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On fractional integro-differentialinclusions with state-dependent delayin Banach spacesMouffak Benchohra a , Sara Litimein a & Gaston N'Guérékata ba Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, BP89, 22000 Sidi Bel-Abbès, Algérieb Department of Mathematics, Morgan State University, 1700 E.Cold Spring Lane, Baltimore, MD 21252, USA

Version of record first published: 26 Sep 2011.

To cite this article: Mouffak Benchohra, Sara Litimein & Gaston N'Guérékata (2011): On fractionalintegro-differential inclusions with state-dependent delay in Banach spaces, Applicable Analysis: AnInternational Journal, DOI:10.1080/00036811.2011.616496

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Applicable Analysis2011, 1–16, iFirst

On fractional integro-differential inclusions with state-dependent

delay in Banach spaces

Mouffak Benchohraa, Sara Litimeina and Gaston N’Guerekatab*

aLaboratoire de Mathematiques, Universite de Sidi Bel-Abbes, BP 89,22000 Sidi Bel-Abbes, Algerie; bDepartment of Mathematics, Morgan State University,

1700 E. Cold Spring Lane, Baltimore, MD 21252, USA

Communicated by X. Zou

(Received 17 February 2011; final version received 12 August 2011)

In this article we investigate the existence of solutions on a compact intervalfor the fractional integro-differential inclusions with state-dependent delayin Banach spaces when the delay is infinite. We consider the cases when themultivalued nonlinear term takes convex values as well as nonconvexvalues.

Keywords: differential inclusions; integral resolvent family; mild solution;fixed points; state-dependent delay; infinite delay

AMS Subject Classifications: 26A30; 34A60; 34G20; 34G25

1. Introduction

This article studies the existence of mild solutions for semilinear integro-differentialinclusions of fractional order of the form

y0ðtÞ �

Z t

0

ðt� sÞ��2

�ð�� 1ÞAyðsÞds2Fðt, y�ðt,ytÞÞ, a.e. t2 J ¼ ½0, b� ð1Þ

y0 ¼ �2B, ð2Þ

where 1<�< 2 and A: D(A)�E!E is the generator of a solution operator definedon a complex Banach space (E, j � j), the convolution integral in the equation isknown as the Riemann–Liouville fractional integral and F: [0, b]�B!P(E ) is amultivalued map (P(E ) is the family of nonempty subsets of E ). For any continuousfunction y defined on (�1, b] and any t� 0, we denote by yt the element of B definedby yt(�)¼ y(tþ �) for �2 (�1, 0]. Here yt(�) represents the history of the state fromeach time � 2 (�1, 0] up to the present time t. We assume that the histories ytbelongs to some abstract phase space B, to be specified later.

*Corresponding author. Email: Gaston.N’Guerekata@morgan.edu

ISSN 0003–6811 print/ISSN 1563–504X online

� 2011 Taylor & Francis

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The theory of fractional differential equations has been emerging as an importantarea of investigation in recent years. Let us mention that this theory has manyapplications in describing numerous real world events and problems. For example,fractional differential equations are often applicable in engineering, physics,chemistry and biology. Fractional differential equations are also considered in themonographs of Kilbas et al. [1], Lakshmikantham et al. [2] and Podlubny [3], and thepapers [4–6].

Differential inclusions are generalization of differential equations, therefore allproblems considered for differential equations, that is, existence of solutions,continuation of solutions, dependence on initial conditions and parameters, arepresent in the theory of differential inclusions. Since a differential inclusion usuallyhas many solutions starting at a given point, new issues appear such as investigationof topological properties of the set of solutions, and selection of solutions with givenproperties. The literature related to partial functional differential inclusions withstate-dependent delay is limited.

On the other hand, functional differential equations with state-dependent delayappear frequently in applications as model of equations and for this reason the studyof this type of equations has received a significant amount of attention in the pastseveral years (we refer to [7–15] and the references therein). The literature related tofunctional differential inclusions with state-dependent delay remains limited [16,17].

In part, the Cauchy problem for abstract differential equations involvingRiemann–Liouville fractional integral in the linear part have been treated by Cuevasand de Souza in [18,19], where they studied S-asymptotically w-periodic solutions.Wang and Chen [20] considered a Cauchy problem for fractional integro-differentialequations with time delay and nonlocal initial conditions. Uniqueness and existenceresults of mild solutions on a semi-infinite interval have been established byBenchohra and Litimein [21]. The existence of mild solutions for the class offractional integro-differential inclusions with state-dependent delay of the form(1)–(2) seems to be an untreated topic.

This article is organized as follows. In Section 2, we introduce some preliminaryresults needed in the sequel. In Section 3, we present two results for the problem(1)–(2) when the right-hand side is convex valued. The first one is based upon a fixedpoint theorem of Bohnenblust–Karlin [22], and the second one on the nonlinearalternative of Leray Schauder type [23]. Another existence result is given for anonconvex-valued right-hand side by using a fixed point theorem for contractionmultivalued maps due to Covitz and Nadler [24]. Finally in Section 4, we present anexample to illustrate the abstract results.

2. Preliminaries

We present in this section the notation, definitions and preliminary facts frommultivalued analysis which will be used throughout this article.

Let C(J,E ) be the Banach space of all continuous functions from J into E withthe norm

k yk1 :¼ supfj yðtÞj: t2 Jg:

Let B(E ) denote the Banach space of all bounded linear operators from E into E.

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A measurable function y: J!E is Bochner integrable if and only if jyj isLebesgue integrable. (For properties of the Bochner integral see Yosida [25]).

Let L1(J,E ) denote the Banach space of all continuous functions y: J!E whichare Bochner integrable and have norm

k ykL1 ¼

Z b

0

j yðtÞjdt for all y2L1ðJ,E Þ:

In this article, we will use an axiomatic definition of the phase space B introducedby Hale and Kato in [26] and follow the terminology in [27]. Thus, (B, k�kB) will be aseminormed linear space of functions mapping (�1, 0] into E, and satisfying thefollowing axioms:

(A1) If y: (�1, b)!E, b> 0, is continuous on J and y02B, then for every t2 J thefollowing conditions hold:

(i) yt2B.(ii) There exists a positive constant H such that jy(t)j �HkytkB.(iii) There exist two functions K(�), M(�): Rþ!Rþ independent of y with K

continuous and M locally bounded such that:

k ytkB � KðtÞ supfj yðsÞj: 0 � s � tg þMðtÞk y0kB:

(A2) For the function y in (A1), yt is a B-valued continuous function on J.(A3) The space B is complete.

Denote Kb¼ sup{K(t): t2 J} and Mb¼ sup{M(t): t2 J}.

Remark 2.1

(1) (A1)(ii) is equivalent to j�(0)j �Hk�kB for every �2B.(2) Since k�kB is a seminorm, two elements �, 2B can verify k�� kB¼ 0

without necessarily �(�)¼ (�) for all �� 0.(3) From the equivalence of in the first remark, we can see that for all �, 2B

such that k�� kB¼ 0: we necessarily have that �(0)¼ (0).

We now recall some examples of phase spaces. For other details we refer forinstance to [27,28].

Example 2.2 Let:

BC the space of all bounded continuous functions defined from (�1, 0] to E;BUC the space of all bounded uniformly continuous functions defined from (�1, 0]to E;

½C1� :¼ �2BC : lim�!�1

�ð�Þ exist in E

� �;

½C0� :¼��2BC: lim

�!�1�ð�Þ ¼ 0

�, endowed with the uniform norm

k�k ¼ sup j�ð�Þj: � � 0� �

:

We have that the spaces BUC, C1 and C0 satisfy conditions (A1)� (A3). However,BC satisfies (A1), (A3) but (A2) is not satisfied.

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Example 2.3 Let g be a positive continuous function on (�1, 0]. We define:

½Cg� :¼ �2Cðð�1, 0�,E Þ:�ð�Þ

gð�Þis bounded on ð�1, 0�

� �;

½C0g� :¼ �2Cg : lim

�!�1

�ð�Þ

gð�Þ¼ 0

� �, endowed with the uniform norm

k�k ¼ supj�ð�Þj

gð�Þ: � � 0

� �:

Then we have that the spaces Cg and C0g satisfy conditions (A3). We consider the

following condition on the function g.

(g1) For all a4 0, sup0�t�a

supgðtþ �Þ

gð�Þ: �15 � � �t

� �51:

They satisfy conditions (A1) and (A2) if (g1) holds.

Example 2.4 The space C�.For any real positive constant �, we define the functional space C� by

C� :¼ �2Cðð�1, 0�,E Þ : lim�!�1

e���ð�Þ exists in E

� �endowed with the following norm:

k�k ¼ supfe��j�ð�Þj : � � 0g:

Then in the space C� the axioms (A1)–(A3) are satisfied.

2.1. Solution operator

The Laplace transformation of a function f2L1locðIRþ,E Þ is defined by

Lð f Þð�Þ :¼:bfð�Þ ¼ Z 10

e��tf ðtÞdt, Reð�Þ4!,

if the integral is absolutely convergent for Re(�)>!. In order to define the mildsolution of the problems (1)–(2) we recall the following definition.

Definition 2.5 Let A be a closed and linear operator with domain D(A) defined on aBanach space E. We call A the generator of a solution operator if there exists !> 0and a strongly continuous function S: IRþ!B(E ) such that

f��: Reð�Þ4!g � �ðAÞ,

and

���1ð�� � AÞ�1x ¼

Z 10

e��tSðtÞx dt, Re �4!, x2E:

In this case, S(t) is called the solution operator generated by A.

The following result is a direct consequence of [29, Proposition 3.1 andLemma 2.2].

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PROPOSITION 2.6 Let {S(t)}t�0�B(E ) be the solution operator with generator A.Then the following conditions are satisfied:

(a) S(t) is strongly continuous for t� 0 and S(0)¼ I;(b) S(t)D(A)�D(A) and AS(t)x¼S(t)Ax for all x2D(A), t� 0;(c) for every x2D(A) and t� 0,

SðtÞx ¼ xþ

Z t

0

ðt� sÞ��1

�ð�ÞASðsÞx ds:

(d) Let x2D(A). ThenR t0ðt�sÞ��1

�ð�Þ SðsÞx ds2DðAÞ and

SðtÞx ¼ xþ A

Z t

0

ðt� sÞ��1

�ð�ÞSðsÞxds:

Remark 2.7 The concept of a solution operator, as defined above, is closely relatedto the concept of a resolvent family [30]. Because of the uniqueness of the Laplacetransform, in the border case �¼ 1, the family S(t) corresponds to a C0 semigroup[31,32], whereas in the case �¼ 2 a solution operator corresponds to the concept ofcosine family [33].

More information on the C0-semigroups and sine families can be found in[32,34,35].

Definition 2.8 A solution operator {S(t)}t>0 is called uniformly continuous if

limt!skSðtÞ � SðsÞkBðEÞ ¼ 0:

2.2. Multivalued analysis

Denote by P(E )¼ {Y�E: Y 6¼ ;, Pcl(E )¼ {Y2P(E ): Y closed}, Pb(E )¼ {Y2P(E ):Y bounded}, Pcv(E )¼ {Y2P(E ): Y convex}, Pcp(E )¼ {Y2P(E ): Y compact}.

Let (X, k�k) be a Banach space. A multivalued map G: X!P(X ) is convex(closed) valued if G(x) is convex (closed) for all x2X. G is bounded on bounded setsif G(B)¼[x2BG(x) is bounded in X for any bounded set B of P(X ) (i.e.supx2B{sup{kyk: y2G(x)}}<1).

G is called upper semicontinuous (u.s.c.) on X if for each x 2X the set G(x) is anonempty, closed subset of X, and if for each open set B of X containing G(x), thereexists an open neighbourhood V of x such that G(V)B.

G is said to be completely continuous if G(B) is relatively compact for everybounded subset BX. If the multivalued map G is completely continuous withnonempty compact values, then G is u.s.c. if and only if G has a closed graph(i.e. xn!x, yn! y, yn2Gxn imply y 2Gx). G has a fixed point if there is x2Xsuch that x2Gx.

A multivalued map G: J!Pcl(E ) is said to be measurable if for each y2E thefunction Y: J!R defined by

YðtÞ ¼ d ð y,GðtÞÞ ¼ inffj y� zj : z2GðtÞg

belongs to L1(J,R).

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Definition 2.9 A multivalued map F: J�E!P(E ) is Caratheodory if

(i) t � F(t, u) is measurable for each u2E, and(ii) u � F(t, u) is u.s.c. for almost all t2 J.

For each y2C(J,E ), define the set of selections for F by

SF,y ¼ fv2L1ðJ,E Þ : vðtÞ 2Fðt, yðtÞÞ for a.e. t2 Jg:

Let (X, d ) be a metric space induced by the normed space (X, j�j). Consider Hd:

P(X )�P(X )!Rþ[ {1} given by

HdðA,BÞ ¼ max supa2A

d ða,BÞ, supb2B

d ðA, bÞ

� �,

where d(A, b)¼ infa2Ad(a, b) and d(a,B)¼ infb2Bd(a, b). Then (Pb,cl(X ),Hd) is a

metric space and (Pcl(X ),Hd) is a generalized metric space [36].

Definition 2.10 A multivalued operator N: X!Pcl(X ) is called:

(a) �-Lipschitz if there exists � > 0 such that

Hd ðNðxÞ,Nð yÞÞ � �d ðx, yÞ for all x, y2X;

(b) a contraction if it is �-Lipschitz with � < 1.

The following lemma will be used in the sequel.

LEMMA 2.11 [37] Given a Banach space E, let F: J�E!Pcp,cv(E ) be an

L1-Caratheodory multivalued map, and let � be a linear continuous mapping from

L1(J,E ) into C(J,E ). Then the operator

� � SF : CðJ,E Þ �!Pcp,cvðCðJ,E ÞÞ,

y 7 �! ð� � SFÞð yÞ :¼ �ðSF,yÞ

has a closed graph in C(J,E )�C(J,E ).

LEMMA 2.12 [22] Let X be a Banach space and K2Pcl,c(X ) and suppose that the

operator G: K!Pcl,c(K) is u.s.c. and the set G(K) is relatively compact in X. Then G

has a fixed point in K.

LEMMA 2.13 [24] Let (X, d ) be a complete metric space. If N: X!Pcl(X ) is a

contraction, then FixN 6¼ ;.

For more details on multivalued maps see the books of Aubin and Cellina [38],

Deimling [39] and Hu and Papageorgiou [40].

3. The main results

In this section, we are concerned with the existence of solutions for the

problem (1)–(2).

Definition 3.1 We say that the function y: (�1, b]!E is a mild solution of (1)–(2)

if y(t)¼�(t) for all t� 0, the restriction of y(�) to the interval [0, b] is continuous and

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there exists v(�)2L1(J,E ), such that vðtÞ 2 f ðt, y�ðt,ytÞÞ a.e t2 ½0, b�, and y satisfies the

following integral equation

yðtÞ ¼ SðtÞ�ð0Þ þ

Z t

0

Sðt� sÞvðsÞds for each t2 ½0, b�: ð3Þ

Set

Rð��Þ ¼ f�ðs, ’Þ : ðs, ’Þ 2 J� B, �ðs,’Þ � 0g:

We always assume that �: J�B! (�1, b] is continuous. Additionally, we state

following hypothesis:

(H�) The function t!�t is continuous from R(��) into B and there exists a

continuous and bounded function L�: R(��)! (0,1) such that

k�tkB � L�ðtÞk�kB for every t2Rð��Þ:

Remark 3.2 The condition (H�), is frequently verified by continuous and bounded

functions. For more details, see for instance [27].

LEMMA 3.3 [14, Lemma 2.4] If y: (�1, b]!E is a function such that y0¼�, then

k yskB � ðMb þ L�Þk�kB þ Kb supfj yð�Þj; � 2 ½0, maxf0, sg�g, s2Rð��Þ [ J,

where L� ¼ supt2Rð��Þ L�ðtÞ.

3.1. The convex case

In this section, we are concerned with the existence of solutions for the problem

(1)–(2) when the right-hand side has convex values. Initially, we assume that F is a

compact and convex valued multivalued map. Our first result is based on the

Bohnenblust-Karlin fixed point theorem [22]. We assume the following hypotheses:

(H1) The operator solution S(t)t2J is compact for t> 0, and there is M> 0 such that

kSðtÞkBðEÞ �M, for each t2 J:

(H2) The multivalued map F: J�B!Pcp,cv(E ) is Caratheodory.(H3) For each r> 0, there exists a positive function hr2L

1(J; Rþ) such that

kFðt, zÞk :¼ supfkvk : v2Fðt, zÞg � hrðtÞ a.e. t2 J,

and for z2B with kzkB � r; and

lim infr!1

1

r

Z b

0

hrðsÞds �1

M:

(H4) For each t2 J and �> 0 the set {F(t,�): k�kB � �} is relatively compact in E.

THEOREM 3.4 Assume that (H�), (H1)–(H4) hold. Then the problem (1)–(2) has a

mild solution on (�1,b].

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Proof We will transform the problem (1)–(2) into a fixed point problem. To this

end we set

B ¼ y : ð�1, b� ! E : yjJ is continuous and y0 2B� �

,

where yjJ is the restriction of y to the real compact interval J. We transform the

problem (1)–(2) into a fixed-point problem. Consider the multivalued operator N:

B!P(B) defined by N(h)¼ {h2B} with

hðtÞ ¼

�ðtÞ, if t � 0;

SðtÞ �ð0Þ þ

Z t

0

Sðt� sÞvðsÞds, v2SF,y�ðs,ys Þ , if t2 J:

8<: ð4Þ

For �2B, we will define the function x(�): (�1, b]!E by

xðtÞ ¼�ðtÞ, if t � 0;

SðtÞ �ð0Þ, if t2 J:

(Then x0¼�. For each function z2B with z0¼ 0, we denote by z the function

defined by

zðtÞ ¼0, if t � 0;zðtÞ, if t2 J:

�If y(�) satisfies (3), we can decompose it as yðtÞ ¼ zðtÞ þ xðtÞ, which implies

yt¼ ztþ xt, for every t2 J, and the function z(�) satisfies

zðtÞ ¼

Z t

0

Sðt� sÞvðsÞds t2 J,

where vðsÞ 2SF,z�ðs,zsþxs Þþx�ðs,zsþxs Þ .Let

B0b ¼ z2B : z0 ¼ 02Bf g:

For any z2B0b we have

kzkb ¼ kz0kB þ supfjzðsÞj : 0 � s5 bg ¼ supfjzðsÞj : 0 � s5 bg:

Thus ðB0b, k�kbÞ is a Banach space. We define the operator P: B0

b! PðB0bÞ by:

PðzÞ :¼ fh2B0bg with

hðtÞ ¼

Z t

0

Sðt� sÞvðsÞds, vðsÞ 2SF,z�ðs,zsþxs Þþx�ðs,zsþxs Þ, t2 J: ð5Þ

Obviously the operator N has a fixed point is equivalent to P has one, so it turns to

prove that P has a fixed point. We shall show that the operators P satisfy all

conditions of Lemma 2.12. For better readability, we break the proof into a sequence

of steps.Let

Kr ¼ fz2B0b : zð0Þ ¼ 0, kzkb � rg,

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where r is the constant given by (H3). It is clear that Kr is a closed, convex, bounded

set in B0b.

Step 1 P(z) is convex for each z2B0b.

Indeed, if h1 and h2 belong to P(z), then there exist v1, v2 2SF,z�ðs,zsþxs Þþx�ðs,zsþxs Þsuch

that, for t2 J, we have

hiðtÞ ¼

Z t

0

Sðt� sÞviðsÞds, i ¼ 1, 2:

Let 0� d� 1. Then, for each t2 J, we have

ðdh1 þ ð1� d Þh2ÞðtÞ ¼

Z t

0

Sðt� sÞ½dv1ðsÞ þ ð1� d Þv2ðsÞ�ds:

Since SF,z�ðs,zsþxs Þþx�ðs,zsþxsÞis convex (because F has convex values), we have

dh1 þ ð1� d Þh2 2PðzÞ:

Step 2 P(Kr)�Kr.

If it is not true, then for each positive number r, there exists a function zr2Kr and

h2P(zr) such that kh(t)k> r for some t2 J. Then we have

r5 jhðtÞj �Z t

0

jSðt� sÞjBðEÞjvðsÞjds

�M

Z b

0

hrðsÞds:

Dividing both sides by r and taking the lower limit and from (H3), we get

15 limr!1

M inf1

r

Z b

0

hrðsÞds

� 1,

which yields to a contradiction. Hence there exists a positive number r0 such that

PðKr0 Þ � Kr0 :

Step 3 PðKr0 Þ is relatively compact.

Since Kr0 is bounded and PðKr0 Þ � Kr0 , it is clear that PðKr0 Þ is bounded. It remains

to show that PðKr0 Þ is equicontinuous.Let �1, �22 J with �2>�1 and z2PðKr0 Þ. Then

jhð�2Þ � hð�1Þj ���� Z �1

0

½Sð�2 � sÞ � Sð�1 � sÞ�vðsÞds���

þ

��� Z �2

�1

Sð�2 � sÞjvðsÞjds���

�

Z �1

0

jSð�2 � sÞ � Sð�1 � sÞjhr0 ðsÞds

þ

Z �2

�1

jSð�2 � sÞjhr0 ðsÞds:

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The right-hand side of the above inequality tends to zero independently of z2Kr0 as

�2� �1! 0, since S(t) is uniformly continuous. As a consequence of Steps 1–3, (H4)

together with the Arzela–Ascoli theorem the operator P is completely continuous.

Step 4 P has a closed graph.

Let zn! z, hn2P(zn), and hn! h. We need to show that h 2P(z). Now,

hn2P(zn) implies there exists vn 2SF,zn�ðs,znsþxsÞþx�ðs,znsþxsÞsuch that, for each t2 J,

hnðtÞ ¼

Z t

0

Sðt� sÞvnðsÞds:

We must show that there exists v 2SF,z�ðs,zsþxsÞþx�ðs,zsþxsÞsuch that for each t2 J,

hðtÞ ¼

Z t

0

Sðt� sÞvðsÞds:

Consider the linear continuous operator �: L1ðJ,E Þ �!B0b defined by

ð�vÞðtÞ ¼

Z t

0

Sðt� sÞvðsÞds:

From the definition of �, we know that

hnðtÞ 2�ðSF,zn�ðs,znsþxs Þþx�ðs,znsþxsÞ Þ:

Since zn! z and � �SP is a closed graph operator by Lemma 2.11, then there exists

v 2SF,z�ðs,zsþxs Þþx�ðs,zsþxs Þ such that

hðtÞ ¼

Z t

0

Sðt� sÞvðsÞds:

Hence h 2P(z),As a consequence of Lemma 2.12, we deduce that P has a fixed point z on the

interval (�1, b], so y ¼ zþ x is a fixed point of the operator N which is the mild

solution of problem (1)–(2).

THEOREM 3.5 Assume that (H1), (H2), (H4) are satisfied then a slight modification of

the proof (i.e. use the usual Leray–Schauder alternative) guarantees that (H3) could be

replaced by

(H3) There exists a function p2L1(J; Rþ) and a continuous nondecreasing function

F: Rþ! (0,1) such that

jFðt, uÞj � pðtÞ ðkukBÞ for a.e. t2 J and each u2B,

with Z þ1cb

ds

ðsÞ4KbM

Z b

0

pðsÞds, ð6Þ

where

cb ¼ ðMb þ L� þ KbMHÞk�kB:

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Let z be solutions of the inclusion z2 �P(z), for some �2 (0, 1), then there exists

v2SF,z�ðs,zsþxsÞþx�ðs,zsþxs Þsuch that

jzðtÞj �

Z t

0

kSðt� sÞkBðEÞ jjFðs, z�ðs,zsþxsÞ þ x�ðs,zsþxsÞÞjds

�M

Z t

0

pðsÞ kz�ðs,zsþxsÞ þ x�ðs,zsþxsÞkB� �

ds

�M

Z t

0

pðsÞ KbjzðsÞj þ ðMb þ L� þ KbMHÞk�kB� �

ds:

Set cb :¼ (MbþL�þKb MH)k�kB. Then, we have

jzðtÞj �M

Z t

0

pðsÞ KbjzðsÞj þ cbð Þds:

Then

KbjzðtÞj þ cb � cb þ KbM

Z t

0

pðsÞ KbjzðsÞj þ cbð Þds:

We consider the function defined by

ðtÞ :¼ supfKbjzðsÞj þ cb: 0 � s � t g, t2 J:

Let t�2 [0, t] be such that (t)¼Kbjz(t�)j þ cbk�kB. By the previous inequality,

we have

ðtÞ � cb þ KbM

Z t

0

pðsÞ ððsÞÞds for t2 J:

Let us take the right-hand side of the above inequality as v(t). Then, we have

ðtÞ � vðtÞ for all t2 J:

From the definition of v, we have

vð0Þ ¼ cb and v0ðtÞ ¼ KbMpðtÞ ððtÞÞ a.e. t2 J:

Using the nondecreasing character of , we get

v0ðtÞ � KbM pðtÞ ðvðtÞÞ a.e. t2 J:

Using the condition (6), this implies that for each t2 J we haveZ vðtÞ

cb

ds

ðsÞ� KbM

Z t

0

pðsÞds � KbM

Z b

0

pðsÞds5Z þ1cb

ds

ðsÞ:

Thus, for every t2 J, there exists a constant � such that v(t)�� and hence (t)��.

Since kzkb�(t), we have kzkb��. Set

U ¼ fz2B0b : kzk15�þ 1g:

As in Theorem 3.4, the operator P: U! PðZÞ is completely continuous. From the

choice of U, there is no z2 @U such that z¼ �P(z), for �2 (0, 1). As a consequence of

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the nonlinear alternative of Leray–Schauder type [23], we deduce that F has a fixedpoint z in U, then the problem (1)–(2) has at least one mild solution on (�1, b].

3.2. The nonconvex case

This section is devoted to proving the existence of solutions for (1)–(2) with anonconvex valued right-hand side. Our result is based on the fixed point theorem forcontraction multivalued maps given by Covitz and Nadler [24].

THEOREM 3.6 Assume that the following hypotheses hold:

(H5) F: J�B!Pcp(E ) has the property that F(�, u): J!Pcp(E ) is measurable,for each u2B.

(H6) There exists l2L1(J,Rþ) such that

Hd ðFðt, uÞ,Fðt, uÞÞ � l ðtÞku� uk for every u, u2B,

and

d ð0,Fðt, 0ÞÞ � l ðtÞ a.e. t2 J:

Then (1)–(2) has at least one mild solution on (�1, b].

Remark 3.7 For each z2B0b, the set SF,z is nonempty since, by (H5), F has a

measurable selection [41, Theorem III.6].

Proof Let P: B0b! PðB

0bÞ where P is defined in Theorem 3.4 be solutions of the

problem (1)–(2). We shall show that P satisfies the assumptions of Lemma 2.13.The proof will be given in two steps.

Step 1 PðzÞ 2PclðB0bÞ for all z2B

0b.

Let (zn)n�02P(z) be such that zn! ~z2B0b. Then there exists

vn 2SF,z�ðs,zsþxs Þþx�ðs,zsþxs Þsuch that, for each t2 J,

znðtÞ ¼

Z t

0

Sðt� sÞvnðsÞds:

Using the fact that F has compact values, we may pass to a subsequence if necessaryto obtain that vn converges to v in L1(J,E ) and hence v2SF,z�ðs,zsþxs Þþx�ðs,zsþxs Þ

. Thus, foreach t2 J,

znðtÞ ! ~zðtÞ ¼

Z t

0

Sðt� sÞvðsÞds,

so ~z2PðzÞ.

Step 2 There exists � < 1 such that

Hd ðFðzÞ,FðzÞÞ � �kz� zk1 for all z, z 2B0b:

Let z, z 2B0b and h2P(z). Then, there exists vðtÞ 2Fðt, z�ðs,zsþxsÞ þ x�ðs,zsþxsÞÞ such

that, for each t2 J,

hðtÞ ¼

Z t

0

Sðt� sÞvðsÞds:

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From (H6) it follows that

Hd ðFðt, z�ðs,zsþxsÞ þ x�ðs,zsþxsÞÞ,Fðt, z�ðs,zsþxsÞ þ x�ðs,zsþxsÞÞ � l ðtÞjzðtÞ � zðtÞj:

Hence, there exists w2Fðt, z�ðs,zsþxsÞ þ x�ðs,zsþxsÞÞ such that

jvðtÞ � wj � l ðtÞjzðtÞ � zðtÞj, t2 J:

Consider U: J!P(E ) given by

UðtÞ ¼ fw2E : jvðtÞ � wj � l ðtÞjzðtÞ � zðtÞjg:

Since the multivalued operator VðtÞ ¼ UðtÞ \ Fðt, z�ðs,zsþxsÞ þ x�ðs,zsþxsÞÞÞ is measur-

able [41, Proposition III.4], there exists a function v2(t) which is a measurable

selection for V. Thus, vðtÞ 2Fðt, z�ðs,zsþxsÞ þ x�ðs,zsþxsÞÞ, and for each t2 J,

jvðtÞ � vðtÞj � l ðtÞjzðtÞ � zðtÞj:

For each t2 J, define

hðtÞ ¼

Z t

0

Sðt� sÞvðsÞds:

Then, for t2 J,

jhðtÞ � hðtÞj �

Z t

0

jjSðs� tÞjjBðEÞjvðsÞ � vðsÞjds

�M

Z t

0

l ðsÞkz� zk

�

Z t

0

lðsÞkz� zkds

�1

�e�LðtÞkz� zkB,

where � > 1, LðtÞ ¼R t0 Ml ðsÞds and k:kB is the Bielecki-type norm on B0

b defined by

kzkB ¼ supfe��LðtÞkzðtÞk : t2 Jg:

Therefore,

kh� hkB �1

�kz� zkB:

By an analogous relation, obtained by interchanging the roles of z and z,

it follows that

Hd ðPðzÞ,PðzÞÞ �1

�kz� zkB:

So, P is a contraction, and thus, by Lemma 2.13 it has a fixed point z, which is a mild

solution to (1)–(2).

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4. An example

To apply our abstract results, we consider the fractional differential equation with

state dependent delay of the form

@u

@tðt, Þ �

1

�ð� 1Þ

Z 0

t

ðt� sÞ�2Luðs, Þds

2 ½ f1ðt, uðt� �ðuðt, 0ÞÞ, ÞÞ, f2ðt, uðt� �ðuðt, 0ÞÞ, ÞÞ�, t2 J :¼ ½0, b�, 2 ½0,��,

uðt, 0Þ ¼ uðt,�Þ ¼ 0, t2 J,

uð�, Þ ¼ u0ð�, Þ, � 2 ð�1, 0�, 2 ½0,��,

8>>>>>>>><>>>>>>>>:ð7Þ

where 1<< 2, � 2C(IR, [0,1)), � > 0, L stands for the operator with respect to

the spatial variable which is given by

L ¼@2

@2� r, with r4 0,

f1, f2: J�B! IR are measurable in t and continuous in y. We assume that for each

t2 J, f1(t, �) is lower semicontinuous (i.e. the set {y2B: f1(t, y)> } is open for all

2 IR), and assume that for each t2 J, f2(t, �) is u.s.c. (i.e. the set {y2B: f2(t, y)< } isopen for each 2 IR).

Take E¼L2([0,�],IR) and the operator A :¼L: D(A)�E!E with domain

DðAÞ :¼ fu2E: u00 2E, uð0Þ ¼ uð�Þ ¼ 0g:

Clearly A is densely defined in E and is sectorial. Hence A is a generator of a solution

operator on E. For the phase space, we choose B¼B� defined by

B� ¼ �2Cðð�1, 0�, IRÞ : lim�!�1

e���ð�Þ exists

� �with the norm

k�k� ¼ sup� 2 ð�1,0�

e��j�ð�Þj:

Notice that the phase space B� satisfies axioms (A1), (A2) and (A3) (see [27] for more

details). Set

yðtÞðÞ ¼ uðt, Þ, t2 ½0, b�, 2 ½0,��,

�ð�ÞðÞ ¼ u0ð�, Þ, t2 ½0, b�, � � 0,

Fðt, ’ÞðÞ ¼ ½ f1ðt, ’ð0, ÞÞ, f2ðt, ’ð0, ÞÞ�, t2 ½0, b�, 2 ½0,��,

�ðt, ’Þ ¼ t� �ð’ð0, 0ÞÞ:

The mutivalued map F is u.s.c. with compact convex values [39]. Hence (H1) and

(H2) are satisfied.An application of Theorem 3.5 yields the following result.

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THEOREM 4.1 Let �2B� be such that (H�) holds, and let t!�t be continuous onR(��). Moreover we assume that (H3) is satisfied. Then there exists at least one mildsolution of (7).

COROLLARY 4.2 Let �2B� be continuous and bounded. Assume that (H3) holds.Then there exists at least one mild solution of (7) on (�1, b].

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