573(119862) for any bounded closed subset 119862 sube 119882 where 119884 is aBanach space

Lemma 9 ([34] Darbo-Sadovskii) If 119882 sube 119884 is boundedclosed and convex the continuous map 119876 119882 rarr 119882 is a120573-contraction then the map 119876 has at least one fixed point in119882

3 Compactness of Cauchy Operators

Let Cauchy operator 119866 119862([0 119887] 119883) rarr 119862([0 119887] 119883) be def-ined by

(119866119891) (119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904 119905 isin [0 119887] (9)

If 119878120572(119905) is a compact 119862

0-semigroup it is well known that

119866 is compact However it is unknown in case of compactresolvent The main difficulty is that the resolvent does nothave the property of semigroups Thus it seems to be morecomplicated to prove the compactness of Cauchy operatorHere we will first discuss the continuity of resolvent in theuniform operator topology Then we can give the positiveanswer to the above problem

In the remainder of this paper we always assume that119872 = sup

119905isin[0119887]119878120572(119905) lt +infin

Lemma 10 Suppose 119878120572(119905) is a compact analytic resolvent of

analyticity type (1205960 1205790) Then the following are hold

(i) limℎrarr0

119878120572(119905 + ℎ) minus 119878

120572(119905) = 0 for 119905 gt 0

(ii) limℎrarr0

+119878120572(119905 + ℎ) minus 119878

120572(ℎ)119878120572(119905) = 0 for 119905 gt 0

(iii) limℎrarr0

+119878120572(119905) minus 119878

120572(ℎ)119878120572(119905 minus ℎ) = 0 for 119905 gt 0

Proof (i) Let 119878120572(119905) be an analytic resolvent of analyticity type

(1205960 1205790) and let 119905 gt 0 be given Then by means of Cauchy

integral formula we have

1198781015840

120572(119905) =

1

2120587119894int|119911minus119905|=119905 sin 120579

119878120572(119911)

(119911 minus 119905)2d119911 (10)

Thus for any 120596 gt 1205960 0 lt 120579 lt 120579

0 there exists a constant

1198721= 1198721(120596 120579) such that

100381710038171003817100381710038171198781015840

120572(119905)

10038171003817100381710038171003817le 1198721(2120587)minus1

int

120587

minus120587

119890120596(119905+119905 sin 120579 cos 120574)

(119905 sin 120579)minus1d120574

le 11987211198902120596119905

(119905 sin 120579)minus1

(11)

Now let 119909 le 1 and |ℎ| lt 119905 It follows from (11) that thereexists a constant 1198721015840 such that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(119905) 119909

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

int

119905+ℎ

119905

1198781015840

120572(119904) 119909 d119904

100381710038171003817100381710038171003817100381710038171003817

le 1198721015840

119909

100381610038161003816100381610038161003816100381610038161003816

int

119905+ℎ

119905

119904minus1 d119904

100381610038161003816100381610038161003816100381610038161003816

le 1198721015840

|ln (119905 + ℎ) minus ln 119905| 997888rarr 0 as ℎ 997888rarr 0

(12)

which implies that 119878120572(119905) is continuous in the uniform opera-

tor topology for 119905 gt 0 that is

limℎrarr0

1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878120572(119905)

1003817100381710038171003817 = 0 for 119905 gt 0 (13)

(ii) Let 119909 le 1 119905 gt 0 and 120576 gt 0 be given Since119878120572(119905) is compact the set 119882

119905= 119878

120572(119905)119909 119909 le 1 is

also compact Thus there exists a finite family 119878120572(119905)1199091

119878120572(119905)1199092 119878

120572(119905)119909119898 sub 119882

119905such that for any 119909 with 119909 le 1

there exists 119909119894(1 le 119894 le 119898) such that

1003817100381710038171003817119878120572 (119905) 119909 minus 119878120572(119905) 119909119894

1003817100381710038171003817 le120576

3 (119872 + 1) (14)

From the strong continuity of 119878120572(119905) 119905 ge 0 there exists 0 lt

ℎ1lt min119905 119887 such that

1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817 le120576

3 forall0 le ℎ le ℎ

1 1 le 119894 le 119898

(15)

On the other hand from (i) there exists 0 lt ℎ2

lt min119905 119887such that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(119905) 119909

1003817100381710038171003817 le120576

3 forall0 le ℎ le ℎ

2 119909 le 1

(16)

4 International Journal of Differential Equations

Thus for 0 le ℎ le minℎ1 ℎ2 and 119909 le 1 it follows from

(14)ndash(16) that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+ (119872 + 1)1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

le120576

3+

120576

3+

120576

3

le 120576

(17)

which implies that limℎrarr0

+119878120572(119905+ℎ)minus119878

120572(ℎ)119878120572(119905) = 0 for all

119905 gt 0(iii) Let 119905 gt 0 and 0 lt ℎ lt min119905 119887 Then there exists

119872 gt 0 such that

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) minus 119878

120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+ 1198721003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 minus ℎ)

1003817100381710038171003817

(18)

It follows from (i) and (ii) that

limℎrarr0

+

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817 = 0 for 119905 gt 0 (19)

This completes the proof

Lemma 11 Suppose 119878120572(119905) is a compact analytic resolvent of

analyticity type (1205960 1205790) Then Cauchy operator 119866 defined by

(9) is a compact operator

Proof We will show 119866 119862([0 119887] 119883) rarr 119862([0 119887] 119883) iscompact by using the Arzela-Ascoli theorem Let 119863

119897=

119891 isin 119862([0 119887] 119883) 119891 le 119897 be any bounded subset of119862([0 119887] 119883)

First we claim that the set 119866119863119897is equicontinuous on

119862([0 119887] 119883) In fact let 0 le 1199051

le 1199052

le 119887 and 119891 isin 119863119897 then

we have

119868 =1003817100381710038171003817(119866119891) (119905

2) minus (119866119891) (119905

1)1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199052

0

119878120572(1199052minus 119904) 119891 (119904) d119904 minus int

1199051

0

119878120572(1199051minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

1003817100381710038171003817119891 (119904)1003817100381710038171003817 d119904

+ int

1199052

1199051

1003817100381710038171003817119878120572 (1199052 minus 119904) 119891 (119904)1003817100381710038171003817 d119904

le 119897 int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

(20)

If 1199051= 0 it is easy to see that

lim1199052rarr0

119868 = 0 uniformly for 119891 isin 119863119897 (21)

If 0 lt 1199051lt 119887 for 0 lt 120575 lt 119905

1 we have

119868 le 119897 int

1199051minus120575

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

1199051

1199051minus120575

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

le 119887119897 sup119904isin[01199051minus120575]

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

+ 2119872119897120575 + 119872119897 (1199052minus 1199051)

(22)

Note that from Lemma 10(i) we know 119878120572(119905) is operator norm

continuous uniformly for 119905 isin [120575 119887] Combining this andthe arbitrariness of 120575 with the above estimation on 119868 we canconclude that

lim|1199051minus1199052|rarr 0

119868 = 0 uniformly for 119891 isin 119863119897 (23)

Thus 119866119863119897is equicontinuous on 119862([0 119887] 119883)

Next we will show that the set (119866119891)(119905) 119891 isin 119863119897 is

precompact in 119883 for every 119905 isin [0 119887] It is easy to see that theset (119866119891)(0) 119891 isin 119863

119897 is precompact in 119883 Now let 0 lt 119905 le 119887

be given and 0 lt 120576 lt 119905 Then

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 119891 isin 119863

119897 (24)

International Journal of Differential Equations 5

is precompact since 119878120572(120576) is compact Moreover for arbitrary

120576 lt 120575 lt 119905 we have10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le 119897 int

119905minus120576

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

119905minus120576

119905minus120575

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904 + 119897120575 (1198722

+ 119872)

(25)

From Lemma 10(iii) we know

119878120572(120576) 119878120572(119905 minus 119904 minus 120576) minus 119878

120572(119905 minus 119904) 997888rarr 0

as 120576 997888rarr 0 for 119904 isin [0 119905 minus 120575]

(26)

Then it follows from the Lebesgue dominated convergencetheorem and the arbitrariness of 120575 that

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(27)

On the other hand10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+ 119897120576119872

(28)

Thus

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(29)

which implies that (119866119891)(119905) 119891 isin 119863119897 is precompact in 119883 by

using the total boundedness and therefore 119866 is compact inview of Arzela-Ascoli theorem

4 Nonlocal Problems

In this section we always assume that 1199060

isin 119883 and that theoperator 119860 generates a compact analytic resolvent 119878

120572(119905) of

analyticity type (1205960 1205790) and we will prove the existence of

mild solutions of (1) when the nonlocal item 119892 is assumed tobe Lipschitz continuous and neither Lipschitz nor compactrespectively

Let 119903 be a fixed positive real number and

119882119903= 119906 isin 119862 ([0 119887] 119883) 119906 le 119903 (30)

Clearly119882119903is a bounded closed and convex set We make the

following assumptions

(H1) 119891 [0 119887] times 119883 rarr 119883 is continuous

(H2) 119892 119862([0 119887] 119883) rarr 119883 is Lipschitz continuous withLipschitz constant 119896 such that 119872119896 lt 1

Under these assumptions we can prove the first mainresult in this paper

Theorem 12 Assume that conditions (H1) and (H2) aresatisfied Then the nonlocal problem (1) has at least one mildsolution provided that

119872[10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817] le 119903

(31)

Proof We consider the solution operator 119876 119862([0 119887] 119883) rarr

119862([0 119887] 119883) defined by

(119876119906) (119905) = 119878120572(119905) [1199060minus 119892 (119906)] + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119906 (119904)) d119904

= (1198761119906) (119905) + (119876

2119906) (119905) 119905 isin [0 119887]

(32)

It is easy to see that the fixed point of 119876 is the mild solutionof nonlocal Cauchy problem (1) Subsequently we will provethat119876 has a fixed point by using Lemma 9 (Darbo-Sadovskiirsquosfixed point theorem)

Firstly we prove that the mapping 119876 is continuous on119862([0 119887] 119883) For this purpose let 119906

119899119899ge1

be a sequence in119862([0 119887] 119883) with lim

119899rarrinfin119906119899= 119906 in 119862([0 119887] 119883) Then

1003817100381710038171003817119876119906119899minus 119876119906

1003817100381710038171003817

le 119872[1003817100381710038171003817119892 (119906119899)minus119892 (119906)

1003817100381710038171003817 + 119887 sup119904isin[0119887]

1003817100381710038171003817119891 (119904 119906119899(119904))minus119891 (119904 119906 (119904))

1003817100381710038171003817]

(33)

By the continuity of 119891 and 119892 we deduce that119876 is continuouson 119862([0 119887] 119883)

6 International Journal of Differential Equations

Secondly we claim that 119876119882119903sube 119882119903 where 119882

119903is defined

by (30) In fact for any 119906 isin 119882119903 by (31) we have

119876119906 le 119872(10038171003817100381710038171199060

1003817100381710038171003817 +1003817100381710038171003817119892 (119906)

1003817100381710038171003817 + int

119887

0

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817 d119904)

le 119872(10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817)

le 119903

(34)

which implies that 119876 maps 119882119903into itself

Now according to Lemma 9 it remains to prove that119876 isa 120573-contraction in 119882

119903 From condition (H2) we get that 119876

1

119882119903

rarr 119862([0 119887] 119883) is Lipschitz continuous with constant119872119896 In fact for 119906 V isin 119882

119903 by (H2) we have

10038171003817100381710038171198761119906 minus 1198761V1003817100381710038171003817 le 119872

1003817100381710038171003817119892 (119906) minus 119892 (V)1003817100381710038171003817 le 119872119896 119906 minus V (35)

Thus it follows from Lemma 8-(30) that 120573(1198761119882119903) le

119872119896120573(119882119903)

For operator 1198762

119882119903

rarr 119862([0 119887] 119883) we have 1198762119906 =

119866119891119906 where 119891

119906(sdot) = 119891(sdot 119906(sdot)) is continuous on [0 119887] and

119866 is the Cauchy operator defined by (9) Thus in view ofLemma 11 we know that 119876

2is compact on 119862([0 119887] 119883) and

hence 120573(1198762119882119903) = 0 Consequently

120573 (119876119882119903) le 120573 (119876

1119882119903) + 120573 (119876

2119882119903) le 119872119896120573 (119882

119903) (36)

Since 119872119896 lt 1 the mapping 119876 is a 120573-contraction on 119882119903 By

Darbo-Sadovskiirsquos fixed point theorem the operator 119876 has afixed point in 119882

119903 which is just the mild solution of nonlocal

Cauchy problem (1)

Now we give the following technical condition on func-tion 119892

(H) 119892 119862([0 119887] 119883) rarr 119883 is continuous and the set119892(conv119876119882

119903) is precompact where conv119861 denotes the

convex closed hull of set 119861 sube 119862([0 119879] 119883) and 119876 isgiven by (32)

Remark 13 It is easy to see that condition (H) is weaker thanthe compactness and convexity of 119892 The same hypothesiscan be seen from [24 30] where the authors considered theexistence of mild solutions for semilinear nonlocal problemsof integer order when 119860 is a linear densely defined operatoron119883 which generates a 119862

0-semigroup After the proof of our

main results we will give some special types of nonlocal item119892 which is neither Lipschitz nor compact but satisfies thecondition (H) in the next Corollaries

Theorem 14 Assume that conditions (H1) and (H) are sat-isfied Then the nonlocal problem (1) has at least one mildsolution provided that (31) holds

Proof We will prove that 119876 has a fixed point by usingSchauderrsquos fixed point theorem According to the proof ofTheorem 12 we have proven that 119876 119882

119903rarr 119882

119903is

continuous Next we will prove that there exists a set119882 sube 119882119903

such that 119876 119882 rarr 119882 is compactFor this purpose let 0 lt 119905 le 119887 and 0 lt 120575 lt 119905 It is easy to

see that the set

119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)] 119906 isin 119882

119903 (37)

is precompact since 119878120572(120575) is compact On the other hand by

Lemma 10(iii) we obtain

lim120575rarr0

1003817100381710038171003817119878120572 (119905) [1199060 minus 119892 (119906)]

minus119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)]

1003817100381710038171003817 = 0 forall119906 isin 119882119903

(38)

which implies that 1198761119882119903(119905) is precompact in 119883 by using the

total boundedness Next we claim 1198761119882119903is equicontinuous

on [120578 119887] for any small positive number 120578 In fact for 119906 isin 119882119903

and 120578 le 1199051le 1199052le 119887 we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817

le1003817100381710038171003817119878120572 (1199052) minus 119878

120572(1199051)1003817100381710038171003817 (

100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817119892 (119906)1003817100381710038171003817)

(39)

By Lemma 10(i) 119878120572(119905) is operator norm continuous for 119905 gt

0 Thus 119878120572(119905) is operator norm continuous uniformly for 119905 isin

[120578 119887] and hence

lim|1199052minus1199051|rarr 0

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 = 0

uniformly for 119906 isin 119882119903

(40)

that is 1198761119882119903is equicontinuous on [120578 119887] Note that 119876

2is

compact by Lemma 11Therefore we have proven that119876119882119903(119905)

is precompact for every 119905 isin (0 119887] and119876119882119903is equicontinuous

on [120578 119887] for any small positive number 120578Now let 119882 = conv119876119882

119903 we get that 119882 is a bounded

closed and convex subset of 119862([0 119887] 119883) and 119876119882 sube 119882 It iseasy to see that119876119882(119905) is precompact in119883 for every 119905 isin (0 119887]

and 119876119882 is equicontinuous on [120578 119887] for any small positivenumber 120578 Moreover we have that 119892(119882) = 119892(conv119876119882

119903) is

precompact due to condition (H)Thus we can now claim that 119876 119882 rarr 119882 is compact In

fact it is easy to see that 1198761119882(0) = 119906

0minus 119892(119882) is precompact

since119892(119882) = 119892(conv119876119882119903) is precompact It remains to prove

that 1198761119882 is equicontinuous on [0 119887] To this end let 119906 isin 119882

and 0 le 1199051lt 1199052le 119887 then we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 le

1003817100381710038171003817119878120572 (1199052) minus 119878120572(1199051)1003817100381710038171003817

10038171003817100381710038171199060 minus 119892 (119906)1003817100381710038171003817

(41)

In view of the compactness of 119892(119882) and the strong continuityof 119878120572(119905) on [0 119887] we obtain the equicontinuous of 119876

1119882 on

[0 119887] Thus 1198761

119882 rarr 119862([0 119887] 119883) is compact by Arzela-Ascoli theorem and hence 119876 119882 rarr 119882 is also compactNow Schauderrsquos fixed point theorem implies that 119876 has afixed point on 119882 which gives rise to a mild solution ofnonlocal problem (1)

The following theorem is a direct consequence ofTheorem 14

International Journal of Differential Equations 7

Theorem 15 Assume that conditions (H1) and (H) are satis-fied for each 119903 gt 0 If

1003817100381710038171003817119892 (119906)1003817100381710038171003817

119906997888rarr 0 119906 997888rarr infin

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817

119909997888rarr 0 119909 997888rarr infin

(42)

uniformly for 119905 isin [0 119887] then the nonlocal problem (1) has atleast one mild solution

Remark 16 It is easy to see that if there exist constants 1198711

1198712gt 0 and 120574

1 1205742isin [0 1) such that1003817100381710038171003817119892 (119906)

1003817100381710038171003817 le 1198711(1 + 119906)

1205741

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817 le 1198712(1 + 119909)

1205742

(43)

for 119905 isin [0 119887] then conditions (42) are satisfied

Next we will give special types of nonlocal item 119892 whichis neither Lipschitz nor compact but satisfies condition (H)

We give the following assumptions

(H3) 119892 119862([0 119887] 119883) rarr 119883 is a continuous mapping whichmaps119882

119903into a bounded set and there is a 120575 = 120575(119903) isin

(0 119887) such that 119892(119906) = 119892(V) for any 119906 V isin 119882119903with

119906(119904) = V(119904) 119904 isin [120575 119887](H4) 119892 (119862([0 119887] 119883) sdot

1198711) rarr 119883 is continuous

Corollary 17 Assume that conditions (H1) and (H3) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(44)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) Moreover by condition (H3)119892(conv119876119882

119903) = 119892(conv(119876119882

119903)120575) is also precompact in

119862([0 119887] 119883) Thus all the hypotheses in Theorem 14 aresatisfied Therefore there is at least one mild solution ofnonlocal problem (1)

Corollary 18 Let condition (H1) be satisfied Suppose that119892(119906) = sum

119901

119895=1119888119895119906(119905119895) where 119888

119895are given positive constants and

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119901le 119887 Then the nonlocal problem (1) has

at least one mild solution on [0 119887] provided that

119872[

[

100381710038171003817100381711990601003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (45)

Proof It is easy to see that the mapping 119892 with 119892(119906) =

sum119901

119895=1119888119895119906(119905119895) satisfies condition (H3) And all the conditions

in Corollary 17 are satisfied So the conclusion holds

Corollary 19 Assume that conditions (H1) and (H4) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof According to Theorem 14 it is sufficient to prove thatthe hypothesis (H) is satisfied For arbitrary 120598 gt 0 there exists0 lt 120575 lt 119887 such that int120575

0

119906(119904)d119904 lt 120598 for all 119906 isin 119876119882119903 Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(46)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) which implies that (119876119882119903)120575is

precompact in 1198711

([0 119887] 119883) Thus 119876119882119903is precompact in

1198711

([0 119887] 119883) as it has an 120598-net (119876119882119903)120575 By condition 119892

(119862([0 119887] 119883) sdot 1198711) rarr 119883 is continuous and conv119876119882

119903sube

(119871)conv119876119882119903 it follows that condition (H) is satisfied where

(119871)conv119861 denotes the convex and closed hull of 119861 in1198711

([0 119887] 119883) Therefore the nonlocal problem (1) has at leastone mild solution on [0 119887]

Finally we give a simple example to illustrate our theory

Example 20 Consider the following fractional partial differ-ential heat equation in R

119863120572

119905119908 (119905 119909)

=1205972

119908 (119905 119909)

1205971199092

+ 1198691minus120572

119905119865 (119905 119908 (119905 119909)) 0 lt 119909 lt 1 0 lt 119905 lt 1

119908 (119905 0) = 119908 (119905 1) 1199081015840

119909(119905 0) = 119908

1015840

119909(119905 1) 0 le 119905 le 1

119908 (0 119909) = 1199080(119909) minus

119901

sum

119895=1

119888119895119908(119905119895 119909) 0 lt 119909 lt 1 0 lt 119905

119895le 1

(47)

where 0 lt 120572 lt 1 and 119863120572

119905denotes the Caputo fractional

derivativeWe introduce the abstract frame as follows Let 119883 = 119906 isin

119862([0 1]R) 119906(0) = 119906(1) Let 119860 be the linear operatorin 119883 defined by 119863(119860) = 119906 isin 119883 119906

1015840

11990610158401015840

isin 119883 andfor 119906 isin 119863(119860) 119860119906 = 119906

10158401015840 Then it is well known that 119860

generates a compact 1198620semigroup 119879(119905) for 119905 gt 0 on119883 From

the subordination principle [6 Theorems 31 and 33] 119860 alsogenerates a compact resolvent 119878

120572(119905) for 119905 gt 0

Assume that 119891 [0 1] times 119883 rarr 119883 is a continuousfunction defined by 119891(119905 119911)(119909) = 119865(119905 119911(119909)) 0 le 119909 le 1 and119892 119862([0 1] 119883) rarr 119883 is also a continuous function definedby 119892(119906)(119909) = 119908

0(119909) minus sum

119901

119895=1119888119895119906(119905119895)(119909) 0 le 119905 119909 le 1 where

119906(119905)(119909) = 119908(119905 119909)

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 3

Lemma 8 (see [34]) Let 119884 be a real Banach space and let119861 119862 sube 119884 be bounded then the following properties aresatisfied

(1) 119861 is precompact if and only if 120573(119861) = 0

(2) 120573(119861) = 120573(119861) = 120573(conv 119861) where 119861 and conv 119861 meanthe closure and convex hull of 119861 respectively

(3) 120573(119861) le 120573(119862) when 119861 sube 119862

(4) 120573(119861 + 119862) le 120573(119861) + 120573(119862) where 119861 + 119862 = 119909 + 119910 119909 isin

119861 119910 isin 119862

(5) 120573(119861 cup 119862) le max120573(119861) 120573(119862)

(6) 120573(120582119861) = |120582|120573(119861) for any 120582 isin R

(7) if the map119876 119863(119876) sube 119884 rarr 119885 is Lipschitz continuouswith constant 119896 then 120573(119876119861) le 119896120573(119861) for any boundedsubset 119861 sube 119863(119876) where 119885 is a Banach space

The map 119876 119882 sube 119884 rarr 119884 is said to be a 120573-contractionif there exists a positive constant 119896 lt 1 such that 120573(119876119862) le

119896120573(119862) for any bounded closed subset 119862 sube 119882 where 119884 is aBanach space

Lemma 9 ([34] Darbo-Sadovskii) If 119882 sube 119884 is boundedclosed and convex the continuous map 119876 119882 rarr 119882 is a120573-contraction then the map 119876 has at least one fixed point in119882

3 Compactness of Cauchy Operators

Let Cauchy operator 119866 119862([0 119887] 119883) rarr 119862([0 119887] 119883) be def-ined by

(119866119891) (119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904 119905 isin [0 119887] (9)

If 119878120572(119905) is a compact 119862

0-semigroup it is well known that

119866 is compact However it is unknown in case of compactresolvent The main difficulty is that the resolvent does nothave the property of semigroups Thus it seems to be morecomplicated to prove the compactness of Cauchy operatorHere we will first discuss the continuity of resolvent in theuniform operator topology Then we can give the positiveanswer to the above problem

In the remainder of this paper we always assume that119872 = sup

119905isin[0119887]119878120572(119905) lt +infin

Lemma 10 Suppose 119878120572(119905) is a compact analytic resolvent of

analyticity type (1205960 1205790) Then the following are hold

(i) limℎrarr0

119878120572(119905 + ℎ) minus 119878

120572(119905) = 0 for 119905 gt 0

(ii) limℎrarr0

+119878120572(119905 + ℎ) minus 119878

120572(ℎ)119878120572(119905) = 0 for 119905 gt 0

(iii) limℎrarr0

+119878120572(119905) minus 119878

120572(ℎ)119878120572(119905 minus ℎ) = 0 for 119905 gt 0

Proof (i) Let 119878120572(119905) be an analytic resolvent of analyticity type

(1205960 1205790) and let 119905 gt 0 be given Then by means of Cauchy

integral formula we have

1198781015840

120572(119905) =

1

2120587119894int|119911minus119905|=119905 sin 120579

119878120572(119911)

(119911 minus 119905)2d119911 (10)

Thus for any 120596 gt 1205960 0 lt 120579 lt 120579

0 there exists a constant

1198721= 1198721(120596 120579) such that

100381710038171003817100381710038171198781015840

120572(119905)

10038171003817100381710038171003817le 1198721(2120587)minus1

int

120587

minus120587

119890120596(119905+119905 sin 120579 cos 120574)

(119905 sin 120579)minus1d120574

le 11987211198902120596119905

(119905 sin 120579)minus1

(11)

Now let 119909 le 1 and |ℎ| lt 119905 It follows from (11) that thereexists a constant 1198721015840 such that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(119905) 119909

1003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817

int

119905+ℎ

119905

1198781015840

120572(119904) 119909 d119904

100381710038171003817100381710038171003817100381710038171003817

le 1198721015840

119909

100381610038161003816100381610038161003816100381610038161003816

int

119905+ℎ

119905

119904minus1 d119904

100381610038161003816100381610038161003816100381610038161003816

le 1198721015840

|ln (119905 + ℎ) minus ln 119905| 997888rarr 0 as ℎ 997888rarr 0

(12)

which implies that 119878120572(119905) is continuous in the uniform opera-

tor topology for 119905 gt 0 that is

limℎrarr0

1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878120572(119905)

1003817100381710038171003817 = 0 for 119905 gt 0 (13)

(ii) Let 119909 le 1 119905 gt 0 and 120576 gt 0 be given Since119878120572(119905) is compact the set 119882

119905= 119878

120572(119905)119909 119909 le 1 is

also compact Thus there exists a finite family 119878120572(119905)1199091

119878120572(119905)1199092 119878

120572(119905)119909119898 sub 119882

119905such that for any 119909 with 119909 le 1

there exists 119909119894(1 le 119894 le 119898) such that

1003817100381710038171003817119878120572 (119905) 119909 minus 119878120572(119905) 119909119894

1003817100381710038171003817 le120576

3 (119872 + 1) (14)

From the strong continuity of 119878120572(119905) 119905 ge 0 there exists 0 lt

ℎ1lt min119905 119887 such that

1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817 le120576

3 forall0 le ℎ le ℎ

1 1 le 119894 le 119898

(15)

On the other hand from (i) there exists 0 lt ℎ2

lt min119905 119887such that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(119905) 119909

1003817100381710038171003817 le120576

3 forall0 le ℎ le ℎ

2 119909 le 1

(16)

4 International Journal of Differential Equations

Thus for 0 le ℎ le minℎ1 ℎ2 and 119909 le 1 it follows from

(14)ndash(16) that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+ (119872 + 1)1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

le120576

3+

120576

3+

120576

3

le 120576

(17)

which implies that limℎrarr0

+119878120572(119905+ℎ)minus119878

120572(ℎ)119878120572(119905) = 0 for all

119905 gt 0(iii) Let 119905 gt 0 and 0 lt ℎ lt min119905 119887 Then there exists

119872 gt 0 such that

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) minus 119878

120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+ 1198721003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 minus ℎ)

1003817100381710038171003817

(18)

It follows from (i) and (ii) that

limℎrarr0

+

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817 = 0 for 119905 gt 0 (19)

This completes the proof

Lemma 11 Suppose 119878120572(119905) is a compact analytic resolvent of

analyticity type (1205960 1205790) Then Cauchy operator 119866 defined by

(9) is a compact operator

Proof We will show 119866 119862([0 119887] 119883) rarr 119862([0 119887] 119883) iscompact by using the Arzela-Ascoli theorem Let 119863

119897=

119891 isin 119862([0 119887] 119883) 119891 le 119897 be any bounded subset of119862([0 119887] 119883)

First we claim that the set 119866119863119897is equicontinuous on

119862([0 119887] 119883) In fact let 0 le 1199051

le 1199052

le 119887 and 119891 isin 119863119897 then

we have

119868 =1003817100381710038171003817(119866119891) (119905

2) minus (119866119891) (119905

1)1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199052

0

119878120572(1199052minus 119904) 119891 (119904) d119904 minus int

1199051

0

119878120572(1199051minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

1003817100381710038171003817119891 (119904)1003817100381710038171003817 d119904

+ int

1199052

1199051

1003817100381710038171003817119878120572 (1199052 minus 119904) 119891 (119904)1003817100381710038171003817 d119904

le 119897 int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

(20)

If 1199051= 0 it is easy to see that

lim1199052rarr0

119868 = 0 uniformly for 119891 isin 119863119897 (21)

If 0 lt 1199051lt 119887 for 0 lt 120575 lt 119905

1 we have

119868 le 119897 int

1199051minus120575

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

1199051

1199051minus120575

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

le 119887119897 sup119904isin[01199051minus120575]

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

+ 2119872119897120575 + 119872119897 (1199052minus 1199051)

(22)

Note that from Lemma 10(i) we know 119878120572(119905) is operator norm

continuous uniformly for 119905 isin [120575 119887] Combining this andthe arbitrariness of 120575 with the above estimation on 119868 we canconclude that

lim|1199051minus1199052|rarr 0

119868 = 0 uniformly for 119891 isin 119863119897 (23)

Thus 119866119863119897is equicontinuous on 119862([0 119887] 119883)

Next we will show that the set (119866119891)(119905) 119891 isin 119863119897 is

precompact in 119883 for every 119905 isin [0 119887] It is easy to see that theset (119866119891)(0) 119891 isin 119863

119897 is precompact in 119883 Now let 0 lt 119905 le 119887

be given and 0 lt 120576 lt 119905 Then

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 119891 isin 119863

119897 (24)

International Journal of Differential Equations 5

is precompact since 119878120572(120576) is compact Moreover for arbitrary

120576 lt 120575 lt 119905 we have10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le 119897 int

119905minus120576

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

119905minus120576

119905minus120575

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904 + 119897120575 (1198722

+ 119872)

(25)

From Lemma 10(iii) we know

119878120572(120576) 119878120572(119905 minus 119904 minus 120576) minus 119878

120572(119905 minus 119904) 997888rarr 0

as 120576 997888rarr 0 for 119904 isin [0 119905 minus 120575]

(26)

Then it follows from the Lebesgue dominated convergencetheorem and the arbitrariness of 120575 that

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(27)

On the other hand10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+ 119897120576119872

(28)

Thus

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(29)

which implies that (119866119891)(119905) 119891 isin 119863119897 is precompact in 119883 by

using the total boundedness and therefore 119866 is compact inview of Arzela-Ascoli theorem

4 Nonlocal Problems

In this section we always assume that 1199060

isin 119883 and that theoperator 119860 generates a compact analytic resolvent 119878

120572(119905) of

analyticity type (1205960 1205790) and we will prove the existence of

mild solutions of (1) when the nonlocal item 119892 is assumed tobe Lipschitz continuous and neither Lipschitz nor compactrespectively

Let 119903 be a fixed positive real number and

119882119903= 119906 isin 119862 ([0 119887] 119883) 119906 le 119903 (30)

Clearly119882119903is a bounded closed and convex set We make the

following assumptions

(H1) 119891 [0 119887] times 119883 rarr 119883 is continuous

(H2) 119892 119862([0 119887] 119883) rarr 119883 is Lipschitz continuous withLipschitz constant 119896 such that 119872119896 lt 1

Under these assumptions we can prove the first mainresult in this paper

Theorem 12 Assume that conditions (H1) and (H2) aresatisfied Then the nonlocal problem (1) has at least one mildsolution provided that

119872[10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817] le 119903

(31)

Proof We consider the solution operator 119876 119862([0 119887] 119883) rarr

119862([0 119887] 119883) defined by

(119876119906) (119905) = 119878120572(119905) [1199060minus 119892 (119906)] + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119906 (119904)) d119904

= (1198761119906) (119905) + (119876

2119906) (119905) 119905 isin [0 119887]

(32)

It is easy to see that the fixed point of 119876 is the mild solutionof nonlocal Cauchy problem (1) Subsequently we will provethat119876 has a fixed point by using Lemma 9 (Darbo-Sadovskiirsquosfixed point theorem)

Firstly we prove that the mapping 119876 is continuous on119862([0 119887] 119883) For this purpose let 119906

119899119899ge1

be a sequence in119862([0 119887] 119883) with lim

119899rarrinfin119906119899= 119906 in 119862([0 119887] 119883) Then

1003817100381710038171003817119876119906119899minus 119876119906

1003817100381710038171003817

le 119872[1003817100381710038171003817119892 (119906119899)minus119892 (119906)

1003817100381710038171003817 + 119887 sup119904isin[0119887]

1003817100381710038171003817119891 (119904 119906119899(119904))minus119891 (119904 119906 (119904))

1003817100381710038171003817]

(33)

By the continuity of 119891 and 119892 we deduce that119876 is continuouson 119862([0 119887] 119883)

6 International Journal of Differential Equations

Secondly we claim that 119876119882119903sube 119882119903 where 119882

119903is defined

by (30) In fact for any 119906 isin 119882119903 by (31) we have

119876119906 le 119872(10038171003817100381710038171199060

1003817100381710038171003817 +1003817100381710038171003817119892 (119906)

1003817100381710038171003817 + int

119887

0

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817 d119904)

le 119872(10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817)

le 119903

(34)

which implies that 119876 maps 119882119903into itself

Now according to Lemma 9 it remains to prove that119876 isa 120573-contraction in 119882

119903 From condition (H2) we get that 119876

1

119882119903

rarr 119862([0 119887] 119883) is Lipschitz continuous with constant119872119896 In fact for 119906 V isin 119882

119903 by (H2) we have

10038171003817100381710038171198761119906 minus 1198761V1003817100381710038171003817 le 119872

1003817100381710038171003817119892 (119906) minus 119892 (V)1003817100381710038171003817 le 119872119896 119906 minus V (35)

Thus it follows from Lemma 8-(30) that 120573(1198761119882119903) le

119872119896120573(119882119903)

For operator 1198762

119882119903

rarr 119862([0 119887] 119883) we have 1198762119906 =

119866119891119906 where 119891

119906(sdot) = 119891(sdot 119906(sdot)) is continuous on [0 119887] and

119866 is the Cauchy operator defined by (9) Thus in view ofLemma 11 we know that 119876

2is compact on 119862([0 119887] 119883) and

hence 120573(1198762119882119903) = 0 Consequently

120573 (119876119882119903) le 120573 (119876

1119882119903) + 120573 (119876

2119882119903) le 119872119896120573 (119882

119903) (36)

Since 119872119896 lt 1 the mapping 119876 is a 120573-contraction on 119882119903 By

Darbo-Sadovskiirsquos fixed point theorem the operator 119876 has afixed point in 119882

119903 which is just the mild solution of nonlocal

Cauchy problem (1)

Now we give the following technical condition on func-tion 119892

(H) 119892 119862([0 119887] 119883) rarr 119883 is continuous and the set119892(conv119876119882

119903) is precompact where conv119861 denotes the

convex closed hull of set 119861 sube 119862([0 119879] 119883) and 119876 isgiven by (32)

Remark 13 It is easy to see that condition (H) is weaker thanthe compactness and convexity of 119892 The same hypothesiscan be seen from [24 30] where the authors considered theexistence of mild solutions for semilinear nonlocal problemsof integer order when 119860 is a linear densely defined operatoron119883 which generates a 119862

0-semigroup After the proof of our

main results we will give some special types of nonlocal item119892 which is neither Lipschitz nor compact but satisfies thecondition (H) in the next Corollaries

Theorem 14 Assume that conditions (H1) and (H) are sat-isfied Then the nonlocal problem (1) has at least one mildsolution provided that (31) holds

Proof We will prove that 119876 has a fixed point by usingSchauderrsquos fixed point theorem According to the proof ofTheorem 12 we have proven that 119876 119882

119903rarr 119882

119903is

continuous Next we will prove that there exists a set119882 sube 119882119903

such that 119876 119882 rarr 119882 is compactFor this purpose let 0 lt 119905 le 119887 and 0 lt 120575 lt 119905 It is easy to

see that the set

119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)] 119906 isin 119882

119903 (37)

is precompact since 119878120572(120575) is compact On the other hand by

Lemma 10(iii) we obtain

lim120575rarr0

1003817100381710038171003817119878120572 (119905) [1199060 minus 119892 (119906)]

minus119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)]

1003817100381710038171003817 = 0 forall119906 isin 119882119903

(38)

which implies that 1198761119882119903(119905) is precompact in 119883 by using the

total boundedness Next we claim 1198761119882119903is equicontinuous

on [120578 119887] for any small positive number 120578 In fact for 119906 isin 119882119903

and 120578 le 1199051le 1199052le 119887 we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817

le1003817100381710038171003817119878120572 (1199052) minus 119878

120572(1199051)1003817100381710038171003817 (

100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817119892 (119906)1003817100381710038171003817)

(39)

By Lemma 10(i) 119878120572(119905) is operator norm continuous for 119905 gt

0 Thus 119878120572(119905) is operator norm continuous uniformly for 119905 isin

[120578 119887] and hence

lim|1199052minus1199051|rarr 0

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 = 0

uniformly for 119906 isin 119882119903

(40)

that is 1198761119882119903is equicontinuous on [120578 119887] Note that 119876

2is

compact by Lemma 11Therefore we have proven that119876119882119903(119905)

is precompact for every 119905 isin (0 119887] and119876119882119903is equicontinuous

on [120578 119887] for any small positive number 120578Now let 119882 = conv119876119882

119903 we get that 119882 is a bounded

closed and convex subset of 119862([0 119887] 119883) and 119876119882 sube 119882 It iseasy to see that119876119882(119905) is precompact in119883 for every 119905 isin (0 119887]

and 119876119882 is equicontinuous on [120578 119887] for any small positivenumber 120578 Moreover we have that 119892(119882) = 119892(conv119876119882

119903) is

precompact due to condition (H)Thus we can now claim that 119876 119882 rarr 119882 is compact In

fact it is easy to see that 1198761119882(0) = 119906

0minus 119892(119882) is precompact

since119892(119882) = 119892(conv119876119882119903) is precompact It remains to prove

that 1198761119882 is equicontinuous on [0 119887] To this end let 119906 isin 119882

and 0 le 1199051lt 1199052le 119887 then we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 le

1003817100381710038171003817119878120572 (1199052) minus 119878120572(1199051)1003817100381710038171003817

10038171003817100381710038171199060 minus 119892 (119906)1003817100381710038171003817

(41)

In view of the compactness of 119892(119882) and the strong continuityof 119878120572(119905) on [0 119887] we obtain the equicontinuous of 119876

1119882 on

[0 119887] Thus 1198761

119882 rarr 119862([0 119887] 119883) is compact by Arzela-Ascoli theorem and hence 119876 119882 rarr 119882 is also compactNow Schauderrsquos fixed point theorem implies that 119876 has afixed point on 119882 which gives rise to a mild solution ofnonlocal problem (1)

The following theorem is a direct consequence ofTheorem 14

International Journal of Differential Equations 7

Theorem 15 Assume that conditions (H1) and (H) are satis-fied for each 119903 gt 0 If

1003817100381710038171003817119892 (119906)1003817100381710038171003817

119906997888rarr 0 119906 997888rarr infin

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817

119909997888rarr 0 119909 997888rarr infin

(42)

uniformly for 119905 isin [0 119887] then the nonlocal problem (1) has atleast one mild solution

Remark 16 It is easy to see that if there exist constants 1198711

1198712gt 0 and 120574

1 1205742isin [0 1) such that1003817100381710038171003817119892 (119906)

1003817100381710038171003817 le 1198711(1 + 119906)

1205741

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817 le 1198712(1 + 119909)

1205742

(43)

for 119905 isin [0 119887] then conditions (42) are satisfied

Next we will give special types of nonlocal item 119892 whichis neither Lipschitz nor compact but satisfies condition (H)

We give the following assumptions

(H3) 119892 119862([0 119887] 119883) rarr 119883 is a continuous mapping whichmaps119882

119903into a bounded set and there is a 120575 = 120575(119903) isin

(0 119887) such that 119892(119906) = 119892(V) for any 119906 V isin 119882119903with

119906(119904) = V(119904) 119904 isin [120575 119887](H4) 119892 (119862([0 119887] 119883) sdot

1198711) rarr 119883 is continuous

Corollary 17 Assume that conditions (H1) and (H3) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(44)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) Moreover by condition (H3)119892(conv119876119882

119903) = 119892(conv(119876119882

119903)120575) is also precompact in

119862([0 119887] 119883) Thus all the hypotheses in Theorem 14 aresatisfied Therefore there is at least one mild solution ofnonlocal problem (1)

Corollary 18 Let condition (H1) be satisfied Suppose that119892(119906) = sum

119901

119895=1119888119895119906(119905119895) where 119888

119895are given positive constants and

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119901le 119887 Then the nonlocal problem (1) has

at least one mild solution on [0 119887] provided that

119872[

[

100381710038171003817100381711990601003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (45)

Proof It is easy to see that the mapping 119892 with 119892(119906) =

sum119901

119895=1119888119895119906(119905119895) satisfies condition (H3) And all the conditions

in Corollary 17 are satisfied So the conclusion holds

Corollary 19 Assume that conditions (H1) and (H4) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof According to Theorem 14 it is sufficient to prove thatthe hypothesis (H) is satisfied For arbitrary 120598 gt 0 there exists0 lt 120575 lt 119887 such that int120575

0

119906(119904)d119904 lt 120598 for all 119906 isin 119876119882119903 Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(46)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) which implies that (119876119882119903)120575is

precompact in 1198711

([0 119887] 119883) Thus 119876119882119903is precompact in

1198711

([0 119887] 119883) as it has an 120598-net (119876119882119903)120575 By condition 119892

(119862([0 119887] 119883) sdot 1198711) rarr 119883 is continuous and conv119876119882

119903sube

(119871)conv119876119882119903 it follows that condition (H) is satisfied where

(119871)conv119861 denotes the convex and closed hull of 119861 in1198711

([0 119887] 119883) Therefore the nonlocal problem (1) has at leastone mild solution on [0 119887]

Finally we give a simple example to illustrate our theory

Example 20 Consider the following fractional partial differ-ential heat equation in R

119863120572

119905119908 (119905 119909)

=1205972

119908 (119905 119909)

1205971199092

+ 1198691minus120572

119905119865 (119905 119908 (119905 119909)) 0 lt 119909 lt 1 0 lt 119905 lt 1

119908 (119905 0) = 119908 (119905 1) 1199081015840

119909(119905 0) = 119908

1015840

119909(119905 1) 0 le 119905 le 1

119908 (0 119909) = 1199080(119909) minus

119901

sum

119895=1

119888119895119908(119905119895 119909) 0 lt 119909 lt 1 0 lt 119905

119895le 1

(47)

where 0 lt 120572 lt 1 and 119863120572

119905denotes the Caputo fractional

derivativeWe introduce the abstract frame as follows Let 119883 = 119906 isin

119862([0 1]R) 119906(0) = 119906(1) Let 119860 be the linear operatorin 119883 defined by 119863(119860) = 119906 isin 119883 119906

1015840

11990610158401015840

isin 119883 andfor 119906 isin 119863(119860) 119860119906 = 119906

10158401015840 Then it is well known that 119860

generates a compact 1198620semigroup 119879(119905) for 119905 gt 0 on119883 From

the subordination principle [6 Theorems 31 and 33] 119860 alsogenerates a compact resolvent 119878

120572(119905) for 119905 gt 0

Assume that 119891 [0 1] times 119883 rarr 119883 is a continuousfunction defined by 119891(119905 119911)(119909) = 119865(119905 119911(119909)) 0 le 119909 le 1 and119892 119862([0 1] 119883) rarr 119883 is also a continuous function definedby 119892(119906)(119909) = 119908

0(119909) minus sum

119901

119895=1119888119895119906(119905119895)(119909) 0 le 119905 119909 le 1 where

119906(119905)(119909) = 119908(119905 119909)

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Differential Equations

Thus for 0 le ℎ le minℎ1 ℎ2 and 119909 le 1 it follows from

(14)ndash(16) that

1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905 + ℎ) 119909 minus 119878

120572(119905) 119909

1003817100381710038171003817

+ (119872 + 1)1003817100381710038171003817119878120572 (119905) 119909 minus 119878

120572(119905) 119909119894

1003817100381710038171003817

+1003817100381710038171003817119878120572 (119905) 119909119894 minus 119878

120572(ℎ) 119878120572(119905) 119909119894

1003817100381710038171003817

le120576

3+

120576

3+

120576

3

le 120576

(17)

which implies that limℎrarr0

+119878120572(119905+ℎ)minus119878

120572(ℎ)119878120572(119905) = 0 for all

119905 gt 0(iii) Let 119905 gt 0 and 0 lt ℎ lt min119905 119887 Then there exists

119872 gt 0 such that

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+1003817100381710038171003817119878120572 (ℎ) 119878120572 (119905) minus 119878

120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817

le1003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 + ℎ)

1003817100381710038171003817 +1003817100381710038171003817119878120572 (119905 + ℎ) minus 119878

120572(ℎ) 119878120572(119905)

1003817100381710038171003817

+ 1198721003817100381710038171003817119878120572 (119905) minus 119878

120572(119905 minus ℎ)

1003817100381710038171003817

(18)

It follows from (i) and (ii) that

limℎrarr0

+

1003817100381710038171003817119878120572 (119905) minus 119878120572(ℎ) 119878120572(119905 minus ℎ)

1003817100381710038171003817 = 0 for 119905 gt 0 (19)

This completes the proof

Lemma 11 Suppose 119878120572(119905) is a compact analytic resolvent of

analyticity type (1205960 1205790) Then Cauchy operator 119866 defined by

(9) is a compact operator

Proof We will show 119866 119862([0 119887] 119883) rarr 119862([0 119887] 119883) iscompact by using the Arzela-Ascoli theorem Let 119863

119897=

119891 isin 119862([0 119887] 119883) 119891 le 119897 be any bounded subset of119862([0 119887] 119883)

First we claim that the set 119866119863119897is equicontinuous on

119862([0 119887] 119883) In fact let 0 le 1199051

le 1199052

le 119887 and 119891 isin 119863119897 then

we have

119868 =1003817100381710038171003817(119866119891) (119905

2) minus (119866119891) (119905

1)1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

int

1199052

0

119878120572(1199052minus 119904) 119891 (119904) d119904 minus int

1199051

0

119878120572(1199051minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

1003817100381710038171003817119891 (119904)1003817100381710038171003817 d119904

+ int

1199052

1199051

1003817100381710038171003817119878120572 (1199052 minus 119904) 119891 (119904)1003817100381710038171003817 d119904

le 119897 int

1199051

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

(20)

If 1199051= 0 it is easy to see that

lim1199052rarr0

119868 = 0 uniformly for 119891 isin 119863119897 (21)

If 0 lt 1199051lt 119887 for 0 lt 120575 lt 119905

1 we have

119868 le 119897 int

1199051minus120575

0

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

1199051

1199051minus120575

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817 d119904 + 119872119897 (1199052minus 1199051)

le 119887119897 sup119904isin[01199051minus120575]

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572(1199051minus 119904)

1003817100381710038171003817

+ 2119872119897120575 + 119872119897 (1199052minus 1199051)

(22)

Note that from Lemma 10(i) we know 119878120572(119905) is operator norm

continuous uniformly for 119905 isin [120575 119887] Combining this andthe arbitrariness of 120575 with the above estimation on 119868 we canconclude that

lim|1199051minus1199052|rarr 0

119868 = 0 uniformly for 119891 isin 119863119897 (23)

Thus 119866119863119897is equicontinuous on 119862([0 119887] 119883)

Next we will show that the set (119866119891)(119905) 119891 isin 119863119897 is

precompact in 119883 for every 119905 isin [0 119887] It is easy to see that theset (119866119891)(0) 119891 isin 119863

119897 is precompact in 119883 Now let 0 lt 119905 le 119887

be given and 0 lt 120576 lt 119905 Then

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 119891 isin 119863

119897 (24)

International Journal of Differential Equations 5

is precompact since 119878120572(120576) is compact Moreover for arbitrary

120576 lt 120575 lt 119905 we have10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le 119897 int

119905minus120576

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

119905minus120576

119905minus120575

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904 + 119897120575 (1198722

+ 119872)

(25)

From Lemma 10(iii) we know

119878120572(120576) 119878120572(119905 minus 119904 minus 120576) minus 119878

120572(119905 minus 119904) 997888rarr 0

as 120576 997888rarr 0 for 119904 isin [0 119905 minus 120575]

(26)

Then it follows from the Lebesgue dominated convergencetheorem and the arbitrariness of 120575 that

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(27)

On the other hand10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+ 119897120576119872

(28)

Thus

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(29)

which implies that (119866119891)(119905) 119891 isin 119863119897 is precompact in 119883 by

using the total boundedness and therefore 119866 is compact inview of Arzela-Ascoli theorem

4 Nonlocal Problems

In this section we always assume that 1199060

isin 119883 and that theoperator 119860 generates a compact analytic resolvent 119878

120572(119905) of

analyticity type (1205960 1205790) and we will prove the existence of

mild solutions of (1) when the nonlocal item 119892 is assumed tobe Lipschitz continuous and neither Lipschitz nor compactrespectively

Let 119903 be a fixed positive real number and

119882119903= 119906 isin 119862 ([0 119887] 119883) 119906 le 119903 (30)

Clearly119882119903is a bounded closed and convex set We make the

following assumptions

(H1) 119891 [0 119887] times 119883 rarr 119883 is continuous

(H2) 119892 119862([0 119887] 119883) rarr 119883 is Lipschitz continuous withLipschitz constant 119896 such that 119872119896 lt 1

Under these assumptions we can prove the first mainresult in this paper

Theorem 12 Assume that conditions (H1) and (H2) aresatisfied Then the nonlocal problem (1) has at least one mildsolution provided that

119872[10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817] le 119903

(31)

Proof We consider the solution operator 119876 119862([0 119887] 119883) rarr

119862([0 119887] 119883) defined by

(119876119906) (119905) = 119878120572(119905) [1199060minus 119892 (119906)] + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119906 (119904)) d119904

= (1198761119906) (119905) + (119876

2119906) (119905) 119905 isin [0 119887]

(32)

It is easy to see that the fixed point of 119876 is the mild solutionof nonlocal Cauchy problem (1) Subsequently we will provethat119876 has a fixed point by using Lemma 9 (Darbo-Sadovskiirsquosfixed point theorem)

Firstly we prove that the mapping 119876 is continuous on119862([0 119887] 119883) For this purpose let 119906

119899119899ge1

be a sequence in119862([0 119887] 119883) with lim

119899rarrinfin119906119899= 119906 in 119862([0 119887] 119883) Then

1003817100381710038171003817119876119906119899minus 119876119906

1003817100381710038171003817

le 119872[1003817100381710038171003817119892 (119906119899)minus119892 (119906)

1003817100381710038171003817 + 119887 sup119904isin[0119887]

1003817100381710038171003817119891 (119904 119906119899(119904))minus119891 (119904 119906 (119904))

1003817100381710038171003817]

(33)

By the continuity of 119891 and 119892 we deduce that119876 is continuouson 119862([0 119887] 119883)

6 International Journal of Differential Equations

Secondly we claim that 119876119882119903sube 119882119903 where 119882

119903is defined

by (30) In fact for any 119906 isin 119882119903 by (31) we have

119876119906 le 119872(10038171003817100381710038171199060

1003817100381710038171003817 +1003817100381710038171003817119892 (119906)

1003817100381710038171003817 + int

119887

0

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817 d119904)

le 119872(10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817)

le 119903

(34)

which implies that 119876 maps 119882119903into itself

Now according to Lemma 9 it remains to prove that119876 isa 120573-contraction in 119882

119903 From condition (H2) we get that 119876

1

119882119903

rarr 119862([0 119887] 119883) is Lipschitz continuous with constant119872119896 In fact for 119906 V isin 119882

119903 by (H2) we have

10038171003817100381710038171198761119906 minus 1198761V1003817100381710038171003817 le 119872

1003817100381710038171003817119892 (119906) minus 119892 (V)1003817100381710038171003817 le 119872119896 119906 minus V (35)

Thus it follows from Lemma 8-(30) that 120573(1198761119882119903) le

119872119896120573(119882119903)

For operator 1198762

119882119903

rarr 119862([0 119887] 119883) we have 1198762119906 =

119866119891119906 where 119891

119906(sdot) = 119891(sdot 119906(sdot)) is continuous on [0 119887] and

119866 is the Cauchy operator defined by (9) Thus in view ofLemma 11 we know that 119876

2is compact on 119862([0 119887] 119883) and

hence 120573(1198762119882119903) = 0 Consequently

120573 (119876119882119903) le 120573 (119876

1119882119903) + 120573 (119876

2119882119903) le 119872119896120573 (119882

119903) (36)

Since 119872119896 lt 1 the mapping 119876 is a 120573-contraction on 119882119903 By

Darbo-Sadovskiirsquos fixed point theorem the operator 119876 has afixed point in 119882

119903 which is just the mild solution of nonlocal

Cauchy problem (1)

Now we give the following technical condition on func-tion 119892

(H) 119892 119862([0 119887] 119883) rarr 119883 is continuous and the set119892(conv119876119882

119903) is precompact where conv119861 denotes the

convex closed hull of set 119861 sube 119862([0 119879] 119883) and 119876 isgiven by (32)

Remark 13 It is easy to see that condition (H) is weaker thanthe compactness and convexity of 119892 The same hypothesiscan be seen from [24 30] where the authors considered theexistence of mild solutions for semilinear nonlocal problemsof integer order when 119860 is a linear densely defined operatoron119883 which generates a 119862

0-semigroup After the proof of our

main results we will give some special types of nonlocal item119892 which is neither Lipschitz nor compact but satisfies thecondition (H) in the next Corollaries

Theorem 14 Assume that conditions (H1) and (H) are sat-isfied Then the nonlocal problem (1) has at least one mildsolution provided that (31) holds

Proof We will prove that 119876 has a fixed point by usingSchauderrsquos fixed point theorem According to the proof ofTheorem 12 we have proven that 119876 119882

119903rarr 119882

119903is

continuous Next we will prove that there exists a set119882 sube 119882119903

such that 119876 119882 rarr 119882 is compactFor this purpose let 0 lt 119905 le 119887 and 0 lt 120575 lt 119905 It is easy to

see that the set

119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)] 119906 isin 119882

119903 (37)

is precompact since 119878120572(120575) is compact On the other hand by

Lemma 10(iii) we obtain

lim120575rarr0

1003817100381710038171003817119878120572 (119905) [1199060 minus 119892 (119906)]

minus119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)]

1003817100381710038171003817 = 0 forall119906 isin 119882119903

(38)

which implies that 1198761119882119903(119905) is precompact in 119883 by using the

total boundedness Next we claim 1198761119882119903is equicontinuous

on [120578 119887] for any small positive number 120578 In fact for 119906 isin 119882119903

and 120578 le 1199051le 1199052le 119887 we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817

le1003817100381710038171003817119878120572 (1199052) minus 119878

120572(1199051)1003817100381710038171003817 (

100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817119892 (119906)1003817100381710038171003817)

(39)

By Lemma 10(i) 119878120572(119905) is operator norm continuous for 119905 gt

0 Thus 119878120572(119905) is operator norm continuous uniformly for 119905 isin

[120578 119887] and hence

lim|1199052minus1199051|rarr 0

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 = 0

uniformly for 119906 isin 119882119903

(40)

that is 1198761119882119903is equicontinuous on [120578 119887] Note that 119876

2is

compact by Lemma 11Therefore we have proven that119876119882119903(119905)

is precompact for every 119905 isin (0 119887] and119876119882119903is equicontinuous

on [120578 119887] for any small positive number 120578Now let 119882 = conv119876119882

119903 we get that 119882 is a bounded

closed and convex subset of 119862([0 119887] 119883) and 119876119882 sube 119882 It iseasy to see that119876119882(119905) is precompact in119883 for every 119905 isin (0 119887]

and 119876119882 is equicontinuous on [120578 119887] for any small positivenumber 120578 Moreover we have that 119892(119882) = 119892(conv119876119882

119903) is

precompact due to condition (H)Thus we can now claim that 119876 119882 rarr 119882 is compact In

fact it is easy to see that 1198761119882(0) = 119906

0minus 119892(119882) is precompact

since119892(119882) = 119892(conv119876119882119903) is precompact It remains to prove

that 1198761119882 is equicontinuous on [0 119887] To this end let 119906 isin 119882

and 0 le 1199051lt 1199052le 119887 then we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 le

1003817100381710038171003817119878120572 (1199052) minus 119878120572(1199051)1003817100381710038171003817

10038171003817100381710038171199060 minus 119892 (119906)1003817100381710038171003817

(41)

In view of the compactness of 119892(119882) and the strong continuityof 119878120572(119905) on [0 119887] we obtain the equicontinuous of 119876

1119882 on

[0 119887] Thus 1198761

119882 rarr 119862([0 119887] 119883) is compact by Arzela-Ascoli theorem and hence 119876 119882 rarr 119882 is also compactNow Schauderrsquos fixed point theorem implies that 119876 has afixed point on 119882 which gives rise to a mild solution ofnonlocal problem (1)

The following theorem is a direct consequence ofTheorem 14

International Journal of Differential Equations 7

Theorem 15 Assume that conditions (H1) and (H) are satis-fied for each 119903 gt 0 If

1003817100381710038171003817119892 (119906)1003817100381710038171003817

119906997888rarr 0 119906 997888rarr infin

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817

119909997888rarr 0 119909 997888rarr infin

(42)

uniformly for 119905 isin [0 119887] then the nonlocal problem (1) has atleast one mild solution

Remark 16 It is easy to see that if there exist constants 1198711

1198712gt 0 and 120574

1 1205742isin [0 1) such that1003817100381710038171003817119892 (119906)

1003817100381710038171003817 le 1198711(1 + 119906)

1205741

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817 le 1198712(1 + 119909)

1205742

(43)

for 119905 isin [0 119887] then conditions (42) are satisfied

Next we will give special types of nonlocal item 119892 whichis neither Lipschitz nor compact but satisfies condition (H)

We give the following assumptions

(H3) 119892 119862([0 119887] 119883) rarr 119883 is a continuous mapping whichmaps119882

119903into a bounded set and there is a 120575 = 120575(119903) isin

(0 119887) such that 119892(119906) = 119892(V) for any 119906 V isin 119882119903with

119906(119904) = V(119904) 119904 isin [120575 119887](H4) 119892 (119862([0 119887] 119883) sdot

1198711) rarr 119883 is continuous

Corollary 17 Assume that conditions (H1) and (H3) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(44)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) Moreover by condition (H3)119892(conv119876119882

119903) = 119892(conv(119876119882

119903)120575) is also precompact in

119862([0 119887] 119883) Thus all the hypotheses in Theorem 14 aresatisfied Therefore there is at least one mild solution ofnonlocal problem (1)

Corollary 18 Let condition (H1) be satisfied Suppose that119892(119906) = sum

119901

119895=1119888119895119906(119905119895) where 119888

119895are given positive constants and

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119901le 119887 Then the nonlocal problem (1) has

at least one mild solution on [0 119887] provided that

119872[

[

100381710038171003817100381711990601003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (45)

Proof It is easy to see that the mapping 119892 with 119892(119906) =

sum119901

119895=1119888119895119906(119905119895) satisfies condition (H3) And all the conditions

in Corollary 17 are satisfied So the conclusion holds

Corollary 19 Assume that conditions (H1) and (H4) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof According to Theorem 14 it is sufficient to prove thatthe hypothesis (H) is satisfied For arbitrary 120598 gt 0 there exists0 lt 120575 lt 119887 such that int120575

0

119906(119904)d119904 lt 120598 for all 119906 isin 119876119882119903 Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(46)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) which implies that (119876119882119903)120575is

precompact in 1198711

([0 119887] 119883) Thus 119876119882119903is precompact in

1198711

([0 119887] 119883) as it has an 120598-net (119876119882119903)120575 By condition 119892

(119862([0 119887] 119883) sdot 1198711) rarr 119883 is continuous and conv119876119882

119903sube

(119871)conv119876119882119903 it follows that condition (H) is satisfied where

(119871)conv119861 denotes the convex and closed hull of 119861 in1198711

([0 119887] 119883) Therefore the nonlocal problem (1) has at leastone mild solution on [0 119887]

Finally we give a simple example to illustrate our theory

Example 20 Consider the following fractional partial differ-ential heat equation in R

119863120572

119905119908 (119905 119909)

=1205972

119908 (119905 119909)

1205971199092

+ 1198691minus120572

119905119865 (119905 119908 (119905 119909)) 0 lt 119909 lt 1 0 lt 119905 lt 1

119908 (119905 0) = 119908 (119905 1) 1199081015840

119909(119905 0) = 119908

1015840

119909(119905 1) 0 le 119905 le 1

119908 (0 119909) = 1199080(119909) minus

119901

sum

119895=1

119888119895119908(119905119895 119909) 0 lt 119909 lt 1 0 lt 119905

119895le 1

(47)

where 0 lt 120572 lt 1 and 119863120572

119905denotes the Caputo fractional

derivativeWe introduce the abstract frame as follows Let 119883 = 119906 isin

119862([0 1]R) 119906(0) = 119906(1) Let 119860 be the linear operatorin 119883 defined by 119863(119860) = 119906 isin 119883 119906

1015840

11990610158401015840

isin 119883 andfor 119906 isin 119863(119860) 119860119906 = 119906

10158401015840 Then it is well known that 119860

generates a compact 1198620semigroup 119879(119905) for 119905 gt 0 on119883 From

the subordination principle [6 Theorems 31 and 33] 119860 alsogenerates a compact resolvent 119878

120572(119905) for 119905 gt 0

Assume that 119891 [0 1] times 119883 rarr 119883 is a continuousfunction defined by 119891(119905 119911)(119909) = 119865(119905 119911(119909)) 0 le 119909 le 1 and119892 119862([0 1] 119883) rarr 119883 is also a continuous function definedby 119892(119906)(119909) = 119908

0(119909) minus sum

119901

119895=1119888119895119906(119905119895)(119909) 0 le 119905 119909 le 1 where

119906(119905)(119909) = 119908(119905 119909)

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 5

is precompact since 119878120572(120576) is compact Moreover for arbitrary

120576 lt 120575 lt 119905 we have10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le 119897 int

119905minus120576

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

+ 119897 int

119905minus120576

119905minus120575

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904

le 119897 int

119905minus120575

0

1003817100381710038171003817119878120572 (120576) 119878120572 (119905 minus 119904 minus 120576) minus 119878120572(119905 minus 119904)

1003817100381710038171003817 d119904 + 119897120575 (1198722

+ 119872)

(25)

From Lemma 10(iii) we know

119878120572(120576) 119878120572(119905 minus 119904 minus 120576) minus 119878

120572(119905 minus 119904) 997888rarr 0

as 120576 997888rarr 0 for 119904 isin [0 119905 minus 120575]

(26)

Then it follows from the Lebesgue dominated convergencetheorem and the arbitrariness of 120575 that

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(27)

On the other hand10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817

int

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904 minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905minus120576

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

+ 119897120576119872

(28)

Thus

lim120576rarr0

10038171003817100381710038171003817100381710038171003817

119878120572(120576) int

119905minus120576

0

119878120572(119905 minus 119904 minus 120576) 119891 (119904) d119904

minusint

119905

0

119878120572(119905 minus 119904) 119891 (119904) d119904

10038171003817100381710038171003817100381710038171003817

= 0

(29)

which implies that (119866119891)(119905) 119891 isin 119863119897 is precompact in 119883 by

using the total boundedness and therefore 119866 is compact inview of Arzela-Ascoli theorem

4 Nonlocal Problems

In this section we always assume that 1199060

isin 119883 and that theoperator 119860 generates a compact analytic resolvent 119878

120572(119905) of

analyticity type (1205960 1205790) and we will prove the existence of

mild solutions of (1) when the nonlocal item 119892 is assumed tobe Lipschitz continuous and neither Lipschitz nor compactrespectively

Let 119903 be a fixed positive real number and

119882119903= 119906 isin 119862 ([0 119887] 119883) 119906 le 119903 (30)

Clearly119882119903is a bounded closed and convex set We make the

following assumptions

(H1) 119891 [0 119887] times 119883 rarr 119883 is continuous

(H2) 119892 119862([0 119887] 119883) rarr 119883 is Lipschitz continuous withLipschitz constant 119896 such that 119872119896 lt 1

Under these assumptions we can prove the first mainresult in this paper

Theorem 12 Assume that conditions (H1) and (H2) aresatisfied Then the nonlocal problem (1) has at least one mildsolution provided that

119872[10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817] le 119903

(31)

Proof We consider the solution operator 119876 119862([0 119887] 119883) rarr

119862([0 119887] 119883) defined by

(119876119906) (119905) = 119878120572(119905) [1199060minus 119892 (119906)] + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119906 (119904)) d119904

= (1198761119906) (119905) + (119876

2119906) (119905) 119905 isin [0 119887]

(32)

It is easy to see that the fixed point of 119876 is the mild solutionof nonlocal Cauchy problem (1) Subsequently we will provethat119876 has a fixed point by using Lemma 9 (Darbo-Sadovskiirsquosfixed point theorem)

Firstly we prove that the mapping 119876 is continuous on119862([0 119887] 119883) For this purpose let 119906

119899119899ge1

be a sequence in119862([0 119887] 119883) with lim

119899rarrinfin119906119899= 119906 in 119862([0 119887] 119883) Then

1003817100381710038171003817119876119906119899minus 119876119906

1003817100381710038171003817

le 119872[1003817100381710038171003817119892 (119906119899)minus119892 (119906)

1003817100381710038171003817 + 119887 sup119904isin[0119887]

1003817100381710038171003817119891 (119904 119906119899(119904))minus119891 (119904 119906 (119904))

1003817100381710038171003817]

(33)

By the continuity of 119891 and 119892 we deduce that119876 is continuouson 119862([0 119887] 119883)

6 International Journal of Differential Equations

Secondly we claim that 119876119882119903sube 119882119903 where 119882

119903is defined

by (30) In fact for any 119906 isin 119882119903 by (31) we have

119876119906 le 119872(10038171003817100381710038171199060

1003817100381710038171003817 +1003817100381710038171003817119892 (119906)

1003817100381710038171003817 + int

119887

0

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817 d119904)

le 119872(10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817)

le 119903

(34)

which implies that 119876 maps 119882119903into itself

Now according to Lemma 9 it remains to prove that119876 isa 120573-contraction in 119882

119903 From condition (H2) we get that 119876

1

119882119903

rarr 119862([0 119887] 119883) is Lipschitz continuous with constant119872119896 In fact for 119906 V isin 119882

119903 by (H2) we have

10038171003817100381710038171198761119906 minus 1198761V1003817100381710038171003817 le 119872

1003817100381710038171003817119892 (119906) minus 119892 (V)1003817100381710038171003817 le 119872119896 119906 minus V (35)

Thus it follows from Lemma 8-(30) that 120573(1198761119882119903) le

119872119896120573(119882119903)

For operator 1198762

119882119903

rarr 119862([0 119887] 119883) we have 1198762119906 =

119866119891119906 where 119891

119906(sdot) = 119891(sdot 119906(sdot)) is continuous on [0 119887] and

119866 is the Cauchy operator defined by (9) Thus in view ofLemma 11 we know that 119876

2is compact on 119862([0 119887] 119883) and

hence 120573(1198762119882119903) = 0 Consequently

120573 (119876119882119903) le 120573 (119876

1119882119903) + 120573 (119876

2119882119903) le 119872119896120573 (119882

119903) (36)

Since 119872119896 lt 1 the mapping 119876 is a 120573-contraction on 119882119903 By

Darbo-Sadovskiirsquos fixed point theorem the operator 119876 has afixed point in 119882

119903 which is just the mild solution of nonlocal

Cauchy problem (1)

Now we give the following technical condition on func-tion 119892

(H) 119892 119862([0 119887] 119883) rarr 119883 is continuous and the set119892(conv119876119882

119903) is precompact where conv119861 denotes the

convex closed hull of set 119861 sube 119862([0 119879] 119883) and 119876 isgiven by (32)

Remark 13 It is easy to see that condition (H) is weaker thanthe compactness and convexity of 119892 The same hypothesiscan be seen from [24 30] where the authors considered theexistence of mild solutions for semilinear nonlocal problemsof integer order when 119860 is a linear densely defined operatoron119883 which generates a 119862

0-semigroup After the proof of our

main results we will give some special types of nonlocal item119892 which is neither Lipschitz nor compact but satisfies thecondition (H) in the next Corollaries

Theorem 14 Assume that conditions (H1) and (H) are sat-isfied Then the nonlocal problem (1) has at least one mildsolution provided that (31) holds

Proof We will prove that 119876 has a fixed point by usingSchauderrsquos fixed point theorem According to the proof ofTheorem 12 we have proven that 119876 119882

119903rarr 119882

119903is

continuous Next we will prove that there exists a set119882 sube 119882119903

such that 119876 119882 rarr 119882 is compactFor this purpose let 0 lt 119905 le 119887 and 0 lt 120575 lt 119905 It is easy to

see that the set

119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)] 119906 isin 119882

119903 (37)

is precompact since 119878120572(120575) is compact On the other hand by

Lemma 10(iii) we obtain

lim120575rarr0

1003817100381710038171003817119878120572 (119905) [1199060 minus 119892 (119906)]

minus119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)]

1003817100381710038171003817 = 0 forall119906 isin 119882119903

(38)

which implies that 1198761119882119903(119905) is precompact in 119883 by using the

total boundedness Next we claim 1198761119882119903is equicontinuous

on [120578 119887] for any small positive number 120578 In fact for 119906 isin 119882119903

and 120578 le 1199051le 1199052le 119887 we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817

le1003817100381710038171003817119878120572 (1199052) minus 119878

120572(1199051)1003817100381710038171003817 (

100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817119892 (119906)1003817100381710038171003817)

(39)

By Lemma 10(i) 119878120572(119905) is operator norm continuous for 119905 gt

0 Thus 119878120572(119905) is operator norm continuous uniformly for 119905 isin

[120578 119887] and hence

lim|1199052minus1199051|rarr 0

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 = 0

uniformly for 119906 isin 119882119903

(40)

that is 1198761119882119903is equicontinuous on [120578 119887] Note that 119876

2is

compact by Lemma 11Therefore we have proven that119876119882119903(119905)

is precompact for every 119905 isin (0 119887] and119876119882119903is equicontinuous

on [120578 119887] for any small positive number 120578Now let 119882 = conv119876119882

119903 we get that 119882 is a bounded

closed and convex subset of 119862([0 119887] 119883) and 119876119882 sube 119882 It iseasy to see that119876119882(119905) is precompact in119883 for every 119905 isin (0 119887]

and 119876119882 is equicontinuous on [120578 119887] for any small positivenumber 120578 Moreover we have that 119892(119882) = 119892(conv119876119882

119903) is

precompact due to condition (H)Thus we can now claim that 119876 119882 rarr 119882 is compact In

fact it is easy to see that 1198761119882(0) = 119906

0minus 119892(119882) is precompact

since119892(119882) = 119892(conv119876119882119903) is precompact It remains to prove

that 1198761119882 is equicontinuous on [0 119887] To this end let 119906 isin 119882

and 0 le 1199051lt 1199052le 119887 then we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 le

1003817100381710038171003817119878120572 (1199052) minus 119878120572(1199051)1003817100381710038171003817

10038171003817100381710038171199060 minus 119892 (119906)1003817100381710038171003817

(41)

In view of the compactness of 119892(119882) and the strong continuityof 119878120572(119905) on [0 119887] we obtain the equicontinuous of 119876

1119882 on

[0 119887] Thus 1198761

119882 rarr 119862([0 119887] 119883) is compact by Arzela-Ascoli theorem and hence 119876 119882 rarr 119882 is also compactNow Schauderrsquos fixed point theorem implies that 119876 has afixed point on 119882 which gives rise to a mild solution ofnonlocal problem (1)

The following theorem is a direct consequence ofTheorem 14

International Journal of Differential Equations 7

Theorem 15 Assume that conditions (H1) and (H) are satis-fied for each 119903 gt 0 If

1003817100381710038171003817119892 (119906)1003817100381710038171003817

119906997888rarr 0 119906 997888rarr infin

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817

119909997888rarr 0 119909 997888rarr infin

(42)

uniformly for 119905 isin [0 119887] then the nonlocal problem (1) has atleast one mild solution

Remark 16 It is easy to see that if there exist constants 1198711

1198712gt 0 and 120574

1 1205742isin [0 1) such that1003817100381710038171003817119892 (119906)

1003817100381710038171003817 le 1198711(1 + 119906)

1205741

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817 le 1198712(1 + 119909)

1205742

(43)

for 119905 isin [0 119887] then conditions (42) are satisfied

Next we will give special types of nonlocal item 119892 whichis neither Lipschitz nor compact but satisfies condition (H)

We give the following assumptions

(H3) 119892 119862([0 119887] 119883) rarr 119883 is a continuous mapping whichmaps119882

119903into a bounded set and there is a 120575 = 120575(119903) isin

(0 119887) such that 119892(119906) = 119892(V) for any 119906 V isin 119882119903with

119906(119904) = V(119904) 119904 isin [120575 119887](H4) 119892 (119862([0 119887] 119883) sdot

1198711) rarr 119883 is continuous

Corollary 17 Assume that conditions (H1) and (H3) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(44)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) Moreover by condition (H3)119892(conv119876119882

119903) = 119892(conv(119876119882

119903)120575) is also precompact in

119862([0 119887] 119883) Thus all the hypotheses in Theorem 14 aresatisfied Therefore there is at least one mild solution ofnonlocal problem (1)

Corollary 18 Let condition (H1) be satisfied Suppose that119892(119906) = sum

119901

119895=1119888119895119906(119905119895) where 119888

119895are given positive constants and

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119901le 119887 Then the nonlocal problem (1) has

at least one mild solution on [0 119887] provided that

119872[

[

100381710038171003817100381711990601003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (45)

Proof It is easy to see that the mapping 119892 with 119892(119906) =

sum119901

119895=1119888119895119906(119905119895) satisfies condition (H3) And all the conditions

in Corollary 17 are satisfied So the conclusion holds

Corollary 19 Assume that conditions (H1) and (H4) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof According to Theorem 14 it is sufficient to prove thatthe hypothesis (H) is satisfied For arbitrary 120598 gt 0 there exists0 lt 120575 lt 119887 such that int120575

0

119906(119904)d119904 lt 120598 for all 119906 isin 119876119882119903 Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(46)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) which implies that (119876119882119903)120575is

precompact in 1198711

([0 119887] 119883) Thus 119876119882119903is precompact in

1198711

([0 119887] 119883) as it has an 120598-net (119876119882119903)120575 By condition 119892

(119862([0 119887] 119883) sdot 1198711) rarr 119883 is continuous and conv119876119882

119903sube

(119871)conv119876119882119903 it follows that condition (H) is satisfied where

(119871)conv119861 denotes the convex and closed hull of 119861 in1198711

([0 119887] 119883) Therefore the nonlocal problem (1) has at leastone mild solution on [0 119887]

Finally we give a simple example to illustrate our theory

Example 20 Consider the following fractional partial differ-ential heat equation in R

119863120572

119905119908 (119905 119909)

=1205972

119908 (119905 119909)

1205971199092

+ 1198691minus120572

119905119865 (119905 119908 (119905 119909)) 0 lt 119909 lt 1 0 lt 119905 lt 1

119908 (119905 0) = 119908 (119905 1) 1199081015840

119909(119905 0) = 119908

1015840

119909(119905 1) 0 le 119905 le 1

119908 (0 119909) = 1199080(119909) minus

119901

sum

119895=1

119888119895119908(119905119895 119909) 0 lt 119909 lt 1 0 lt 119905

119895le 1

(47)

where 0 lt 120572 lt 1 and 119863120572

119905denotes the Caputo fractional

derivativeWe introduce the abstract frame as follows Let 119883 = 119906 isin

119862([0 1]R) 119906(0) = 119906(1) Let 119860 be the linear operatorin 119883 defined by 119863(119860) = 119906 isin 119883 119906

1015840

11990610158401015840

isin 119883 andfor 119906 isin 119863(119860) 119860119906 = 119906

10158401015840 Then it is well known that 119860

generates a compact 1198620semigroup 119879(119905) for 119905 gt 0 on119883 From

the subordination principle [6 Theorems 31 and 33] 119860 alsogenerates a compact resolvent 119878

120572(119905) for 119905 gt 0

Assume that 119891 [0 1] times 119883 rarr 119883 is a continuousfunction defined by 119891(119905 119911)(119909) = 119865(119905 119911(119909)) 0 le 119909 le 1 and119892 119862([0 1] 119883) rarr 119883 is also a continuous function definedby 119892(119906)(119909) = 119908

0(119909) minus sum

119901

119895=1119888119895119906(119905119895)(119909) 0 le 119905 119909 le 1 where

119906(119905)(119909) = 119908(119905 119909)

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Differential Equations

Secondly we claim that 119876119882119903sube 119882119903 where 119882

119903is defined

by (30) In fact for any 119906 isin 119882119903 by (31) we have

119876119906 le 119872(10038171003817100381710038171199060

1003817100381710038171003817 +1003817100381710038171003817119892 (119906)

1003817100381710038171003817 + int

119887

0

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817 d119904)

le 119872(10038171003817100381710038171199060

1003817100381710038171003817 + sup119906isin119882119903

1003817100381710038171003817119892 (119906)1003817100381710038171003817 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817)

le 119903

(34)

which implies that 119876 maps 119882119903into itself

Now according to Lemma 9 it remains to prove that119876 isa 120573-contraction in 119882

119903 From condition (H2) we get that 119876

1

119882119903

rarr 119862([0 119887] 119883) is Lipschitz continuous with constant119872119896 In fact for 119906 V isin 119882

119903 by (H2) we have

10038171003817100381710038171198761119906 minus 1198761V1003817100381710038171003817 le 119872

1003817100381710038171003817119892 (119906) minus 119892 (V)1003817100381710038171003817 le 119872119896 119906 minus V (35)

Thus it follows from Lemma 8-(30) that 120573(1198761119882119903) le

119872119896120573(119882119903)

For operator 1198762

119882119903

rarr 119862([0 119887] 119883) we have 1198762119906 =

119866119891119906 where 119891

119906(sdot) = 119891(sdot 119906(sdot)) is continuous on [0 119887] and

119866 is the Cauchy operator defined by (9) Thus in view ofLemma 11 we know that 119876

2is compact on 119862([0 119887] 119883) and

hence 120573(1198762119882119903) = 0 Consequently

120573 (119876119882119903) le 120573 (119876

1119882119903) + 120573 (119876

2119882119903) le 119872119896120573 (119882

119903) (36)

Since 119872119896 lt 1 the mapping 119876 is a 120573-contraction on 119882119903 By

Darbo-Sadovskiirsquos fixed point theorem the operator 119876 has afixed point in 119882

119903 which is just the mild solution of nonlocal

Cauchy problem (1)

Now we give the following technical condition on func-tion 119892

(H) 119892 119862([0 119887] 119883) rarr 119883 is continuous and the set119892(conv119876119882

119903) is precompact where conv119861 denotes the

convex closed hull of set 119861 sube 119862([0 119879] 119883) and 119876 isgiven by (32)

Remark 13 It is easy to see that condition (H) is weaker thanthe compactness and convexity of 119892 The same hypothesiscan be seen from [24 30] where the authors considered theexistence of mild solutions for semilinear nonlocal problemsof integer order when 119860 is a linear densely defined operatoron119883 which generates a 119862

0-semigroup After the proof of our

main results we will give some special types of nonlocal item119892 which is neither Lipschitz nor compact but satisfies thecondition (H) in the next Corollaries

Theorem 14 Assume that conditions (H1) and (H) are sat-isfied Then the nonlocal problem (1) has at least one mildsolution provided that (31) holds

Proof We will prove that 119876 has a fixed point by usingSchauderrsquos fixed point theorem According to the proof ofTheorem 12 we have proven that 119876 119882

119903rarr 119882

119903is

continuous Next we will prove that there exists a set119882 sube 119882119903

such that 119876 119882 rarr 119882 is compactFor this purpose let 0 lt 119905 le 119887 and 0 lt 120575 lt 119905 It is easy to

see that the set

119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)] 119906 isin 119882

119903 (37)

is precompact since 119878120572(120575) is compact On the other hand by

Lemma 10(iii) we obtain

lim120575rarr0

1003817100381710038171003817119878120572 (119905) [1199060 minus 119892 (119906)]

minus119878120572(120575) 119878120572(119905 minus 120575) [119906

0minus 119892 (119906)]

1003817100381710038171003817 = 0 forall119906 isin 119882119903

(38)

which implies that 1198761119882119903(119905) is precompact in 119883 by using the

total boundedness Next we claim 1198761119882119903is equicontinuous

on [120578 119887] for any small positive number 120578 In fact for 119906 isin 119882119903

and 120578 le 1199051le 1199052le 119887 we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817

le1003817100381710038171003817119878120572 (1199052) minus 119878

120572(1199051)1003817100381710038171003817 (

100381710038171003817100381711990601003817100381710038171003817 +

1003817100381710038171003817119892 (119906)1003817100381710038171003817)

(39)

By Lemma 10(i) 119878120572(119905) is operator norm continuous for 119905 gt

0 Thus 119878120572(119905) is operator norm continuous uniformly for 119905 isin

[120578 119887] and hence

lim|1199052minus1199051|rarr 0

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 = 0

uniformly for 119906 isin 119882119903

(40)

that is 1198761119882119903is equicontinuous on [120578 119887] Note that 119876

2is

compact by Lemma 11Therefore we have proven that119876119882119903(119905)

is precompact for every 119905 isin (0 119887] and119876119882119903is equicontinuous

on [120578 119887] for any small positive number 120578Now let 119882 = conv119876119882

119903 we get that 119882 is a bounded

closed and convex subset of 119862([0 119887] 119883) and 119876119882 sube 119882 It iseasy to see that119876119882(119905) is precompact in119883 for every 119905 isin (0 119887]

and 119876119882 is equicontinuous on [120578 119887] for any small positivenumber 120578 Moreover we have that 119892(119882) = 119892(conv119876119882

119903) is

precompact due to condition (H)Thus we can now claim that 119876 119882 rarr 119882 is compact In

fact it is easy to see that 1198761119882(0) = 119906

0minus 119892(119882) is precompact

since119892(119882) = 119892(conv119876119882119903) is precompact It remains to prove

that 1198761119882 is equicontinuous on [0 119887] To this end let 119906 isin 119882

and 0 le 1199051lt 1199052le 119887 then we have

1003817100381710038171003817(1198761119906) (1199052) minus (1198761119906) (1199051)1003817100381710038171003817 le

1003817100381710038171003817119878120572 (1199052) minus 119878120572(1199051)1003817100381710038171003817

10038171003817100381710038171199060 minus 119892 (119906)1003817100381710038171003817

(41)

In view of the compactness of 119892(119882) and the strong continuityof 119878120572(119905) on [0 119887] we obtain the equicontinuous of 119876

1119882 on

[0 119887] Thus 1198761

119882 rarr 119862([0 119887] 119883) is compact by Arzela-Ascoli theorem and hence 119876 119882 rarr 119882 is also compactNow Schauderrsquos fixed point theorem implies that 119876 has afixed point on 119882 which gives rise to a mild solution ofnonlocal problem (1)

The following theorem is a direct consequence ofTheorem 14

International Journal of Differential Equations 7

Theorem 15 Assume that conditions (H1) and (H) are satis-fied for each 119903 gt 0 If

1003817100381710038171003817119892 (119906)1003817100381710038171003817

119906997888rarr 0 119906 997888rarr infin

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817

119909997888rarr 0 119909 997888rarr infin

(42)

uniformly for 119905 isin [0 119887] then the nonlocal problem (1) has atleast one mild solution

Remark 16 It is easy to see that if there exist constants 1198711

1198712gt 0 and 120574

1 1205742isin [0 1) such that1003817100381710038171003817119892 (119906)

1003817100381710038171003817 le 1198711(1 + 119906)

1205741

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817 le 1198712(1 + 119909)

1205742

(43)

for 119905 isin [0 119887] then conditions (42) are satisfied

Next we will give special types of nonlocal item 119892 whichis neither Lipschitz nor compact but satisfies condition (H)

We give the following assumptions

(H3) 119892 119862([0 119887] 119883) rarr 119883 is a continuous mapping whichmaps119882

119903into a bounded set and there is a 120575 = 120575(119903) isin

(0 119887) such that 119892(119906) = 119892(V) for any 119906 V isin 119882119903with

119906(119904) = V(119904) 119904 isin [120575 119887](H4) 119892 (119862([0 119887] 119883) sdot

1198711) rarr 119883 is continuous

Corollary 17 Assume that conditions (H1) and (H3) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(44)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) Moreover by condition (H3)119892(conv119876119882

119903) = 119892(conv(119876119882

119903)120575) is also precompact in

119862([0 119887] 119883) Thus all the hypotheses in Theorem 14 aresatisfied Therefore there is at least one mild solution ofnonlocal problem (1)

Corollary 18 Let condition (H1) be satisfied Suppose that119892(119906) = sum

119901

119895=1119888119895119906(119905119895) where 119888

119895are given positive constants and

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119901le 119887 Then the nonlocal problem (1) has

at least one mild solution on [0 119887] provided that

119872[

[

100381710038171003817100381711990601003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (45)

Proof It is easy to see that the mapping 119892 with 119892(119906) =

sum119901

119895=1119888119895119906(119905119895) satisfies condition (H3) And all the conditions

in Corollary 17 are satisfied So the conclusion holds

Corollary 19 Assume that conditions (H1) and (H4) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof According to Theorem 14 it is sufficient to prove thatthe hypothesis (H) is satisfied For arbitrary 120598 gt 0 there exists0 lt 120575 lt 119887 such that int120575

0

119906(119904)d119904 lt 120598 for all 119906 isin 119876119882119903 Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(46)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) which implies that (119876119882119903)120575is

precompact in 1198711

([0 119887] 119883) Thus 119876119882119903is precompact in

1198711

([0 119887] 119883) as it has an 120598-net (119876119882119903)120575 By condition 119892

(119862([0 119887] 119883) sdot 1198711) rarr 119883 is continuous and conv119876119882

119903sube

(119871)conv119876119882119903 it follows that condition (H) is satisfied where

(119871)conv119861 denotes the convex and closed hull of 119861 in1198711

([0 119887] 119883) Therefore the nonlocal problem (1) has at leastone mild solution on [0 119887]

Finally we give a simple example to illustrate our theory

Example 20 Consider the following fractional partial differ-ential heat equation in R

119863120572

119905119908 (119905 119909)

=1205972

119908 (119905 119909)

1205971199092

+ 1198691minus120572

119905119865 (119905 119908 (119905 119909)) 0 lt 119909 lt 1 0 lt 119905 lt 1

119908 (119905 0) = 119908 (119905 1) 1199081015840

119909(119905 0) = 119908

1015840

119909(119905 1) 0 le 119905 le 1

119908 (0 119909) = 1199080(119909) minus

119901

sum

119895=1

119888119895119908(119905119895 119909) 0 lt 119909 lt 1 0 lt 119905

119895le 1

(47)

where 0 lt 120572 lt 1 and 119863120572

119905denotes the Caputo fractional

derivativeWe introduce the abstract frame as follows Let 119883 = 119906 isin

119862([0 1]R) 119906(0) = 119906(1) Let 119860 be the linear operatorin 119883 defined by 119863(119860) = 119906 isin 119883 119906

1015840

11990610158401015840

isin 119883 andfor 119906 isin 119863(119860) 119860119906 = 119906

10158401015840 Then it is well known that 119860

generates a compact 1198620semigroup 119879(119905) for 119905 gt 0 on119883 From

the subordination principle [6 Theorems 31 and 33] 119860 alsogenerates a compact resolvent 119878

120572(119905) for 119905 gt 0

Assume that 119891 [0 1] times 119883 rarr 119883 is a continuousfunction defined by 119891(119905 119911)(119909) = 119865(119905 119911(119909)) 0 le 119909 le 1 and119892 119862([0 1] 119883) rarr 119883 is also a continuous function definedby 119892(119906)(119909) = 119908

0(119909) minus sum

119901

119895=1119888119895119906(119905119895)(119909) 0 le 119905 119909 le 1 where

119906(119905)(119909) = 119908(119905 119909)

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 7

Theorem 15 Assume that conditions (H1) and (H) are satis-fied for each 119903 gt 0 If

1003817100381710038171003817119892 (119906)1003817100381710038171003817

119906997888rarr 0 119906 997888rarr infin

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817

119909997888rarr 0 119909 997888rarr infin

(42)

uniformly for 119905 isin [0 119887] then the nonlocal problem (1) has atleast one mild solution

Remark 16 It is easy to see that if there exist constants 1198711

1198712gt 0 and 120574

1 1205742isin [0 1) such that1003817100381710038171003817119892 (119906)

1003817100381710038171003817 le 1198711(1 + 119906)

1205741

1003817100381710038171003817119891 (119905 119909)1003817100381710038171003817 le 1198712(1 + 119909)

1205742

(43)

for 119905 isin [0 119887] then conditions (42) are satisfied

Next we will give special types of nonlocal item 119892 whichis neither Lipschitz nor compact but satisfies condition (H)

We give the following assumptions

(H3) 119892 119862([0 119887] 119883) rarr 119883 is a continuous mapping whichmaps119882

119903into a bounded set and there is a 120575 = 120575(119903) isin

(0 119887) such that 119892(119906) = 119892(V) for any 119906 V isin 119882119903with

119906(119904) = V(119904) 119904 isin [120575 119887](H4) 119892 (119862([0 119887] 119883) sdot

1198711) rarr 119883 is continuous

Corollary 17 Assume that conditions (H1) and (H3) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(44)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) Moreover by condition (H3)119892(conv119876119882

119903) = 119892(conv(119876119882

119903)120575) is also precompact in

119862([0 119887] 119883) Thus all the hypotheses in Theorem 14 aresatisfied Therefore there is at least one mild solution ofnonlocal problem (1)

Corollary 18 Let condition (H1) be satisfied Suppose that119892(119906) = sum

119901

119895=1119888119895119906(119905119895) where 119888

119895are given positive constants and

0 lt 1199051lt 1199052lt sdot sdot sdot lt 119905

119901le 119887 Then the nonlocal problem (1) has

at least one mild solution on [0 119887] provided that

119872[

[

100381710038171003817100381711990601003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + 119887 sup119904isin[0119887]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (45)

Proof It is easy to see that the mapping 119892 with 119892(119906) =

sum119901

119895=1119888119895119906(119905119895) satisfies condition (H3) And all the conditions

in Corollary 17 are satisfied So the conclusion holds

Corollary 19 Assume that conditions (H1) and (H4) aresatisfied Then the nonlocal problem (1) has at least one mildsolution on [0 119887] provided that (31) holds

Proof According to Theorem 14 it is sufficient to prove thatthe hypothesis (H) is satisfied For arbitrary 120598 gt 0 there exists0 lt 120575 lt 119887 such that int120575

0

119906(119904)d119904 lt 120598 for all 119906 isin 119876119882119903 Let

(119876119882119903)120575= 119906 isin 119862 ([0 119887] 119883) 119906 (119905) = V (119905) for 119905 isin [120575 119887]

119906 (119905) = 119906 (120575) for 119905 isin [0 120575) where V isin 119876119882119903

(46)

From the proof of Theorem 14 we know that (119876119882119903)120575is

precompact in 119862([0 119887] 119883) which implies that (119876119882119903)120575is

precompact in 1198711

([0 119887] 119883) Thus 119876119882119903is precompact in

1198711

([0 119887] 119883) as it has an 120598-net (119876119882119903)120575 By condition 119892

(119862([0 119887] 119883) sdot 1198711) rarr 119883 is continuous and conv119876119882

119903sube

(119871)conv119876119882119903 it follows that condition (H) is satisfied where

(119871)conv119861 denotes the convex and closed hull of 119861 in1198711

([0 119887] 119883) Therefore the nonlocal problem (1) has at leastone mild solution on [0 119887]

Finally we give a simple example to illustrate our theory

Example 20 Consider the following fractional partial differ-ential heat equation in R

119863120572

119905119908 (119905 119909)

=1205972

119908 (119905 119909)

1205971199092

+ 1198691minus120572

119905119865 (119905 119908 (119905 119909)) 0 lt 119909 lt 1 0 lt 119905 lt 1

119908 (119905 0) = 119908 (119905 1) 1199081015840

119909(119905 0) = 119908

1015840

119909(119905 1) 0 le 119905 le 1

119908 (0 119909) = 1199080(119909) minus

119901

sum

119895=1

119888119895119908(119905119895 119909) 0 lt 119909 lt 1 0 lt 119905

119895le 1

(47)

where 0 lt 120572 lt 1 and 119863120572

119905denotes the Caputo fractional

derivativeWe introduce the abstract frame as follows Let 119883 = 119906 isin

119862([0 1]R) 119906(0) = 119906(1) Let 119860 be the linear operatorin 119883 defined by 119863(119860) = 119906 isin 119883 119906

1015840

11990610158401015840

isin 119883 andfor 119906 isin 119863(119860) 119860119906 = 119906

10158401015840 Then it is well known that 119860

generates a compact 1198620semigroup 119879(119905) for 119905 gt 0 on119883 From

the subordination principle [6 Theorems 31 and 33] 119860 alsogenerates a compact resolvent 119878

120572(119905) for 119905 gt 0

Assume that 119891 [0 1] times 119883 rarr 119883 is a continuousfunction defined by 119891(119905 119911)(119909) = 119865(119905 119911(119909)) 0 le 119909 le 1 and119892 119862([0 1] 119883) rarr 119883 is also a continuous function definedby 119892(119906)(119909) = 119908

0(119909) minus sum

119901

119895=1119888119895119906(119905119895)(119909) 0 le 119905 119909 le 1 where

119906(119905)(119909) = 119908(119905 119909)

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Differential Equations

Under these assumptions the fractional partial differen-tial heat equation (47) can be reformulated as the abstractproblem (1) If the inequality

119872[

[

100381710038171003817100381711990801003817100381710038171003817 +

119901

sum

119895=1

119888119895119903 + sup119904isin[01]119906isin119882119903

1003817100381710038171003817119891 (119904 119906 (119904))1003817100381710038171003817]

]

le 119903 (48)

holds for some constant 119903 gt 0 there exists at least one mildsolution for fractional equation (47) in view of Corollary 18

Acknowledgments

This work was completed when the first author visited thePennsylvania State University The author is grateful to Pro-fessor Alberto Bressan and the Department of Mathematicsfor their hospitality and providing good working conditionsThe work was supported by the NSF of China (11001034)and Jiangsu Overseas Research amp Training Program forUniversity Prominent Young amp Middle-aged Teachers andPresidents

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] J Pruss Evolutionary Integral Equations and Applications vol87 of Monographs in Mathematics Birkhauser Basel Switzer-land 1993

[3] R P Agarwal Y Zhou and Y He ldquoExistence of fractional neu-tral functional differential equationsrdquo Computers amp Mathemat-ics with Applications vol 59 no 3 pp 1095ndash1100 2010

[4] R P Agarwal M Belmekki and M Benchohra ldquoExistenceresults for semilinear functional differential inclusions involv-ing Riemann-Liouville fractional derivativerdquo Dynamics of Con-tinuous Discrete amp Impulsive Systems vol 17 no 3 pp 347ndash3612010

[5] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis vol 72 no 6 pp 2859ndash28622010

[6] E Bajlekova Fractional evolution equations in Banach spaces[PhD thesis] University Press Facilities Eindhoven Universityof Technology 2001

[7] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[8] M Benchohra and B A Slimani ldquoExistence and uniquenessof solutions to impulsive fractional differential equationsrdquoElectronic Journal of Differential Equations no 10 pp 1ndash11 2009

[9] M Feckan J Wang and Y Zhou ldquoControllability of fractionalfunctional evolution equations of Sobolev type via character-istic solution operatorsrdquo Journal of Optimization Theory andApplications vol 156 no 1 pp 79ndash95 2013

[10] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquo Nonlinear Analysis vol 73 no 10pp 3462ndash3471 2010

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoExistenceresults for abstract fractional differential equations with nonlo-cal conditions via resolvent operatorsrdquo Indagationes Mathemat-icae vol 24 no 1 pp 68ndash82 2013

[12] C Lizama ldquoRegularized solutions for abstract Volterra equa-tionsrdquo Journal of Mathematical Analysis and Applications vol243 no 2 pp 278ndash292 2000

[13] C Lizama ldquoAn operator theoretical approach to a class offractional order differential equationsrdquo Applied MathematicsLetters vol 24 no 2 pp 184ndash190 2011

[14] C Lizama and G M NrsquoGuerekata ldquoBounded mild solutionsfor semilinear integro differential equations in Banach spacesrdquoIntegral Equations and Operator Theory vol 68 no 2 pp 207ndash227 2010

[15] M Li C Chen and F-B Li ldquoOn fractional powers of generatorsof fractional resolvent familiesrdquo Journal of Functional Analysisvol 259 no 10 pp 2702ndash2726 2010

[16] K Li J Peng and J Jia ldquoCauchy problems for fractional differ-ential equations with Riemann-Liouville fractional derivativesrdquoJournal of Functional Analysis vol 263 no 2 pp 476ndash510 2012

[17] G M Mophou and G M NrsquoGuerekata ldquoExistence of mildsolutions of some semilinear neutral fractional functionalevolution equations with infinite delayrdquo Applied Mathematicsand Computation vol 216 no 1 pp 61ndash69 2010

[18] G M Mophou and G M NrsquoGuerekata ldquoA note on a semilinearfractional differential equation of neutral type with infinitedelayrdquo Advances in Difference Equations vol 2010 Article ID674630 8 pages 2010

[19] J Wang Z Fan and Y Zhou ldquoNonlocal controllability of semi-linear dynamic systems with fractional derivative in Banachspacesrdquo Journal of Optimization Theory and Applications vol154 no 1 pp 292ndash302 2012

[20] J Wang Y Zhou andM Feckan ldquoAbstract Cauchy problem forfractional differential equationsrdquo Nonlinear Dynamics vol 71no 4 pp 685ndash700 2013

[21] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquoApplicable Analysis vol 40no 1 pp 11ndash19 1991

[22] L Chen and Z Fan ldquoOnmild solutions to fractional differentialequations with nonlocal conditionsrdquo Electronic Journal of Qual-itative Theory of Differential Equations no 53 pp 1ndash13 2011

[23] Z Fan ldquoExistence of nondensely defined evolution equationswith nonlocal conditionsrdquoNonlinear Analysis vol 70 no 11 pp3829ndash3836 2009

[24] Z Fan ldquoImpulsive problems for semilinear differential equa-tions with nonlocal conditionsrdquo Nonlinear Analysis vol 72 no2 pp 1104ndash1109 2010

[25] Z Fan and G Li ldquoExistence results for semilinear differentialequations with nonlocal and impulsive conditionsrdquo Journal ofFunctional Analysis vol 258 no 5 pp 1709ndash1727 2010

[26] E Hernandez J S dos Santos and K A G Azevedo ldquoExistenceof solutions for a class of abstract differential equations withnonlocal conditionsrdquo Nonlinear Analysis vol 74 no 7 pp2624ndash2634 2011

[27] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems vol 15 no 6 pp 815ndash824 2008

[28] J Liang J H Liu and T-J Xiao ldquoNonlocal impulsive problemsfor nonlinear differential equations in Banach spacesrdquo Mathe-matical and Computer Modelling vol 49 no 3-4 pp 798ndash8042009

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 9

[29] J Liang and Z Fan ldquoNonlocal impulsive Cauchy problemsfor evolution equationsrdquo Advances in Difference Equations vol2011 Article ID 784161 17 pages 2011

[30] X M Xue ldquoExistence of semilinear differential equations withnonlocal initial conditionsrdquo Acta Mathematica Sinica vol 23no 6 pp 983ndash988 2007

[31] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquoNonlinear Analysis vol 11 no 5 pp 4465ndash4475 2010

[32] L Zhu and G Li ldquoExistence results of semilinear differentialequations with nonlocal initial conditions in Banach spacesrdquoNonlinear Analysis vol 74 no 15 pp 5133ndash5140 2011

[33] L Zhu Q Huang and G Li ldquoExistence and asymptotic proper-ties of solutions of nonlinear multivalued differential inclusionswith nonlocal conditionsrdquo Journal ofMathematical Analysis andApplications vol 390 no 2 pp 523ndash534 2012

[34] J Banas and K GoebelMeasures of Noncompactness in BanachSpaces vol 60 ofLectureNotes in Pure andAppliedMathematicsMarcel Dekker New York NY USA 1980

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of