Mild Solutions for Nonlocal Impulsive Fractional Semilinear … · 2013-12-24 · Ahmed Gamal Ibrahim, Nawal Abdulwahab Al Sarori Mathematics Department, Faculty of Science, King

Applied Mathematics, 2013, 4, 40-56 http://dx.doi.org/10.4236/am.2013.47A008 Published Online July 2013 (http://www.scirp.org/journal/am)

Mild Solutions for Nonlocal Impulsive Fractional Semilinear Differential Inclusions with Delay

in Banach Spaces

Ahmed Gamal Ibrahim, Nawal Abdulwahab Al Sarori Mathematics Department, Faculty of Science, King Faisal University, Al-Ahsa, KSA

Email: [email protected], [email protected]

Received March 19, 2013; revised April 19, 2013; accepted April 26, 2013

Copyright © 2013 Ahmed Gamal Ibrahim, Nawal Abdulwahab Al Sarori. This is an open access article distributed under the Crea-tive Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this paper, we give various existence results concerning the existence of mild solutions for nonlocal impulsive differential inclusions with delay and of fractional order in Caputo sense in Banach space. We consider the case when the values of the orient field are convex as well as nonconvex. Our obtained results improve and generalize many results proved in recent papers. Keywords: Fractional Differential Inclusions; Impulsive Semilinear Functional Differential Inclusions; The

Infinitesimal Generator of a Semigroup; Nonlocal Conditions; Mild Solutions

1. Introduction

During the past two decades, fractional differential equations and fractional differential inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics and engineering. For some of these applications, one can see [1-4] and the references therein. El Sayed et al. [5] initiated the study of fractional multi-valued differential inclusions. Re- cently, some basic theory for initial—value problems for fractional differential equations and inclusions was dis- cussed by [6-14].

The theory of impulsive differential equations and impulsive differential inclusions has been an object in- terest because of its wide applications in physics, biology, engineering, medical fields, industry and technology. The reason for this applicability arises from the fact that impulsive differential problems are an appropriate model for describing process which at certain moments change their state rapidly and which cannot described using the classical differential problems. For some of these applications we refer to [15-17]. During the last ten years, impulsive differential inclusions with different conditions have intensely student by many mathematicians. At present, the foundations of the general theory of impulsive differential equations and inclusions are already laid, and many of them are investigated in details in the book of

Benchohra et al. [18]. Moreover, a strong motivation for investigating the

nonlocal Cauchy problems, which is a generalization for the classical Cauchy problems with initial condition, comes from physical problems. For example, it used to determine the unknown physical parameters in some in- verse heat condition problems. The nonlocal condition can be applied in physics with better effect than the classical initial condition 00 .x x For example, g x may be given by

1

,i m

i ii

g x c x t

where 1, 2, , ic i m are given constants and 10 t

2 nt t b . For the applications of nonlocal conditions problems we refer to [19,20]. In the few past years, several papers have been devoted to study the existence of solutions for differential equations or differential inclusions with nonlocal conditions [21-23]. For impulsive differential equation or inclusions with nonlocal conditions of order one we refer to [22,23]. For impulsive differential equation or inclusions of fractional order we refer to [10,24-27] and the references therein.

In this paper we are concerned with the existence of mild solution to the following nonlocal impulsive semilinear differential inclusions with delay and of order

0,1 of the type

Copyright © 2013 SciRes. AM

A. G. IBRAHIM, N. A. AL SARORI 41

1 2, , . . on , , ,

, ,0 ,

, 1, 2, , ,

cm

i i i i

D x t Ax t F t t x a e J t t t

x t t g x t r

x t x t I x t i m

,

(1.1)

where 0, , , 0J b r b

,, is the Caputo de-

rivative of order cD x t

:A D A E E is the infinitesimal generator of a semigroup 0C , 0T t t on

a real separable Banach space , E :F 2EJ be a multi-function,

0 1 m m is a given continuous function,

10 t t t t :

, : ,0 Eb

r,g E

: 1, 2, ,i

is a nonlinear function related to the nonlocal condition at the origin, I E E i m

,i

impulsive functions which characterize the jump of the solutions at impulse points, and x t it

are the right and left limits of x at the point respectively. Finally, for any it ,t J

defined by :t

, ,0 ,x x t r x ,

where and will define in the next section. To study the theory of abstract impulsive differential

inclusions with fractional order, the first step is how to define the mild solution. Mophou [24] firstly introduced a concept on a mild solution which was inspired by Jara- dat et al. [25]. However, it does not incorporate the mem- ory effects involved in fractional calculus and impulsive conditions. Wang et al. [10] introduced a new concept of PC-mild solutions for (1.1) without delay and derived existence and uniqueness results concerning the PC-mild solutions for (1.1) when F is a Lipschitz single-valued function or continuous and maps bounded sets into bounded sets and is compact. T t , t 0

In order to do a comparison between our obtained results in this paper and the known recent results in the same domain, we refer to: Ouahab [9] proved a version of Fillippov’s theorem for (1.1) without impulse, without delay and A is an almost sectorial operator, Wang et al. [11] proved existence and controllability results for (1.1) without impulse, without delay and with local condition, Zhang et al. [12] considered the problem (1.1) without impulse, without delay, F is a single-valued function and is strongly equicontinuous C0-semigroup, Zhou et al. [13,14] introduced a suitable definition of mild solution for (1.1) based on Laplace transforma- tion and probability density functions for (1.1) when

, t 0T t

F is single-valued function and without impulse, Cardinali et al. [22] proved the existence of mild solutions to the problem (1.1) without delay, when 1 and the multivalued function F satisfies the lower Scorza-Dragoni property and

0t is a family of linear operator,

generating a strongly continuous evolution operators, Fan [23] studied a nonlocal Cauchy problem in the presence

of impulses, governed by autonomous semilinear differential equation, Dads et al. [26] and Henderson et al. [27] considered the problem (1.1) when Among the previous works, little is concerned with nonlocal fractional differential inclusions with impulses and with delay.

A t

0.A

In Section 3 in this paper, motivated by the works mentioned above, we derive various existence results of mild solutions for (1.1) when the values of the orient field are convex as well as non-convex.

The paper is organized as follows: In Section 2, we collect some background material and lemmas to be used later. In Section 3, we prove three existence results for (1.1). We adopt the definition of mild solution introduced by Wang et al. [10]. Our basic tools are the properties of multi-functions, methods and results for semilinear differential inclusions, and fixed point techniques.

2. Preliminaries and Notations

Let ,C J E the space of -valued continuous functions on

EJ with the uniform norm

1sup , , , x x t t J L J E the space of E-val-

ued Bochner integrable functions on J with the norm

1 , 0d

b

L J Ef f t t , bP E = { : B is non- B E

empty and bounded}, clP E = { : B is non- B Eempty and closed}, EkP = { : B is nonempty B E

and compact}, ,cl cvP E = { : B is nonempty, closed and convex},

B E EckP

conv = { : B is nonempty,

convex and compact}, (respectively, B E

B conv B ) be the convex hull (respectively, convex

closed hull in ) of a subset . E BDefinition 1 ([28]). A semigroup , 0T t ,t of

bounded linear operators on a Banach space X is said to be

1) uniformly continuous if

0

Lim 0,t

T t I

where I is the identity operator. 2) strongly continuous if

0

lim , for everyt

T t x x

.x X

A strongly continuous semigroup of bounded linear operators on X will be called a semigroup of class

or simply a 0 -semigroup. It is known that if 0C C , 0 tT t

0 is a -semigroup, then there exist

constants 0C

and 1M such that



e , for 0 .tT t M t

A 0 semigroup -C , 0T t t T t

is called compact if for every is compact. It is known that ([28], Theorem 3.2) every compact semigroup is uniformly continuous.

0,t 0 -C

Definition 2 ([28]). Let , 0 ,T t t be a semigroup of bounded linear operators on a Banach space

.X The linear operator A defined by

0

: lim existst

T t x xD A x X

t

and

0

limt

T t x xAx

t

is called the infinitesimal generator of the semigroup is the domain of ,T t D A .A

Definition 3 ([29-33]). Let X and Y be two topological spaces. A multifunction is said to be upper semicontinuous (u.s.c.) if

:G X P Y

1 :G V x X G x V is an open subset of X

for every open V Y . is said to be lower semicontinuous if

.)c

G1G V( . .l s :x X G x V

is an open subset of X for every open is called closed if its graph

.V Y G

, :G x y X Y y G x is closed subset of the

topological space X Y G BB

. is said to be completely continuous if is relatively compact for every bounded subset of

G

.X If the multifunction is completely continuous with non empty compact values, then is u.s.c. if and only if is closed.

G

G GLemma 1 ([29] Theorem 8.2.8). Let , , ,A be a

complete finite measure space, X a complete separable metric space and be a measurable multivalued function with non empty closed images. Consider a multivalued function G from :

:F 2X

X to is a complete separable metric space such

that for every ,P Y Y

x X the multivalued function is measurable and for every ,w xw G w the

multivalued function , x G w x is continuous.

Then the multivalued function ,w G w F w is

measurable. In particular for every measurable single- valued function the multivalued function

is measurable and for every Cara- theodory single-valued function

:z

,X ,w G w z w

: ,X Y the

multivalued function ,w w F w is measurable.

Definition 4 A nonempty subset 1 ,M L J E,

is said to be decomposable provided for every f g M and each Lebesgue measurable set Z in ,J

,Z J Zf g M where Z is the characteristic

function of the set .Z Definition 5 A sequence 1: ,n f n L J E is

said to be semi-compact if: 1) It is integrably bounded, i.e. there is 1 ,q L J

such that

. . .nf t q t a e t J

2)The set :nf t n is relatively compact in E. . .a e t J We recall one fundamental result which follows from

Dunford-Pettis Theorem. Lemma 2 ([33]). Every semi-compact sequence in 1 ,L J E is weakly compact in 1 , .L J EFor more about multifunctions we refer to [29-33]. Lemma 3 ([11], lemma 2.10). For 0,1 and

0 e c , we have e c c e .

Definition 6 According to the Riemann -Liouville a- pproach, the fractional integral of order 0,1 of a function 1 ,f L J E is defined by

1

0 d , 0,

t t sI f t f s s t

provided the right side is defined on J , where is the

Euler gamma function defined by 1

0e d .tt t

Definition 7 The Caputo derivative of order 0,1

f J E

of a continuously differentiable function is defined by

:

1 1 1

0

1d .

1

tc D f t t s f s s I f t

Note that the integrals appear in the two previous defi- nitions are taken in Bochner’ sense and c D I f t

f t for all .t J For more informations about the fractional calculus we refer to [2,4].

Definition 8 ([14], Lemma 3.1 and Definition 3.1, see also [11-13]). Let A function :h J E . ,x C J E is said to be a mild solution of the following system:

0

, ,

0 ,

c D x t A t x t h t t J

x x E

(2.1)

if it satisfies the following integral equation

1

1 0 20d ,

,

tx t K t x t s K t s h s s

t J

(2.2)

where

1 20

0

1 11

d ,

d ,

10,

K t T t K t

T t

w


A. G. IBRAHIM, N. A. AL SARORI


43

1 11

111 si

!n n

n

nw n

n

n ,

1) For any fixed are linear bound-

ed operators. 1 20, ,t K t K t

2) For 0

10,1 , d .

1

0, and is a probability density function de-

fined on that is Note that the 0, , 0

d 1

.3) If , 0T t M t , then for any

function must be chosen such that the integral appears in (2.2) is well be defined. 1,x E K t x M x and 2 .

MK t x x

Remark 1 Since 1 2. , .K K are associated with

the numbrer , there are no analogue of the semigroup property, i.e. 1 1 1 ,K t s K t K

s

.

4) For any fixed 1 20, ,t K t K t are strongly continuous.

5) If , 0T t t is compact, then 1K t and 2K t are compact.

2 2 2K t s K t K s In the following we recall the properties of 1 . ,K

. 2 .K In order to define the concept of mild solution of (1.1), let 0 10, ,J t 1, ,i i iJ t t 1, , ,i m 2 sider the set of functions:

Lemma 4 ([14], Lemma 3.2, Lemma 3.3 and Lemma 3.5)

and con

: ,0 : is continuous everywhere except for a finite number

of points at which and exist and ,

r E

s s s s s

|, : : , , 0,1,2, , and and exist for each 1, 2, , ,iJ i i iPC J E x J E x C J E i m x t x t i m

and

|,0: , : , , , 0,1, 2, , and and exist for each 1,2, , .

iJ i i irx r b E x x C J E i m x t x t i m

For any x and any the element of ,t J t x

defined by It is easy to check that are are Banach spaces

endowed with the norms

max : ,0 ,x t t r , ,t x x t r 0 .

Here t x represents the history of the state time t r up the present time For any subset and for any

.t B 0,1,2, , ,i m let

and

max : , .x x t t r b

| : : , and , , 0,1, , .

i i i i iJB x J E x t x t t J x t x t x B i m

Of course 0 0| |

: .J J

B x x B Let us recall the concept of mild solutions, introduced

by Wang et al. [10], for the impulsive fractional evolution equation:

1 2

0

, , . . on , , ,

0 ,

, 1,2, , ,

cm

i i i

D x t Ax t f t x t a e J t t t

x x

x t x t y i m

,

,

(2.3)

where ,h PC J E and It is easily observe that

.iy Ex can be decomposed to where v w ,J Ev C is the continuous mild solution for

where : , , 1, 2,if I E E y E i m . At first Wang et al. [10] considered the following non-

homogeneous impulsive fractional equation

1 2

0

, , , ,

0 ,

, 1, 2, , ,

cm

i i i

D x t Ax t h t t J t t t

x x

x t x t y i m

0

,

0 ,

c D v t Av t h t t J

v x

,

(2.4)

,

(2.5)

and ,w PC J E is the mild solution for the impul-


sive evolution equation

1 2, , , ,

0 0,

, 1,2, , .

cm

i i i

D w t Aw t t J t t t

w

w t w t y i m

,

m

(2.6)

Indeed, by adding together (2.5) with (2.6), it follows (2.4). Note is continuous, so v ,i iv t v t

1, 2, ,i . On the other hand, any solution of (2.4) can be decomposed to (2.5) and (2.6). By Definition 9, a mild solution of (2.5) is given by

1 0

1

20d , .

t

v t K t x

t s K t s h s s t J

(2.7)

Now we rewrite system (2.6) in the equivalent integral equation

1

00

1

1 10

1

1 2 20

1

01

1d , ,

1d , ,

1d , ,

1d , .

t

t

t

i m t

i mi

t s Aw s s t J

y t s Aw s s t J

w t y y t s Aw s s t J

y t s Aw s s t

J

(2.8)

The above equation can be expressed as

1

1

0

1d , ,

i m

i ii

t

w t y t

t s Aw s s t J

(2.9)

where

0, for 0,

1, for , .i

ii

t tt

t t b

We apply the Laplace transform for (2.8) to get (see, [25])

1

e 1,

iti m

ii

u y A u

which implies

11

1

e .ii m

ti

i

u I

A y

Note that the Laplace transform for 1 iK t y is

11 .iI A y Thus we can derive the mild solu-

tion of (2.6) as

11

.i m

i ii

w t t K t t y

i (2.10)

By (2.7) and (2.10), the mild solution of (2.4) is given by

1

1 0 1 201

d , .i m t

i i ii

x t K t x t K t t y t s K t s h s s t J

By using the above results, we can write the following

definition of mild solution of the system (2.3). Definition 9 ([10], Definition 3.1). By a mild solution

of the system (2.3) we mean a function ,x PC J E which satisfies the following integral equation

1

1 0 2 00

1

1 0 1 1 1 2 10

1

1 0 1 201

, d , ,

, d , ,

, d ,

t

t

i m t

i i mi

K t x t s K t s f s x s s t J

K t x K t t y t s K t s f s x s s t Jx t

.K t x K t t y t s K t s f s x s s t J



Now we can give the concept of mild solution for our

considered problem (1.1). Definition 10 By a mild solution for (1.1), we mean a

function x which satisfies the following integral equation

1

1 2 00

1

1 1 1 1 20

=1

1 1 20=1

, ,0

0 d , ,

0 d

0 d

t

t

i m t

i i mi

t g x t r

K t g x t s K t s f s s t J

x t K t g x K t t y t s K t s f s s t J1, ,

, ,K t g x K t t y t s K t s f s s t

J

(2.11)

where and , 1, ,i i iy I x t i m f is an integra-

ble selection for ., .F x .

Remark 2 It is easily to see that the solution given by (2.11) satisfies the relation

, 1,2, ,i i i ix t x t I x t i m .

Remark 3 If for all and if there is no delay then Formula (2.11) will take the form

0,i iI x t 1, , , 0i m g

1

1 0 20d , .

tx t K t x t s K t s f s s t J

2

This means that when there is no neither impulse nor delay in the problem (1.1), its solution is equal to the formula given in (2.2).

Theorem 1 ([34]). Let be a nonempty subset of a Banach space , which is bounded, closed and convex. Suppose

WE

: ER W is u.s.c. with closed, convex values, and such that and is compact. Then has a fixed point

R W W R WR

The following fixed point theorem for contraction multivalued is proved by Govitz and Nadler [35].

Theorem 2 Let , X d

be a complete metric space. If is contraction, then has a fixed point.

: clR X P X R

Theorem 3 ([36], Corollary 3.3.1) (Schauder fixed point theorem). Let be a Banach space, a nonempty, convex, closed and bounded subset of and

E BE

:f B B be continuous. If f is compact or is compact, then

Bf has a fixed point.

3. Existence Results for the Problem (1.1)

In this section, we give the main results of mild solutions of (1.1).

3.1. Convex Case

In the following Theorem we derive the first existence result concerning the mild solution for the problem (1.1).

Theorem 4 Let : ckF J P E be a multifunction. Assume the following conditions:

(H1) A is the infinitesimal generator of a semi- 0 -C

group : 0T t t and is compact. , 0T t t

(H2) For every is measurable, for almost

, ,h t F t h , ,F t ht J h

,x is upper semi-continuous and

for each the set

1 1,

, : , , . .F x

S f L J E f t F t t x a e is

nonempty.

(H3) There exist a function 1

,qL J ,

0 q such that for any x

sup : , 1 0 , . . .z z F t x t x a e t J (3.1)

(H4) :g E is continuous, compact and there exist two positive numbers such that ,a d

,g h a h d h . (3.2)

(H5) For every 1,2, ,i m , iI is continuous and compact and there exists a positive constant such that

ih

,i i .I x h x x E (3.3)

Then, for a given continuous function : ,0r E , the problem (1.1) has a mild solution provided that there is such that 0r

1

1 .

M a r d

M h r r r

(3.4)

where, such that 0M supt J

T t M

,

1

11 ,,

1q

qi m

i qi L J

bh h

and 1

.1 q

Proof. In view of (H2), for each the set ,x



1 1,

, : , , . .F x

S f L J E f t F t t x a e,

is nonempty. So, we can define a multifunction as follows: : 2G y G x if and only if

1

1 20

1 11

1

20

, ,0 ,

0 d

0

d , , 1,2, , .

t

k i

k k kk

t

i

t g x t r

0, ,K t g x t s K t s f s s t

y tK t g x K t t I x t

t s K t s f s s t J i m

J

where 1

,.

F xf S Obviously, every fixed point for

G is a mild solution for the problem (1.1). So, our goal is to apply Theorem 1. The proof will be given in several steps.

Step 1. The values of are convex and closed subset in

G.

Since the values of F are convex, it is easily to see

that the values of are convex. In order to prove that Gthe values of are closed, let and G x , 1ny n

be a sequence in G x such that in

Then, according to the definition of there is a se- ny

,

y .G

quence 1,n n

f

in 1

., F t xS such that for any ,it J

0,1, ,i m

1

1 20

1 11

1

20

, ,0 ,

0 d

0

d , , 1, 2, , ..

t

n

k in

k k kk

t

n i

t g x t r




J

(3.5)

Not that, from (3.1), for any for almost 1,n s J

1 0 1 1nf s s s x s x s s x

This show that the set . is integrably bound-

ed. Moreover, because : 1nf n

, : 1 ,nf t n F t t x for a.e. the set ,t J

: 1nf t n is relativity compact in for a.e. E .t J Therefore, the set is semi-compact and then, by Lemma 2 it is weakly compact in So, without loss of generality we can assume that n

: 1nf n 1 ,L J .

f converges weakly to a function From Mazur’s lemma, there is a sequence such

,J ,ng n

.1

1f L

that : 1n n ; : 1g t n Conv f t n t J and ng

converges strongly to f . Since, the values of F are

convex, 1

.,n F t xg S and hence, by the compactness of

, ,F t t x 1

.,.F t x

f S Moreover, for every

, ,t s J s 0,t and for every 1,n

1

2

1 1

1 0,

nt s K t s f s

Mt s s x L t R

, .

Therefore, by passing to the limit as in (3.5), we obtain from the Lebesgue dominated convergence theorem that, for every

n

0,1, , ,i m

1

1 20

1 11

1

20

, ,0 ,

0 d

0

d , , 1,2, , .

t

k i

k k kk

t

i

t g x t r




J



Then .y G x

Step 2. We claim that where ,r rG B x B x

, ,

0 , .

t t rx t

t J

0

and :rB x x x x r

rx B x , y G x and If .t J ,0 ,t r then by (3.2)

.

y t x t g x

a x d a r d

(3.6)

For 0t J . By using Lemma 4(3), (3.1), (3.2) and (3.5) we get

. To prove that, let

1

1

1

0

11

1, 0

1 ,

0 0 1 d

0 1

1

1

1 .

q

q

t

qt

qL J

q

q L J

My t x t M a r d x t s s s

MM a r d r t s s

b M rM a r d

M a r d r

d (3.7)

Similarly, by using Lemma 4(3), (3.1), (3.2),(3.3) and (3.5) we have for it J , 1, 2, , ,i m

1y t M a r d h r r . (3.8)

Therefore, from (3.4).(3.6).(3.7) and (3.8), we con-

clude that . r rG B x B x Step 3. Let .rB G B x We claim that is B

equicontinuous, let x B and y G x . According to the definition of we have G

1

1 2 00

1

1 1 201

, ,0 ,

0 d , ,

0 d

t

k i t

k k k ik

t g x t r

y t K t g x t s K t s f s s t J

, , 1,K t g x K t t I x t t s K t s f s s t J i

where 1

,.

F xf S By the continuity of , we can

see easily that if , r

0

lim 0.y t y t

To show that | ,JB it suffices to verify that | iJ

B is equicontinuous for every 0,1, ,i m , where

,0 ,t t then

1, : , , , ,i i i i iB x C J E x t x t t J t t x t x t x B .i

We consider the following cases: Case 1. Let 10, 0, .t t In view of Holder’s inequality we get

1

1

1

1 1 20

11

11 1 , 0

1 1 1,

0

0 0 0

10 0 d

10 0 .

1

q

q

q

qL J

q

qL J

y t y t y t y t y y

k g x k g x s k s f

M rk k g x s s

M rk k g x

ds s

(3.9)



Since T is compact, 1 K is also, (see, Lemma

4(v)), and hence, 1 K h is uniformly continuous on J (see [28]). Therefore, the last inequality tends to zero as

0, independently of .x Case 2. Let ,t t be two points in 10, t , then

1 1

1 1

2 20 0

1 2 3 4

0 0

d d

,

t t

y t y t y t y t

k t g x k t g x

t s k t s f s s t s k t s f s s

G G G G

(3.10)

where

1 1 10 0G k t g x k t g x ,

1

2 2 d ,t

tG t s k t s f s

s

1 1

3 20d ,

tG t s t s k t s f s

1

4 2 20d .

tG t s k t s k t s f s

s

We only need to check as 0iG 0 for every 1, 2,3, 4i . At first, we note that, as we mention above

s

nd a

the operators 1 ,K t t 0 are uniformly continuous on

.J So, 10

lim 0,h

G

independently of .rx B x

For by the Holder inequality we have 2 ,G

1 1

1

2 20 0

1 1

0 0

11 11

1, ,0 0

lim lim d

1lim d lim d

1 1lim d lim 0,

1q q

t

t

t t

t t

qqt

qL J L Jt

G t s k t s f s s

M rMt s f s s t s s s

M r M rt s s

(3.11)

dependently of in .rx B x Then

For we note that 3 ,G 1= 1

1 q

,0 , then for

,s t t we have .t s t s By apply-

ing Lemma into account 3 and taking 0,1 we 1 qget

1 1

1

.

q q

q

t s t s

t s t s

1 1

1

.q

t s t s

t s t s

This leads to

11 1 1

.

qt s t s

t s t s

Therefore,

1

1

1

111 1 1

3 ,00 0

1

,00

111 1

,0

1lim lim d

1lim d

1 1lim 0,

1

q

q

q

q

t qL J

qt

L J

q

L J

M rG t s t s s

M rt s t s s

M rt t

(3.12)



independently of .rx B x For by using (H1) and the Lebesgue dominated conver heorem, we get

4 ,Ggence t

1

4 2 200 0

1

2 2 00 0,

lim lim d

lim sup d .

t

t

s t

G t s k t s k t s f s s

K t s K t s t s f s

s

By the uniform continuity of , we con-

lude that indepen

(3.13)

2 , 0K t t dently of c 4

0lim 0,G .rx B x

1, 2, ,i m Case 3. When , let ,t t be two po n ints i .iJ Invoking to the definition of G we have

1 1

1 11

1 1

2 20 0

0 0

d d

k i

k k k k k kk

t t

y t y t y t y t K t g x K t g x

K t t I x t K t t I x t

t s k t s f s s t s k t s f s s

.

Arguing as in the first case we get

0

lim 0.y t y t

(3.14)

Case 4. When , let it t , 1, ,i m 0

1i i it t t , then we have

limi

i i it

y t y t y t y

be such that i it J and 0 such that According the definition of we get

G

1 1

1 1

1

20 0

0 0

d d .i

i i

k i

i k k k k k k

t

y t y k t g x k g x

K t t I x t K t I x t

s f s s s k s f s s

at

1k

1

2i it s k t

Arguing as in the first case we can see th

0

lim 0.

i

i

t

y t y

(3.15)

From (3.9) (3.15) we conclude that | iJ

B is equi-

continuous for every Step 4. Our aim in this step is to show that for any

0,1, ,i m .

,t r b

x x : : , rt y t y G x B

is relatively compact in Let us introduce the following maps:

,

where

E .

1 2 3, : , : 2G G G

1

, ,

0, 0, ,

t g x t rG x t

t b

0 ,

, the set

21

1

k i

k k k

G x t1, ,t r t0,

, , 1, 2, , ,ik

K t t I x t

t J i m

and 3

y G x if and only if

1

1 20

0, ,0 ,

0 d ,t

y m

t ry t

1, , .K t g x t s K t s f s s t J t

t

where

,.y F x

f S rB x Obviously, 3. Because 1 2G G G G is a bounded subset in and g



is compact, the set

relatively compact in . Also, since the functions

1 1 : rt G x t x B x

is E,k 1,2, ,I k m are compact, the set

is relatively compact in It remains to show that the set

2 2 : rt G x t x B x

.E

3 3: , rt y t y G x x B x

is relatively compact in For each .E 1, , ,mt J t t

arbitrary 0,h t and , we

0,1 define

1

, 00 d d

t h

h y d .y t T t g x t s T t s f s

s

that we can r

Note ewrite ,hy t in the form

,

1

0

0 dh

t h

y t T h T t h g x

T h s

d d .yT t s h f s s t

ator , 0T t t is com g Since the oper pact and is compact on the set

,

, 3: ,h rt y G x x B x y

is relatively compact in . Moreover, by using (H3) and (H4) we get

E

0

1 1

0 0 0

1

0 0

1 1

d

d d d d

0 d d d

d d d d

t t h

y y

t

y

t t h

y

g x

t s T t s f s s t s T t s f s s

M g x t s T t s f s s

t s T t s f s s t s T t s f s

, 0hy t y t T t

0 0 0y

0

1

1

0

1

0 0

0

0 d d d

d d

0 d 1 d d

1 d d

0 d 1 d

t

yt h

t

y

t h

t

t h

s

M a x d t s T t s f s s

t s T t s f s s

M ar a d M t s s x s

M t s s x s

M ar a d M r

1

0 0

t

1

1

0 0

d

1 d d .

t

t

t s s s

M r t s s s

Using Hölder’s inequality to get

1 1

, 0

1 1, , 0

0 d

1 d 11 1

q q

h

q q

qq qL J L J

y t y t M ar a d

h bM r M r

d .

Obviously, by Lemma 4(2), the right hand side of the

previous inequality tend to zero as ., 0h Hence, there exists a relatively compact set that can be arbitrary close to the set 3 , 0,t t b . Hence, this set is rela-



tively compact in . Hence,E , ,t t r b

and 4 with Arzeis relatively compact.

on B .

is relatively compact

As a consequence Steps 3 la-Ascoli theorem we conclude thatStep 5. has a closed graph

Let

of, B

G

,n r nx B x x x in and

, 1n ny G x n that

with in We will show ny y .B

.y G x By recalling the definition of for

any there exists

,G

1n , nn F xf S such that

, , 1, 2, , .

1 2 0

1

20

0 d , ,

0 d

n

n

t

k n k n i

x t

y t K g K t s f s s t J

1

0

1 11

, ,0 ,t

n n

k i

n kk

r

x t s

t g

t

K t g t I x t t s K t s f s s t J i

x K t

nce 1n nf

is se

gence of n

m

(3.16)

Let us show that the seque micom-

pact. From the un conver

iform

x ,x towards for any t J

lim 0.nn

t x t x

Moreover ,.F t

er 0n

is upper semicontinuous with co there exists a atural numb such that fo y

,

mpact values, then for every 0,r evern 0n n

,

, 0,1 , . .

n nf t F t t x

F t t x B a e t J

where 0,1 : 1 .B z E z Then, the compactness of ,F t t x implies that the set is re

: 1f t n n

latively compact for . .a e . In addition, assumption (H3) implies

1 0

1

1 , . .

n n

n

f t t t x

t x

t r a e t

.J

Then, by Lemma 2, , 1nf n guing as in sequence

is semicompact, hence weakly compact. Ar Step 1 from Ma- zur’s theorem, there is a such that

, 1nz n

: 1 : 1 ; n nz t n Conv f t n t J

and converges strongly tonz 1 ,f L J E . Since, the values of F are convex, ., . nn F xz S and hence,

. .,F xf S . By passing to the lim t ini (3.16), with tak-

unt that ing acco into g is obtain continuous, we

,g x t

0

1

1 1 201

,0 ,

d , ,

0 dt

k k k ik

t r

f s s t J

, , 1, 2, , .

1

1 200

t

k i

y t K t g x t s K t s

K t g x K t t I x t t s K t s f s s t J i

m

This proves that the graph of is closed. Now, as a consequence of Step 1 to Step 5, we con-

clude that the multifunction of is a compact multi-valued function, u.s.c with c pact values. By pplying Theorem 1, we can deduce that has a

G

G onvex com

afi

Gxed point x which is a mild s n of Problem

(1.1). In the following Theorems we give ano er version f r

an existe esu

olutio

th once r lt for (1.1).

Theorem 5 Let : ckF J P

.

E multifunc-

tio

be a

n, A is the infinitesimal generator of a 0 -C semi-

group : 0 ,T t t and : g E e suppose the

following assum :

W

ptions(H6) For every , ,x t F t x

1

, , 0qL J q

is measurable.

(H7) There is a function such that

For every ,x y

, , , 0 0 ,

for . . ,

h F t x F t y t x y

a e t J

Sup : ,0 , for . . .z x F t t a e t J

(H ) a positive constant a su8 There is ch that

, for all , .g z g w z w z w a

(H9) For each 1, 2, , ,i m there is 0i such that



, for all , .i i iI x I y x y x y E

(H10)

1,a M a

1

1 ,

where 1

q

q

q L J

b

, 1

1 q

and

1 ii.

i m

mild solution.

Then (1.1) has a Proof. For ,x set

1 , for . .L J F t t x a e t 1., .

, :F x

S f E f t .J

By Lemma 1, (H6) and (H7), ., .F x

osed, it has a m is meas-

urable. Since its values are cl easurable selection (see [29], Theorem 8.1.3) which, by hypothesis

(H7), belongs to Thus 1 , .L J E 1

., .F xS is non-

empty. Let us transform the problem into a fixed point

problem. Consider the multifunction map, defined as follows: for

: 2R ,x R x is the set of all

functions ,y R x su

ch that for each 1, 2, , ,i m

1 0t

y t K t x t s

1

0

, ,0 ,

d , ,

k i

t g x t r

g K t s f s s t J

, 20

1 11

0k i

k kk

1

20d , 1, , .

tK t g x K t t I x

where

1., .

.

t


F xf S It is easy to see that any fixed point

for R is a mild solution for (1.1). So, we shall show th

ps. of are nonemp

at R satisfies the assumptions of Theorem 2. The proof will be given in two ste

Step1. The values ty and closed.

Since 1

., .

R

F xS is non-empty, the values o are f

non-empty. In order to prove the values of are closed,

R

Rlet x and , 1ny n be in a sequence R x

such that

nition of n

,R

y y in . T

there is a

hen, acc h

sequence

ording to t

n

e defi-

in , 1f n

1

., .F x S such that for any , 0it J i ,1, m ,

t

1t

1 2 00

1 11

, ,0 ,

0 d , ,

0 d

n n

k i t

k k k ik

g x t r


1

20,n , 1, 2, , .K t g x K t t I x t J i

t s K t s f s s t m

(3.17)

Since

, 0F t , 0t such

is closed, for any there is that

,t J v t F ,0 .F t

t J 0,v t d In view

and for a.e.of (H9), for every 1,n

, ,0

0, ,0 , ,0

, ,0

1 1

n n

n

f t v t d f t F t

d F t d f t F t

x F t

t x

This show that the set

.Et x t

,

0E

t H F t t

t t t x

: 1nf n is integrably

bounded. Using the fact that F

n

has compact values, the set is relativity compact in for a.e.

ore, the set is -compact

and in

: 1nf t n .t J Theref

Esemi : 1f n

1 , .L J E So, we may

n

Then, by Lemma 2, it is weakly compact. pass to a subsequence if necessary to get that f converges weakly to a function

, .1f L J E From

que

Mazur’s theorem, there is a se- nce , 1nz n such that

: 1 : 1 ;n nConv f t n t J z t n

and z converges strongly to .n f Since, the values of F are convex, 1

., .n F xz S and hence, b - y the com

pactness of ., . ,F x 1

., ..

F xf S Note that for

every , 0 t , and for every t s J s ， 1,n

11 0, , .

nK t s f s

Mt x L t R

1

2

1

t s

t s



Therefore, by means of the Lebesgue dominated convergence Theorem and the continuity of 2 , 0K t t

we obtain from (3.17)

1

1 2 00

, ,0 ,

lim 0 d , ,t

nn

k i

k

t g x t r


2,1 10 k k 1

20d , , 1, , .

t

i1k

K t g x K t t I x t t s K t s f s s t J i m

So, .y R x Step 2. R is contraction. Let 2 1,x x a

1 1

nd .y R x , ther s 1

1., .

Then e iF x

f S such that for any , 0,1, 2, , .it J i m

1

1

1 1 1 2 00

1

1 1 1 201

, ,0 ,

0 d , ,

0 d

t

k i t

k k k ik

t g x t r


, , 1,2, , .K t g x K t t I x t t s K t s f s s t J i

m

(3.18)

Consider the multifunction : 2EZ J defined by

1 2: .Z t u E f t u t x t x t

For each nonempty. Indeed, l ,J Z t is et ,t Jt from (H7), we have

1 2

1 2

1 2

, , ,

0 0

.

h F t t x F t t x

t t x t x

t x t x t

Hence, there exists 2,tu F t x t

1 2 .tu f t t x t x t

Since the functions 1 2, , ,f x x are measurable, Pro- position [30], tells us that the multifunction III.4 in

: ,V t 2t x is mepty and closed

Z t F tvalues are nonem

asurable. Because its there is 2,

1F t t x

h S with

1 2h t x t x t (3.19)

such that

1 2 , . . .

f t t

t x x a e t J

s define

Let u

2 ,t g x t

,0 ,

d

i

r

x s s

.

1

2 1 2 20

1 2 11

0

0

t

k i

k k kk

y t K t g t s K t s h

0

1

20

, ,

d , , 1,2, ,t

t J

K t g K t t I x t

Obviously, 2 2

x

t s K t s h s s t J i m

(3.20)

y R x and if ,0 ,t r then by (H8

)

2 1 1 2 .y t y t a x x

If we get from and (H8)

0 ,t J 3.18 3.20

1

1

2 1 1 2 0

1

1 2 1 2 0

11

11 2 1 2 1 2, 0

d

d

d q

t

t

qt

qL J

My t y t M g x g x t s h s f s s

MMa x x x x t s s s

M MMa x x x x t s s x x Ma

(3.21)



Similarly, if , we get from, 1, ,it J i m 3.18 3.20 , (H8) and (H9)

1

2 1 1 2 1 20d . (3.22)

tMy t y t M a x x t s h s f s s x x M a

By interchanging the role of and , we obtain

rom (3.21), (3.22) and (H ) 2y 1y

f 10

2 1 1 2 .R x R x x x

Therefore, on and thus by Theorem R has a fixed point w ich is a mild solution for (1.1

R is contracti 2h ).

3.2. Nonconvex Case

In the following Theorem we give nonconv rsion for Theorem 4. Our hypothesis on the orient field is the following: (H

ex ve

11) : clF J P unction sucE is a multif h that

1)

, ,t x F t x is graph measurable and

,x F t x is lower semicontinuous.

2) There exists a function 1

,qL J , 0 q

such that for any x

, . . .F ,t x t a e t J

Theorem 6 If the hypotheses (H ), (H ), (Hthen t

1 4 5) and (H11), he problem (1.1) has a mild solution provided that

there is such that the condition (3.4) Proof. Consider the multivalued Nemits

defined by

0r is satisfied. ky operator

1 ,L J EN : 2 ,

1 1., .

, : , , . .F x

N x S f L J E f t F t t x a e t J .

We shall prove that has a nonempty closed de-

composable value and Since N

l.s.c. F has closed values, 1FS is closed ([37]). Because F is integrably bounded, 1FS is nonempty (see, T 37]). It is readily

verified,

heorem 3.2 [

1FS

semi-continuity of

is decomposable. To check

we need to show that, for every

the lower

N ,

, ,J E 1u L ,x d u

om

N x

this end fr Theorem

is upper semicontinu-

2.2 [37], we have

ous. To

10,

0 0

, inf inf d

inf d , , d .

b

Lv N x v t F t t x

b b

d u N x u v u t v t t

u t z t t d u t F t t x t

(3.23)

We shall show that, for any

,z t F t t x

in . Then, for all ,t J nx t x t 0, the set

: ,u x d u N x

is closed. For this purpose, let nx u

nx x

and assume

that in ue H11)(1) the fu.E By virt of ( nction

, ,z d u t F t z is u.s.c. So, via the Fatou Lemma,

and (3.23) we have

0

0 0

lim sup , lim sup , , d

lim sup d d , .

b

n nn n

b b

nn

d u N x d u t F t t x t

t x t d t d u N x

, , , ,d u t F t u t F t t x

Therefore,

x u

This

con

and this proves the lower semicon-

tinuity of allows us to apply Theorem 3 of [38]

nd obtai tinuous map

.N

n a a 1: ,Z L J E

,Z x N x for every Then, .x

such

that

, , . .Z x s .F s s J x a e s

Consider a map : defined by

1

1 2

1

21

, ,0 ,

0 )

)

t

kk

t g x t r

x t K t g x t s K t s Z 0d , ,x s s t J

0

1 10k i

k k 0

d , , 1,2, , .t

iK t g x K t t I x t s K t s Z x

t s s t J i m



ng as in the pr an show

that satie eorem 3 (Schau- der d poi such that

Argui oof of Theorem 4, we c

fixesfies all the conditions of Thnt theorem). Thus, there is x

.x t x t This means that x i

condit n (3.4) will be satisfied if

Indeed, condition (3.4) can be written as

s a mild solution for (1.1).

Remark 4 The io

1.a M a h

1 1 1.

M a d Mh Mr

1 a M a h

Conclusion

this paper, existence problems of nonlocal fractional-

4.

In order impulsive semi-linear differential inclusions with delay have been considered. We have been considered the case when the values of the orient field are convex as well as non-convex. Some sufficient conditions have been obtained, as pointed in the first section, theses conditions are strictly weaker than the most of the existing on ue al o discuss som

ES

] W. H. Glocke and T. F. Nonnemacher, “A Fractional Cal- Culus Approach of Self-Similar Protein Dynamics,” Bio- physical Journal, Vol. 68, No. 1, 1995, pp. 46-53. doi:10.1016/S0006-3495(95)80157-8

es. In addition, our techniq lows us te fractional differential inclusions with delay.

REFERENC

[1

] R. Hilfer, “Applications of Fractional Calculus in Phys-

ppudies, 204,” Elsev

ence, Publishers BV, Amsterdam, 2006.

[4] K. S. Miller and B. Ross, “An Introduction to the Frac- tional Calculus and Differential Equations,” John Wiley, New York, 1993.

[5] A. M. A. El-Sayed and A. G. Ibrahim, “Multivalued Dif- Ferential Inclusions,” Applied Mathematics and Compu- tation, Vol. 68, No. 1, 1995, pp. 15-25. doi:10.1016/0096-3003(94)00080-N

[2ics,” World Scientific, Singapore City, 1999.

[3] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and A lications of Fractional Differential Equations, in: North Holland Mathematics St ier Sci-

[6]In

pp. 805-813

Phys., Vol. 44, 2008, pp. 1-21.

[8] ce Results for ferential Equa-

tions with Nondense Domain,” Nonlinear Analysis, Vol. 72, No. 2, 2010, pp. 925-932.

B. Ahmed, “Existence Results for Fractional Differential clusions with Separated Boundary Conditions,” Bulletin

of the Korean Mathematical Society, Vol. 47, No. 4, 2010, .

[7] R. P. Agarwal, M. Benchohra and B. A. Slimani, “Exis- tence Results for Differential Equations with FractionalOrder and Impulses,” Mem. Differential Equations, Math.

M. Belmekki and M. Benchohra, “ExistenFractional Order Semilinear Functional Dif

doi:10.1016/j.na.2009.07.034

[9] A. Ouahab, “Fractional Semilinear Differential Inclusions,” Computer and Mathematics with Applications, Vol. 64,

[10] J.-R. Wang, Zhou, “On the New Con- cept of Solu ence Results for Impulsive Fractional Evolutions,” Dynamics of PDE, Vol. 8, No. 4, 2011, pp. 345-361.

[11] J. R. Wang and Y. Zhou, “Existence and Controllability Results for Fractional Sem near Differential Inclusions,” Nonlinear Analysis: Real World Applications, Vol. 12, No. 6, 2011, pp. 3642-3653.

No. 10, 2012, pp. 3235-3252.

M. Feckan and Y. tions and Exist

ili

doi:10.1016/j.nonrwa.2011.06.021

[12] Z. Zhang and B Liu, “Existence of Mild Solutions for Fractional Evolu ns Equations,” Journal of Fractional

l. 2, No. 10, 2012, pp. 1-10.

[13] Y. Zhou and F. Jiao, “Nonlocal Cauchy Problem for Frac- tional Evolution Equations,” Nonlinear Analysis: RWA

. tio

Calculus and Applications, Vo

Vol. 11, No. 5, 2010, pp. 4465-4475.

[14] Y. Zhou and F. Jiao, “Nonlocal Ctional Natural Evolution Equatiothematics with Applications, Vol. 59, No. 3, 2010, pp. 1063- 1077. doi:10.1016/j.camwa.2009.06.026

,

auchy Problem for Frac- ns,” Computer and Ma-

Z. Agur, jo and YL. ion across AgeS Academy of Sci- ences of the Unit

doi:10.1073/pnas.90.24.11698

[15] L. Co caru, G. Mazaur, R. M. Anderson . Danon, “Pulse Mass Measles Vaccinat

horts,” Proceedings of the National ed States of America, Vol. 90, No. 24,

1993, pp. 11698-11702.

d X. Liu, “Boundedness for Impulsive Delay Differential Equations and Applications in Popula- tions Growth Models,” Nonlinear Analysis, Vol. 53, No. 7-8, 2003, pp. 1041-1062. doi:10.1016/S0362-546X(03)00041-5

[16] G. Ballinger an

[17] A. D. Onofrio, “On Pulse Vaccination Strategy in the SIR idemic Model Trans hematics Letters, Vol. 18, No. 7, 2005

doi:10.1016/j.aml.2004.05.012

Ep with Vertical mission,” AppliedMat , pp 9-732. . 72

[1 s, “Impulsive Differential Equations and Inclusions,” Hindawi Publish- ing, New York, Egypt, 2007.

[19] J. M. Ball, “Initial Boundary Value Problems for an Ex- tensible Beam,” Journal of Mathematical Analysis and Applications, Vol. 42, No. 1, 1973, pp. 16-90. doi:10.1016/0022-247X(73)90121-2

8] M. Benchohra, J. Henderson and S. Ntouya

[20] W. E. Fitzgibbon, “Global Existence and Boundedness of Solutions to the Extensible Beam Equation,” SIAM Jour- nal on Mathematical Analysis, Vol. 13, No. 5, 1982, pp. 739-745. doi:10.1137/0513050

[21] R. A. Al-Omair and A. G. Ibrahim, “Existence of Mild Solutions of a Semilinear Evolution Differential Inclu- sions with Nonlocal Conditions,” EJDE, 2009, pp. 1-11.

d P. Rubbioni, “Impulsive Mild Solution [22] T. Cardinali anfor Semilinear Differential Inclusions with Nonlocal Con- ditions in Banach Spaces,” Nonlinear Analysis, Vol. 75, No. 2, 2012, pp. 871-879. doi:10.1016/j.na.2011.09.023

[23] Z. Fan, “Impulsive Problems for Semilinear Differential Equations with Nonlocal Conditions,” Nonlinear Analysis, Vol. 72, No. 2, 2010, pp. 1104-1109.



doi:10.1016/j.na.2009.07.049

[24] G. M. Mophou, “Existence and Uniqueness of Mild Solu- tion to Impulsive Fractional Differential Equations,” Nonlinear Analysis: Theory, Methods & ApplicVol. 72, No. 3-4, 2010, pp. 1604-1615.

ations

[25] O. K. Jaradat, A. ni, “Existence ofthe Mild Solut inear Initial Value

,

Al-Omari and S. Momaion for Fractional Semi-L

problems,” Nonlinear Analysis: Theory, Methods & Ap- plications, Vol. 69, No. 1, 2008, pp. 3153-3159. doi:10.1016/j.na.2007.09.008

[26] E. A. Ddas, M. Benchohra and S. Hamani, “Impulsive Fractional Differential Inclusions Involving the Caputo Fractional Derivative,” Fractional Calculus & Applied Analysis, Vol. 12, No. 1, 2009, pp. 15-36.

[27] J. Henderson and A. Ouahab, “Impulsive Differential Inclusions with Fractional Order,” Computers & Mathe-matics with Applications, Vol. 59, No. 3, 2010, pp. 1191- 1226. doi:10.1016/j.camwa.2009.05.011

[28] A. Pazy, “Semigroups of Linear Operators and Applica- tions to Partial Differential Equations,” Springer Verlag, New York, 1983.

[29] J. P. Aubin and H. Frankoeska, “Set-Valued Analysis,” Birkhäuser, Boston, Basel, Berlin, 1990.

[30] C. Castaing and M. Valadier, “Convex Analysis and Meas-

lisher, Dordrecht,

, Walter, Berlin, 2001.

Theorem of Ville,

Space,” Israel Journal of

003.

, pp. 149-

urable Multifunctions,” Lecture Notes in Mathematics, Springer Verlag, Berlin and New York, 1977.

[31] S. Hu and N. S. Papageorgiou, “Handbook of Multival-

ued Analysis. Vol. I: Theory in Mathematics and Its Ap- plications, Vol. 419,” Kluwer Academic Pub1979.

[32] S. Hu and N. S. Papageorgiou, “Handbook of Multival- ued Analysis. Vol. II: Theory in Mathematics and Its Ap- Plications, Vol. 500, Kluwer Academic Publisher, Dor- drecht, 2000.

[33] M. Kamenskii, V. Obukhowskii and P. Zecca, “Condens- ing Multivalued Maps and Semilinear Differential Inclu- sions in Banach Spaces,” De Gruyter Series in Nonlinear Analysis and Applications 7

[34] H. F. Bohnenblust and S. Karlin, “On ain: Contribution to the Theory of Games,” Princeton Uni- versity Press, Princeton, 1950, pp. 155-160.

[35] H. Covitz and S. B. Nadler, “Multivalued ContractionMapping in Generalized MetricMathematics, Vol. 8, No. 1, 1970, pp. 5-11.

[36] A. Granass and J. Dugundij, “Fixed Point Theorems,” Springer-Verlag, New York, 2

[37] F. Hiai and H. Umegaki, “Integrals, Conditional Expecta- tion, and Martingales of Multivalued Functions,” Journal of Multivariate Analysis, Vol. 7, No. 1, 1977182.

[38] A. Bressan and G. Coombo, “Extensions and Selections of Maps with Decomposable Values,” Studia Mathe- matica, Vol. 90, 1, 1988, pp. 69-86.


http://dx.doi.org/10.1016/S0006-3495(95)80157-8

http://dx.doi.org/10.1016/S0006-3495(95)80157-8

Documents

Mild Solutions for Nonlocal Impulsive Fractional Semilinear … · 2013-12-24 · Ahmed Gamal Ibrahim, Nawal Abdulwahab Al Sarori Mathematics Department, Faculty of Science, King