683–699 DOI: 10.18514/MMN.2019.2241 CAPUTO FRACTIONAL DIFFERENTIAL INCLUSIONS OF ARBITRARY ORDER WITH NONLOCAL INTEGRO-MULTIPOINT BOUNDARY CONDITIONS BASHIR AHMAD, DOA’A GAROUT, SOTIRIS K

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 2, pp. 683–699 DOI: 10.18514/MMN.2019.2241

CAPUTO FRACTIONAL DIFFERENTIAL INCLUSIONS OF ARBITRARY ORDER WITH NONLOCAL

INTEGRO-MULTIPOINT BOUNDARY CONDITIONS

BASHIR AHMAD, DOA’A GAROUT, SOTIRIS K. NTOUYAS, AND AHMED ALSAEDI Received 18 February, 2017

Abstract. We study a new class of boundary value problems of Caputo type fractional differential inclusions supplemented with nonlocal integro-multipoint boundary conditions. An existence result for the problem with convex valued (multivalued) map is obtained via nonlinear alternative of Leray-Schauder type, while the existence of solutions for the problem involving nonconvex valued map is established by means of Wegrzyk’s fixed point theorem. Our results are well illustrated with examples.

2010Mathematics Subject Classification: 34A60; 34A08; 34A12; 34B15

Keywords: Caputo fractional derivative, fractional differential inclusions, existence, fixed point theorems

1. INTRODUCTION

In this paper, we obtain sufficient criteria for the existence of solutions for a Caputo type fractional differential inclusion

cD^qx.t /2F .t; x.t //; n 1 < qn; t2Œ0; 1; (1.1) equipped with the boundary conditions

8 ˆˆ

<

ˆˆ :

x.0/D0; x⁰.0/D0; x⁰⁰.0/D0; : : : ; x^{.n 2/}.0/D0;

x.1/Da Z

0

x.s/dsCb

m 2

X

iD1

˛ix.i/; 0 < < 1< 2< : : : < m 2< 1;

(1.2) where^cD^qdenotes the Caputo fractional derivative of orderq,F WŒ0; 1R!P.R/

is a multi-valued map, P.R/is a family of all nonempty subsets ofR, aandb are real constants and˛i; iD1; : : : ; m 2;are positive real constants.

Here we remark that the last condition in (1.2) connecting the nonlocal multi-point and strip conditions can be interpreted as the linear combination of the values of the unknown function at nonlocal pointsi 2.0; 1/together with the strip contribution of the unknown function on an arbitrary segment .0; /Œ0; 1 is proportional to

c 2019 Miskolc University Press

the value of the unknown function at t D1:The multi-point-strip boundary data, occurring in certain problems of thermodynamics, elasticity and wave propagation [3,8,27], correspond to the situation when the controllers at the end points of the interval dissipate or add energy according to censors located at interior positions (finite many points and strip) of the domain.

In recent years, the topic of initial and boundary value problems involving frac- tional differential equations and inclusions has received great attention and many au- thors have contributed to its advancement by producing a variety of results [2,4–7,9, 15,18,23,25,28,34]. The interest in this topic owes to its extensive applications in the mathematical modeling of scientific and applied problems occurring in various dis- ciplines such as ecology, acoustics, viscoelasticity, electromagnetics, control theory and material sciences. An important characteristic, distinguishing fractional order operators from their classical counterparts, is their nonlocal nature that can take care of the past history of the processes and phenomena involved in the problem. For some specific examples, we refer the reader to the works [20,35] and the references cited therein. In [29], the authors studied the existence of solutions for one-dimensional higher-order semi-linear Caputo type fractional differential equations supplemented with nonlocal multi-point discrete and integral boundary conditions.

Differential inclusions (generalization of differential equations and inequalities) are found to be of great utility in studying dynamical systems and stochastic pro- cesses. An important application of differential inclusions can be found in the area of sweeping processes. In fact, evolution differential inclusions appear in the math- ematical modelling of sweeping processes. For a detailed account of this subject, we refer the reader to the monograph [26] and research articles [1,16]. In [31], it has been shown that the existence and uniqueness of BV continuous sweeping processes can be easily reduced to the Lipschitz continuous case by means of a suitable repara- metrization of the associated moving convex set. Differential inclusions also play a key role in the study of granular systems [30,32], nonlinear dynamics of wheeled vehicles [10], control problems [21], etc. In [24], one can find a detailed description of pressing issues in stochastic processes, control, differential games, optimization and their application in finance, manufacturing, queueing networks, and climate con- trol. For application of fractional differential inclusions in synchronization processes, we refer the reader to the paper [13].

The aim of the present work is to develop the existence theory for a multivalued analog of the problem addressed in [29]. The first result dealing with the convex valued maps involved in the given problem is obtained via nonlinear alternative of Leray-Schauder type, while the existence of solutions for the nonconvex valued maps is shown by applying Wegrzyk’s fixed point theorem. The main results are presented in Section 3. The preliminary material needed to execute the main work is outlined in Section 2.

2. PRELIMINARIES

First of all, we recall some definitions of fractional calculus [20,35].

Definition 1. The Riemann-Liouville fractional integral of order˛ > 0of a func- tiongW.0;1/!Ris defined by

J^˛g.t /D Z t

0

.t s/^{˛ 1}

.˛/ g.s/ds;

provided the right-hand side is point-wise defined on.0;1/, where is the Gamma function.

Definition 2. The Riemann-Liouville fractional derivative of order ˛ > 0 of a continuous functiongW.0;1/!Ris defined by

D^˛g.t /D 1 .n ˛/

d dt

nZ t 0

g.s/

.t s/^{˛ nC1}ds; n 1 < ˛ < n;

wherenDŒ˛C1,Œ˛denotes the integer part of real number˛, provided the right- hand side is point-wise defined on.0;1/.

Definition 3. The Caputo derivative of orderqfor a functionf WŒ0;1/!Rcan be written as

cD^qf .t /DD^q f .t /

n 1

X

kD0

t^k

kŠf^.k/.0/

!

; t > 0; n 1 < q < n:

Remark1. Iff .t /2CⁿŒ0;1/;then

cD^qf .t /D 1 .n q/

Z t 0

f^.n/.s/

.t s/^{qC1 n}dsDI^{n q}f^.n/.t /; t > 0; n 1 < q < n:

In order to define a solution of the given problem, we need the following known result [29].

Lemma 1. Let!2C.Œ0; 1;R/and D1 aⁿ

n b

m 2

X

iD1

˛i^{n 1}_i ¤0: (2.1)

The boundary value problem 8

ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

cD^qx.t /D!.t /; t2Œ0; 1;

x.0/D0; x⁰.0/D0; x⁰⁰.0/D0; : : : ; x^{.n 2/}.0/D0;

x.1/Da Z

0

x.s/dsCb

m 2

X

iD1

˛ix.i/; 0 < < 1< 2< : : : < m 2< 1;

(2.2)

is equivalent to the following integral equation

x.t /D Z t

0

.t s/^{q 1}

.q/ !.s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/!.s/ds Cb

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1} .q/ !.s/ds

Z 1 0

.1 s/^{q 1} .q/ !.s/ds

!

: (2.3)

Next we fix our terminology and outline some basic concepts of multivalued ana- lysis [14,19].

Denote byC.Œ0; 1;R/the Banach space of all continuous functions fromŒ0; 1into Rendowed with the normkxk Dsupfjx.t /j; t 2Œ0; 1g:ByL¹.Œ0; 1;R/we denote the space of Lebesgue measurable and integrable functionsxWŒ0; 1!Rsuch that kxkL¹DR1

0 jx.t /jdt:

For a normed space.X;k k/, letP_cl.X /D fY 2P.X /WY is closedg;P_b.X /D fY 2P.X /WY is boundedg;P_cl;b.X /D fY 2P.X /WY is closed and boundedg; Pcp.X /D fY 2P.X /WY is compactg;andPcp;c.X /D fY 2P.X /WY is compact and convexg:

A multi-valued mapHWX !P.X /W

(i) isconvex (closed) valuedifH.x/is convex (closed) for allx2X:

(ii) isboundedon bounded sets ifH.Y /D [^x2YH.x/is bounded inX for all Y 2Pb.X /(i.e. sup_x2Yfsupfjyj Wy2H.x/gg<1/:

(iii) is called upper semi-continuous (u.s.c.) on X if for each x02X; the set H.x0/is a nonempty closed subset ofX, and if for each open setN ofX containing H.x0/; there exists an open neighborhood N0 ofx0 such that H.N0/N:

(iv) Gislower semi-continuous (l.s.c.)if the setfy2XWH.y/\Y ¤¿gis open for any open setY inX:

(v) is said to be completely continuousif H.B/is relatively compact for every B2P_b.X /IIf the multi-valued mapH is completely continuous with nonempty compact values, thenH is u.s.c. if and only ifH has a closed graph, i.e., xn!x; yn!y; yn2H.xn/implyy2H.x/:

(vi) is said to bemeasurableif for everyy2X, the function t7 !d.y;H.t //Dinffjy ´j W´2H.t /g is measurable.

(vii) has a fixed pointif there isx2X such thatx2H.x/:The fixed point set of the multivalued operatorH will be denoted byFixH:

3. EXISTENCE RESULTS

By Lemma1, we can define a solution of problem (1.1)-(1.2) as follows.

Definition 4. A function x2Cⁿ.Œ0; 1;R/is a solution of problem (1.1)-(1.2) if there exists a functionf 2L¹.Œ0; 1;R/such thatf .t /2F .t; x.t //a.e. onŒ0; 1and

x.t /D Z t

0

.t s/^{q 1}

.q/ f .s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/f .s/ds Cb

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1} .q/ f .s/ds

Z 1 0

.1 s/^{q 1} .q/ f .s/ds

!

: (3.1)

Next, we define an operatorKWC.Œ0; 1;R/!P.C.Œ0; 1;R//by K.x/D

(

´2C.Œ0; 1;R/W´.t /D Z t

0

.t s/^{q 1} .q/ f .s/ds Ct^{n 1}

a

Z 0

. s/^q

.qC1/f .s/dsCb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1} .q/ f .s/ds Z 1

0

.1 s/^{q 1}

.q/ f .s/ds

!

; f 2SF ;x

)

: (3.2)

Definition 5. A multi-valued mapF WŒ0; 1R!P.R/is said to be Carath´eodory if

(i) t7!F .t; x/is measurable for eachx2Rand

(ii) x7!F .t; x/is upper semicontinuous for almost allt2Œ0; 1.

Further, a Caratheodory functionF is calledL¹ Carath´eodory if (iii) for eacha > 0;there exists'a2L¹.Œ0; 1;R^C/such that

kF .t; x/k Dsupfjvj Wv2F .t; x/g 'a.t / for allkxk aand for a.et2Œ0; 1:

For eachy2C.Œ0; 1;R/;we define the set of selections ofF by

S_{F ;y}WD fv2L¹.Œ0; 1;R/Wv.t /2F .t; y.t // f or a:e: t2Œ0; 1g: 3.1. The convex-valued (Carath´eodory) case

In this subsection, we prove an existence result for problem (1.1)-(1.2), assuming thatF is Carath´eodory (F has convex values).

Before presenting the main result, let us state the auxiliary lemmas.

Lemma 2(Nonlinear alternative of Leray-Schauder type [17]). LetC be a convex set in a normed space, and E C be open subset with 02E: Then each upper semicontinuous and compact mappingG WE!P .C /with compact convex values that is fixed point free on@Ehas at least one of the following two properties:

(a) G has a fixed point inE;or

(b) there existx2@E and2.0; 1/such thatx2G.x/:

Lemma 3 ([22]). Let X be a separable Banach space. Let F W Œ0; 1R! Pcp;c.X / be anL¹ Caratheodory multi-valued map, and let be a linear con- tinuous mapping fromL¹.Œ0; 1; X /toC.Œ0; 1; X /:Then the operator

ıS_F WC.Œ0; 1; X / !Pcp;c.C.Œ0; 1; X //; x7!.ıS_F/.x/D.S_{F ;x}/;

is a closed graph operator inC.Œ0; 1; X /C.Œ0; 1; X /:

Theorem 1. Assume that

.H1/ F WŒ0; 1R!P.R/ is L¹ Carath´eodory and has compact and convex values;

.H2/ there exist a continuous nondecreasing functionWŒ0;1/!.0;1/and a functionk2C.Œ0; 1;R^C/such that

kF .t; x/kP WDsupfjyj Wy2F .t; x/g k.t /.kxk/ f or al l .t; x/2Œ0; 1RI .H3/ There exists a numberN > 0such that

N .N /kkk

( 1 .qC1/

"

1C 1

jj 1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

!#) 1

> 1;

whereis given by (2.1).

Then the problem (1.1)-(1.2) has at least one solution onŒ0; 1.

Proof. We transform the problem (1.1)-(1.2) into a fixed point theorem by consid- ering the operatorK defined by (3.2). It is obvious that the fixed points ofK are solutions of the boundary value problem (1.1)-(1.2). We will show thatK satisfies the assumptions of Leray-Schauder nonlinear alternative. In the first step, we will show that K.x/ is convex for eachx2C.Œ0; 1;R/:Let´1; ´22K.x/, then there existf1; f22SF ;xsuch that for eacht2Œ0; 1, we have

´i.t /D Z t

0

.t s/^{q 1}

.q/ fi.s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/fi.s/ds Cb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1}

.q/ fi.s/ds Z 1

0

.1 s/^{q 1}

.q/ fi.s/ds

!

; iD1; 2:

Set0ı1, then for eacht2Œ0; 1;we have Œı´1C.1 ı/´2.t /D

Z t 0

.t s/^{q 1}

.q/ Œıf1.s/C.1 ı/f2.s/ds Ct^{n 1}

a

Z 0

. s/^q

.qC1/Œıf1.s/C.1 ı/f2.s/ds

Cb

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1}

.q/ Œıf1.s/C.1 ı/f2.s/ds Z 1

0

.1 s/^{q 1}

.q/ Œıf1.s/C.1 ı/f2.s/ds

! : Hence, by the convexity ofSF ;x, it follows thatı´1C.1 ı/´22K.x/:

Now, we show that K maps bounded sets into bounded sets in C.Œ0; 1;R/:For > 0;letBD fx2C.Œ0; 1;R/W kxk gbe a bounded set inC.Œ0; 1;R/:Thus, for each´2K.x/; x2B;there existsf 2SF ;xsuch that

´.t /D Z t

0

.t s/^{q 1}

.q/ f .s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/f .s/ds Cb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1}

.q/ f .s/ds Z 1

0

.1 s/^{q 1}

.q/ f .s/ds

!

: (3.3) Then, forx2B;in view of.H2/;we obtain

j´.t /j

Z t 0

.t s/^{q 1}

.q/ k.s/.kxk/dsCt^{n 1} jj jaj

Z 0

. s/^q

.qC1/k.s/.kxk/ds C jbj

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1}

.q/ k.s/.kxk/dsC Z 1

0

.1 s/^{q 1}

.q/ k.s/.kxk/ds

!

; which implies that

k´k ./kkk

.qC1/ 1C 1 jj

"

1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

#!

:

Next,we show thatKmaps bounded sets into equicontinuous sets inC.Œ0; 1;R/:

Lett1; t22Œ0; 1witht1< t2andx2B;then, we obtain for each´2K.x/

j´.t2/ ´.t1/j D

ˇ ˇ ˇ ˇ ˇ

Z t2

0

.t2 s/^{q 1}

.q/ f .s/ds Z t1

0

.t1 s/^{q 1}

.q/ f .s/dsC.t₂^{n 1} t₁^{n 1}/

"

a Z

0

. s/^q

.qC1/f .s/dsCb

m 2

X

iD1

˛_i Z _i

0

.i s/^{q 1}

.q/ f .s/ds

Z 1 0

.1 s/^{q 1} .q/ f .s/ds

#ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ

Z t1

0

.t2 s/^{q 1} .t1 s/^{q 1}

.q/ f .s/dsC

Z t2

t1

.t2 s/^{q 1}

.q/ f .s/ds ˇ ˇ ˇ ˇ ˇ C

ˇ ˇ ˇ ˇ ˇ

t₂^{n 1} t₁^{n 1}

ˇ ˇ ˇ ˇ ˇ

"

jaj Z

0

. s/^q

.qC1/jf .s/jds C jbj

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1}

.q/ jf .s/jdsC Z 1

0

.1 s/^{q 1}

.q/ jf .s/jds

#

./kkk

"

jt₂^q t₁^qj C2.t2 t1/^q .qC1/

Cjt₂^{n 1} t₁^{n 1}j

.qC1/jj 1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

!#

;

which tends to zero independent ofx2B as.t2 t1/!0:Consequently, by the Arzel´a-Ascoli theorem, the operatorK is completely continuous.

SinceK is completely continuous, in order to prove that it is u.s.c. it is enough to prove that it has a closed graph. Thus, now,we want to show thatK has a closed graph. Letxn! Ox,´n2K.xn/;and´n! O´. We have to show that´O 2K.x/:O So, for´n2K.xn/;there existsfn2SF ;xnsuch that for allt2Œ0; 1, we have

´n.t /D Z t

0

.t s/^{q 1}

.q/ fn.s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/fn.s/ds Cb

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1}

.q/ fn.s/ds Z 1

0

.1 s/^{q 1}

.q/ fn.s/ds

! :

Thus, we have to show that there existsfO2S_{F ;xO} such that for eacht2Œ0; 1;

O

´.t /D Z t

0

.t s/^{q 1}

.q/ f .s/dsO Ct^{n 1}

a

Z 0

. s/^q

.qC1/f .s/dsO Cb

m 2

X

iD1

˛_i Z _i

0

.i s/^{q 1}

.q/ f .s/dsO Z 1

0

.1 s/^{q 1}

.q/ f .s/dsO

! :

Consider the continuous linear operator˚WL¹.Œ0; 1;R/!C.Œ0; 1;R/given by f 7!˚.f /.t /D

Z t 0

.t s/^{q 1}

.q/ f .s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/f .s/ds Cb

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1} .q/ f .s/ds

Z 1 0

.1 s/^{q 1}

.q/ f .s/ds

! : Observe that

k´n.t / ´.t /O k D

Z t 0

.t s/^{q 1}

.q/ .fn.s/ f .s//dsO Ct^{n 1}

a

Z 0

. s/^q

.qC1/.fn.s/ f .s//dsO Cb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1}

.q/ .fn.s/ f .s//dsO Z 1

0

.1 s/^{q 1}

.q/ .fn.s/ f .s//dsO

!

tends to0asn! 1. Thus, it follows by Lemma3 that˚ıSF is a closed graph operator. Moreover, we have´n.t /2˚.SF ;xn/:Sincexn! Ox;we have then

O

´.t /D Z t

0

.t s/^{q 1}

.q/ f .s/dsO Ct^{n 1}

a

Z 0

. s/^q

.qC1/f .s/dsO Cb

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1}

.q/ f .s/dsO Z 1

0

.1 s/^{q 1}

.q/ f .s/dsO

!

; for somefO2S_{F ;xO}:

Finally, we show there exists an open setV C.Œ0; 1;R/ with x…K.x/ for any2.0; 1/ and allx2@V: Letxbe a solution of (1.1)-(1.2). Then there exists f 2L¹.Œ0; 1;R/withf 2SF ;xsuch that fort2Œ0; 1, we have

x.t /D Z t

0

.t s/^{q 1}

.q/ f .s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/f .s/ds Cb

m 2

X

iD1

˛_i Z _i

0

.i s/^{q 1}

.q/ f .s/ds Z 1

0

.1 s/^{q 1}

.q/ f .s/ds

! :

Using.H2/, we obtain jx.t /j .kxk/kkk

.qC1/ 1C 1 jj

"

1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

#!

; which implies

kxk .kxk/kkk

"

1 .qC1/

( 1C 1

jj 1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

!)# 1

1:

By the assumption.H3/;there existsN > 0such thatkxk ¤N. Let us take V D fx2C.Œ0; 1;R/W kxk< NC1g: Note that the operatorK WV !P.C.Œ0; 1;R//

is upper semicontinuous and completely continuous. From the choice ofV;there is nox2@V such thatx2K.x/for some2.0; 1/. Therefore, by Leray-Schauder alternative, it follows that the operatorK has a fixed pointx2V, which is a solution of the problem (1.1)-(1.2). This completes the proof.

3.2. Nonconvex-valued (Lipschitz) case

In this part, we discuss the existence of solutions for the inclusion problem (1.1)- (1.2) with the right-hand side being nonconvex set-valued map by applying Wegrzyk’s fixed point theorem.

Let.X; d /be a metric space induced from the normed space.XI k k/. Consider Q_d WP.X /P.X /!RS

f1ggiven by Q_d.A; B/Dmaxfsup

a2A

d.a; B/;sup

b2B

d.b; A/g;

whered.a; B/Dinfb2Bd.a; b/:The mapQdis the (generalized) Pompeiu-Hausdorff functional. Clearly,.P_b;cl.X /; Q_d/is a metric space and.P_d.X /; Q_d/is a general- ized metric space.

Definition 6. A function˝WR^C!R^Cis said to be a strict comparison function if it is continuous strictly increasing andP₁

nD1˝ⁿ.t / <1for allt > 0.

Definition 7. A multi-valued operatorAWX !P_cl.X /is called

(a) -Lipchitz if and only if there exists > 0such that Q_d.A.X /;A.Y //

d.x; y/for eachx; y2X:

(b) a contraction if and only if it is Lipschitz with < 1I

(c) a generalized contraction if and only if there is a strict comparison function

˝WR^C!R^Csuch thatQd.A.x/;A.y//˝.d.x; y//for eachx; y2X:

Lemma 4(Wegrzyk’s fixed point theorem [33]). Let.X; d /be a complete metric space. IfAWX !P_cl.X /is a generalized contraction, then the operatorAhas at least one fixed point.

Theorem 2. Assume that the following conditions hold:

.H4/ F WŒ0; 1R!Pcp.R/is such thatF .:; x/WŒ0; 1!Pcp.R/is measurable for eachx2R:

.H5/ Qd.F .t; x/; F .t;x//N .t /˝.kx xNk/for almost allt2Œ0; 1andx;xN2R with a function 2C.Œ0; 1;R^C/andd.0; F .t; 0//.t /for almost allt 2 Œ0; 1;where˝WR^C!R^Cis strictly increasing.

Then the problem (1.1)-(1.2) has at least one solution onŒ0; 1if ˝WR^C!R^Cis a strict comparison function, where

D kk

.qC1/ 1C 1 jj

"

1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

#!

:

Proof. Assume that ˝WR^C!R^Cis a strict comparison function. Then, in view of.H4/and.H5/,F .; x.//is measurable and has a measurable selection./(see Theorem III.6 [11]). Also,2C.Œ0; 1;R^C/and we have

j.t /j d.0; F .t; 0//CQ_d.F .t; 0/; F .t; x.t ///

.t /C.t /˝.jx.t /j/.1C˝.kxk//.t /:

Hence, the setSF ;xis nonempty for eachx2C.Œ0; 1;R/:Now, we will show that the operatorKwhich is given by (3.2), satisfies the assumptions of Wegrzyk’s fixed point Theorem. First, we show that K.x/2P_cl.C.Œ0; 1;R// for each x2C.Œ0; 1;R/.

So, let, fgngⁿ02K.x/be such thatgn!g asn! 1inC.Œ0; 1;R/:Thusg2 C.Œ0; 1;R/;and there existsn2SF ;xn;such that, for allt2Œ0; 1;we have

gn.t /D Z t

0

.t s/^{q 1}

.q/ n.s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/n.s/ds Cb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1}

.q/ n.s/ds Z 1

0

.1 s/^{q 1}

.q/ n.s/ds

! : SinceF has compact values, we take a subsequence to obtain that_nconverges to inL¹.Œ0; 1;R/:Hence,2SF ;x;and for eacht2Œ0; 1;we have,

gn.t /!g.t /D Z t

0

.t s/^{q 1}

.q/ .s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/.s/ds Cb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1} .q/ .s/ds

Z 1 0

.1 s/^{q 1} .q/ .s/ds

! : Thus,g2K.x/:

Now, we show thatQd.K.x/;K.x//N ˝.kx xNk/;for allx;xN2C.Œ0; 1;R/:

Letx; xN 2C.Œ0; 1;R/and´12K.x/;then, there exists1.t /2SF ;xsuch that, for allt2Œ0; 1;we have

´1.t /D Z t

0

.t s/^{q 1}

.q/ 1.s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/1.s/ds Cb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1}

.q/ 1.s/ds Z 1

0

.1 s/^{q 1}

.q/ 1.s/ds

! : In view of the assumption.H5/;we have Q_d.F .t; x/; F .t;x//N .t /˝.jx.t / N

x.t /j/:Thus, there exists2F .t;x.t //such thatN

j1.t / j .t /˝.jx.t / x.t /N j/; t2Œ0; 1:

Let us define the map,W WŒ0; 1!P.R/by

W .t /D f2RW j1.t / j .t /˝.jx.t / x.t /N j/g: Because the nonempty closed set-valued operator W .t /T

F .t;x.t //N is measurable (Proposition III.4 [11]), there exists a function2.t /which is a measurable selection forW .t /T

F .t;x.t //:N Hence,2.t /2F .t;x.t //;N andj1.t / 2.t /j .t /˝.jx.t / N

x.t /j/for eacht2Œ0; 1:

Define for eacht2Œ0; 1;

´2.t /D Z t

0

.t s/^{q 1}

.q/ 2.s/dsCt^{n 1}

a

Z 0

. s/^q

.qC1/2.s/ds Cb

m 2

X

iD1

˛i

Z i

0

.i s/^{q 1}

.q/ 2.s/ds Z 1

0

.1 s/^{q 1}

.q/ 2.s/ds

! : So, by the assumption.H5/, we obtain

j´1.t / ´2.t /j

Z t 0

.t s/^{q 1}

.q/ j1.t / 2.t /jdsCt^{n 1} jj jaj

Z 0

. s/^q

.qC1/j1.t / 2.t /jds C jbj

m 2

X

iD1

˛i

Z _i 0

.i s/^{q 1}

.q/ j1.t / 2.t /jds C

Z 1 0

.1 s/^{q 1}

.q/ j1.t / 2.t /jds

!

;

which implies that

k´1 ´2k kk˝.kx xNk/

.qC1/ 1C 1 jj

"

1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

#!

: Interchanging the roles ofxandx;N we obtain

Qd.K.x/;K.x//N ˝.kx xNk/ D kk˝.kx xNk/

.qC1/ 1C 1 jj

"

1C jaj ^qC1 .qC1/C jbj

m 2

X

iD1

˛iiq

#!

;

for allx; xN 2C.Œ0; 1;R/:Thus, the operatorK is a generalized contraction. There- fore, by Wegrzyk’s fixed point Theorem, the operatorK has at least one fixed point x, which is a solution of the problem (1.1)-(1.2). This completes the proof.

Remark2. We emphasize that the existence result (Theorem2) obtained by apply- ing Wegrzyk’s theorem holds for several choices of the strictly increasing function

˝involved in its hypothesis. For instance, by taking˝.x/Dx;we obtain a special case which is usually obtained by applying a fixed point theorem due to Covitz and Nadler [12].

3.3. Examples

Example1. (Convex-valued case) Consider the following multi-valued fractional boundary value problem:

8 ˆ<

ˆ:

cD⁵⁼⁴2F .t; x.t //; t2Œ0; 1;

x.0/D0; x.1/D Z 1=8

0

x.s/dsC

4

X

iD1

˛ix.i/; (3.4) whereqD5=4; aDbD1; D1=8; 1D1=6; 2D1=3; 3D1=2; 4D2=3; ˛1D 1=7; ˛2D2=7; ˛3D3=7; ˛4D4=7and

F .t; x.t //D

"

jxj

9.jxj C5/C 1

18cos²t ; 1

2e ^tsinx

#

: (3.5)

Using the given data, we get jj D ˇ ˇ ˇ ˇ ˇ

1 a²

2 b

4

X

iD1

˛ii

ˇ ˇ ˇ ˇ ˇ

D0:2779;

1 .qC1/

( 1C 1

jj 1C jaj^qC1 qC1C jbj

m 2

X

iD1

˛_ii^q

!)

D6:0155:

Moreover, forf 2F, we have jfj max

n jxj

9.jxj C5/C 1

18cos²t ; 1

2e ^tsinxo

; x2R; t 2Œ0; 1:

Thus

kF .t; x/kP WDsupfjyj Wy2F .t; x/g 1

2; x2R;

with k.t /D1; .kxk/D1=2. Next, using the condition .H3/, we find thatN >

N1'3:0078:Thus, all the conditions of Theorem1are satisfied and consequently, the problem (3.4) withF .t; x/given by (3.5), has at least one solution onŒ0; 1:

Example2. (Nonconvex-valued case) Consider the problem (3.4) that is given in Example1, with

F .t; x.t //D

"

p 1

625Ct² ; sinxC²tan ¹x 2.4Ct .1 t //² C 4

81

#

: (3.6)

So, we find that

supfjgj Wg2F .t; x/g 1

.4Ct .1 t //²C 4 81; Q_d.F .t; x/; F .t;x//N 1

.4Ct .1 t //²

h.2C/

2 kx xNki :

Put.t /D 1

.4Ct .1 t //², so,kk D1=16:Thus, we get D6:0155kk D6:0155=16'0:37597;

andQ_d.F .t; x/; F .t;x//N .t /˝.kx xNk/;where˝.kx xNk/D.2C/

2 kx xNk. Hence, all the conditions of Theorem2 are satisfied. Therefore, the problem (3.4) withF .t; x/given by (3.6), has at least one solution onŒ0; 1:

ACKNOWLEDGEMENT

We appreciate the reviewer for his/her constructive remarks that led to the im- provement of the original manuscript.

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Authors’ addresses

Bashir Ahmad

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathem- atics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

E-mail address:bashirahmad qau@yahoo.com

Doa’a Garout

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathem- atics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

E-mail address:dgarout@kau.edu.sa

Sotiris K. Ntouyas

University of Ioannina, Department of Mathematics, 451 10 Ioannina, Greece and, Nonlinear Ana- lysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Sci- ence, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

E-mail address:sntouyas@uoi.gr

Ahmed Alsaedi

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathem- atics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

E-mail address:aalsaedi@hotmail.com