Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 292643 9 pageshttpdxdoiorg1011552013292643

Research ArticleExtremal Solutions and Relaxation Problems forFractional Differential Inclusions

Juan J Nieto12 Abdelghani Ouahab3 and P Prakash14

1 Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela15782 Santiago de Compostela Spain

2Department of Mathematics King Abdulaziz University Jeddah 21589 Saudi Arabia3 Laboratory of Mathematics Sidi-Bel-Abbes University PO Box 89 22000 Sidi-Bel-Abbes Algeria4Department of Mathematics Periyar University Salem 636 011 India

Correspondence should be addressed to Juan J Nieto juanjosenietoroigusces

Received 10 May 2013 Accepted 31 July 2013

Academic Editor Daniel C Biles

Copyright copy 2013 Juan J Nieto et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present the existence of extremal solution and relaxation problem for fractional differential inclusion with initial conditions

1 Introduction

Differential equations with fractional order have recentlyproved to be valuable tools in the modeling of many physicalphenomena [1ndash9] There has also been a significant theoreti-cal development in fractional differential equations in recentyears see themonographs of Kilbas et al [10]Miller andRoss[11] Podlubny [12] and Samko et al [13] and the papers ofKilbas and Trujillo [14] Nahusev [15] Podlubny et al [16]and Yu and Gao [17]

Recently some basic theory for initial value problems forfractional differential equations and inclusions involving theRiemann-Liouville differential operator was discussed forexample by Lakshmikantham [18] and Chalco-Cano et al[19]

Applied problems requiring definitions of fractionalderivatives are those that are physically interpretable for ini-tial conditions containing 119910(0) 1199101015840

(0) and so forthThe samerequirements are true for boundary conditions Caputorsquosfractional derivative satisfies these demands Formore detailson the geometric and physical interpretation for fractionalderivatives of both Riemann-Liouville and Caputo types seePodlubny [12]

Fractional calculus has a long history We refer the readerto [20]

Recently fractional functional differential equations andinclusions and impulsive fractional differential equations

and inclusions with standard Riemann-Liouville and Caputoderivatives with differences conditions were studied byAbbaset al [21 22] Benchohra et al [23] Henderson and Ouahab[24 25] Jiao and Zhou [26] and Ouahab [27ndash29] and in thereferences therein

In this paper we will be concerned with the existence ofsolutions Filippovrsquos theorem and the relaxation theorem ofabstract fractional differential inclusions More precisely wewill consider the following problem

119888119863

120572119910 (119905) isin 119865 (119905 119910 (119905)) ae 119905 isin 119869 = [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(1)

119888119863

120572119910 (119905) isin ext119865 (119905 119910 (119905)) ae 119905 isin 119869 = [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(2)

where 119888119863

120572 is the Caputo fractional derivatives 120572 isin (1 2]119865 119869 times R119873

rarr P(R119873) is a multifunction and ext119865(119905 119910)

represents the set of extreme points of 119865(119905 119910) (P(R119873) is the

family of all nonempty subsets of R119873During the last couple of years the existence of extremal

solutions and relaxation problem for ordinary differentialinclusionswas studied bymany authors for example see [30ndash34] and the references therein

2 Abstract and Applied Analysis

The paper is organized as follows We first collect somebackground material and basic results from multivaluedanalysis and give some results on fractional calculus inSections 2 and 3 respectivelyThenwewill be concernedwiththe existence of solution for extremal problemThis is the aimof Section 4 In Section 5 we prove the relaxation problem

2 Preliminaries

The reader is assumed to be familiar with the theory of multi-valued analysis and differential inclusions in Banach spacesas presented in Aubin et al [35 36] Hu and Papageorgiou[37] Kisielewicz [38] and Tolstonogov [32]

Let (119883 sdot ) be a real Banach space [0 119887] an interval in 119877and 119862([0 119887] 119883) the Banach space of all continuous functionsfrom 119869 into119883 with the norm

10038171003817100381710038171199101003817100381710038171003817infin = sup 1003817100381710038171003817119910 (119905)

1003817100381710038171003817 0 le 119905 le 119887 (3)

A measurable function 119910 [0 119887] rarr 119883 is Bochnerintegrable if 119910 is Lebesgue integrable In what follows1198711([0 119887] 119883) denotes the Banach space of functions 119910

[0 119887] rarr 119883 which are Bochner integrable with norm

100381710038171003817100381711991010038171003817100381710038171 = int

119887

0

1003817100381710038171003817119910 (119905)1003817100381710038171003817 119889119905

(4)

Denote by 1198711

119908([0 119887] 119883) the space of equivalence classes of

Bochner integrable function 119910 [0 119887] rarr 119883 with the norm

10038171003817100381710038171199101003817100381710038171003817119908 = sup

119905isin[0119905]

10038171003817100381710038171003817100381710038171003817int

119905

0

119910 (119904) 119889119904

10038171003817100381710038171003817100381710038171003817 (5)

The norm sdot 119908is weaker than the usual norm sdot

1 and for a

broad class of subsets of 1198711([0 119887] 119883) the topology defined by

the weak norm coincides with the usual weak topology (see[37 Proposition 414 page 195]) Denote by

P (119883) = 119884 sub 119883 119884 = 0

Pcl (119883) = 119884 isin P (119883) 119884 closed

P119887(119883) = 119884 isin P (119883) 119884 bounded

Pcv (119883) = 119884 isin P (119883) 119884 convex

Pcp (119883) = 119884 isin P (119883) 119884 compact

(6)

A multivalued map 119866 119883 rarr P(119883) has convex (closed)values if 119866(119909) is convex (closed) for all 119909 isin 119883 We say that 119866is bounded on bounded sets if 119866(119861) is bounded in 119883 for eachbounded set 119861 of119883 (ie sup

119909isin119861sup119910 119910 isin 119866(119909) lt infin)

Definition 1 A multifunction 119865 119883 rarr P(119884) is said to beupper semicontinuous at the point 119909

0isin 119883 if for every open

119882 sube 119884 such that 119865(1199090) sub 119882 there exists a neighborhood

119881(1199090) of 119909

0such that 119865(119881(119909

0)) sub 119882

A multifunction is called upper semicontinuous (usc forshort) on119883 if for each 119909 isin 119883 it is usc at 119909

Definition 2 A multifunction 119865 119883 rarr P(119884) is said to belower continuous at the point 119909

0isin 119883 if for every open119882 sube

119884 such that 119865(1199090)cap119882 = 0 there exists a neighborhood119881(119909

0)

of 1199090with property that 119865(119909) cap 119882 = 0 for all 119909 isin 119881(119909

0)

A multifunction is called lower semicontinuous (lsc forshort) provided that it is lower semicontinuous at every point119909 isin 119883

Lemma 3 (see [39 Lemma 32]) Let 119865 [0 119887] rarr P(119884)

be a measurable multivalued map and 119906 [119886 119887] rarr 119884 ameasurable function Then for any measurable V [119886 119887] rarr

(0 +infin) there exists a measurable selection 119891V of 119865 such thatfor ae 119905 isin [119886 119887]

1003817100381710038171003817119906 (119905) minus 119891V (119905)1003817100381710038171003817 le 119889 (119906 (119905) 119865 (119905)) + V (119905) (7)

First consider the Hausdorff pseudometric

119867119889 P (119864) timesP (119864) 997888rarr R

+cup infin (8)

defined by

119867119889(119860 119861) = maxsup

119886isin119860

119889 (119886 119861) sup119887isin119861

119889 (119860 119887) (9)

where 119889(119860 119887) = inf119886isin119860

119889(119886 119887) and 119889(119886 119861) = inf119887isin119861

119889(119886 119887)(P

119887cl(119864)119867119889) is a metric space and (Pcl(119883)119867119889

) is a gener-alized metric space

Definition 4 A multifunction 119865 119884 rarr P(119883) is calledHausdorff lower semicontinuous at the point 119910

0isin 119884 if for

any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910

0

such that

119865 (1199100) sub 119865 (119910) + 120598119861 (0 1) for every 119910 isin 119880 (119910

0) (10)

where 119861(0 1) is the unite ball in119883

Definition 5 A multifunction 119865 119884 rarr P(119883) is calledHausdorff upper semicontinuous at the point 119910

0isin 119884 if for

any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910

0

such that

119865 (119910) sub 119865 (1199100) + 120598119861 (0 1) for every 119910 isin 119880 (119910

0) (11)

119865 is called continuous if it is Hausdorff lower and uppersemicontinuous

Definition 6 Let 119883 be a Banach space a subset 119860 sub

1198711([0 119887] 119883) is decomposable if for all 119906 V isin 119860 and for every

Lebesgue measurable set 119868 sub 119869 one has

119906120594119868+ V120594

[0119887]119868isin 119860 (12)

where 120594119860stands for the characteristic function of the set 119860

We denote by Dco(1198711([0 119887] 119883)) the family of decomposable

sets

Abstract and Applied Analysis 3

Let 119865 [0 119887] times 119883 rarr P(119883) be a multivalued map withnonempty closed values Assign to119865 themultivalued operatorF 119862([0 119887] 119883) rarr P(119871

1([0 119887] 119883)) defined by

F (119910) = V isin 1198711

([0 119887] 119883) V (119905) isin 119865 (119905 119910 (119905))

ae 119905 isin [0 119887] (13)

The operator F is called the Nemytsrsquokiı operator associatedto 119865

Definition 7 Let 119865 [0 119887] times 119883 rarr P(119883) be a multivaluedmapwith nonempty compact valuesWe say that119865 is of lowersemicontinuous type (lsc type) if its associated Nemytsrsquokiıoperator F is lower semicontinuous and has nonemptyclosed and decomposable values

Next we state a classical selection theorem due to Bressanand Colombo

Lemma 8 (see [40]) Let119883 be a separable metric space and let119864 be a Banach spaceThen every lsc multivalued operator119873

119883 rarr P119888119897(119871

1([0 119887] 119864)) with closed decomposable values has

a continuous selection that is there exists a continuous single-valued function 119891 119883 rarr 119871

1([0 119887] 119864) such that 119891(119909) isin 119873(119909)

for every 119909 isin 119883

Let us introduce the following hypothesis

(H1) 119865 [0 119887]times119883 rarr P(119883) is a nonempty compact valuedmultivalued map such that

(a) the mapping (119905 119910) 997891rarr 119865(119905 119910) is L otimes Bmeasurable

(b) the mapping 119910 997891rarr 119865(119905 119910) is lower semicontinu-ous for ae 119905 isin [0 119887]

Lemma 9 (see eg [41]) Let 119865 119869 times 119883 rarr P119888119901(119864) be an

integrably bounded multivalued map satisfying (H1) Then 119865

is of lower semicontinuous type

Define

119865 (119870) = 119891 isin 1198711

([0 119887] 119883) 119891 (119905) isin 119870 ae 119905 isin [0 119887]

119870 sub 119883

(14)

where119883 is a Banach space

Lemma 10 (see [37]) Let 119870 sub 119883 be a weakly compactsubset of 119883 Then 119865(119870) is relatively weakly compact subset of1198711([0 119887] 119883) Moreover if 119870 is convex then 119865(119870) is weakly

compact in 1198711([0 119887] 119883)

Definition 11 A multifunction 119865 [0 119887] times 119884 rarr P119908cpcv(119883)

possesses the Scorza-Dragoni property (S-D property) if foreach 120598 gt 0 there exists a closed set 119869

120598sub [0 119887]whose Lebesgue

measure 120583(119869120598) le 120598 and such that 119865 [0 119887] 119869

120598times 119884 rarr 119883 is

continuous with respect to the metric 119889119883(sdot sdot)

Remark 12 It is well known that if the map 119865 [0 119887] times 119884 rarr

P119908cpcv(119883) is continuous with respect to 119910 for almost every

119905 isin [0 119887] and is measurable with respect to 119905 for every 119910 isin 119884then it possesses the S-D property

In what follows we present some definitions and proper-ties of extreme points

Definition 13 Let119860 be a nonempty subset of a real or complexlinear vector space An extreme point of a convex set 119860 is apoint 119909 isin 119860 with the property that if 119909 = 120582119910 + (1 minus 120582)119911 with119910 119911 isin 119860 and 120582 isin [0 1] then 119910 = 119909 andor 119911 = 119909 ext(119860) willdenote the set of extreme points of 119860

In other words an extreme point is a point that is not aninterior point of any line segment lying entirely in 119860

Lemma 14 (see [42]) A nonempty compact set in a locallyconvex linear topological space has extremal points

Let 1199091015840

119899119899isinN be a denumerable dense (in 120590(1198831015840

119883) topol-ogy) subset of the set 119909 isin 119883 119909 le 1 For any 119860 isin

Pcpcv(119883) and 1199091015840

119899define the function

119889119899

(119860 119906) = max ⟨119910 minus 119911 1199091015840

119899⟩ 119910 119911 isin 119860 119906 =

119910 + 119911

2

(15)

Lemma 15 (see [33]) 119906 isin ext(119860) if and only if 119889119899(119860 119906) = 0

for all 119899 ge 1

In accordance with Krein-Milman and Trojansky theo-rem [43] the set ext(119878

119865) is nonempty and co(ext(119878

119865)) = 119878

119865

Lemma 16 (see [33]) Let 119865 [0 119887] rarr P119908119888119901119888V(119883) be a

measurable integrably bounded map Then

ext (119878119865) sube 119878

119865 (16)

where ext (119878119865) is the closure of set ext (119878

119865) in the topology of

the space 1198711([0 119887] 119883)

Theorem 17 (see [33]) Let 119865 [0 119887] times 119884 rarr P119908119888119901119888V(119883)

be a multivalued map that has the 119878-119863 property and let it beintegrable bounded on compacts from 119884 Consider a compactsubset 119870 sub 119862([0 119887] 119883) and define the multivalued map 119866

119870 rarr 1198711([0 119887] 119883) by

119866 (119910 (sdot))

= 119891 isin 1198711

([0 119887] 119883) 119891 (119905) isin 119865 (119905 119910 (119905)) 119886119890 119900119899 [0 119887]

119910 isin 119870

(17)

Then for every 119870 compact in 119862([0 119887] 119883) 120598 gt 0 and anycontinuous selection 119891 119870 rarr 119871

1([0 119887] 119883) there exists a

continuous selector119892 119870 rarr 1198711([0 119887] 119883) of themap ext (119866)

119870 rarr 1198711([0 119887] 119883) such that for all 119910 isin 119862([0 119887] 119883) one has

sup119905isin[0119887]

10038171003817100381710038171003817100381710038171003817int

119905

0

((119891119910) (119904) minus (119892119910) (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817le 120598 (18)

4 Abstract and Applied Analysis

For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]

3 Fractional Calculus

According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads

119868119899119891 (119905) =

1

(119899 minus 1)int

119905

0

(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N

(19)

Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871

1([119886 119887]R) is defined by

119868120572

119886+119891 (119905) = int

119905

119886

(119905 minus 119904)120572minus1

Γ (120572)119891 (119904) 119889119904 (20)

where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601

120572(119905) where 120601

120572(119905) = 119905

(120572minus1)Γ(120572) for 119905 gt 0 and

we write 120601120572(119905) = 0 for 119905 le 0 and 120601

120572rarr 120575(119905) as 120572 rarr 0 where

120575 is the delta function and Γ is the Euler gamma functiondefined by

Γ (120572) = int

infin

0

119905120572minus1

119890minus119905119889119905 120572 gt 0 (21)

For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0

+ this limit may be infinite

After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that

119863119899119868119899= Id 119868

119899119863

119899= Id 119899 isin N (22)

that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that

119868119899119863

119899119891 (119905) = 119891 (119905) minus

119899minus1

sum

119896=0

119891(119896)(119886

+)(119905 minus 119886)

119896

119896 119905 gt 0 (23)

As a consequence we expect that 119863120572 is defined as the leftinverse to 119868

120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0

Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by

119863120572119891 (119905) =

1

Γ (119899 minus 120572)(119889

119889119905)

119899

int

119905

119886

(119905 minus 119904)minus120572+119899minus1

119891 (119904) 119889119904 (24)

where 119899 = [120572] + 1 and [120572] is the integer part of 120572

Defining for consistency 1198630= 119868

0= Id then we easily

recognize that

119863120572119868120572= Id 120572 ge 0 (25)

119863120572119905120574=

Γ (120574 + 1)

Γ (120574 + 1 minus 120572)119905120574minus120572

120572 gt 0 120574 isin (minus1 0) cup (0 +infin) 119905 gt 0

(26)

Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer

Note the remarkable fact that the fractional derivative119863

120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N

In fact (26) with 120574 = 0 illustrates that

1198631205721 =

(119905 minus 119886)minus120572

Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)

It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the

gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional

derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])

Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-

order derivative of 119891 is defined by

(119888

119863120572

119891) (119905) =1

Γ (119899 minus 120572)int

119905

119886

(119905 minus 119904)119899minus120572minus1

119891119899

(119904) 119889119904 (28)

This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572

lowastwe (tacitly) assume that this condition is

met We easily recognize that in general

119863120572119891 (119905) = 119863

119898119868119898minus120572

119891 (119905) = 119869119898minus120572

119863119898119891 (119905) = 119863

120572

lowast119891 (119905) (29)

unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0

119863120572119891 (119905) =

119888

119863120572

119891 (119905) +

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+) (30)

and therefore recalling the fractional derivative of the powerfunction (26) one has

119863120572(119891 (119905) minus

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+)) = 119863

120572

lowast119891 (119905) (31)

Abstract and Applied Analysis 5

The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886

+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero

119888119863

1205721 = 0 120572 gt 0 (32)

We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have

119863120572

(119905 minus 119886)120572minus119895

= 0 for 119895 = 1 2 [120572] + 1 (33)

From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N

119863120572119891 (119905) = 119863

120572119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

120572minus119895

119888119863

120572119891 (119905) =

119888

119863120572

119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

119899minus119895

(34)

In these formulas the coefficients 119888119895are arbitrary constants

For proving allmain results we present the following auxiliarylemmas

Lemma 21 (see [10]) Let 120572 gt 0 and let 119910 isin 119871infin(119886 119887) or

119862([119886 119887]) Then

(119888

119863120572

119868120572119910) (119905) = 119910 (119905) (35)

Lemma 22 (see [10]) Let 120572 gt 0 and 119899 = [120572] + 1 If 119910 isin

119860119862119899[119886 119887] or 119910 isin 119862119899

[119886 119887] then

(119868120572 119888

119863120572119910) (119905) = 119910 (119905) minus

119899minus1

sum

119896=0

119910(119896)(119886)

119896(119905 minus 119886)

119896 (36)

For further readings and details on fractional calculus werefer to the books and papers by Kilbas [10] Podlubny [12]Samko [13] and Caputo [46ndash48]

4 Existence Result

Definition 23 A function 119910 isin 119862([0 119887]R119873) is called mild

solution of problem (1) if there exist 119891 isin 1198711(119869R119873

) such that

119910 (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)1minus120572

119891 (119904) 119889119904 119905 isin [0 119887]

(37)

where 119891 isin 119878119865119910

= V isin 1198711([0 119887]R119873

) 119891(119905) isin 119865(119905 119910(119905)) aeon [0 119887]

We will impose the following conditions on 119865

(H1) The function 119865 119869 timesR119873

rarr Pcpcv(R119873) such that

(a) for all 119909 isin R119873 the map 119905 997891rarr 119865(119905 119909) ismeasurable

(b) for every 119905 isin [0 119887] the multivalued map 119909 rarr

119865(119905 119909) is119867119889continuous

(H2) There exist 119901 isin 119871

1(119869R+

) and a continuous nonde-creasing function 120595 [0infin) rarr (0infin) such that

119865 (119905 119909)P = sup V V isin 119865 (119905 119909) le 119901 (119905) 120595 (119909)

for ae 119905 isin [0 119887] and each 119909 isin R119873

(38)

with

int

119887

0

119901 (119904) 119889119904 lt int

infin

1199100+119887119910

1

119889119906

120595 (119906) (39)

Theorem24 Assume that the conditions (H1)-(H

2) and then

the problem (2) have at least one solution

Proof From (H2) there exists 119872 gt 0 such that 119910

infinle 119872

for each 119910 isin 119878119888

Let

1198651(119905 119910) =

119865 (119905 119910) if 10038171003817100381710038171199101003817100381710038171003817 le 119872

119865(119905119872119910

10038171003817100381710038171199101003817100381710038171003817

) if 10038171003817100381710038171199101003817100381710038171003817 ge 119872

(40)

We consider119888

119863120572

119910 (119905) isin 1198651(119905 119910 (119905)) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(41)

It is clear that all the solutions of (41) are solutions of (2)Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595lowast

(119905)

120595lowast(119905) = 119901 (119905) 120595 (119872)

(42)

It is clear that 119881 is weakly compact in 1198711([0 119887]R119873

) Remarkthat for every 119891 isin 119881 there exists a unique solution 119871(119891) ofthe following problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (119905) = 1199100 119910

1015840

(0) = 1199101

(43)

this solution is defined by

119871 (119891) (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (44)

We claim that 119871 is continuous Indeed let 119891119899rarr 119891 converge

in 1198711([0 119887]R119873

) as 119899 rarr infin set 119910119899= 119871(119891

119899) 119899 isin N It is clear

6 Abstract and Applied Analysis

that 119910119899 119899 isin N is relatively compact in 119862([0 119887]R119873

) and 119910119899

converge to 119910 isin 119862([0 119887]R119873) Let

119911 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 isin [0 119887]

(45)

Then

1003817100381710038171003817119910119899minus 119911

1003817100381710038171003817infin le119887120572

Γ (120572)int

119887

0

1003817100381710038171003817119891119899 (119904) minus 119891 (119904)1003817100381710038171003817 119889119904 997888rarr 0

as 119899 997888rarr infin

(46)

Hence 119870 = 119871(119881) is compact and convex subset of119862([0 119887]R119873

) Let 119878119865 119870 rarr Pclcv(119871

1([0 119887]R119873

)) be themultivalued Nemitsky operator defined by

1198781198651

(119910) = 119891 isin 1198711([0 119887] R

119873) 119891 (119905) isin 119865

1(119905 119910 (119905))

ae 119905 isin [0 119887] = 1198781198651119910

(47)

It is clear that 1198651(sdot sdot) is 119867

119889continuous and 119865

1(sdot sdot) isin

P119908119896cpcv(R

119873) and is integrably bounded then byTheorem 17

(see also Theorem 65 in [32] or Theorem 11 in [34]) we canfind a continuous function 119892 119870 rarr 119871

1

119908([0 119887]R119873

) such that

119892 (119909) isin ext 1198781198651

(119910) forall119910 isin 119870 (48)

From Benamara [49] we know that

ext 1198781198651

(119910) = 119878ext 1198651(sdot119910(sdot))

forall119910 isin 119870 (49)

Setting119873 = 119871 ∘ 119892 and letting 119910 isin 119870 then

119892 (119910) isin 1198651(sdot 119910 (sdot)) 997904rArr 119892 (119910) isin 119881 997904rArr 119873(119910)

= 119871 (119892 (119910)) isin 119870

(50)

Now we prove that 119873 is continuous Indeed let 119910119899isin 119870

converge to 119910 in 119862([0 119887]R119873)

Then

119892 (119910119899) converge weakly to 119892 (119910) as 119899 997888rarr infin (51)

Since119873(119910119899) = 119871(119892(119910

119899)) isin 119870 and 119892(119910

119899)(sdot) isin 119865(119905 119910

119899(119905)) then

119892 (119910119899) (sdot) isin 119865 (sdot 119861

119872) isin Pcp (R

119873) (52)

FromLemma 10 119892(119910119899) converge weakly to119910 in 1198711

([0 119887]R119873)

as 119899 rarr infin By the definition of119873 we have

119873(119910119899) = 119910

0+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910119899) (119904) 119889119904

119905 isin [0 119887]

119873 (119910) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910) (119904) 119889119904

119905 isin [0 119887]

(53)

Since 119873(119910119899) 119899 isin N sub 119870 then there exists subsequence of

119873(119910119899) converge in 119862([0 119887]R119873

) Then

119873(119910119899) (119905) 997888rarr 119873(119910) (119905) forall119905 isin [0 119887] as 119899 997888rarr infin

(54)

This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)

5 The Relaxed Problem

In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]

Now we present our main result of this section

Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873

) be amultifunction satisfying the following hypotheses

(H3) The function 119865 [0 119887]timesR119873

rarr P119888119901119888V(R

119873) such that

for all 119909 isin R119873 the map

119905 997891997888rarr 119865 (119905 119909) (55)

is measurable(H

4) There exists 119901 isin 1198711

(119869R+) such that

119867119889(119865 (119905 119909) 119865 (119905 119910)) le 119901 (119905)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

for ae 119905 isin [0 119887] and each 119909 119910 isin R119873

119867119889(119865 (119905 0) 0) le 119901 (119905) for ae 119905 isin [0 119887]

(56)

Then 119878119890= 119878

119888

Proof By Coviz and Nadlar fixed point theorem we caneasily prove that 119878

119888= 0 and since 119865 has compact and convex

valued then 119878119888is compact in 119862([0 119887]R119873

) For more infor-mation we see [25 27ndash29 51 52]

Let 119910 isin 119878119888 then there exists 119891 isin 119878

119865119910such that

119910 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (57)

Let119870 be a compact and convex set in 119862([0 119887]R119873) such that

119878119888sub 119870 Given that 119910

lowastisin 119870 and 120598 gt 0 we define the following

multifunction 119880120598 [0 119887] rarr P(R119873

) by

119880120598(119905) = 119906 isin R

1198731003817100381710038171003817119891 (119905) minus 119906

1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598

119906 isin 119865 (119905 119910lowast(119905))

(58)

Abstract and Applied Analysis 7

The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr

119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact

values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873

) Then by Lemma 3 there exists a measurableselection V

1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that

1003817100381710038171003817119891 (119905) minus V1(119905)1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598 (59)

this implies that 119880120598(sdot) = 0 We consider 119866

120598 119870 rarr P(119871

1(119869

R119873) defined by

119866120598(119910) = 119891

lowastisin F (119910)

1003817100381710038171003817119891 (119905) minus 119891lowast (119905)1003817100381710038171003817

lt 120598 + 119889 (119891lowast(119905) 119865 (119905 119910 (119905)))

(60)

Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866

120598(119910) is lsc with decomposable

values By Lemma 8 there exists a continuous selection 119891120598

119862([0 119887]R119873) rarr 119871

1(119869R119873

) such that

119891120598(119910) isin 119866

120598(119910) forall119910 isin 119862 ([0 119887] R

119873) (61)

FromTheorem 17 there exists function 119892120598 119870 rarr 119871

119908([0 119887]

R119873) such that

119892120598(119910) isin ext 119878

119865(119910) = 119878ext 119865(sdot119910(sdot)) forall119910 isin 119870

119892120598(119910) minus 119891

120598(119910)

119908le 120598 forall119910 isin 119870

(62)

From (H3) we can prove that there exists119872 gt 0 such that

10038171003817100381710038171199101003817100381710038171003817infin le 119872 forall119910 isin 119878

119888 (63)

Consider the sequence 120598119899rarr 0 as 119899 rarr infin and set119892

119899= 119892

120598119899

119891119899= 119891

120598119899

Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595 (119905) ae 119905 isin [0 119887]

120595 (119905) = (1 +119872)119901 (119905)

(64)

Let 119871 119881 rarr 119862([0 119887]R119873) be the map such that each 119891 isin 119881

assigns the unique solution of the problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(65)

As in Theorem 24 we can prove that 119871(119881) is compact in119862([0 119887]R119873

) and the operator 119873119899= 119871 ∘ 119892

119899 119870 rarr 119870 is

compact then by Schauderrsquos fixed point there exists 119910119899isin 119870

such that 119910119899isin 119878

119890and

119910119899(119905) = 119910

0+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892119899(119910

119899) (119904) 119889119904

ae 119905 isin [0 119887] 119899 isin N

(66)

Hence

1003817100381710038171003817119910 (119905) minus 119910119899(119905)1003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

1003817100381710038171003817119891119899 (119910119899) (119904) minus 119891 (119904)

1003817100381710038171003817 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119889 (119891 (119904) 119891

119899(119910

119899) (119904))) 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119867

119889(119865 (119904 119910 (119904)) 119865 (119904 119910

119899(119904)))) 119889119904

le119887120572+1

Γ (120572 + 1)120598119899+119887120572+1

Γ (120572)120598119899+ int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910119899

(119904)1003817100381710038171003817

(67)

Let 119910(sdot) be a limit point of the sequence 119910119899(sdot) Then it follows

that from the above inequality one has

1003817100381710038171003817119910 (119905) minus 119910 (119905)1003817100381710038171003817 le int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910 (119904)

1003817100381710038171003817 119889119904(68)

which implies 119910(sdot) = 119910(sdot) Consequently 119910 isin 119878119888is a unique

limit point of 119910119899(sdot) isin 119878

119890

Example 26 Let 119865 119869 timesR119873rarr Pcpcv(R

119873) with

119865 (119905 119910) = 119861 (1198911(119905 119910) 119891

2(119905 119910)) (69)

where 1198911 119891

2 119869 times R119873

rarr R119873 are Caratheodory functionsand bounded

Then (2) is solvable

Example 27 If in addition to the conditions on 119865 ofExample 26 119891

1and 119891

2are Lipschitz functions then 119878

119890= 119878

119888

Acknowledgments

This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER

8 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

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

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2 Abstract and Applied Analysis

The paper is organized as follows We first collect somebackground material and basic results from multivaluedanalysis and give some results on fractional calculus inSections 2 and 3 respectivelyThenwewill be concernedwiththe existence of solution for extremal problemThis is the aimof Section 4 In Section 5 we prove the relaxation problem

2 Preliminaries

The reader is assumed to be familiar with the theory of multi-valued analysis and differential inclusions in Banach spacesas presented in Aubin et al [35 36] Hu and Papageorgiou[37] Kisielewicz [38] and Tolstonogov [32]

Let (119883 sdot ) be a real Banach space [0 119887] an interval in 119877and 119862([0 119887] 119883) the Banach space of all continuous functionsfrom 119869 into119883 with the norm

10038171003817100381710038171199101003817100381710038171003817infin = sup 1003817100381710038171003817119910 (119905)

1003817100381710038171003817 0 le 119905 le 119887 (3)

A measurable function 119910 [0 119887] rarr 119883 is Bochnerintegrable if 119910 is Lebesgue integrable In what follows1198711([0 119887] 119883) denotes the Banach space of functions 119910

[0 119887] rarr 119883 which are Bochner integrable with norm

100381710038171003817100381711991010038171003817100381710038171 = int

119887

0

1003817100381710038171003817119910 (119905)1003817100381710038171003817 119889119905

(4)

Denote by 1198711

119908([0 119887] 119883) the space of equivalence classes of

Bochner integrable function 119910 [0 119887] rarr 119883 with the norm

10038171003817100381710038171199101003817100381710038171003817119908 = sup

119905isin[0119905]

10038171003817100381710038171003817100381710038171003817int

119905

0

119910 (119904) 119889119904

10038171003817100381710038171003817100381710038171003817 (5)

The norm sdot 119908is weaker than the usual norm sdot

1 and for a

broad class of subsets of 1198711([0 119887] 119883) the topology defined by

the weak norm coincides with the usual weak topology (see[37 Proposition 414 page 195]) Denote by

P (119883) = 119884 sub 119883 119884 = 0

Pcl (119883) = 119884 isin P (119883) 119884 closed

P119887(119883) = 119884 isin P (119883) 119884 bounded

Pcv (119883) = 119884 isin P (119883) 119884 convex

Pcp (119883) = 119884 isin P (119883) 119884 compact

(6)

A multivalued map 119866 119883 rarr P(119883) has convex (closed)values if 119866(119909) is convex (closed) for all 119909 isin 119883 We say that 119866is bounded on bounded sets if 119866(119861) is bounded in 119883 for eachbounded set 119861 of119883 (ie sup

119909isin119861sup119910 119910 isin 119866(119909) lt infin)

Definition 1 A multifunction 119865 119883 rarr P(119884) is said to beupper semicontinuous at the point 119909

0isin 119883 if for every open

119882 sube 119884 such that 119865(1199090) sub 119882 there exists a neighborhood

119881(1199090) of 119909

0such that 119865(119881(119909

0)) sub 119882

A multifunction is called upper semicontinuous (usc forshort) on119883 if for each 119909 isin 119883 it is usc at 119909

Definition 2 A multifunction 119865 119883 rarr P(119884) is said to belower continuous at the point 119909

0isin 119883 if for every open119882 sube

119884 such that 119865(1199090)cap119882 = 0 there exists a neighborhood119881(119909

0)

of 1199090with property that 119865(119909) cap 119882 = 0 for all 119909 isin 119881(119909

0)

A multifunction is called lower semicontinuous (lsc forshort) provided that it is lower semicontinuous at every point119909 isin 119883

Lemma 3 (see [39 Lemma 32]) Let 119865 [0 119887] rarr P(119884)

be a measurable multivalued map and 119906 [119886 119887] rarr 119884 ameasurable function Then for any measurable V [119886 119887] rarr

(0 +infin) there exists a measurable selection 119891V of 119865 such thatfor ae 119905 isin [119886 119887]

1003817100381710038171003817119906 (119905) minus 119891V (119905)1003817100381710038171003817 le 119889 (119906 (119905) 119865 (119905)) + V (119905) (7)

First consider the Hausdorff pseudometric

119867119889 P (119864) timesP (119864) 997888rarr R

+cup infin (8)

defined by

119867119889(119860 119861) = maxsup

119886isin119860

119889 (119886 119861) sup119887isin119861

119889 (119860 119887) (9)

where 119889(119860 119887) = inf119886isin119860

119889(119886 119887) and 119889(119886 119861) = inf119887isin119861

119889(119886 119887)(P

119887cl(119864)119867119889) is a metric space and (Pcl(119883)119867119889

) is a gener-alized metric space

Definition 4 A multifunction 119865 119884 rarr P(119883) is calledHausdorff lower semicontinuous at the point 119910

0isin 119884 if for

any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910

0

such that

119865 (1199100) sub 119865 (119910) + 120598119861 (0 1) for every 119910 isin 119880 (119910

0) (10)

where 119861(0 1) is the unite ball in119883

Definition 5 A multifunction 119865 119884 rarr P(119883) is calledHausdorff upper semicontinuous at the point 119910

0isin 119884 if for

any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910

0

such that

119865 (119910) sub 119865 (1199100) + 120598119861 (0 1) for every 119910 isin 119880 (119910

0) (11)

119865 is called continuous if it is Hausdorff lower and uppersemicontinuous

Definition 6 Let 119883 be a Banach space a subset 119860 sub

1198711([0 119887] 119883) is decomposable if for all 119906 V isin 119860 and for every

Lebesgue measurable set 119868 sub 119869 one has

119906120594119868+ V120594

[0119887]119868isin 119860 (12)

where 120594119860stands for the characteristic function of the set 119860

We denote by Dco(1198711([0 119887] 119883)) the family of decomposable

sets

Abstract and Applied Analysis 3

Let 119865 [0 119887] times 119883 rarr P(119883) be a multivalued map withnonempty closed values Assign to119865 themultivalued operatorF 119862([0 119887] 119883) rarr P(119871

1([0 119887] 119883)) defined by

F (119910) = V isin 1198711

([0 119887] 119883) V (119905) isin 119865 (119905 119910 (119905))

ae 119905 isin [0 119887] (13)

The operator F is called the Nemytsrsquokiı operator associatedto 119865

Definition 7 Let 119865 [0 119887] times 119883 rarr P(119883) be a multivaluedmapwith nonempty compact valuesWe say that119865 is of lowersemicontinuous type (lsc type) if its associated Nemytsrsquokiıoperator F is lower semicontinuous and has nonemptyclosed and decomposable values

Next we state a classical selection theorem due to Bressanand Colombo

Lemma 8 (see [40]) Let119883 be a separable metric space and let119864 be a Banach spaceThen every lsc multivalued operator119873

119883 rarr P119888119897(119871

1([0 119887] 119864)) with closed decomposable values has

a continuous selection that is there exists a continuous single-valued function 119891 119883 rarr 119871

1([0 119887] 119864) such that 119891(119909) isin 119873(119909)

for every 119909 isin 119883

Let us introduce the following hypothesis

(H1) 119865 [0 119887]times119883 rarr P(119883) is a nonempty compact valuedmultivalued map such that

(a) the mapping (119905 119910) 997891rarr 119865(119905 119910) is L otimes Bmeasurable

(b) the mapping 119910 997891rarr 119865(119905 119910) is lower semicontinu-ous for ae 119905 isin [0 119887]

Lemma 9 (see eg [41]) Let 119865 119869 times 119883 rarr P119888119901(119864) be an

integrably bounded multivalued map satisfying (H1) Then 119865

is of lower semicontinuous type

Define

119865 (119870) = 119891 isin 1198711

([0 119887] 119883) 119891 (119905) isin 119870 ae 119905 isin [0 119887]

119870 sub 119883

(14)

where119883 is a Banach space

Lemma 10 (see [37]) Let 119870 sub 119883 be a weakly compactsubset of 119883 Then 119865(119870) is relatively weakly compact subset of1198711([0 119887] 119883) Moreover if 119870 is convex then 119865(119870) is weakly

compact in 1198711([0 119887] 119883)

Definition 11 A multifunction 119865 [0 119887] times 119884 rarr P119908cpcv(119883)

possesses the Scorza-Dragoni property (S-D property) if foreach 120598 gt 0 there exists a closed set 119869

120598sub [0 119887]whose Lebesgue

measure 120583(119869120598) le 120598 and such that 119865 [0 119887] 119869

120598times 119884 rarr 119883 is

continuous with respect to the metric 119889119883(sdot sdot)

Remark 12 It is well known that if the map 119865 [0 119887] times 119884 rarr

P119908cpcv(119883) is continuous with respect to 119910 for almost every

119905 isin [0 119887] and is measurable with respect to 119905 for every 119910 isin 119884then it possesses the S-D property

In what follows we present some definitions and proper-ties of extreme points

Definition 13 Let119860 be a nonempty subset of a real or complexlinear vector space An extreme point of a convex set 119860 is apoint 119909 isin 119860 with the property that if 119909 = 120582119910 + (1 minus 120582)119911 with119910 119911 isin 119860 and 120582 isin [0 1] then 119910 = 119909 andor 119911 = 119909 ext(119860) willdenote the set of extreme points of 119860

In other words an extreme point is a point that is not aninterior point of any line segment lying entirely in 119860

Lemma 14 (see [42]) A nonempty compact set in a locallyconvex linear topological space has extremal points

Let 1199091015840

119899119899isinN be a denumerable dense (in 120590(1198831015840

119883) topol-ogy) subset of the set 119909 isin 119883 119909 le 1 For any 119860 isin

Pcpcv(119883) and 1199091015840

119899define the function

119889119899

(119860 119906) = max ⟨119910 minus 119911 1199091015840

119899⟩ 119910 119911 isin 119860 119906 =

119910 + 119911

2

(15)

Lemma 15 (see [33]) 119906 isin ext(119860) if and only if 119889119899(119860 119906) = 0

for all 119899 ge 1

In accordance with Krein-Milman and Trojansky theo-rem [43] the set ext(119878

119865) is nonempty and co(ext(119878

119865)) = 119878

119865

Lemma 16 (see [33]) Let 119865 [0 119887] rarr P119908119888119901119888V(119883) be a

measurable integrably bounded map Then

ext (119878119865) sube 119878

119865 (16)

where ext (119878119865) is the closure of set ext (119878

119865) in the topology of

the space 1198711([0 119887] 119883)

Theorem 17 (see [33]) Let 119865 [0 119887] times 119884 rarr P119908119888119901119888V(119883)

be a multivalued map that has the 119878-119863 property and let it beintegrable bounded on compacts from 119884 Consider a compactsubset 119870 sub 119862([0 119887] 119883) and define the multivalued map 119866

119870 rarr 1198711([0 119887] 119883) by

119866 (119910 (sdot))

= 119891 isin 1198711

([0 119887] 119883) 119891 (119905) isin 119865 (119905 119910 (119905)) 119886119890 119900119899 [0 119887]

119910 isin 119870

(17)

Then for every 119870 compact in 119862([0 119887] 119883) 120598 gt 0 and anycontinuous selection 119891 119870 rarr 119871

1([0 119887] 119883) there exists a

continuous selector119892 119870 rarr 1198711([0 119887] 119883) of themap ext (119866)

119870 rarr 1198711([0 119887] 119883) such that for all 119910 isin 119862([0 119887] 119883) one has

sup119905isin[0119887]

10038171003817100381710038171003817100381710038171003817int

119905

0

((119891119910) (119904) minus (119892119910) (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817le 120598 (18)

4 Abstract and Applied Analysis

For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]

3 Fractional Calculus

According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads

119868119899119891 (119905) =

1

(119899 minus 1)int

119905

0

(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N

(19)

Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871

1([119886 119887]R) is defined by

119868120572

119886+119891 (119905) = int

119905

119886

(119905 minus 119904)120572minus1

Γ (120572)119891 (119904) 119889119904 (20)

where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601

120572(119905) where 120601

120572(119905) = 119905

(120572minus1)Γ(120572) for 119905 gt 0 and

we write 120601120572(119905) = 0 for 119905 le 0 and 120601

120572rarr 120575(119905) as 120572 rarr 0 where

120575 is the delta function and Γ is the Euler gamma functiondefined by

Γ (120572) = int

infin

0

119905120572minus1

119890minus119905119889119905 120572 gt 0 (21)

For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0

+ this limit may be infinite

After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that

119863119899119868119899= Id 119868

119899119863

119899= Id 119899 isin N (22)

that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that

119868119899119863

119899119891 (119905) = 119891 (119905) minus

119899minus1

sum

119896=0

119891(119896)(119886

+)(119905 minus 119886)

119896

119896 119905 gt 0 (23)

As a consequence we expect that 119863120572 is defined as the leftinverse to 119868

120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0

Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by

119863120572119891 (119905) =

1

Γ (119899 minus 120572)(119889

119889119905)

119899

int

119905

119886

(119905 minus 119904)minus120572+119899minus1

119891 (119904) 119889119904 (24)

where 119899 = [120572] + 1 and [120572] is the integer part of 120572

Defining for consistency 1198630= 119868

0= Id then we easily

recognize that

119863120572119868120572= Id 120572 ge 0 (25)

119863120572119905120574=

Γ (120574 + 1)

Γ (120574 + 1 minus 120572)119905120574minus120572

120572 gt 0 120574 isin (minus1 0) cup (0 +infin) 119905 gt 0

(26)

Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer

Note the remarkable fact that the fractional derivative119863

120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N

In fact (26) with 120574 = 0 illustrates that

1198631205721 =

(119905 minus 119886)minus120572

Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)

It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the

gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional

derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])

Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-

order derivative of 119891 is defined by

(119888

119863120572

119891) (119905) =1

Γ (119899 minus 120572)int

119905

119886

(119905 minus 119904)119899minus120572minus1

119891119899

(119904) 119889119904 (28)

This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572

lowastwe (tacitly) assume that this condition is

met We easily recognize that in general

119863120572119891 (119905) = 119863

119898119868119898minus120572

119891 (119905) = 119869119898minus120572

119863119898119891 (119905) = 119863

120572

lowast119891 (119905) (29)

unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0

119863120572119891 (119905) =

119888

119863120572

119891 (119905) +

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+) (30)

and therefore recalling the fractional derivative of the powerfunction (26) one has

119863120572(119891 (119905) minus

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+)) = 119863

120572

lowast119891 (119905) (31)

Abstract and Applied Analysis 5

The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886

+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero

119888119863

1205721 = 0 120572 gt 0 (32)

We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have

119863120572

(119905 minus 119886)120572minus119895

= 0 for 119895 = 1 2 [120572] + 1 (33)

From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N

119863120572119891 (119905) = 119863

120572119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

120572minus119895

119888119863

120572119891 (119905) =

119888

119863120572

119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

119899minus119895

(34)

In these formulas the coefficients 119888119895are arbitrary constants

For proving allmain results we present the following auxiliarylemmas

Lemma 21 (see [10]) Let 120572 gt 0 and let 119910 isin 119871infin(119886 119887) or

119862([119886 119887]) Then

(119888

119863120572

119868120572119910) (119905) = 119910 (119905) (35)

Lemma 22 (see [10]) Let 120572 gt 0 and 119899 = [120572] + 1 If 119910 isin

119860119862119899[119886 119887] or 119910 isin 119862119899

[119886 119887] then

(119868120572 119888

119863120572119910) (119905) = 119910 (119905) minus

119899minus1

sum

119896=0

119910(119896)(119886)

119896(119905 minus 119886)

119896 (36)

For further readings and details on fractional calculus werefer to the books and papers by Kilbas [10] Podlubny [12]Samko [13] and Caputo [46ndash48]

4 Existence Result

Definition 23 A function 119910 isin 119862([0 119887]R119873) is called mild

solution of problem (1) if there exist 119891 isin 1198711(119869R119873

) such that

119910 (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)1minus120572

119891 (119904) 119889119904 119905 isin [0 119887]

(37)

where 119891 isin 119878119865119910

= V isin 1198711([0 119887]R119873

) 119891(119905) isin 119865(119905 119910(119905)) aeon [0 119887]

We will impose the following conditions on 119865

(H1) The function 119865 119869 timesR119873

rarr Pcpcv(R119873) such that

(a) for all 119909 isin R119873 the map 119905 997891rarr 119865(119905 119909) ismeasurable

(b) for every 119905 isin [0 119887] the multivalued map 119909 rarr

119865(119905 119909) is119867119889continuous

(H2) There exist 119901 isin 119871

1(119869R+

) and a continuous nonde-creasing function 120595 [0infin) rarr (0infin) such that

119865 (119905 119909)P = sup V V isin 119865 (119905 119909) le 119901 (119905) 120595 (119909)

for ae 119905 isin [0 119887] and each 119909 isin R119873

(38)

with

int

119887

0

119901 (119904) 119889119904 lt int

infin

1199100+119887119910

1

119889119906

120595 (119906) (39)

Theorem24 Assume that the conditions (H1)-(H

2) and then

the problem (2) have at least one solution

Proof From (H2) there exists 119872 gt 0 such that 119910

infinle 119872

for each 119910 isin 119878119888

Let

1198651(119905 119910) =

119865 (119905 119910) if 10038171003817100381710038171199101003817100381710038171003817 le 119872

119865(119905119872119910

10038171003817100381710038171199101003817100381710038171003817

) if 10038171003817100381710038171199101003817100381710038171003817 ge 119872

(40)

We consider119888

119863120572

119910 (119905) isin 1198651(119905 119910 (119905)) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(41)

It is clear that all the solutions of (41) are solutions of (2)Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595lowast

(119905)

120595lowast(119905) = 119901 (119905) 120595 (119872)

(42)

It is clear that 119881 is weakly compact in 1198711([0 119887]R119873

) Remarkthat for every 119891 isin 119881 there exists a unique solution 119871(119891) ofthe following problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (119905) = 1199100 119910

1015840

(0) = 1199101

(43)

this solution is defined by

119871 (119891) (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (44)

We claim that 119871 is continuous Indeed let 119891119899rarr 119891 converge

in 1198711([0 119887]R119873

) as 119899 rarr infin set 119910119899= 119871(119891

119899) 119899 isin N It is clear

6 Abstract and Applied Analysis

that 119910119899 119899 isin N is relatively compact in 119862([0 119887]R119873

) and 119910119899

converge to 119910 isin 119862([0 119887]R119873) Let

119911 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 isin [0 119887]

(45)

Then

1003817100381710038171003817119910119899minus 119911

1003817100381710038171003817infin le119887120572

Γ (120572)int

119887

0

1003817100381710038171003817119891119899 (119904) minus 119891 (119904)1003817100381710038171003817 119889119904 997888rarr 0

as 119899 997888rarr infin

(46)

Hence 119870 = 119871(119881) is compact and convex subset of119862([0 119887]R119873

) Let 119878119865 119870 rarr Pclcv(119871

1([0 119887]R119873

)) be themultivalued Nemitsky operator defined by

1198781198651

(119910) = 119891 isin 1198711([0 119887] R

119873) 119891 (119905) isin 119865

1(119905 119910 (119905))

ae 119905 isin [0 119887] = 1198781198651119910

(47)

It is clear that 1198651(sdot sdot) is 119867

119889continuous and 119865

1(sdot sdot) isin

P119908119896cpcv(R

119873) and is integrably bounded then byTheorem 17

(see also Theorem 65 in [32] or Theorem 11 in [34]) we canfind a continuous function 119892 119870 rarr 119871

1

119908([0 119887]R119873

) such that

119892 (119909) isin ext 1198781198651

(119910) forall119910 isin 119870 (48)

From Benamara [49] we know that

ext 1198781198651

(119910) = 119878ext 1198651(sdot119910(sdot))

forall119910 isin 119870 (49)

Setting119873 = 119871 ∘ 119892 and letting 119910 isin 119870 then

119892 (119910) isin 1198651(sdot 119910 (sdot)) 997904rArr 119892 (119910) isin 119881 997904rArr 119873(119910)

= 119871 (119892 (119910)) isin 119870

(50)

Now we prove that 119873 is continuous Indeed let 119910119899isin 119870

converge to 119910 in 119862([0 119887]R119873)

Then

119892 (119910119899) converge weakly to 119892 (119910) as 119899 997888rarr infin (51)

Since119873(119910119899) = 119871(119892(119910

119899)) isin 119870 and 119892(119910

119899)(sdot) isin 119865(119905 119910

119899(119905)) then

119892 (119910119899) (sdot) isin 119865 (sdot 119861

119872) isin Pcp (R

119873) (52)

FromLemma 10 119892(119910119899) converge weakly to119910 in 1198711

([0 119887]R119873)

as 119899 rarr infin By the definition of119873 we have

119873(119910119899) = 119910

0+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910119899) (119904) 119889119904

119905 isin [0 119887]

119873 (119910) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910) (119904) 119889119904

119905 isin [0 119887]

(53)

Since 119873(119910119899) 119899 isin N sub 119870 then there exists subsequence of

119873(119910119899) converge in 119862([0 119887]R119873

) Then

119873(119910119899) (119905) 997888rarr 119873(119910) (119905) forall119905 isin [0 119887] as 119899 997888rarr infin

(54)

This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)

5 The Relaxed Problem

In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]

Now we present our main result of this section

Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873

) be amultifunction satisfying the following hypotheses

(H3) The function 119865 [0 119887]timesR119873

rarr P119888119901119888V(R

119873) such that

for all 119909 isin R119873 the map

119905 997891997888rarr 119865 (119905 119909) (55)

is measurable(H

4) There exists 119901 isin 1198711

(119869R+) such that

119867119889(119865 (119905 119909) 119865 (119905 119910)) le 119901 (119905)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

for ae 119905 isin [0 119887] and each 119909 119910 isin R119873

119867119889(119865 (119905 0) 0) le 119901 (119905) for ae 119905 isin [0 119887]

(56)

Then 119878119890= 119878

119888

Proof By Coviz and Nadlar fixed point theorem we caneasily prove that 119878

119888= 0 and since 119865 has compact and convex

valued then 119878119888is compact in 119862([0 119887]R119873

) For more infor-mation we see [25 27ndash29 51 52]

Let 119910 isin 119878119888 then there exists 119891 isin 119878

119865119910such that

119910 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (57)

Let119870 be a compact and convex set in 119862([0 119887]R119873) such that

119878119888sub 119870 Given that 119910

lowastisin 119870 and 120598 gt 0 we define the following

multifunction 119880120598 [0 119887] rarr P(R119873

) by

119880120598(119905) = 119906 isin R

1198731003817100381710038171003817119891 (119905) minus 119906

1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598

119906 isin 119865 (119905 119910lowast(119905))

(58)

Abstract and Applied Analysis 7

The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr

119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact

values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873

) Then by Lemma 3 there exists a measurableselection V

1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that

1003817100381710038171003817119891 (119905) minus V1(119905)1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598 (59)

this implies that 119880120598(sdot) = 0 We consider 119866

120598 119870 rarr P(119871

1(119869

R119873) defined by

119866120598(119910) = 119891

lowastisin F (119910)

1003817100381710038171003817119891 (119905) minus 119891lowast (119905)1003817100381710038171003817

lt 120598 + 119889 (119891lowast(119905) 119865 (119905 119910 (119905)))

(60)

Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866

120598(119910) is lsc with decomposable

values By Lemma 8 there exists a continuous selection 119891120598

119862([0 119887]R119873) rarr 119871

1(119869R119873

) such that

119891120598(119910) isin 119866

120598(119910) forall119910 isin 119862 ([0 119887] R

119873) (61)

FromTheorem 17 there exists function 119892120598 119870 rarr 119871

119908([0 119887]

R119873) such that

119892120598(119910) isin ext 119878

119865(119910) = 119878ext 119865(sdot119910(sdot)) forall119910 isin 119870

119892120598(119910) minus 119891

120598(119910)

119908le 120598 forall119910 isin 119870

(62)

From (H3) we can prove that there exists119872 gt 0 such that

10038171003817100381710038171199101003817100381710038171003817infin le 119872 forall119910 isin 119878

119888 (63)

Consider the sequence 120598119899rarr 0 as 119899 rarr infin and set119892

119899= 119892

120598119899

119891119899= 119891

120598119899

Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595 (119905) ae 119905 isin [0 119887]

120595 (119905) = (1 +119872)119901 (119905)

(64)

Let 119871 119881 rarr 119862([0 119887]R119873) be the map such that each 119891 isin 119881

assigns the unique solution of the problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(65)

As in Theorem 24 we can prove that 119871(119881) is compact in119862([0 119887]R119873

) and the operator 119873119899= 119871 ∘ 119892

119899 119870 rarr 119870 is

compact then by Schauderrsquos fixed point there exists 119910119899isin 119870

such that 119910119899isin 119878

119890and

119910119899(119905) = 119910

0+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892119899(119910

119899) (119904) 119889119904

ae 119905 isin [0 119887] 119899 isin N

(66)

Hence

1003817100381710038171003817119910 (119905) minus 119910119899(119905)1003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

1003817100381710038171003817119891119899 (119910119899) (119904) minus 119891 (119904)

1003817100381710038171003817 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119889 (119891 (119904) 119891

119899(119910

119899) (119904))) 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119867

119889(119865 (119904 119910 (119904)) 119865 (119904 119910

119899(119904)))) 119889119904

le119887120572+1

Γ (120572 + 1)120598119899+119887120572+1

Γ (120572)120598119899+ int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910119899

(119904)1003817100381710038171003817

(67)

Let 119910(sdot) be a limit point of the sequence 119910119899(sdot) Then it follows

that from the above inequality one has

1003817100381710038171003817119910 (119905) minus 119910 (119905)1003817100381710038171003817 le int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910 (119904)

1003817100381710038171003817 119889119904(68)

which implies 119910(sdot) = 119910(sdot) Consequently 119910 isin 119878119888is a unique

limit point of 119910119899(sdot) isin 119878

119890

Example 26 Let 119865 119869 timesR119873rarr Pcpcv(R

119873) with

119865 (119905 119910) = 119861 (1198911(119905 119910) 119891

2(119905 119910)) (69)

where 1198911 119891

2 119869 times R119873

rarr R119873 are Caratheodory functionsand bounded

Then (2) is solvable

Example 27 If in addition to the conditions on 119865 ofExample 26 119891

1and 119891

2are Lipschitz functions then 119878

119890= 119878

119888

Acknowledgments

This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER

8 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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OptimizationJournal of

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Let 119865 [0 119887] times 119883 rarr P(119883) be a multivalued map withnonempty closed values Assign to119865 themultivalued operatorF 119862([0 119887] 119883) rarr P(119871

1([0 119887] 119883)) defined by

F (119910) = V isin 1198711

([0 119887] 119883) V (119905) isin 119865 (119905 119910 (119905))

ae 119905 isin [0 119887] (13)

The operator F is called the Nemytsrsquokiı operator associatedto 119865

Definition 7 Let 119865 [0 119887] times 119883 rarr P(119883) be a multivaluedmapwith nonempty compact valuesWe say that119865 is of lowersemicontinuous type (lsc type) if its associated Nemytsrsquokiıoperator F is lower semicontinuous and has nonemptyclosed and decomposable values

Next we state a classical selection theorem due to Bressanand Colombo

Lemma 8 (see [40]) Let119883 be a separable metric space and let119864 be a Banach spaceThen every lsc multivalued operator119873

119883 rarr P119888119897(119871

1([0 119887] 119864)) with closed decomposable values has

a continuous selection that is there exists a continuous single-valued function 119891 119883 rarr 119871

1([0 119887] 119864) such that 119891(119909) isin 119873(119909)

for every 119909 isin 119883

Let us introduce the following hypothesis

(H1) 119865 [0 119887]times119883 rarr P(119883) is a nonempty compact valuedmultivalued map such that

(a) the mapping (119905 119910) 997891rarr 119865(119905 119910) is L otimes Bmeasurable

(b) the mapping 119910 997891rarr 119865(119905 119910) is lower semicontinu-ous for ae 119905 isin [0 119887]

Lemma 9 (see eg [41]) Let 119865 119869 times 119883 rarr P119888119901(119864) be an

integrably bounded multivalued map satisfying (H1) Then 119865

is of lower semicontinuous type

Define

119865 (119870) = 119891 isin 1198711

([0 119887] 119883) 119891 (119905) isin 119870 ae 119905 isin [0 119887]

119870 sub 119883

(14)

where119883 is a Banach space

Lemma 10 (see [37]) Let 119870 sub 119883 be a weakly compactsubset of 119883 Then 119865(119870) is relatively weakly compact subset of1198711([0 119887] 119883) Moreover if 119870 is convex then 119865(119870) is weakly

compact in 1198711([0 119887] 119883)

Definition 11 A multifunction 119865 [0 119887] times 119884 rarr P119908cpcv(119883)

possesses the Scorza-Dragoni property (S-D property) if foreach 120598 gt 0 there exists a closed set 119869

120598sub [0 119887]whose Lebesgue

measure 120583(119869120598) le 120598 and such that 119865 [0 119887] 119869

120598times 119884 rarr 119883 is

continuous with respect to the metric 119889119883(sdot sdot)

Remark 12 It is well known that if the map 119865 [0 119887] times 119884 rarr

P119908cpcv(119883) is continuous with respect to 119910 for almost every

119905 isin [0 119887] and is measurable with respect to 119905 for every 119910 isin 119884then it possesses the S-D property

In what follows we present some definitions and proper-ties of extreme points

Definition 13 Let119860 be a nonempty subset of a real or complexlinear vector space An extreme point of a convex set 119860 is apoint 119909 isin 119860 with the property that if 119909 = 120582119910 + (1 minus 120582)119911 with119910 119911 isin 119860 and 120582 isin [0 1] then 119910 = 119909 andor 119911 = 119909 ext(119860) willdenote the set of extreme points of 119860

In other words an extreme point is a point that is not aninterior point of any line segment lying entirely in 119860

Lemma 14 (see [42]) A nonempty compact set in a locallyconvex linear topological space has extremal points

Let 1199091015840

119899119899isinN be a denumerable dense (in 120590(1198831015840

119883) topol-ogy) subset of the set 119909 isin 119883 119909 le 1 For any 119860 isin

Pcpcv(119883) and 1199091015840

119899define the function

119889119899

(119860 119906) = max ⟨119910 minus 119911 1199091015840

119899⟩ 119910 119911 isin 119860 119906 =

119910 + 119911

2

(15)

Lemma 15 (see [33]) 119906 isin ext(119860) if and only if 119889119899(119860 119906) = 0

for all 119899 ge 1

In accordance with Krein-Milman and Trojansky theo-rem [43] the set ext(119878

119865) is nonempty and co(ext(119878

119865)) = 119878

119865

Lemma 16 (see [33]) Let 119865 [0 119887] rarr P119908119888119901119888V(119883) be a

measurable integrably bounded map Then

ext (119878119865) sube 119878

119865 (16)

where ext (119878119865) is the closure of set ext (119878

119865) in the topology of

the space 1198711([0 119887] 119883)

Theorem 17 (see [33]) Let 119865 [0 119887] times 119884 rarr P119908119888119901119888V(119883)

be a multivalued map that has the 119878-119863 property and let it beintegrable bounded on compacts from 119884 Consider a compactsubset 119870 sub 119862([0 119887] 119883) and define the multivalued map 119866

119870 rarr 1198711([0 119887] 119883) by

119866 (119910 (sdot))

= 119891 isin 1198711

([0 119887] 119883) 119891 (119905) isin 119865 (119905 119910 (119905)) 119886119890 119900119899 [0 119887]

119910 isin 119870

(17)

Then for every 119870 compact in 119862([0 119887] 119883) 120598 gt 0 and anycontinuous selection 119891 119870 rarr 119871

1([0 119887] 119883) there exists a

continuous selector119892 119870 rarr 1198711([0 119887] 119883) of themap ext (119866)

119870 rarr 1198711([0 119887] 119883) such that for all 119910 isin 119862([0 119887] 119883) one has

sup119905isin[0119887]

10038171003817100381710038171003817100381710038171003817int

119905

0

((119891119910) (119904) minus (119892119910) (119904)) 119889119904

10038171003817100381710038171003817100381710038171003817le 120598 (18)

4 Abstract and Applied Analysis

For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]

3 Fractional Calculus

According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads

119868119899119891 (119905) =

1

(119899 minus 1)int

119905

0

(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N

(19)

Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871

1([119886 119887]R) is defined by

119868120572

119886+119891 (119905) = int

119905

119886

(119905 minus 119904)120572minus1

Γ (120572)119891 (119904) 119889119904 (20)

where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601

120572(119905) where 120601

120572(119905) = 119905

(120572minus1)Γ(120572) for 119905 gt 0 and

we write 120601120572(119905) = 0 for 119905 le 0 and 120601

120572rarr 120575(119905) as 120572 rarr 0 where

120575 is the delta function and Γ is the Euler gamma functiondefined by

Γ (120572) = int

infin

0

119905120572minus1

119890minus119905119889119905 120572 gt 0 (21)

For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0

+ this limit may be infinite

After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that

119863119899119868119899= Id 119868

119899119863

119899= Id 119899 isin N (22)

that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that

119868119899119863

119899119891 (119905) = 119891 (119905) minus

119899minus1

sum

119896=0

119891(119896)(119886

+)(119905 minus 119886)

119896

119896 119905 gt 0 (23)

As a consequence we expect that 119863120572 is defined as the leftinverse to 119868

120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0

Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by

119863120572119891 (119905) =

1

Γ (119899 minus 120572)(119889

119889119905)

119899

int

119905

119886

(119905 minus 119904)minus120572+119899minus1

119891 (119904) 119889119904 (24)

where 119899 = [120572] + 1 and [120572] is the integer part of 120572

Defining for consistency 1198630= 119868

0= Id then we easily

recognize that

119863120572119868120572= Id 120572 ge 0 (25)

119863120572119905120574=

Γ (120574 + 1)

Γ (120574 + 1 minus 120572)119905120574minus120572

120572 gt 0 120574 isin (minus1 0) cup (0 +infin) 119905 gt 0

(26)

Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer

Note the remarkable fact that the fractional derivative119863

120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N

In fact (26) with 120574 = 0 illustrates that

1198631205721 =

(119905 minus 119886)minus120572

Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)

It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the

gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional

derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])

Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-

order derivative of 119891 is defined by

(119888

119863120572

119891) (119905) =1

Γ (119899 minus 120572)int

119905

119886

(119905 minus 119904)119899minus120572minus1

119891119899

(119904) 119889119904 (28)

This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572

lowastwe (tacitly) assume that this condition is

met We easily recognize that in general

119863120572119891 (119905) = 119863

119898119868119898minus120572

119891 (119905) = 119869119898minus120572

119863119898119891 (119905) = 119863

120572

lowast119891 (119905) (29)

unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0

119863120572119891 (119905) =

119888

119863120572

119891 (119905) +

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+) (30)

and therefore recalling the fractional derivative of the powerfunction (26) one has

119863120572(119891 (119905) minus

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+)) = 119863

120572

lowast119891 (119905) (31)

Abstract and Applied Analysis 5

The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886

+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero

119888119863

1205721 = 0 120572 gt 0 (32)

We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have

119863120572

(119905 minus 119886)120572minus119895

= 0 for 119895 = 1 2 [120572] + 1 (33)

From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N

119863120572119891 (119905) = 119863

120572119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

120572minus119895

119888119863

120572119891 (119905) =

119888

119863120572

119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

119899minus119895

(34)

In these formulas the coefficients 119888119895are arbitrary constants

For proving allmain results we present the following auxiliarylemmas

Lemma 21 (see [10]) Let 120572 gt 0 and let 119910 isin 119871infin(119886 119887) or

119862([119886 119887]) Then

(119888

119863120572

119868120572119910) (119905) = 119910 (119905) (35)

Lemma 22 (see [10]) Let 120572 gt 0 and 119899 = [120572] + 1 If 119910 isin

119860119862119899[119886 119887] or 119910 isin 119862119899

[119886 119887] then

(119868120572 119888

119863120572119910) (119905) = 119910 (119905) minus

119899minus1

sum

119896=0

119910(119896)(119886)

119896(119905 minus 119886)

119896 (36)

For further readings and details on fractional calculus werefer to the books and papers by Kilbas [10] Podlubny [12]Samko [13] and Caputo [46ndash48]

4 Existence Result

Definition 23 A function 119910 isin 119862([0 119887]R119873) is called mild

solution of problem (1) if there exist 119891 isin 1198711(119869R119873

) such that

119910 (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)1minus120572

119891 (119904) 119889119904 119905 isin [0 119887]

(37)

where 119891 isin 119878119865119910

= V isin 1198711([0 119887]R119873

) 119891(119905) isin 119865(119905 119910(119905)) aeon [0 119887]

We will impose the following conditions on 119865

(H1) The function 119865 119869 timesR119873

rarr Pcpcv(R119873) such that

(a) for all 119909 isin R119873 the map 119905 997891rarr 119865(119905 119909) ismeasurable

(b) for every 119905 isin [0 119887] the multivalued map 119909 rarr

119865(119905 119909) is119867119889continuous

(H2) There exist 119901 isin 119871

1(119869R+

) and a continuous nonde-creasing function 120595 [0infin) rarr (0infin) such that

119865 (119905 119909)P = sup V V isin 119865 (119905 119909) le 119901 (119905) 120595 (119909)

for ae 119905 isin [0 119887] and each 119909 isin R119873

(38)

with

int

119887

0

119901 (119904) 119889119904 lt int

infin

1199100+119887119910

1

119889119906

120595 (119906) (39)

Theorem24 Assume that the conditions (H1)-(H

2) and then

the problem (2) have at least one solution

Proof From (H2) there exists 119872 gt 0 such that 119910

infinle 119872

for each 119910 isin 119878119888

Let

1198651(119905 119910) =

119865 (119905 119910) if 10038171003817100381710038171199101003817100381710038171003817 le 119872

119865(119905119872119910

10038171003817100381710038171199101003817100381710038171003817

) if 10038171003817100381710038171199101003817100381710038171003817 ge 119872

(40)

We consider119888

119863120572

119910 (119905) isin 1198651(119905 119910 (119905)) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(41)

It is clear that all the solutions of (41) are solutions of (2)Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595lowast

(119905)

120595lowast(119905) = 119901 (119905) 120595 (119872)

(42)

It is clear that 119881 is weakly compact in 1198711([0 119887]R119873

) Remarkthat for every 119891 isin 119881 there exists a unique solution 119871(119891) ofthe following problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (119905) = 1199100 119910

1015840

(0) = 1199101

(43)

this solution is defined by

119871 (119891) (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (44)

We claim that 119871 is continuous Indeed let 119891119899rarr 119891 converge

in 1198711([0 119887]R119873

) as 119899 rarr infin set 119910119899= 119871(119891

119899) 119899 isin N It is clear

6 Abstract and Applied Analysis

that 119910119899 119899 isin N is relatively compact in 119862([0 119887]R119873

) and 119910119899

converge to 119910 isin 119862([0 119887]R119873) Let

119911 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 isin [0 119887]

(45)

Then

1003817100381710038171003817119910119899minus 119911

1003817100381710038171003817infin le119887120572

Γ (120572)int

119887

0

1003817100381710038171003817119891119899 (119904) minus 119891 (119904)1003817100381710038171003817 119889119904 997888rarr 0

as 119899 997888rarr infin

(46)

Hence 119870 = 119871(119881) is compact and convex subset of119862([0 119887]R119873

) Let 119878119865 119870 rarr Pclcv(119871

1([0 119887]R119873

)) be themultivalued Nemitsky operator defined by

1198781198651

(119910) = 119891 isin 1198711([0 119887] R

119873) 119891 (119905) isin 119865

1(119905 119910 (119905))

ae 119905 isin [0 119887] = 1198781198651119910

(47)

It is clear that 1198651(sdot sdot) is 119867

119889continuous and 119865

1(sdot sdot) isin

P119908119896cpcv(R

119873) and is integrably bounded then byTheorem 17

(see also Theorem 65 in [32] or Theorem 11 in [34]) we canfind a continuous function 119892 119870 rarr 119871

1

119908([0 119887]R119873

) such that

119892 (119909) isin ext 1198781198651

(119910) forall119910 isin 119870 (48)

From Benamara [49] we know that

ext 1198781198651

(119910) = 119878ext 1198651(sdot119910(sdot))

forall119910 isin 119870 (49)

Setting119873 = 119871 ∘ 119892 and letting 119910 isin 119870 then

119892 (119910) isin 1198651(sdot 119910 (sdot)) 997904rArr 119892 (119910) isin 119881 997904rArr 119873(119910)

= 119871 (119892 (119910)) isin 119870

(50)

Now we prove that 119873 is continuous Indeed let 119910119899isin 119870

converge to 119910 in 119862([0 119887]R119873)

Then

119892 (119910119899) converge weakly to 119892 (119910) as 119899 997888rarr infin (51)

Since119873(119910119899) = 119871(119892(119910

119899)) isin 119870 and 119892(119910

119899)(sdot) isin 119865(119905 119910

119899(119905)) then

119892 (119910119899) (sdot) isin 119865 (sdot 119861

119872) isin Pcp (R

119873) (52)

FromLemma 10 119892(119910119899) converge weakly to119910 in 1198711

([0 119887]R119873)

as 119899 rarr infin By the definition of119873 we have

119873(119910119899) = 119910

0+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910119899) (119904) 119889119904

119905 isin [0 119887]

119873 (119910) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910) (119904) 119889119904

119905 isin [0 119887]

(53)

Since 119873(119910119899) 119899 isin N sub 119870 then there exists subsequence of

119873(119910119899) converge in 119862([0 119887]R119873

) Then

119873(119910119899) (119905) 997888rarr 119873(119910) (119905) forall119905 isin [0 119887] as 119899 997888rarr infin

(54)

This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)

5 The Relaxed Problem

In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]

Now we present our main result of this section

Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873

) be amultifunction satisfying the following hypotheses

(H3) The function 119865 [0 119887]timesR119873

rarr P119888119901119888V(R

119873) such that

for all 119909 isin R119873 the map

119905 997891997888rarr 119865 (119905 119909) (55)

is measurable(H

4) There exists 119901 isin 1198711

(119869R+) such that

119867119889(119865 (119905 119909) 119865 (119905 119910)) le 119901 (119905)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

for ae 119905 isin [0 119887] and each 119909 119910 isin R119873

119867119889(119865 (119905 0) 0) le 119901 (119905) for ae 119905 isin [0 119887]

(56)

Then 119878119890= 119878

119888

Proof By Coviz and Nadlar fixed point theorem we caneasily prove that 119878

119888= 0 and since 119865 has compact and convex

valued then 119878119888is compact in 119862([0 119887]R119873

) For more infor-mation we see [25 27ndash29 51 52]

Let 119910 isin 119878119888 then there exists 119891 isin 119878

119865119910such that

119910 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (57)

Let119870 be a compact and convex set in 119862([0 119887]R119873) such that

119878119888sub 119870 Given that 119910

lowastisin 119870 and 120598 gt 0 we define the following

multifunction 119880120598 [0 119887] rarr P(R119873

) by

119880120598(119905) = 119906 isin R

1198731003817100381710038171003817119891 (119905) minus 119906

1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598

119906 isin 119865 (119905 119910lowast(119905))

(58)

Abstract and Applied Analysis 7

The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr

119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact

values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873

) Then by Lemma 3 there exists a measurableselection V

1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that

1003817100381710038171003817119891 (119905) minus V1(119905)1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598 (59)

this implies that 119880120598(sdot) = 0 We consider 119866

120598 119870 rarr P(119871

1(119869

R119873) defined by

119866120598(119910) = 119891

lowastisin F (119910)

1003817100381710038171003817119891 (119905) minus 119891lowast (119905)1003817100381710038171003817

lt 120598 + 119889 (119891lowast(119905) 119865 (119905 119910 (119905)))

(60)

Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866

120598(119910) is lsc with decomposable

values By Lemma 8 there exists a continuous selection 119891120598

119862([0 119887]R119873) rarr 119871

1(119869R119873

) such that

119891120598(119910) isin 119866

120598(119910) forall119910 isin 119862 ([0 119887] R

119873) (61)

FromTheorem 17 there exists function 119892120598 119870 rarr 119871

119908([0 119887]

R119873) such that

119892120598(119910) isin ext 119878

119865(119910) = 119878ext 119865(sdot119910(sdot)) forall119910 isin 119870

119892120598(119910) minus 119891

120598(119910)

119908le 120598 forall119910 isin 119870

(62)

From (H3) we can prove that there exists119872 gt 0 such that

10038171003817100381710038171199101003817100381710038171003817infin le 119872 forall119910 isin 119878

119888 (63)

Consider the sequence 120598119899rarr 0 as 119899 rarr infin and set119892

119899= 119892

120598119899

119891119899= 119891

120598119899

Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595 (119905) ae 119905 isin [0 119887]

120595 (119905) = (1 +119872)119901 (119905)

(64)

Let 119871 119881 rarr 119862([0 119887]R119873) be the map such that each 119891 isin 119881

assigns the unique solution of the problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(65)

As in Theorem 24 we can prove that 119871(119881) is compact in119862([0 119887]R119873

) and the operator 119873119899= 119871 ∘ 119892

119899 119870 rarr 119870 is

compact then by Schauderrsquos fixed point there exists 119910119899isin 119870

such that 119910119899isin 119878

119890and

119910119899(119905) = 119910

0+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892119899(119910

119899) (119904) 119889119904

ae 119905 isin [0 119887] 119899 isin N

(66)

Hence

1003817100381710038171003817119910 (119905) minus 119910119899(119905)1003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

1003817100381710038171003817119891119899 (119910119899) (119904) minus 119891 (119904)

1003817100381710038171003817 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119889 (119891 (119904) 119891

119899(119910

119899) (119904))) 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119867

119889(119865 (119904 119910 (119904)) 119865 (119904 119910

119899(119904)))) 119889119904

le119887120572+1

Γ (120572 + 1)120598119899+119887120572+1

Γ (120572)120598119899+ int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910119899

(119904)1003817100381710038171003817

(67)

Let 119910(sdot) be a limit point of the sequence 119910119899(sdot) Then it follows

that from the above inequality one has

1003817100381710038171003817119910 (119905) minus 119910 (119905)1003817100381710038171003817 le int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910 (119904)

1003817100381710038171003817 119889119904(68)

which implies 119910(sdot) = 119910(sdot) Consequently 119910 isin 119878119888is a unique

limit point of 119910119899(sdot) isin 119878

119890

Example 26 Let 119865 119869 timesR119873rarr Pcpcv(R

119873) with

119865 (119905 119910) = 119861 (1198911(119905 119910) 119891

2(119905 119910)) (69)

where 1198911 119891

2 119869 times R119873

rarr R119873 are Caratheodory functionsand bounded

Then (2) is solvable

Example 27 If in addition to the conditions on 119865 ofExample 26 119891

1and 119891

2are Lipschitz functions then 119878

119890= 119878

119888

Acknowledgments

This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER

8 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

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

4 Abstract and Applied Analysis

For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]

3 Fractional Calculus

According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads

119868119899119891 (119905) =

1

(119899 minus 1)int

119905

0

(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N

(19)

Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871

1([119886 119887]R) is defined by

119868120572

119886+119891 (119905) = int

119905

119886

(119905 minus 119904)120572minus1

Γ (120572)119891 (119904) 119889119904 (20)

where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601

120572(119905) where 120601

120572(119905) = 119905

(120572minus1)Γ(120572) for 119905 gt 0 and

we write 120601120572(119905) = 0 for 119905 le 0 and 120601

120572rarr 120575(119905) as 120572 rarr 0 where

120575 is the delta function and Γ is the Euler gamma functiondefined by

Γ (120572) = int

infin

0

119905120572minus1

119890minus119905119889119905 120572 gt 0 (21)

For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0

+ this limit may be infinite

After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that

119863119899119868119899= Id 119868

119899119863

119899= Id 119899 isin N (22)

that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that

119868119899119863

119899119891 (119905) = 119891 (119905) minus

119899minus1

sum

119896=0

119891(119896)(119886

+)(119905 minus 119886)

119896

119896 119905 gt 0 (23)

As a consequence we expect that 119863120572 is defined as the leftinverse to 119868

120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0

Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by

119863120572119891 (119905) =

1

Γ (119899 minus 120572)(119889

119889119905)

119899

int

119905

119886

(119905 minus 119904)minus120572+119899minus1

119891 (119904) 119889119904 (24)

where 119899 = [120572] + 1 and [120572] is the integer part of 120572

Defining for consistency 1198630= 119868

0= Id then we easily

recognize that

119863120572119868120572= Id 120572 ge 0 (25)

119863120572119905120574=

Γ (120574 + 1)

Γ (120574 + 1 minus 120572)119905120574minus120572

120572 gt 0 120574 isin (minus1 0) cup (0 +infin) 119905 gt 0

(26)

Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer

Note the remarkable fact that the fractional derivative119863

120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N

In fact (26) with 120574 = 0 illustrates that

1198631205721 =

(119905 minus 119886)minus120572

Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)

It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the

gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional

derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])

Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-

order derivative of 119891 is defined by

(119888

119863120572

119891) (119905) =1

Γ (119899 minus 120572)int

119905

119886

(119905 minus 119904)119899minus120572minus1

119891119899

(119904) 119889119904 (28)

This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572

lowastwe (tacitly) assume that this condition is

met We easily recognize that in general

119863120572119891 (119905) = 119863

119898119868119898minus120572

119891 (119905) = 119869119898minus120572

119863119898119891 (119905) = 119863

120572

lowast119891 (119905) (29)

unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0

119863120572119891 (119905) =

119888

119863120572

119891 (119905) +

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+) (30)

and therefore recalling the fractional derivative of the powerfunction (26) one has

119863120572(119891 (119905) minus

119898minus1

sum

119896=0

(119905 minus 119886)119896minus120572

Γ (119896 minus 120572 + 1)119891

(119896)(119886

+)) = 119863

120572

lowast119891 (119905) (31)

Abstract and Applied Analysis 5

The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886

+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero

119888119863

1205721 = 0 120572 gt 0 (32)

We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have

119863120572

(119905 minus 119886)120572minus119895

= 0 for 119895 = 1 2 [120572] + 1 (33)

From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N

119863120572119891 (119905) = 119863

120572119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

120572minus119895

119888119863

120572119891 (119905) =

119888

119863120572

119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

119899minus119895

(34)

In these formulas the coefficients 119888119895are arbitrary constants

For proving allmain results we present the following auxiliarylemmas

Lemma 21 (see [10]) Let 120572 gt 0 and let 119910 isin 119871infin(119886 119887) or

119862([119886 119887]) Then

(119888

119863120572

119868120572119910) (119905) = 119910 (119905) (35)

Lemma 22 (see [10]) Let 120572 gt 0 and 119899 = [120572] + 1 If 119910 isin

119860119862119899[119886 119887] or 119910 isin 119862119899

[119886 119887] then

(119868120572 119888

119863120572119910) (119905) = 119910 (119905) minus

119899minus1

sum

119896=0

119910(119896)(119886)

119896(119905 minus 119886)

119896 (36)

For further readings and details on fractional calculus werefer to the books and papers by Kilbas [10] Podlubny [12]Samko [13] and Caputo [46ndash48]

4 Existence Result

Definition 23 A function 119910 isin 119862([0 119887]R119873) is called mild

solution of problem (1) if there exist 119891 isin 1198711(119869R119873

) such that

119910 (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)1minus120572

119891 (119904) 119889119904 119905 isin [0 119887]

(37)

where 119891 isin 119878119865119910

= V isin 1198711([0 119887]R119873

) 119891(119905) isin 119865(119905 119910(119905)) aeon [0 119887]

We will impose the following conditions on 119865

(H1) The function 119865 119869 timesR119873

rarr Pcpcv(R119873) such that

(a) for all 119909 isin R119873 the map 119905 997891rarr 119865(119905 119909) ismeasurable

(b) for every 119905 isin [0 119887] the multivalued map 119909 rarr

119865(119905 119909) is119867119889continuous

(H2) There exist 119901 isin 119871

1(119869R+

) and a continuous nonde-creasing function 120595 [0infin) rarr (0infin) such that

119865 (119905 119909)P = sup V V isin 119865 (119905 119909) le 119901 (119905) 120595 (119909)

for ae 119905 isin [0 119887] and each 119909 isin R119873

(38)

with

int

119887

0

119901 (119904) 119889119904 lt int

infin

1199100+119887119910

1

119889119906

120595 (119906) (39)

Theorem24 Assume that the conditions (H1)-(H

2) and then

the problem (2) have at least one solution

Proof From (H2) there exists 119872 gt 0 such that 119910

infinle 119872

for each 119910 isin 119878119888

Let

1198651(119905 119910) =

119865 (119905 119910) if 10038171003817100381710038171199101003817100381710038171003817 le 119872

119865(119905119872119910

10038171003817100381710038171199101003817100381710038171003817

) if 10038171003817100381710038171199101003817100381710038171003817 ge 119872

(40)

We consider119888

119863120572

119910 (119905) isin 1198651(119905 119910 (119905)) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(41)

It is clear that all the solutions of (41) are solutions of (2)Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595lowast

(119905)

120595lowast(119905) = 119901 (119905) 120595 (119872)

(42)

It is clear that 119881 is weakly compact in 1198711([0 119887]R119873

) Remarkthat for every 119891 isin 119881 there exists a unique solution 119871(119891) ofthe following problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (119905) = 1199100 119910

1015840

(0) = 1199101

(43)

this solution is defined by

119871 (119891) (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (44)

We claim that 119871 is continuous Indeed let 119891119899rarr 119891 converge

in 1198711([0 119887]R119873

) as 119899 rarr infin set 119910119899= 119871(119891

119899) 119899 isin N It is clear

6 Abstract and Applied Analysis

that 119910119899 119899 isin N is relatively compact in 119862([0 119887]R119873

) and 119910119899

converge to 119910 isin 119862([0 119887]R119873) Let

119911 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 isin [0 119887]

(45)

Then

1003817100381710038171003817119910119899minus 119911

1003817100381710038171003817infin le119887120572

Γ (120572)int

119887

0

1003817100381710038171003817119891119899 (119904) minus 119891 (119904)1003817100381710038171003817 119889119904 997888rarr 0

as 119899 997888rarr infin

(46)

Hence 119870 = 119871(119881) is compact and convex subset of119862([0 119887]R119873

) Let 119878119865 119870 rarr Pclcv(119871

1([0 119887]R119873

)) be themultivalued Nemitsky operator defined by

1198781198651

(119910) = 119891 isin 1198711([0 119887] R

119873) 119891 (119905) isin 119865

1(119905 119910 (119905))

ae 119905 isin [0 119887] = 1198781198651119910

(47)

It is clear that 1198651(sdot sdot) is 119867

119889continuous and 119865

1(sdot sdot) isin

P119908119896cpcv(R

119873) and is integrably bounded then byTheorem 17

(see also Theorem 65 in [32] or Theorem 11 in [34]) we canfind a continuous function 119892 119870 rarr 119871

1

119908([0 119887]R119873

) such that

119892 (119909) isin ext 1198781198651

(119910) forall119910 isin 119870 (48)

From Benamara [49] we know that

ext 1198781198651

(119910) = 119878ext 1198651(sdot119910(sdot))

forall119910 isin 119870 (49)

Setting119873 = 119871 ∘ 119892 and letting 119910 isin 119870 then

119892 (119910) isin 1198651(sdot 119910 (sdot)) 997904rArr 119892 (119910) isin 119881 997904rArr 119873(119910)

= 119871 (119892 (119910)) isin 119870

(50)

Now we prove that 119873 is continuous Indeed let 119910119899isin 119870

converge to 119910 in 119862([0 119887]R119873)

Then

119892 (119910119899) converge weakly to 119892 (119910) as 119899 997888rarr infin (51)

Since119873(119910119899) = 119871(119892(119910

119899)) isin 119870 and 119892(119910

119899)(sdot) isin 119865(119905 119910

119899(119905)) then

119892 (119910119899) (sdot) isin 119865 (sdot 119861

119872) isin Pcp (R

119873) (52)

FromLemma 10 119892(119910119899) converge weakly to119910 in 1198711

([0 119887]R119873)

as 119899 rarr infin By the definition of119873 we have

119873(119910119899) = 119910

0+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910119899) (119904) 119889119904

119905 isin [0 119887]

119873 (119910) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910) (119904) 119889119904

119905 isin [0 119887]

(53)

Since 119873(119910119899) 119899 isin N sub 119870 then there exists subsequence of

119873(119910119899) converge in 119862([0 119887]R119873

) Then

119873(119910119899) (119905) 997888rarr 119873(119910) (119905) forall119905 isin [0 119887] as 119899 997888rarr infin

(54)

This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)

5 The Relaxed Problem

In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]

Now we present our main result of this section

Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873

) be amultifunction satisfying the following hypotheses

(H3) The function 119865 [0 119887]timesR119873

rarr P119888119901119888V(R

119873) such that

for all 119909 isin R119873 the map

119905 997891997888rarr 119865 (119905 119909) (55)

is measurable(H

4) There exists 119901 isin 1198711

(119869R+) such that

119867119889(119865 (119905 119909) 119865 (119905 119910)) le 119901 (119905)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

for ae 119905 isin [0 119887] and each 119909 119910 isin R119873

119867119889(119865 (119905 0) 0) le 119901 (119905) for ae 119905 isin [0 119887]

(56)

Then 119878119890= 119878

119888

Proof By Coviz and Nadlar fixed point theorem we caneasily prove that 119878

119888= 0 and since 119865 has compact and convex

valued then 119878119888is compact in 119862([0 119887]R119873

) For more infor-mation we see [25 27ndash29 51 52]

Let 119910 isin 119878119888 then there exists 119891 isin 119878

119865119910such that

119910 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (57)

Let119870 be a compact and convex set in 119862([0 119887]R119873) such that

119878119888sub 119870 Given that 119910

lowastisin 119870 and 120598 gt 0 we define the following

multifunction 119880120598 [0 119887] rarr P(R119873

) by

119880120598(119905) = 119906 isin R

1198731003817100381710038171003817119891 (119905) minus 119906

1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598

119906 isin 119865 (119905 119910lowast(119905))

(58)

Abstract and Applied Analysis 7

The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr

119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact

values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873

) Then by Lemma 3 there exists a measurableselection V

1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that

1003817100381710038171003817119891 (119905) minus V1(119905)1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598 (59)

this implies that 119880120598(sdot) = 0 We consider 119866

120598 119870 rarr P(119871

1(119869

R119873) defined by

119866120598(119910) = 119891

lowastisin F (119910)

1003817100381710038171003817119891 (119905) minus 119891lowast (119905)1003817100381710038171003817

lt 120598 + 119889 (119891lowast(119905) 119865 (119905 119910 (119905)))

(60)

Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866

120598(119910) is lsc with decomposable

values By Lemma 8 there exists a continuous selection 119891120598

119862([0 119887]R119873) rarr 119871

1(119869R119873

) such that

119891120598(119910) isin 119866

120598(119910) forall119910 isin 119862 ([0 119887] R

119873) (61)

FromTheorem 17 there exists function 119892120598 119870 rarr 119871

119908([0 119887]

R119873) such that

119892120598(119910) isin ext 119878

119865(119910) = 119878ext 119865(sdot119910(sdot)) forall119910 isin 119870

119892120598(119910) minus 119891

120598(119910)

119908le 120598 forall119910 isin 119870

(62)

From (H3) we can prove that there exists119872 gt 0 such that

10038171003817100381710038171199101003817100381710038171003817infin le 119872 forall119910 isin 119878

119888 (63)

Consider the sequence 120598119899rarr 0 as 119899 rarr infin and set119892

119899= 119892

120598119899

119891119899= 119891

120598119899

Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595 (119905) ae 119905 isin [0 119887]

120595 (119905) = (1 +119872)119901 (119905)

(64)

Let 119871 119881 rarr 119862([0 119887]R119873) be the map such that each 119891 isin 119881

assigns the unique solution of the problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(65)

As in Theorem 24 we can prove that 119871(119881) is compact in119862([0 119887]R119873

) and the operator 119873119899= 119871 ∘ 119892

119899 119870 rarr 119870 is

compact then by Schauderrsquos fixed point there exists 119910119899isin 119870

such that 119910119899isin 119878

119890and

119910119899(119905) = 119910

0+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892119899(119910

119899) (119904) 119889119904

ae 119905 isin [0 119887] 119899 isin N

(66)

Hence

1003817100381710038171003817119910 (119905) minus 119910119899(119905)1003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

1003817100381710038171003817119891119899 (119910119899) (119904) minus 119891 (119904)

1003817100381710038171003817 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119889 (119891 (119904) 119891

119899(119910

119899) (119904))) 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119867

119889(119865 (119904 119910 (119904)) 119865 (119904 119910

119899(119904)))) 119889119904

le119887120572+1

Γ (120572 + 1)120598119899+119887120572+1

Γ (120572)120598119899+ int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910119899

(119904)1003817100381710038171003817

(67)

Let 119910(sdot) be a limit point of the sequence 119910119899(sdot) Then it follows

that from the above inequality one has

1003817100381710038171003817119910 (119905) minus 119910 (119905)1003817100381710038171003817 le int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910 (119904)

1003817100381710038171003817 119889119904(68)

which implies 119910(sdot) = 119910(sdot) Consequently 119910 isin 119878119888is a unique

limit point of 119910119899(sdot) isin 119878

119890

Example 26 Let 119865 119869 timesR119873rarr Pcpcv(R

119873) with

119865 (119905 119910) = 119861 (1198911(119905 119910) 119891

2(119905 119910)) (69)

where 1198911 119891

2 119869 times R119873

rarr R119873 are Caratheodory functionsand bounded

Then (2) is solvable

Example 27 If in addition to the conditions on 119865 ofExample 26 119891

1and 119891

2are Lipschitz functions then 119878

119890= 119878

119888

Acknowledgments

This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER

8 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

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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

Abstract and Applied Analysis 5

The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886

+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero

119888119863

1205721 = 0 120572 gt 0 (32)

We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have

119863120572

(119905 minus 119886)120572minus119895

= 0 for 119895 = 1 2 [120572] + 1 (33)

From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N

119863120572119891 (119905) = 119863

120572119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

120572minus119895

119888119863

120572119891 (119905) =

119888

119863120572

119892 (119905) lArrrArr 119891 (119905) = 119892 (119905) +

119898

sum

119895=1

119888119895(119905 minus 119886)

119899minus119895

(34)

In these formulas the coefficients 119888119895are arbitrary constants

For proving allmain results we present the following auxiliarylemmas

Lemma 21 (see [10]) Let 120572 gt 0 and let 119910 isin 119871infin(119886 119887) or

119862([119886 119887]) Then

(119888

119863120572

119868120572119910) (119905) = 119910 (119905) (35)

Lemma 22 (see [10]) Let 120572 gt 0 and 119899 = [120572] + 1 If 119910 isin

119860119862119899[119886 119887] or 119910 isin 119862119899

[119886 119887] then

(119868120572 119888

119863120572119910) (119905) = 119910 (119905) minus

119899minus1

sum

119896=0

119910(119896)(119886)

119896(119905 minus 119886)

119896 (36)

For further readings and details on fractional calculus werefer to the books and papers by Kilbas [10] Podlubny [12]Samko [13] and Caputo [46ndash48]

4 Existence Result

Definition 23 A function 119910 isin 119862([0 119887]R119873) is called mild

solution of problem (1) if there exist 119891 isin 1198711(119869R119873

) such that

119910 (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)1minus120572

119891 (119904) 119889119904 119905 isin [0 119887]

(37)

where 119891 isin 119878119865119910

= V isin 1198711([0 119887]R119873

) 119891(119905) isin 119865(119905 119910(119905)) aeon [0 119887]

We will impose the following conditions on 119865

(H1) The function 119865 119869 timesR119873

rarr Pcpcv(R119873) such that

(a) for all 119909 isin R119873 the map 119905 997891rarr 119865(119905 119909) ismeasurable

(b) for every 119905 isin [0 119887] the multivalued map 119909 rarr

119865(119905 119909) is119867119889continuous

(H2) There exist 119901 isin 119871

1(119869R+

) and a continuous nonde-creasing function 120595 [0infin) rarr (0infin) such that

119865 (119905 119909)P = sup V V isin 119865 (119905 119909) le 119901 (119905) 120595 (119909)

for ae 119905 isin [0 119887] and each 119909 isin R119873

(38)

with

int

119887

0

119901 (119904) 119889119904 lt int

infin

1199100+119887119910

1

119889119906

120595 (119906) (39)

Theorem24 Assume that the conditions (H1)-(H

2) and then

the problem (2) have at least one solution

Proof From (H2) there exists 119872 gt 0 such that 119910

infinle 119872

for each 119910 isin 119878119888

Let

1198651(119905 119910) =

119865 (119905 119910) if 10038171003817100381710038171199101003817100381710038171003817 le 119872

119865(119905119872119910

10038171003817100381710038171199101003817100381710038171003817

) if 10038171003817100381710038171199101003817100381710038171003817 ge 119872

(40)

We consider119888

119863120572

119910 (119905) isin 1198651(119905 119910 (119905)) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(41)

It is clear that all the solutions of (41) are solutions of (2)Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595lowast

(119905)

120595lowast(119905) = 119901 (119905) 120595 (119872)

(42)

It is clear that 119881 is weakly compact in 1198711([0 119887]R119873

) Remarkthat for every 119891 isin 119881 there exists a unique solution 119871(119891) ofthe following problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (119905) = 1199100 119910

1015840

(0) = 1199101

(43)

this solution is defined by

119871 (119891) (119905) = 1199100+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (44)

We claim that 119871 is continuous Indeed let 119891119899rarr 119891 converge

in 1198711([0 119887]R119873

) as 119899 rarr infin set 119910119899= 119871(119891

119899) 119899 isin N It is clear

6 Abstract and Applied Analysis

that 119910119899 119899 isin N is relatively compact in 119862([0 119887]R119873

) and 119910119899

converge to 119910 isin 119862([0 119887]R119873) Let

119911 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 isin [0 119887]

(45)

Then

1003817100381710038171003817119910119899minus 119911

1003817100381710038171003817infin le119887120572

Γ (120572)int

119887

0

1003817100381710038171003817119891119899 (119904) minus 119891 (119904)1003817100381710038171003817 119889119904 997888rarr 0

as 119899 997888rarr infin

(46)

Hence 119870 = 119871(119881) is compact and convex subset of119862([0 119887]R119873

) Let 119878119865 119870 rarr Pclcv(119871

1([0 119887]R119873

)) be themultivalued Nemitsky operator defined by

1198781198651

(119910) = 119891 isin 1198711([0 119887] R

119873) 119891 (119905) isin 119865

1(119905 119910 (119905))

ae 119905 isin [0 119887] = 1198781198651119910

(47)

It is clear that 1198651(sdot sdot) is 119867

119889continuous and 119865

1(sdot sdot) isin

P119908119896cpcv(R

119873) and is integrably bounded then byTheorem 17

(see also Theorem 65 in [32] or Theorem 11 in [34]) we canfind a continuous function 119892 119870 rarr 119871

1

119908([0 119887]R119873

) such that

119892 (119909) isin ext 1198781198651

(119910) forall119910 isin 119870 (48)

From Benamara [49] we know that

ext 1198781198651

(119910) = 119878ext 1198651(sdot119910(sdot))

forall119910 isin 119870 (49)

Setting119873 = 119871 ∘ 119892 and letting 119910 isin 119870 then

119892 (119910) isin 1198651(sdot 119910 (sdot)) 997904rArr 119892 (119910) isin 119881 997904rArr 119873(119910)

= 119871 (119892 (119910)) isin 119870

(50)

Now we prove that 119873 is continuous Indeed let 119910119899isin 119870

converge to 119910 in 119862([0 119887]R119873)

Then

119892 (119910119899) converge weakly to 119892 (119910) as 119899 997888rarr infin (51)

Since119873(119910119899) = 119871(119892(119910

119899)) isin 119870 and 119892(119910

119899)(sdot) isin 119865(119905 119910

119899(119905)) then

119892 (119910119899) (sdot) isin 119865 (sdot 119861

119872) isin Pcp (R

119873) (52)

FromLemma 10 119892(119910119899) converge weakly to119910 in 1198711

([0 119887]R119873)

as 119899 rarr infin By the definition of119873 we have

119873(119910119899) = 119910

0+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910119899) (119904) 119889119904

119905 isin [0 119887]

119873 (119910) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910) (119904) 119889119904

119905 isin [0 119887]

(53)

Since 119873(119910119899) 119899 isin N sub 119870 then there exists subsequence of

119873(119910119899) converge in 119862([0 119887]R119873

) Then

119873(119910119899) (119905) 997888rarr 119873(119910) (119905) forall119905 isin [0 119887] as 119899 997888rarr infin

(54)

This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)

5 The Relaxed Problem

In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]

Now we present our main result of this section

Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873

) be amultifunction satisfying the following hypotheses

(H3) The function 119865 [0 119887]timesR119873

rarr P119888119901119888V(R

119873) such that

for all 119909 isin R119873 the map

119905 997891997888rarr 119865 (119905 119909) (55)

is measurable(H

4) There exists 119901 isin 1198711

(119869R+) such that

119867119889(119865 (119905 119909) 119865 (119905 119910)) le 119901 (119905)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

for ae 119905 isin [0 119887] and each 119909 119910 isin R119873

119867119889(119865 (119905 0) 0) le 119901 (119905) for ae 119905 isin [0 119887]

(56)

Then 119878119890= 119878

119888

Proof By Coviz and Nadlar fixed point theorem we caneasily prove that 119878

119888= 0 and since 119865 has compact and convex

valued then 119878119888is compact in 119862([0 119887]R119873

) For more infor-mation we see [25 27ndash29 51 52]

Let 119910 isin 119878119888 then there exists 119891 isin 119878

119865119910such that

119910 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (57)

Let119870 be a compact and convex set in 119862([0 119887]R119873) such that

119878119888sub 119870 Given that 119910

lowastisin 119870 and 120598 gt 0 we define the following

multifunction 119880120598 [0 119887] rarr P(R119873

) by

119880120598(119905) = 119906 isin R

1198731003817100381710038171003817119891 (119905) minus 119906

1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598

119906 isin 119865 (119905 119910lowast(119905))

(58)

Abstract and Applied Analysis 7

The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr

119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact

values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873

) Then by Lemma 3 there exists a measurableselection V

1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that

1003817100381710038171003817119891 (119905) minus V1(119905)1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598 (59)

this implies that 119880120598(sdot) = 0 We consider 119866

120598 119870 rarr P(119871

1(119869

R119873) defined by

119866120598(119910) = 119891

lowastisin F (119910)

1003817100381710038171003817119891 (119905) minus 119891lowast (119905)1003817100381710038171003817

lt 120598 + 119889 (119891lowast(119905) 119865 (119905 119910 (119905)))

(60)

Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866

120598(119910) is lsc with decomposable

values By Lemma 8 there exists a continuous selection 119891120598

119862([0 119887]R119873) rarr 119871

1(119869R119873

) such that

119891120598(119910) isin 119866

120598(119910) forall119910 isin 119862 ([0 119887] R

119873) (61)

FromTheorem 17 there exists function 119892120598 119870 rarr 119871

119908([0 119887]

R119873) such that

119892120598(119910) isin ext 119878

119865(119910) = 119878ext 119865(sdot119910(sdot)) forall119910 isin 119870

119892120598(119910) minus 119891

120598(119910)

119908le 120598 forall119910 isin 119870

(62)

From (H3) we can prove that there exists119872 gt 0 such that

10038171003817100381710038171199101003817100381710038171003817infin le 119872 forall119910 isin 119878

119888 (63)

Consider the sequence 120598119899rarr 0 as 119899 rarr infin and set119892

119899= 119892

120598119899

119891119899= 119891

120598119899

Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595 (119905) ae 119905 isin [0 119887]

120595 (119905) = (1 +119872)119901 (119905)

(64)

Let 119871 119881 rarr 119862([0 119887]R119873) be the map such that each 119891 isin 119881

assigns the unique solution of the problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(65)

As in Theorem 24 we can prove that 119871(119881) is compact in119862([0 119887]R119873

) and the operator 119873119899= 119871 ∘ 119892

119899 119870 rarr 119870 is

compact then by Schauderrsquos fixed point there exists 119910119899isin 119870

such that 119910119899isin 119878

119890and

119910119899(119905) = 119910

0+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892119899(119910

119899) (119904) 119889119904

ae 119905 isin [0 119887] 119899 isin N

(66)

Hence

1003817100381710038171003817119910 (119905) minus 119910119899(119905)1003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

1003817100381710038171003817119891119899 (119910119899) (119904) minus 119891 (119904)

1003817100381710038171003817 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119889 (119891 (119904) 119891

119899(119910

119899) (119904))) 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119867

119889(119865 (119904 119910 (119904)) 119865 (119904 119910

119899(119904)))) 119889119904

le119887120572+1

Γ (120572 + 1)120598119899+119887120572+1

Γ (120572)120598119899+ int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910119899

(119904)1003817100381710038171003817

(67)

Let 119910(sdot) be a limit point of the sequence 119910119899(sdot) Then it follows

that from the above inequality one has

1003817100381710038171003817119910 (119905) minus 119910 (119905)1003817100381710038171003817 le int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910 (119904)

1003817100381710038171003817 119889119904(68)

which implies 119910(sdot) = 119910(sdot) Consequently 119910 isin 119878119888is a unique

limit point of 119910119899(sdot) isin 119878

119890

Example 26 Let 119865 119869 timesR119873rarr Pcpcv(R

119873) with

119865 (119905 119910) = 119861 (1198911(119905 119910) 119891

2(119905 119910)) (69)

where 1198911 119891

2 119869 times R119873

rarr R119873 are Caratheodory functionsand bounded

Then (2) is solvable

Example 27 If in addition to the conditions on 119865 ofExample 26 119891

1and 119891

2are Lipschitz functions then 119878

119890= 119878

119888

Acknowledgments

This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER

8 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

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 Abstract and Applied Analysis

that 119910119899 119899 isin N is relatively compact in 119862([0 119887]R119873

) and 119910119899

converge to 119910 isin 119862([0 119887]R119873) Let

119911 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904 119905 isin [0 119887]

(45)

Then

1003817100381710038171003817119910119899minus 119911

1003817100381710038171003817infin le119887120572

Γ (120572)int

119887

0

1003817100381710038171003817119891119899 (119904) minus 119891 (119904)1003817100381710038171003817 119889119904 997888rarr 0

as 119899 997888rarr infin

(46)

Hence 119870 = 119871(119881) is compact and convex subset of119862([0 119887]R119873

) Let 119878119865 119870 rarr Pclcv(119871

1([0 119887]R119873

)) be themultivalued Nemitsky operator defined by

1198781198651

(119910) = 119891 isin 1198711([0 119887] R

119873) 119891 (119905) isin 119865

1(119905 119910 (119905))

ae 119905 isin [0 119887] = 1198781198651119910

(47)

It is clear that 1198651(sdot sdot) is 119867

119889continuous and 119865

1(sdot sdot) isin

P119908119896cpcv(R

119873) and is integrably bounded then byTheorem 17

(see also Theorem 65 in [32] or Theorem 11 in [34]) we canfind a continuous function 119892 119870 rarr 119871

1

119908([0 119887]R119873

) such that

119892 (119909) isin ext 1198781198651

(119910) forall119910 isin 119870 (48)

From Benamara [49] we know that

ext 1198781198651

(119910) = 119878ext 1198651(sdot119910(sdot))

forall119910 isin 119870 (49)

Setting119873 = 119871 ∘ 119892 and letting 119910 isin 119870 then

119892 (119910) isin 1198651(sdot 119910 (sdot)) 997904rArr 119892 (119910) isin 119881 997904rArr 119873(119910)

= 119871 (119892 (119910)) isin 119870

(50)

Now we prove that 119873 is continuous Indeed let 119910119899isin 119870

converge to 119910 in 119862([0 119887]R119873)

Then

119892 (119910119899) converge weakly to 119892 (119910) as 119899 997888rarr infin (51)

Since119873(119910119899) = 119871(119892(119910

119899)) isin 119870 and 119892(119910

119899)(sdot) isin 119865(119905 119910

119899(119905)) then

119892 (119910119899) (sdot) isin 119865 (sdot 119861

119872) isin Pcp (R

119873) (52)

FromLemma 10 119892(119910119899) converge weakly to119910 in 1198711

([0 119887]R119873)

as 119899 rarr infin By the definition of119873 we have

119873(119910119899) = 119910

0+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910119899) (119904) 119889119904

119905 isin [0 119887]

119873 (119910) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892 (119910) (119904) 119889119904

119905 isin [0 119887]

(53)

Since 119873(119910119899) 119899 isin N sub 119870 then there exists subsequence of

119873(119910119899) converge in 119862([0 119887]R119873

) Then

119873(119910119899) (119905) 997888rarr 119873(119910) (119905) forall119905 isin [0 119887] as 119899 997888rarr infin

(54)

This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)

5 The Relaxed Problem

In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]

Now we present our main result of this section

Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873

) be amultifunction satisfying the following hypotheses

(H3) The function 119865 [0 119887]timesR119873

rarr P119888119901119888V(R

119873) such that

for all 119909 isin R119873 the map

119905 997891997888rarr 119865 (119905 119909) (55)

is measurable(H

4) There exists 119901 isin 1198711

(119869R+) such that

119867119889(119865 (119905 119909) 119865 (119905 119910)) le 119901 (119905)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

for ae 119905 isin [0 119887] and each 119909 119910 isin R119873

119867119889(119865 (119905 0) 0) le 119901 (119905) for ae 119905 isin [0 119887]

(56)

Then 119878119890= 119878

119888

Proof By Coviz and Nadlar fixed point theorem we caneasily prove that 119878

119888= 0 and since 119865 has compact and convex

valued then 119878119888is compact in 119862([0 119887]R119873

) For more infor-mation we see [25 27ndash29 51 52]

Let 119910 isin 119878119888 then there exists 119891 isin 119878

119865119910such that

119910 (119905) = 1199100+ 119910

1119905 +

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119891 (119904) 119889119904

ae 119905 isin [0 119887] (57)

Let119870 be a compact and convex set in 119862([0 119887]R119873) such that

119878119888sub 119870 Given that 119910

lowastisin 119870 and 120598 gt 0 we define the following

multifunction 119880120598 [0 119887] rarr P(R119873

) by

119880120598(119905) = 119906 isin R

1198731003817100381710038171003817119891 (119905) minus 119906

1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598

119906 isin 119865 (119905 119910lowast(119905))

(58)

Abstract and Applied Analysis 7

The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr

119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact

values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873

) Then by Lemma 3 there exists a measurableselection V

1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that

1003817100381710038171003817119891 (119905) minus V1(119905)1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598 (59)

this implies that 119880120598(sdot) = 0 We consider 119866

120598 119870 rarr P(119871

1(119869

R119873) defined by

119866120598(119910) = 119891

lowastisin F (119910)

1003817100381710038171003817119891 (119905) minus 119891lowast (119905)1003817100381710038171003817

lt 120598 + 119889 (119891lowast(119905) 119865 (119905 119910 (119905)))

(60)

Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866

120598(119910) is lsc with decomposable

values By Lemma 8 there exists a continuous selection 119891120598

119862([0 119887]R119873) rarr 119871

1(119869R119873

) such that

119891120598(119910) isin 119866

120598(119910) forall119910 isin 119862 ([0 119887] R

119873) (61)

FromTheorem 17 there exists function 119892120598 119870 rarr 119871

119908([0 119887]

R119873) such that

119892120598(119910) isin ext 119878

119865(119910) = 119878ext 119865(sdot119910(sdot)) forall119910 isin 119870

119892120598(119910) minus 119891

120598(119910)

119908le 120598 forall119910 isin 119870

(62)

From (H3) we can prove that there exists119872 gt 0 such that

10038171003817100381710038171199101003817100381710038171003817infin le 119872 forall119910 isin 119878

119888 (63)

Consider the sequence 120598119899rarr 0 as 119899 rarr infin and set119892

119899= 119892

120598119899

119891119899= 119891

120598119899

Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595 (119905) ae 119905 isin [0 119887]

120595 (119905) = (1 +119872)119901 (119905)

(64)

Let 119871 119881 rarr 119862([0 119887]R119873) be the map such that each 119891 isin 119881

assigns the unique solution of the problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(65)

As in Theorem 24 we can prove that 119871(119881) is compact in119862([0 119887]R119873

) and the operator 119873119899= 119871 ∘ 119892

119899 119870 rarr 119870 is

compact then by Schauderrsquos fixed point there exists 119910119899isin 119870

such that 119910119899isin 119878

119890and

119910119899(119905) = 119910

0+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892119899(119910

119899) (119904) 119889119904

ae 119905 isin [0 119887] 119899 isin N

(66)

Hence

1003817100381710038171003817119910 (119905) minus 119910119899(119905)1003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

1003817100381710038171003817119891119899 (119910119899) (119904) minus 119891 (119904)

1003817100381710038171003817 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119889 (119891 (119904) 119891

119899(119910

119899) (119904))) 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119867

119889(119865 (119904 119910 (119904)) 119865 (119904 119910

119899(119904)))) 119889119904

le119887120572+1

Γ (120572 + 1)120598119899+119887120572+1

Γ (120572)120598119899+ int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910119899

(119904)1003817100381710038171003817

(67)

Let 119910(sdot) be a limit point of the sequence 119910119899(sdot) Then it follows

that from the above inequality one has

1003817100381710038171003817119910 (119905) minus 119910 (119905)1003817100381710038171003817 le int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910 (119904)

1003817100381710038171003817 119889119904(68)

which implies 119910(sdot) = 119910(sdot) Consequently 119910 isin 119878119888is a unique

limit point of 119910119899(sdot) isin 119878

119890

Example 26 Let 119865 119869 timesR119873rarr Pcpcv(R

119873) with

119865 (119905 119910) = 119861 (1198911(119905 119910) 119891

2(119905 119910)) (69)

where 1198911 119891

2 119869 times R119873

rarr R119873 are Caratheodory functionsand bounded

Then (2) is solvable

Example 27 If in addition to the conditions on 119865 ofExample 26 119891

1and 119891

2are Lipschitz functions then 119878

119890= 119878

119888

Acknowledgments

This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER

8 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

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

Abstract and Applied Analysis 7

The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr

119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact

values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873

) Then by Lemma 3 there exists a measurableselection V

1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that

1003817100381710038171003817119891 (119905) minus V1(119905)1003817100381710038171003817 lt 119889 (119891 (119905) 119865 (119905 119910 (119905))) + 120598 (59)

this implies that 119880120598(sdot) = 0 We consider 119866

120598 119870 rarr P(119871

1(119869

R119873) defined by

119866120598(119910) = 119891

lowastisin F (119910)

1003817100381710038171003817119891 (119905) minus 119891lowast (119905)1003817100381710038171003817

lt 120598 + 119889 (119891lowast(119905) 119865 (119905 119910 (119905)))

(60)

Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866

120598(119910) is lsc with decomposable

values By Lemma 8 there exists a continuous selection 119891120598

119862([0 119887]R119873) rarr 119871

1(119869R119873

) such that

119891120598(119910) isin 119866

120598(119910) forall119910 isin 119862 ([0 119887] R

119873) (61)

FromTheorem 17 there exists function 119892120598 119870 rarr 119871

119908([0 119887]

R119873) such that

119892120598(119910) isin ext 119878

119865(119910) = 119878ext 119865(sdot119910(sdot)) forall119910 isin 119870

119892120598(119910) minus 119891

120598(119910)

119908le 120598 forall119910 isin 119870

(62)

From (H3) we can prove that there exists119872 gt 0 such that

10038171003817100381710038171199101003817100381710038171003817infin le 119872 forall119910 isin 119878

119888 (63)

Consider the sequence 120598119899rarr 0 as 119899 rarr infin and set119892

119899= 119892

120598119899

119891119899= 119891

120598119899

Set

119881 = 119891 isin 1198711([0 119887] R

119873)

1003817100381710038171003817119891 (119905)1003817100381710038171003817 le 120595 (119905) ae 119905 isin [0 119887]

120595 (119905) = (1 +119872)119901 (119905)

(64)

Let 119871 119881 rarr 119862([0 119887]R119873) be the map such that each 119891 isin 119881

assigns the unique solution of the problem

119888

119863120572

119910 (119905) = 119891 (119905) ae 119905 isin [0 119887]

119910 (0) = 1199100 119910

1015840

(0) = 1199101

(65)

As in Theorem 24 we can prove that 119871(119881) is compact in119862([0 119887]R119873

) and the operator 119873119899= 119871 ∘ 119892

119899 119870 rarr 119870 is

compact then by Schauderrsquos fixed point there exists 119910119899isin 119870

such that 119910119899isin 119878

119890and

119910119899(119905) = 119910

0+ 119905119910

1+

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119892119899(119910

119899) (119904) 119889119904

ae 119905 isin [0 119887] 119899 isin N

(66)

Hence

1003817100381710038171003817119910 (119905) minus 119910119899(119905)1003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891 (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

1003817100381710038171003817119891119899 (119910119899) (119904) minus 119891 (119904)

1003817100381710038171003817 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119889 (119891 (119904) 119891

119899(119910

119899) (119904))) 119889119904

le1

Γ (120572)

10038171003817100381710038171003817100381710038171003817int

119905

0

(119905 minus 119904)120572minus1

[119892119899(119910

119899) (119904) minus 119891

119899(119910

119899) (119904)] 119889119904

10038171003817100381710038171003817100381710038171003817

+119887120572

Γ (120572)int

119905

0

(120598119899+ 119867

119889(119865 (119904 119910 (119904)) 119865 (119904 119910

119899(119904)))) 119889119904

le119887120572+1

Γ (120572 + 1)120598119899+119887120572+1

Γ (120572)120598119899+ int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910119899

(119904)1003817100381710038171003817

(67)

Let 119910(sdot) be a limit point of the sequence 119910119899(sdot) Then it follows

that from the above inequality one has

1003817100381710038171003817119910 (119905) minus 119910 (119905)1003817100381710038171003817 le int

119905

0

119901 (119904)1003817100381710038171003817119910 (119904) minus 119910 (119904)

1003817100381710038171003817 119889119904(68)

which implies 119910(sdot) = 119910(sdot) Consequently 119910 isin 119878119888is a unique

limit point of 119910119899(sdot) isin 119878

119890

Example 26 Let 119865 119869 timesR119873rarr Pcpcv(R

119873) with

119865 (119905 119910) = 119861 (1198911(119905 119910) 119891

2(119905 119910)) (69)

where 1198911 119891

2 119869 times R119873

rarr R119873 are Caratheodory functionsand bounded

Then (2) is solvable

Example 27 If in addition to the conditions on 119865 ofExample 26 119891

1and 119891

2are Lipschitz functions then 119878

119890= 119878

119888

Acknowledgments

This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER

8 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

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 Abstract and Applied Analysis

References

[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999

[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991

[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997

[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012

[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013

[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011

[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012

[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006

[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993

[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977

[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005

[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008

[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013

[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011

[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013

[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011

[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008

[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009

[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011

[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008

[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009

[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012

[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996

[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993

[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000

[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995

[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991

[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

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

Abstract and Applied Analysis 9

[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990

[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997

[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991

[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991

[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988

[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990

[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958

[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983

[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel

[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012

[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967

[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971

[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975

[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010

[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012

[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010

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