Nonlocal impulsive fractional differential inclusions ... - Semantic Scholar

Applied Mathematics and Computation 257 (2015) 103–118

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Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces q JinRong Wang a,b, Ahmed Gamal Ibrahim c, Michal Fecˇkan d,e,⇑ a

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, PR China School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, PR China c Department of Mathematics, Faculty of Science, King Faisal University, Al-Ahasa 31982, Saudi Arabia d Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia e Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia b

a r t i c l e

i n f o

Keywords: Impulsive fractional differential inclusions Fractional sectorial operators Nonlocal conditions Mild solutions

a b s t r a c t This paper investigates existence of PC-mild solutions of impulsive fractional differential inclusions with nonlocal conditions when the linear part is a fractional sectorial operators like in Bajlekova (2001) [1] on Banach spaces. We derive two existence results of PC-mild solutions when the values of the semilinear term F is convex as well as another existence result when its values are nonconvex. Further, the compactness of the set of solutions is characterized. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Fractional differential equations and fractional differential inclusions arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology, etc., involves derivatives of fractional order. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. For some applications of fractional differential equations, one can see [26,28,33–36,40] and the references therein. It seems that El-Sayed and Ibrahim [22] initiated the study of fractional multivalued differential inclusions. Recently, some basic theory for initial value problems for fractional differential equations and inclusions was discussed by Kilbas et al. [33], Lakshimkantham et al. [34], Miller et al. [36], Podlubny [40] and the papers [1–3,5,12,18,23,27,35,42–49] and the references therein. The theory of impulsive differential equations and impulsive differential inclusions has been an object interest 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 be described using the classical differential problems. For some of these applications we refer to [6,10]. During the last ten years, impulsive differential inclusions with different conditions

q The first author acknowledges the support by National Natural Science Foundation of China (11201091), Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169) and Doctor Project of Guizhou Normal College (13BS010) and Guizhou Province Education Planning Project (2013A062). The third author acknowledges the support by Grants VEGA-MS 1/0071/14 and VEGA-SAV 2/0029/13. ⇑ Corresponding author at: Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia. E-mail addresses: [email protected] (J. Wang), [email protected] (A.G. Ibrahim), [email protected] (M. Fecˇkan).

http://dx.doi.org/10.1016/j.amc.2014.04.093 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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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. [13]. 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 inverse heat condition problems. For the applications of nonlocal conditions problems we refer to [11,25]. 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 [7]. For impulsive differential equation or inclusions with nonlocal conditions of order one we refer to [16,24]. For impulsive differential equation or inclusions of fractional order we refer to [4,21,41,45] and the references therein. In this paper we are concerned with the existence of mild solutions to the following nonlocal impulsive fractional differential inclusions of the type

8c a > < D xðtÞ 2 AxðtÞ þ Fðt; xðtÞÞ; a 2 ð0; 1Þ; a:e: on J  ft 1 ; t2 ; . . . ; tm g; xð0Þ ¼ x0  gðxÞ; > : þ xðt i Þ ¼ xðti Þ þ Ii ðxðti ÞÞ; i ¼ 1; 2; . . . ; m;

ð1Þ

where J :¼ ½0; b with b > 0 is fixed, c Da is the Caputo fractional derivative of the order a 2 ð0; 1Þ with the lower limit zero, A is a fractional sectorial operator like in [1] defined on a separable Banach space E, F : J  E ! 2E  f/g is a multifunction, 0 ¼ t0 < t1 <    < t m < tmþ1 ¼ b; Ii : E ! E ði ¼ 1; 2; . . . ; mÞ are impulsive functions which characterize the jump of the solutions at impulse points t i , g : PCðJ; EÞ!E; is a nonlinear function related to the nonlocal condition at the origin and xðtþ i Þ, xðt i Þ are the right and left limits of x at the point t i respectively and PCðJ; EÞ will be define later. Concerning with the main problem (1), we have to study the following impulsive fractional evolution equations with nonlocal conditions:

8c a > < D xðtÞ ¼ AxðtÞ þ f ðt; xðtÞÞ; a 2 ð0; 1Þ; a:e: on J  ft 1 ; t 2 ; . . . ; t m g; xð0Þ ¼ x0  gðxÞ; > : þ xðt i Þ ¼ xðti Þ þ Ii ðxðti ÞÞ; i ¼ 1; 2; . . . ; m:

ð2Þ

In [44], Wang et al. introduced a new concept of mild solutions for (2) and derived existence and uniqueness results concerning the PC-mild solutions for (2) when f is a Lipschitz single-valued function or continuous and maps bounded sets into bounded sets and A is the infinitesimal generator of a compact semigroup fTðtÞ; t P 0g. After reviewing the previous research on the fractional evolution equations, we find that the operator in the linear part is the infinitesimal generator of a strongly continuous semigroup, an analytic semigroup, or compact semigroup, or a Hille– Yosida operator, much less is known about the fractional evolution (differential) inclusions with sectorial or almost sectorial operators. In order to do a comparison between our obtained results in this paper and the known recent results in the same topic, we would like to mention that, recently, the study of evolution equations involving sectorial or almost sectorial operators has been investigated to a large extent. For example, Shu et al. [42] introduced a new concept of mild solutions for impulsive fractional evolution equations and derived existence results concerning the mild solutions for (2) when F is a completely continuous single-valued function, g ¼ 0 and A is a sectorial operator such that the operators families fSa ðtÞ; t P 0g and fT a ðtÞ; t P 0g are compact. We will explain in Remark 2.21 why does the definition given in [42] not suitable in some sense. So, we will give another definition for PC-mild solutions for (1) based on the definition given by Wang et al. [44]. Periago and Straub [39] gave a functional calculus for almost sectorial operator, and using the semigroup of growth 1 þ c which is defined by this functional calculus, obtained the existence and uniqueness for Cauchy problems of abstract evolution equations involving almost sectorial operator, that is by constructing an evolution process of growth 1 þ c. More recently, Wang et al. [46] considered abstract fractional Cauchy problem when F is a single valued, g ¼ 0 and A is an almost sectorial operator whose resolvent satisfies the estimate of growth c ð1 < c < 0Þ in a sector of the complex plane. Agarwal et al. [3] proved an existence result for (1) without impulses and when A is a sectorial operator and the dimension of E is finite. They studied the dimension of the set of mild solutions. Ouahab [38,11] proved a version of Fillippov’s Theorem for (1) without impulse, g ¼ 0 and A is an almost sectorial operator. The study of differential equations or evolution equations in which the linear part is the infinitesimal generator of C 0 -semigroup has been investigated by many authors. Cardinali and Rubbioni [16] proved the existence of mild solutions to (1) when a ¼ 1 and the multivalued function F satisfies the lower Scorza-Dragoni property and fAðtÞ; t P 0g is a family of linear operator, generating a strongly continuous evolution operators. Henderson and Ouahab [29] considered the problem (1) when A ¼ 0, and Zhou et al. [47,48] introduced a suitable definition of mild solution for (1) based on Laplace transformation and probability density functions for (1) without impulses when A is the infinitesimal generator of C 0 -semigroup, F is single-valued function. Very recently, Wang and Ibrahim [45] proved existence and controllability results for (1) when A is the infinitesimal generator of C 0 -semigroup and fTðtÞ; t > 0g is strongly equicontinuous C 0 -semigroup. In addition, Ibrahim et al. [27] proved the existence of mild solutions to the problem (1) when the multivalued function F satisfies the lower Scorza-Dragoni property and A is the infinitesimal generator of a compact semigroup fTðtÞ; t > 0g.

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In this paper, motivated by the works mentioned above, we derive two existence results of PC-mild solutions for (1) when the values of the semilinear term F are convex as well as nonconvex and linear term A is a fractional sectorial operator like in [1]. 2. Preliminaries and notation Let PCðJ; EÞ be the space of E-valued bounded functions on J with the uniform norm kxk ¼ sup fkxðtÞk; t 2 J g such that 1 xðt þ i Þ exist for any i ¼ 0; . . . ; m and xðtÞ is continuous on J i ; i ¼ 0; . . . ; m, where J i ¼ ðt i ; t iþ1  and t 0 ¼ 0, t mþ1 ¼ b. L ðJ; EÞ be Rb the space of E-valued Bochner integrable functions on J with the norm kf kL1 ðJ;EÞ ¼ 0 kf ðtÞkdt; PðEÞ ¼ fB # E : B is nonemptyg,

P k ðEÞ ¼ fB # E : B is nonempty and compactg, P cl ðEÞ ¼ fB # E : B is nonempty and closedg, P clb ðEÞ ¼ fB # E : B is nonempty, closed and boundedg, Pck ðEÞ ¼ fB # E : B is nonempty, convex and compactg, conv ðBÞ (respectively, conv ðBÞÞ be the convex hull (respectively, convex closed hull in E) of a subset B in E. As usual, for a linear operator A, we denote by DðAÞ the domain of A, by rðAÞ its spectrum, while qðAÞ ¼ C  rðAÞ is the resolvent set of A, and denote by the family Rðz; AÞ ¼ ðzI  AÞ1 ; z 2 qðAÞ of bounded linear operators the resolvent operators of A. Moreover, we denote by LðZ; YÞ the space of all bounded linear operators between two normed spaces Z and Y and its domain is Z. When Y ¼ Z we write LðZÞ. Definition 2.1 ([8,17,30,31]). Let X and Y be two topological spaces. A multifunction G : X ! PðYÞ is said to be upper semicontinuous if

G1 ðVÞ ¼ fx 2 X : GðxÞ # V g is an open subset of X for every open V # Y. G is called closed if its graph CG ¼ fðx; yÞ 2 X  Y : y 2 GðxÞg is closed subset of the topological space X  Y. Gis said to be completely continuous if GðBÞ is relatively compact for every bounded subset B of X. If the multifunction G is completely continuous with non empty compact values, then G is u:s:c. if and only if G is closed. If G is a normed space, then the set

S1G ¼ ff 2 L1 ðJ; EÞ : f ðtÞ 2 GðtÞ;

for a:e: t 2 Jg

is called the set of selections of G. Lemma 2.2 [32, Theorem 1.3.5]. Let X 0 ; X be (not necessarily separable) Banach spaces, and let F : J  X 0 ! P k ðXÞ be such that (i) for every x 2 X 0 the multifunction Fð; xÞ has a strongly measurable selection; (ii) for a.e. t 2 J the multifunction Fðt; Þ is upper semicontinuous. Then for every strongly measurable function z : J ! X 0 there exists a strongly measurable function f : J ! X such that f ðtÞ 2 Fðt; zðtÞÞ a.e. t 2 J. Remark 2.3 ([32, Theorem 1.3.1]). For single-valued or compact-valued multifunctions acting on a separable Banach space the notions measurability and strongly measurable coincide. So, if X 0 ; X are separable Banach spaces we can replace strongly measurable with measurable in the above lemma, and by Theorem 1.3.4 of [32], for every measurable multifunction G : J ! P k ðX 0 Þ the multifunction X : J ! P k ðXÞ; XðtÞ ¼ Fðt; GðtÞÞ is measurable. Lemma 2.4 ([17], Generalized Cantor’s intersection). If fBn gnP1 is a decreasing sequence of nonempty closed subsets of E and T limn!1 vðBn Þ ¼ 0, then 1 n¼1 Bn is nonempty and compact. Definition 2.5 ([8]). A sequence ffn : n 2 Ng L1 ðJ; EÞ is said to be semicompact if (i) It is integrable bounded, i.e., there is q 2 L1 ðJ; Rþ Þ such that kfn ðtÞk 6 qðtÞ a.e. t 2 J. (ii) The set ffn ðtÞ : n 2 Ngis relatively compact in E a.e. t 2 J. We need the following simple result related to [8, Theorem 1.1.4]. Lemma 2.6. Let fZ n gnP1 be a sequence of subsets of E. Suppose there is a compact and convex subset Z  X such that for any T S neighborhood U of Z there is an n so that for any m P n : Z m  U. Then N>0 conv ð nPN Z n Þ  Z. We recall one fundamental result which follows from Dunford–Pettis Theorem. Lemma 2.7 ([32]). Every semicompact sequence in L1 ðJ; EÞ is weakly compact in L1 ðJ; EÞ. For more about multifunctions we refer to [20,30–34].

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Lemma 2.8 ([16]). Let v be the Hausdorff measure of noncompactness on E. If ðBn ÞnP1 is a decreasing sequence of nonempty T closed subsets of E and limn!1 vðBn Þ ¼ 0, then 1 n ¼1 Bn is nonempty and compact. Lemma 2.9 ([14]). Let B be a bounded set in E. Then for every

e > 0 there is a sequence ðxn ÞnP1 in B such that

vðBÞ 6 2vfxn : n P 1g þ e: Lemma 2.10 ([37]). Let t 2 J,

vCðJ;EÞ be the Hausdorff measure of noncompactness on CðJ; EÞ. If W # CðJ; EÞ is bounded, then for every

vðWðtÞÞ 6 vCðJ;EÞ ðWÞ; where WðtÞ ¼ fxðtÞ : x 2 Wg. Furthermore, if W is equicontinuous on J, then the map t ! vfxðtÞ : x 2 Wg is continuous on J and

vCðJ;EÞ ðWÞ ¼ sup vfxðtÞ : x 2 Wg: t2J

Lemma 2.11 ([9, Lemma 4]). Let ffn : n 2 Ng  Lp ðJ; EÞ; p P 1 be an integrable bounded sequence such that

vffn : n P 1g 6 cðtÞ; a:e: t 2 J; where c 2 L1 ðJ; Rþ Þ. Then for each  > 0 there exists a compact K  # E, a measurable set J   J, with measure less than , and a sequence of functions fg n g  Lp ðJ; EÞ such that

fg n ðtÞ : n P 1g # K  ;

t2J

and

kfn ðtÞ  g n ðtÞk < 2cðtÞ þ ;

for every n P 1 and every t 2 J  J  :

Definition 2.12 ([33]). The fractional integral of order c with the lower limit zero for a function f 2 L1 ðJ; EÞ is defined as

Ict f ðtÞ ¼

1 CðcÞ

Z

t

0

f ðsÞ ðt  sÞ1c

ds;

t > 0; c > 0

provided the right side is point-wise defined on ½0; 1Þ, where CðÞ is the gamma function. Definition 2.13 ([33]). The Riemann–Liouville derivative of order c with the lower limit zero for a function f 2 L1 ðJ; EÞ can be written as n

Dct f ðtÞ ¼

1 d Cðn  cÞ dtn

Z

t 0

f ðsÞ ðt  sÞcþ1n

ds;

t > 0; n  1 < c < n:

Definition 2.14 ([33]). The Caputo derivative of order c for a function f 2 L1 ðJ; EÞ can be written as c

Dct f ðtÞ ¼ Dct ðf ðtÞ  f ð0ÞÞ;

t > 0; 0 < c < 1:

For further readings and details on fractional calculus, we refer to the books and papers by Kilbas et al. [33] and Podlubny [40]. Next, we are ready to recall some facts of fractional Cauchy problem. Bajlekova [1] studied the following linear fractional Cauchy problem

c

Da xðtÞ ¼ AxðtÞ;

t P 0;

xð0Þ ¼ x0 2 E;

ð3Þ

where A is linear closed and DðAÞ is dense. Definition 2.15 (see [1, Definition 2.3]). A family fSa ðtÞ : t P 0g  LðEÞ is called a solution operator for (3) if the following conditions are satisfied: (a) Sa ðtÞ is strongly continuous for t P 0 and Sa ð0Þ ¼ I; (b) Sa ðtÞDðAÞ  DðAÞ and ASa ðtÞx ¼ Sa ðtÞAx for all x 2 DðAÞ and t P 0; (c) Sa ðtÞx is a solution of (3) for all x 2 DðAÞ and t P 0.

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Definition 2.16 (see [1, Definition 2.4]). An operator A is said to be belong to ea ðM; xÞ if the solution operator Sa ðÞ of (3) satisfies

kSa ðtÞkLðEÞ 6 Mext ;

tP0

for some constants M P 1 and x P 0. Definition 2.17 (see [1, Definition 2.13]). A solution operator Sa ðtÞ of (3) is called analytic if it admits an analytic extension to a sector Rh0 ¼ fk 2 C  f0g : jarg kj < h0 g for some h0 2 ð0; p2 . An analytic solution operator is said to be of analyticity type ðh0 ; x0 Þ if for each h < h0 and x > x0 there is an M ¼ Mðh; xÞ such that

kSa ðtÞkLðEÞ 6 MexRt ;

t 2 Rh :

Set

ea ðxÞ :¼

[ [ fea ðM; xÞ : M P 1g and ea :¼ fea ðxÞ : x P 0g;

Aa ðh0 ; x0 Þ ¼ fA 2 ea : A generates an analytic solution operator Sa of type ðh0 ; x0 Þg: Remark 2.18 ([1, Theorem 2.14]). Let a 2 ð0; 2Þ. A linear closed densely defined operator A belongs to Aa ðh0 ; x0 Þ if and only if ka 2 qðAÞ for each k 2 Rh0 þp2 ðw0 Þ ¼ fk 2 C  f0g : jargðk  w0 Þj < h0 þ p2 g and for any x > x0 ; h < h0 there is a constant C ¼ Cðh; xÞ such that

kka1 Rðka ; AÞkLðEÞ 6

C jk  xj

for k 2 Rh0 þp2 ðwÞ. According to the proof of Theorem 2.14 in [1], if A 2 Aa ðh0 ; w0 Þ for some h0 2 ð0; pÞ and w0 2 R, the solution operator for the Eq. (3) is given by

Sa ðtÞ ¼

1 2pi

Z

ekt ka1 Rðka ; AÞdk

ð4Þ

C

for a suitable path C. Next following [1,3,42], a mild solution of the Cauchy problem

c

Da xðtÞ ¼ AxðtÞ þ f ðtÞ; xð0Þ ¼ x0 2 E

t 2 J;

can be defined by

uðtÞ ¼ Sa ðtÞx0 þ

Z

t

T a ðt  sÞf ðsÞds;

0

where

T a ðtÞ ¼

1 2pi

Z

ekt Rðka ; AÞdk

C

for a suitable path C and f : J ! E is continuous. We need the following estimates Lemma 2.19 ([1, (2.26)], [42]). If A 2 Aa ðh0 ; x0 Þ then

kSa ðtÞkLðEÞ 6 Mext and kT a ðtÞkLðEÞ 6 Cext ð1 þ t a1 Þ for every t > 0; x > x0 . So putting

MS :¼ sup kSa ðtÞkLðEÞ ; 06t6b

MT :¼ sup Cext ð1 þ t 1a Þ; 06t6b

we get

kSa ðtÞkLðEÞ 6 M S ;

kT a ðtÞkLðEÞ 6 ta1 MT :

Based on the above consideration (see also [44]), we introduce the definition of mild solution for (1). Definition 2.20. Let A 2 Aa ðh0 ; x0 Þ with h0 2 ð0; p2 and x0 2 R. A function x 2 PCðJ; EÞ is called a mild solution of (1) if

ð5Þ

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J. Wang et al. / Applied Mathematics and Computation 257 (2015) 103–118

8 Rt > Sa ðtÞðx0  gðxÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > > Rt > > > S ðtÞðx0  gðxÞÞ þ Sa ðt  t 1 ÞI1 ððt 1 Þ þ 0 T a ðt  sÞf ðsÞds; t 2 J1 ; > < a xðtÞ ¼ .. . > > > > i¼m X > Rt > > Sa ðt  t i ÞIi ðxðti ÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 Jm ; > Sa ðtÞðx0  gðxÞÞ þ :

ð6Þ

i¼1

where f 2

S1Fð;xðÞÞ .

Remark 2.21. (i) In Definition 2.2 of [42], Shu et al. introduced the following definition of mild solutions for (1) when F is a single valued function and g ¼ 0

xðtÞ ¼

8 Rt > Sa ðtÞx0 þ 0 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > > R > > < Sa ðt  t1 Þðxðt1 Þ þ I1 ððt 1 ÞÞ þ tt T a ðt  sÞf ðsÞds; 1 .. > > > . > > > : S ðt  t Þðxðt Þ þ I ððt  ÞÞ þ R t T ðt  sÞf ðsÞds; a m m m m tm a

t 2 J1 ;

ð7Þ

t 2 Jm ;

where f 2 S1Fð;xðÞÞ . (ii) Our Definition 2.20 is more suitable than the definition given by Shu et al. [42]. In fact, if a ¼ 1 then (6) reduces to

8 Rt > TðtÞðx0  gðxÞÞ þ 0 Tðt  sÞf ðsÞds; t 2 J 0 ; > > > Rt >  > > > TðtÞðx0  gðxÞÞ þ Tðt  t 1 ÞI1 ððt 1 Þ þ 0 Tðt  sÞf ðsÞds; t 2 J 1 ; < xðtÞ ¼ .. > >. > > i¼m X > Rt > > TðtÞðx0  gðxÞÞ þ Tðt  ti ÞIi ðxðti ÞÞ þ 0 Tðt  sÞf ðsÞds; t 2 J m ; > : i¼1

which is the standard formula of PC-mild solutions of

8 0 > < x ðtÞ 2 AxðtÞ þ Fðt; xðtÞÞ; a:e: on J  ft 1 ; t 2 ; . . . ; t m g; xð0Þ ¼ x0  gðxÞ; > : þ xðt i Þ ¼ xðti Þ þ Ii ðxðti ÞÞ; i ¼ 1; 2; . . . ; m: (iii) There are also other concepts of solutions for impulsive fractional differential equations, see [50–53]. We are now aware according to the paper [54] that the definition of solutions of impulsive Caputo fractional equations (ICFE) is questionable. We tried to explain our attitude in [55]. It is now clear that the definition of solutions of ICFE is not so simple as for natural-order differential equations with impulses. Our approach is based on the fact the lower limit in the Caputo derivative is given, so it is fixed. This means that a family of solutions is set at the beginning by the Cauchy initial conditions. Then at each impulses the solution is kicked by the impulse on one of these solutions. This was first demonstrated in derivation of formula (10) in [56, Lemma 2.7] for non-homogeneous linear Caputo fractional differential equation. Latter this formula is adapted for determining a mild solution for linear Caputo fractional abstract differential equation with impulses in [44, formula (3.4)] (see also [57, Definition 4.1]). We follow this path also in this paper by introducing Definition 2.20. Summarizing, we do not claim that our definition of solutions for ICFE is the best one, but we tried to follow a similar way as for ordinary differential equations with impulses, since there, for ODE, a solution is kicked by an impulse on a solution with different Cauchy condition at the beginning. The following fixed point theorems are crucial in the proof of our main results. Lemma 2.22 ([32, Corollary 3.3.1]). Let W be a closed convex subset of a Banach space X and R : W ! P ck ðXÞ be a closed multifunction which is v-condensing, where v is a non singular measure of noncompactness defined on subsets of W, then R has a fixed point.

Lemma 2.23 ([32, Proposition 3.5.1]). Let W be a closed subset of a Banach space X and R : W ! Pk ðXÞ be a closed multifunction which is v-condensing on every bounded subset of W, where v is a monotone measure of noncompactness defined on X. If the set of fixed points for R is a bounded subset of X then it is compact.

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109

Lemma 2.24 ([19]). Let ðX; dÞ be a complete metric space. If R : X ! Pclb ðXÞ is contraction, then R has a fixed point. 3. Existence results for convex case In this section, we give some existence results for (1) when A is a sectorial operator. Theorem 3.1. Let A 2 Aa ðh0 ; x0 Þ with h0 2 ð0; p2 and x0 2 R; F : J  E ! P ck ðEÞ a multifunction, g : PCðJ; EÞ ! E and Ii : E ! E ði ¼ 1; 2; . . . ; mÞ. We assume the following conditions: ðHF1Þ For every x 2 E; t!Fðt; xÞ is measurable, for a.e. t 2 J; x!Fðt; xÞ is upper semicontinuous. 1 ðHF2Þ There exists a function u 2 Lq ðJ; Rþ Þ; q 2 ð0; aÞ and a nondecreasing continuous function X : Rþ ! Rþ such that for any x2E

kFðt; xÞk 6 uðtÞXðkxkÞ;

a:e: t 2 J: 1

ðHF3Þ There exists a function b 2 Lq ðJ; Rþ Þ; q 2 ð0; aÞ satisfying

2gMT kbk

1

Lq ðJ;Rþ Þ

aq

where g ¼ -b 1q and

< 1;

ð8Þ

q - ¼ a1q and for every bounded subset D # E,

vðFðt; DÞÞ 6 bðtÞvðDÞ for a.e. t 2 J, where v is the Hausdorff measure of noncompactness in E. ðHgÞ The function g is continuous, compact and there are two positive constants a; d such that

kgðxÞk 6 akxkPCðJ;EÞ þ d;

for all x 2 PCðJ; EÞ:

ðHIÞ For every i ¼ 1; 2; . . . ; m; Ii is continuous and compact and there exists a positive constant hi such that

kIi ðxÞk 6 hi kxk;

x 2 E:

Then the problem (1) has a mild solution provided that there is r > 0 such that

MS ðkx0 k þ ar þ d þ h rÞ þ MT gXðrÞkuk where h ¼

Pi¼m i¼1

1

Lq ðJ;Rþ Þ

6 r;

ð9Þ

hi .

Proof. We turn the problem (1) into fixed point problem and define a multifunction R : PCðJ; EÞ ! 2PCðJ;EÞ as follows: for x 2 PCðJ; EÞ; RðxÞ is the set of all functions y 2 RðxÞ such that

8 Rt > S ðtÞðx0  gðxÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > a > > < .. yðtÞ ¼ . > k¼i X > Rt > > > Sa ðt  t k ÞIk ðxðt k ÞÞ þ 0 T a ðt  sÞf ðsÞds; : Sa ðtÞðx0  gðxÞÞ þ

ð10Þ t 2 J i ; 1 6 i 6 m;

k¼1

where f 2 S1Fð;xðÞÞ . In view of ðHF1Þ the values of R are nonempty. It is easy to see that any fixed point for R is a mild solution for (1). So our goal is to prove, by using Lemma 2.22, that R has a fixed point. The proof will be given in several steps. Step 1. The values of R are closed. Let x 2 PCðJ; EÞ and fyn : n P 1g be a sequence in RðxÞ and converging to y in PCðJ; EÞ. Then, according to the definition of R, there is a sequence ffn : n P 1g in S1Fð;xðÞÞ such that for any t 2 J i ; i ¼ 0; 1; . . . ; m

8 Rt > Sa ðtÞðx0  gðxÞÞ þ 0 T a ðt  sÞfn ðsÞds; t 2 J0 ; > > > > < .. yn ðtÞ ¼ . > k¼i X > Rt > > > Sa ðt  tk ÞIk ðxðtk ÞÞ þ 0 T a ðt  sÞfn ðsÞds; : Sa ðtÞðx0  gðxÞÞ þ k¼1

In view of ðHF2Þ for every n P 1, and for a.e. t 2 J

  kfn ðtÞk 6 uðtÞXðkxðtÞkÞ 6 uðtÞX kxkPCðJ;EÞ :

ð11Þ t 2 J i ; 1 6 i 6 m:

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This show that the set ffn : n P 1g is integrably bounded. Moreover, because ffn ðtÞ : n P 1g  Fðt; xðtÞÞ, for a.e. t 2 J, the set ffn ðtÞ : n P 1g is relativity compact in E for a.e. t 2 J. Therefore, the set ffn : n P 1g is semicompact and then, by Lemma 2.7 it is weakly compact in L1 ðJ; EÞ. So, without loss of generality we can assume that fn converges weakly to a function f 2 L1 ðJ; EÞ. From Mazur’s lemma, for every natural number j there is a natural number k0 ðjÞ > j and a sequence of nonnegative real numPk0 ðjÞ Pk0 ðjÞ bers kj;k ; k ¼ j; . . . ; k0 ðjÞ such that k¼j kj;k ¼ 1 and the sequence of convex combinations zj ¼ k¼j kj;k fk ; j P 1 converges strongly to f in L1 ðJ; EÞ as j ! 1. So we may suppose that zj ðtÞ ! f ðtÞ for a.e. t 2 J. Since F takes convex and closed values, we obtain for a.e. t 2 J

f ðtÞ 2

\

fzk ðtÞ : k P jg #

jP1

\

conv ffk : k P jg # Fðt; xðtÞÞ:

jP1

Note that, by (5) for every t; s 2 J; s 2 ð0; t and every n P 1

kT a ðt  sÞzn ðsÞk 6 jt  sja1 M T uðsÞXðkxkPCðJ;EÞ Þ 2 L1 ðð0; t; Rþ Þ: ~n ðtÞ ¼ Next taking y

Pk0 ðjÞ k¼j

kj;k yk , (11) implies

8 Rt > S ðtÞðx0  gðxÞÞ þ 0 T a ðt  sÞzn ðsÞds; t 2 J 0 ; > > a > > < .. ~n ðtÞ ¼ . y > k¼i X > Rt > > > Sa ðt  t k ÞIk ðxðt k ÞÞ þ 0 T a ðt  sÞzn ðsÞds; : Sa ðtÞðx0  gðxÞÞ þ

ð12Þ t 2 J i ; 1 6 i 6 m:

k¼1

~n ðtÞ ! yðtÞ and zn ðtÞ ! f ðtÞ for a.e. t 2 J, therefore, by passing to the limit as n ! 1 in (12), we obtain from the Lebesgue But y dominated convergence theorem that, for every i ¼ 0; 1; . . . ; m,

8 Rt > S ðtÞðx0  gðxÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > a > > < .. yðtÞ ¼ . > k¼J X > Rt > > > Sa ðt  t k ÞIk ðxðt k ÞÞ þ 0 T a ðt  sÞf ðsÞds; : Sa ðtÞðx0  gðxÞÞ þ

t 2 J i ; 1 6 i 6 m:

k¼1

This proves that RðxÞ is closed. Step 2. Set B0 ¼ fx 2 PCðJ; EÞ : kxk 6 rg. Obviously, B0 is a bounded, closed and convex subset of PCðJ; EÞ. We claim that RðB0 Þ # B0 . To prove that, let x 2 B0 and y 2 RðxÞ. By using (5), (9), (11), ðHF2Þ; ðHgÞ and Hölder’s inequality, we get for t 2 J 0

kyðtÞk 6 MS ðkx0 k þ ar þ dÞ þ M T XðrÞ

Z

t

ðt  sÞa1 uðsÞds

0

6 MS ðkx0 k þ ar þ dÞ þ M T XðrÞkuk ¼ M S ðkx0 k þ ar þ dÞ þ MT gXðrÞkuk

Z 1 Lq ðJ;Rþ Þ 1

Lq ðJ;Rþ Þ

t

0

1q aq a1 b M T XðrÞ ðt  sÞ1q ds 6 M S ðkx0 k þ ar þ dÞ þ kuk 1q

-

6 r:

Similarly, by using ðHIÞ in addition, we get for t 2 J i , i ¼ 1; 2; . . . ; m

kyðtÞk 6 MS ðkx0 k þ ar þ d þ hrÞ þ M T gXðrÞkuk

1

Lq ðJ;Rþ Þ

6 r:

Therefore, RðB0 Þ # B0 . Step 3. Let Z ¼ RðB0 Þ. We claim that the set Z jJi is equicontinuous for every i ¼ 0; 1; 2; . . . ; m; where

Z jJi ¼ fy 2 CðJ i ; EÞ : y ðtÞ ¼ yðtÞ; t 2 J i ; y ðt i Þ ¼ yðt þi Þ; y 2 Zg: Let y 2 Z. Then there is x 2 B0 with y 2 RðxÞ. By recalling the definition of R, there is f 2 S1Fð;xðÞÞ such that

8 Rt > S ðtÞðx0  gðxÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > a > > < .. yðtÞ ¼ . > k¼J X > Rt > > > Sa ðt  t k ÞIk ðxðt k ÞÞ þ 0 T a ðt  sÞf ðsÞds; : Sa ðtÞðx0  gðxÞÞ þ k¼1

We consider the following cases: Case 1. When i ¼ 0, let t; t þ d be two points in J 0 , then

t 2 J i ; 1 6 i 6 m:

1

Lq ðJ;Rþ Þ

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J. Wang et al. / Applied Mathematics and Computation 257 (2015) 103–118

ky ðt þ dÞ  y ðtÞk ¼ kyðt þ dÞ  yðtÞk 6 kSa ðt þ dÞðx0  gðxÞÞ  Sa ðtÞðx0  gðxÞÞk þ

Z

tþd

T a ðt þ d  sÞf ðsÞÞds  0 Z tþd     T ðt þ d  sÞf ðsÞds 6 kSa ðt þ dÞðx0  gðxÞÞ  Sa ðtÞðx0  gðxÞÞk þ  a   t Z t     þ  ½T a ðt þ d  sÞ  T a ðt  sÞf ðsÞds :¼ G1 þ G2 þ G3 ;

Z

t

T a ðt  sÞf ðsÞds

0

ð13Þ

0

where

G1 ¼ kSa ðt þ dÞðx0  gðxÞÞ  Sa ðtÞðx0  gðxÞÞk; Z tþd    ; G2 ¼  T ðt þ d  sÞf ðsÞds a   t Z t    : G3 ¼  ½T ðt þ d  sÞ  T ðt  sÞf ðsÞds a a   0

We only need to check Gi ! 0 as d ! 0 for every i ¼ 1; 2; 3. For G1 we have

lim G1 ¼ limkSa ðt þ dÞðx0  gðxÞÞ  Sa ðtÞðx0  gðxÞÞk 6 limkSa ðt þ dÞ  Sa ðtÞkðkx0 k þ ar þ dÞ ¼ 0 d!0

d!0

d!0

ð14Þ

uniformly for x 2 B0 . For G2 , by the Hölder’s inequality we have

Z  lim G2 ¼ lim  d!0

d!0

t

tþd

 Z  T a ðt þ d  sÞf ðsÞds 6 M lim T  d!0

6 M T XðrÞlim d!0

6 M T XðrÞlim d!0

Z

tþd

ðt þ d  sÞa1 kf ðsÞkds

t

Z ðt þ d  sÞa1 uðsÞds 6 MT XðrÞlim

tþd

d!0

t

 - 1q d

-

kuk

1

Lq ðJ;Rþ Þ

tþd

1q a1 ðt þ d  sÞ1q ds kuk

t

1

Lq ðJ;Rþ Þ

¼0

ð15Þ

uniformly for x 2 B0 . For G3 , by using and the Lebesgue dominated convergence theorem and the definition of T a we get

 Z t Z t    limG3 6 lim limkT a ðt þ d  sÞf ðsÞ  T a ðt  sÞf ðsÞkds ¼ 0;  T a ðt þ d  sÞf ðsÞ  T a ðt  sÞf ðsÞds 6 s!0

d!0

0

0

d!0

ð16Þ

independently of x. Case 2. When i 2 f1; 2; . . . ; mg, let t; t þ d be two points in J i . Invoking to the definition of R, we have

ky ðt þ dÞ  y ðtÞk ¼ kyðt þ dÞ  yðtÞk 6 kSa ðt þ dÞðx0  gðxÞÞ  Sa ðtÞðx0  gðxÞÞk þ Z  þ 

tþd

T a ðt þ d  sÞf ðsÞds 

0

k¼i X kSa ðt þ d  t k ÞJ k ðxðtk ÞÞ  Sa ðt  tk ÞIk ðxðtk ÞÞk k¼1

Z

t 0

  T a ðt  sÞf ðsÞds :

Arguing as in the first case we get

limkyðt þ dÞ  yðtÞk ¼ 0:

ð17Þ

d!0

Case 3. When t ¼ t i , i ¼ 1; . . . ; m, let k > 0 be such that ti þ k 2 J i and 

r > 0 such that ti < r < ti þ d 6 tiþ1 , then we have



ky ðt i þ dÞ  y ðti Þk ¼ limþ kyðti þ dÞ  yðrÞk: r!ti

According the definition of R we get

kyðt i þ dÞ  yðrÞk 6 kSa ðt i þ dÞðx0  gðxÞÞ  Sa ðrÞððx0  gðxÞÞk þ Z  þ 

0

k¼i X kSa ðt i þ d  tk ÞIk ðxðtk ÞÞ  Sa ðr  tk ÞIk ðxðtk ÞÞk k¼1

t i þd

a1

ðt i þ d  sÞ

T a ðti þ d  sÞf ðsÞds 

Z r 0

  ðr  sÞa1 T a ðr  sÞf ðsÞds :

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J. Wang et al. / Applied Mathematics and Computation 257 (2015) 103–118

Arguing as in the first case we can see that

lim kyðt i þ dÞ  yðrÞk ¼ 0:

ð18Þ

d!0;r!t þ i

From (13)–(18) we conclude that Z jJi is equicontinuous for every i ¼ 0; 1; 2; . . . ; m. Now for every n P 1, set Bn ¼ conv RðBn1 Þ. From Step 1, Bn is a nonempty, closed and convex subset of PCðJ; EÞ. Moreover, B1 ¼ conv RðB0 Þ # B0 . Also B2 ¼ conv RðB1 Þ # conv RðB0 Þ # B1 . By induction, the sequence ðBn Þ; n P 1 is a decreasing sequence of nonempty, closed and bounded subsets of PCðJ; EÞ. T Our goal is to show that the subset B ¼ 1 n¼1 Bn is nonempty and compact in PCðJ; EÞ. By Lemma 2.8, it is enough to show that

lim vPC ðBn Þ ¼ 0;

ð19Þ

n!1

where vPC is the Hausdorff measure of noncompactness on PCðJ; EÞ defined in Section 2. In the following step we prove (19). Step 4. Let n P 1 be a fixed natural number and e > 0. In view of Lemma 2.9, there exists a sequence ðyk Þ; k P 1 in RðBn1 Þ such that

vPC ðBn Þ ¼ vPC RðBn1 Þ 6 2vPC fyk : k P 1g þ e: From the definition of

vPC , the above inequality becomes

vPC ðBn Þ 6 2 max vi ðSjJi Þ þ e;

ð20Þ

i¼0;1;...;m

where S ¼ fyk : k P 1g and vi is the Hausdorff measure of noncompactness on CðJ i ; EÞ. Arguing as in the previous step we can show that BnjJi , i ¼ 0; 1; . . . ; m is equicontinuous. Then, by applying Lemma 2.10 we obtain

vi ðSjJi Þ ¼ sup vðSðtÞÞ; t2J i

where

v is the Hausdorff measure of noncompactness on E. Therefore, by using the nonsingularity of v, (20) becomes "

vPC ðBn Þ 6 2 max

i ¼0;1;...;m

#

supvðSðtÞÞ þ e ¼ 2sup vðSðtÞÞ þ e ¼ 2sup vfyk ðtÞ : k P 1g þ e: t2J

t2J i

ð21Þ

t2J

Now, since yk 2 RðBn1 Þ; k P 1 there is xk 2 Bn1 such that yk 2 Rðxk Þ; k P 1. By recalling the definition of R for every k P 1 there is fk 2 S1Fð;xk ðÞÞ such that for every t 2 J,

8 R vfSa ðtÞðx0  gðxk ÞÞ : k P 1g þ vf 0t T a ðt  sÞfk ðsÞds : k P 1g; if t 2 J0 ; > > > > > . > > > .. < p¼i X vfyk ðtÞ : k P 1g 6 > v fSa ðtÞðx0  gðxk ÞÞ : k P 1g þ vfSa ðt  tp ÞIp ðxk ðtp ÞÞ : k P 1g > > > > p¼1 > > > Rt : þvf 0 T a ðt  sÞfk ðsÞds : k P 1g; if t 2 Ji ; i ¼ 1; . . . ; m:

ð22Þ

Since g is compact, the set fgðxk Þ : k P 1g is relatively compact. Hence, for every t 2 J we have

vfSa ðtÞðx0  gðxk ÞÞ : k P 1g ¼ 0:

ð23Þ

Furthermore, condition ðHIÞ implies, for every p ¼ 1; 2; . . . ; m and every t 2 J.

vfSa ðt  tp ÞðIp ðxk ðtp ÞÞÞ : k P 1g ¼ 0:

ð24Þ

In order to estimate

Z

v

t

T a ðt  sÞfk ðsÞds : k P 1 ;

0

we observe that, from ðHF3Þ it holds that for a:e: t 2 J

vðffk ðtÞ : k P 1g 6 vfFðt; xk ðtÞÞ : k P 1g 6 bðtÞ vfxk ðtÞ : k P 1g 6 bðtÞ vðBn1 ðtÞÞ 6 bðtÞ vPC ðBn1 Þ ¼ cðtÞ: 1

Furthermore, for any k P 1, by ðHF2Þ, for almost t 2 J; kfk ðtÞk 6 uðtÞXðrÞ. Consequently, fk 2 Lq ðJ; EÞ; k P 1. Note that 1 q

þ

c 2 L ðJ; R Þ. Then, by virtue of Lemma 2.11, there exists a compact K  # E, a measurable set J  J, with measure less than 1 q

, and a sequence of functions fgk g  L ðJ; EÞ such that for all s 2 J;

fg k ðsÞ : k P 1g # K  , and

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J. Wang et al. / Applied Mathematics and Computation 257 (2015) 103–118

kfk ðsÞ  g k ðsÞk < 2cðsÞ þ ;

for every k P 1 and every s 2 J 0 ¼ J  J  :

ð25Þ

Then using Minkowski’s inequality, we get

 Z "Z #q   1    q ð2cðsÞ þ Þ ds 6 MT gk2cðsÞ þ k 1q þ  T a ðt  sÞðfk ðsÞ  g k ðsÞÞds 6 M T g L ðJ;R Þ   J0 J0     q 6 M T g k2ck 1q þ þ kk 1q þ 6 2M T g kck 1q þ þ b L ðJ;R Þ L ðJ;R Þ L ðJ;R Þ   q ¼ 2M T g vPC ðBn1 Þkbk 1q þ þ b

ð26Þ

L ðJ;R Þ

and Hölder’s inequality

 Z "Z #q   1   q T ðt  sÞf ðsÞds 6 M g X ðrÞ u ðsÞ ds :   T k   J a J

ð27Þ

So by (26) and (27), we derive

Z

t

v

T a ðt  sÞfk ðsÞds : k P 1



6v

0

6v

(Z (Z

þv

)! J0

(Z

þv

T a ðt  sÞfk ðsÞds : k P 1

)! T a ðt  sÞfk ðsÞds : k P 1

J

)! 

T a ðt  sÞðfk ðsÞ  g k ðsÞÞds : k P 1

J0

(Z

þv

Z

v

)! 

T a ðt  sÞg k ðsÞds : k P 1

T a ðt  sÞfk ðsÞds : k P 1



By taking into account that

J 0

)! J

6 2M T g

(Z

q

vPC ðBn1 ÞkbkL1q ðJ;Rþ Þ þ b



þ MT gXðrÞ

"Z

1 q

#q

uðsÞ ds :

J

 is arbitrary, we get for all t 2 J



t

T a ðt  sÞfk ðsÞds : k P 1

0

6 2M T gvPC ðBn1 Þkbk

1

Lq ðJ;Rþ Þ

:

Then, by (22) and (24), for every t 2 J

vfyk ðtÞ : k P 1g 6 2MT gvPC ðBn1 Þ kbkL1q ðJ;Rþ Þ : This inequality with (21) and with the fact that

e is arbitrary, imply

vPC ðBn Þ 6 2MT gvPC ðBn1 Þ kbkL1q ðJ;Rþ Þ : By means of a finite number of steps, we can write

 0 6 vPC ðBn Þ 6 2MT gkbk

n1

1 Lq ðJ;Rþ Þ

vPC ðB1 Þ; for all n P 1:

Since this inequality is true for every n 2 N, by (8) and by passing to the limit as n ! þ1, we obtain (19) and so our aim in this step is verified. T Step 5. At this point, we are in position to apply Lemma 2.8. We claim that the set B ¼ 1 n¼1 Bn is a nonempty and compact subset of PCðJ; EÞ. Moreover, every Bn being bounded, closed and convex, B is also bounded closed and convex. Let us verify that RðBÞ # B. Indeed,

RðBÞ # RðBn Þ # conv RðBn Þ ¼ Bnþ1 for every n P 1. Therefore, RðBÞ 

RðBÞ 

1 \ n¼2

Bn ¼

1 \

T1

n¼2 Bn .

On the other hand Bn  B1 for every n P 1. So,

Bn ¼ B:

n¼1

Step 6. The graph of the multi-valued function RjB : B ! 2B is closed. Consider a sequence fxn gnP1 in B with xn ! x in B and let yn 2 Rðxn Þ with yn ! y in PCðJ; EÞ. We have to show that y 2 RðxÞ. By recalling the definition of R, for any n P 1, there is fn 2 S1Fð;xn ðÞÞ such that

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J. Wang et al. / Applied Mathematics and Computation 257 (2015) 103–118

8 Rt > Sa ðtÞðx0  gðxn ÞÞ þ 0 T a ðt  sÞfn ðsÞds; t 2 J 0 ; > > > > < .. yn ðtÞ ¼ . > k¼i X > Rt > > > Sa ðt  t k ÞIk ðxn ðt k ÞÞ þ 0 T a ðt  sÞfn ðsÞds; : Sa ðtÞðx0  gðxn ÞÞ þ

ð28Þ t 2 J i ; 1 6 i 6 m:

k¼1

Observe that for every n P 1 and for a.e. t 2 J

kfn ðtÞk 6 uðtÞXðkxn ðtÞkÞ 6 uðtÞXðrÞ: This show that the set ffn : n P 1g is integrably bounded. In addition, the set ffn ðtÞ : n P 1g is relatively compact for a.e. t 2 J because assumption ðHF3Þ both with the convergence of fxn gnP1 imply that

vffn ðtÞ : n P 1g 6 vðFðt; fxn ðtÞ : n P 1gg 6 bðtÞvfxn ðtÞ : n P 1g ¼ 0: So, the sequence ffn gnP1 is semi-compact, hence by Lemma 2.7 it is weakly compact in L1 ðJ; EÞ. So, without loss of generality we can assume that fn converges weakly to a function f 2 L1 ðJ; EÞ. From Mazur’s lemma, for every natural number j there is a Pk0 ðjÞ natural number k0 ðjÞ > j and a sequence of nonnegative real numbers kj;k ; k ¼ j; . . . ; k0 ðjÞ such that k¼j kj;k ¼ 1 and the Pk0 ðjÞ 1 sequence of convex combinations zj ¼ k¼j kj;k fk ; j P 1 converges strongly to f in L ðJ; EÞ as j ! 1. So we may suppose that zj ðtÞ ! f ðtÞ for a.e. t 2 J. Let t be such that Fðt; Þ is upper semicontinuous. Then, for any neighborhood U of Fðt; Þ, there is a natural number n0 so that for any n P n0 we have

Fðt; xn ðtÞÞ # U: Because the values of F are convex and compact, Lemma 2.6 tells us that

\

[

conv

jP1

!

Fðt; xn ðtÞÞ # Fðt; xðtÞÞ:

nPj

As in Step 1, from Mazur’s theorem, there is a sequence fzn : n P 1g of convex combinations of fn such that for a.e. t 2 J

f ðtÞ 2

\

fzn ðtÞ : n P jg #

jP1

\

conv ffn ðtÞ : n P jg

jP1

and zn converges strongly to f 2 L1 ðJ; EÞ. Then, for a:e. t 2 J

f ðtÞ 2

\

fzn ðtÞ : n P jg #

j P1

\

conv ffn ðtÞ : n P jg #

j P 1

\ jP1

conv

[

! Fðt; xn ðtÞÞ # Fðt; xðtÞÞ:

nPj

Then, by the continuity of g; Sa ; T a ; Ik ðk ¼ 1; 2; . . . ; mÞ and by the same arguments used in Step 1, we conclude from relation (28) that

8 Rt > S ðtÞðx0  gðxÞÞ þ 0 ðt  sÞa1 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > a > > < .. yðtÞ ¼ . > k¼J X > Rt > > > Sa ðt  t k ÞIk ðxðt k ÞÞ þ 0 ðt  sÞa1 T a ðt  sÞf ðsÞds; : Sa ðtÞðx0  gðxÞÞ þ

t 2 J i ; 1 6 i 6 m:

k¼1

Therefore, y 2 RðxÞ. This show that the graph of R is closed. As a result of the Steps 1–5 the multivalued RjB : B ! 2B is closed and vPC -condensing, with nonempty convex compact values. Applying the fixed point theorem, Lemma 2.22, there is x 2 B such that x 2 RðxÞ. Clearly x is a PC-mild solution for the problem (1). h In the following theorem we prove that the set of mild solutions of (1) is compact. Theorem 3.2. If the function X in ðHF2Þ is given of the form XðrÞ ¼ r þ 1, and under the assumptions ðHAÞ; ðHF1Þ; ðHF3Þ; ðHgÞ; ðHIÞ of Theorem 3.1, then the set of mild solutions of (1) is compact in PCðJ; EÞ provided that

 M S ða þ hÞ þ MT gkuk



1 Lq ðJ;Rþ Þ

< 1:

Proof. Not that by Theorem 3.1 the set of solutions of (1) is nonempty. Indeed, we take

r¼

MS ðkx0 k þ dÞ þ MT gkuk 1q þ L ðJ;R Þ   1  M S ða þ hÞ þ MT gkuk 1q þ L ðJ;R Þ

ð29Þ

115

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in (9). So we have a solution in B0 . Now we show that any mild solution of (1) belongs to B0 . So let x be a mild solution of (1). Then

8 Rt > Sa ðtÞðx0  gðxÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > > > < .. xðtÞ ¼ . > k¼i X > Rt > > > Sa ðt  t k ÞIk ðxðt k ÞÞ þ 0 T a ðt  sÞf ðsÞds; : Sa ðtÞðx0  gðxÞÞ þ

t 2 J i ; 1 6 i 6 m;

k¼1

where f is an integrable selection for Fð; xðÞÞ. By arguing as in Step 2 of Theorem 3.1 we get

kxkPCðJ;EÞ 6 MS ðkx0 k þ akxkPCðJ;EÞ þ d þ hkxkPCðJ;EÞ Þ þ M T gðkxkPCðJ;EÞ þ 1Þkuk

1

Lq ðJ;EÞ

:

Therefore,

kxkPCðJ;EÞ 6

M S ðkx0 k þ dÞ þ M T gkuk 1q L ðJ;EÞ  ¼ r: 1  MS ða þ hÞ þ M T gkuk 1q L ðJ;EÞ

Due to Lemma 2.23, the proof is completed. h Next, we present another existence result for the PC-mild solution of the problem (1). Theorem 3.3. Let E be a separable Banach space, F : J  E ! Pck ðEÞ Ii : E ! E ði ¼ 1; 2; . . . ; mÞ. We assume the following conditions:

be

a

multifunction,

g : PCðJ; EÞ ! E

and

ðHAÞ A 2 Aa ðh0 ; x0 Þ with h0 2 ð0; p2  and x0 2 R. ðH1 Þ For every x 2 E; t!Fðt; xÞ is measurable. 1 ðH2 Þ There is a function 1 2 Lq ðJ; Rþ Þ; q 2 ð0; aÞ such that (i) For every x; y 2 E

hðFðt; xÞ; Fðt; yÞÞ 6 1ðtÞkx  yk;

for a:e: t 2 J;

þ

where h : P clb ðEÞ  Pclb ðEÞ ! R is the Hausdorff distance. (ii) For every x 2 E

supfkxk : x 2 Fðt; 0Þg 6 1ðtÞ;

for a:e: t 2 J:

ðH3 Þ There is a positive constants a such that

kgðx1 Þ  gðx2 Þk 6 akx1  x2 kPCðJ;EÞ ;

for all x1 ; x2 2 PCðJ; EÞ:

ðH4 Þ For each i ¼ 1; 2; . . . ; m, there is ni > 0 such that

kIi ðxÞ  Ii ðyÞk 6 ni kx  yk;

for all x; y 2 E:

ðH5 Þ The following inequality hold

MS ða þ nÞ þ M T k1kL1qðJ;Rþ Þ g < 1;

n¼

i¼m X ni : i¼1

Then the problem (1) has a PC-mild solution. Proof. For x 2 PCðJ; EÞ, set S1Fð;xðÞÞ ¼ ff 2 L1 ðJ; EÞ : f ðtÞ 2 Fðt; xðtÞÞ for a.e. t 2 Jg. By Lemma 2.2, ðH1 Þ and ðH2 Þ(i), [32, Theorems 1.1.9 and 1.3.1] Fð; xðÞÞ has a measurable selection which, by hypothesis ðH1 Þ(ii), belongs to L1 ðJ; EÞ. Thus S1Fð;xðÞÞ is nonempty. Let us transform the problem into a fixed point problem. Consider the multifunction map, R : PCðJ; EÞ ! 2PCðJ;EÞ defined as follows: for x 2 PCðJ; EÞ; RðxÞ is the set of all functions y 2 RðxÞ given by (10). It is easy to see that any fixed point for R is a mild solution for (1). So, we shall show that R satisfies the assumptions of Lemma 2.24. The proof will be given in two steps. Step 1. The values of R are closed. By [32, Theorem 1.1.9], assumptions ðH1 Þ and ðH2 Þ(ii) imply assumption ðHF1Þ of Theorem 3.1. Next, since by ðH2 Þ

kFðt; xÞk ¼ hðFðt; xÞ; f0gÞ 6 hðFðt; xÞ; Fðt; 0ÞÞ þ hðFðt; 0Þ; f0gÞ 6 1ðtÞkxk þ kFðt; 0Þk 6 1ðtÞð1 þ kxkÞ; assumption ðHF2Þ of Theorem 3.1 holds as well. So the statement follows from the 1th step of the proof of Theorem 3.1. Step 2. R is a contraction. Let x1 ; x2 2 PCðJ; EÞ and y1 2 Rðx1 Þ. Then, there is f 2 S1Fð;x1 ðÞÞ such that for any t 2 J i , i ¼ 0; 1; 2 . . . ; m,

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8 Rt > Sa ðtÞðx0  gðx1 ÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J 0 ; > > > Rt > > > S ðtÞðx0  gðx1 ÞÞ þ Sa ðt  t1 ÞI1 ðx1 ðt 1 ÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J 1 ; > < a y1 ðtÞ ¼ .. . > > > > k¼i X > Rt > > Sa ðt  t k ÞIk ðx1 ðt k ÞÞ þ 0 T a ðt  sÞf ðsÞds; t 2 J i ; 1 6 i 6 m: > Sa ðtÞðx0  gðx1 ÞÞ þ :

ð30Þ

k¼1

Consider the multifunction Z : J ! 2E defined by

ZðtÞ ¼ fu 2 E : kf ðtÞ  uk 6 1ðtÞkx2 ðtÞ  x1 ðtÞkg: For each t 2 J; ZðtÞ is nonempty. Indeed, let t 2 J, from ðH2 Þ(i), we have

hðFðt; x2 ðtÞÞ; Fðt; x1 ðtÞÞÞ 6 1ðtÞkx1 ðtÞ  x2 ðtÞk: Hence, there exists u 2 Fð; x2 ðÞÞ such that

kut  f ðtÞk 6 1ðtÞkx1 ðtÞ  x2 ðtÞk: Since the functions f ; 1; x1 ; x2 are measurable, Proposition 3.4 in [17] (or [32, Corollary 1.3.1(a)]), tells us that the multifunction V : t ! ZðtÞ \ Fðt; x2 ðtÞÞ is measurable. Because its values are nonempty and compact, by [30, Theorem 1.3.1], there is h 2 S1Fð;x2 ðÞÞ with

khðtÞ  f ðtÞk 6 1ðtÞkx2 ðtÞ  x1 ðtÞk;

a:e: t 2 J:

ð31Þ

Let us define

8 Rt > Sa ðtÞðx0  gðx2 ÞÞ þ 0 T a ðt  sÞhðsÞds; t 2 J 0 ; > > > Rt > > > S ðtÞðx0  gðx2 ÞÞ þ Sa ðt  t1 ÞI1 ðx2 ðt 1 ÞÞ þ 0 T a ðt  sÞhðsÞds; t 2 J 1 ; > < a y2 ðtÞ ¼ .. . > > > > k¼i X > Rt > > Sa ðt  t k ÞIk ðx2 ðt k ÞÞ þ 0 T a ðt  sÞhðsÞds; t 2 J i ; 1 6 i 6 m: > Sa ðtÞðx0  gðx2 ÞÞ þ :

ð32Þ

k¼1

Obviously, y2 2 Rðx2 Þ and if t 2 J 0 we get from (30)–(32), ðH3 Þ and ðH4 Þ

ky2 ðtÞ  y1 ðtÞk 6 M S kgðx1 Þ  gðx2 Þk þ MT

Z

t

ðt  sÞa1 khðsÞ  f ðsÞkds

0

Z

t

ðt  sÞa1 1ðsÞds h i 6 M S akx1  x2 kPCðJ;EÞ þ M T kx1  x2 kPCðJ;EÞ k1kL1qðJ;Rþ Þ g 6 M S a þ M T k1kL1qðJ;Rþ Þ g kx1  x2 kPCðJ;EÞ : 6 M S akx1  x2 kPCðJ;EÞ þ M T kx1  x2 kPCðJ;EÞ

0

ð33Þ

Similarly, if t 2 J i ; i ¼ 1; . . . ; m, we get from (30)–(32), ðH3 Þ and ðH4 Þ

h i ky2 ðtÞ  y1 ðtÞk 6 MS ða þ nÞ þ MT k1kL1qðJ;Rþ Þ g kx1  x2 kPCðJ;EÞ :

ð34Þ

By interchanging the role of y2 and y1 we obtain from (33) and (34)

h i hðRðx2 Þ; Rðx1 ÞÞ 6 M S ða þ nÞ þ M T k1kL1qðJ;Rþ Þ g kx1  x2 kPCðJ;EÞ :

Therefore, R is a contraction due to ðH5 Þ and thus by Lemma 2.23, R has a fixed point which is a mild solution for (1). This completes the proof. h 4. Existence results for nonconvex case In this section, we give a nonconvex version for Theorem 3.3. Our hypothesis on the orient field is the following: ðHF  Þ F : J  E ! P cl ðEÞ is a multifunction such that (i) ðt; xÞ ! Fðt; xÞ is graph measurable and x ! Fðt; xÞ is lower semicontinuous. 1 (ii) There exists a function u 2 Lq ðJ; Rþ Þ; 0 < q < a such that for any x 2 E,

kFðt; xÞk 6 uðtÞ;

a:e: t 2 J:

 Theorem 4.1. Let A 2 Aa ðh0 ; x0 Þ with h0 2 0; p2 and x0 2 R; F : J  E ! Pck ðEÞ a multifunction, g : PCðJ; EÞ ! E and Ii : E ! E ði ¼ 1; 2; . . . ; mÞ. If the conditions ðHF3Þ; ðHgÞ; ðHIÞ and ðHF  Þ hold, then the problem (1) has a PC-mild solution provided that there is r > 0 provided that

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J. Wang et al. / Applied Mathematics and Computation 257 (2015) 103–118

MS ðkx0 k þ ar þ d þ h rÞ þ MT gkuk

6 r:

1

Lq ðJ;Rþ Þ

ð35Þ L1 ðJ;EÞ

Proof. Consider the multivalued Nemitsky operator N : PCðJ; EÞ ! 2

, defined by

NðxÞ ¼ S1Fð;xðÞÞ ¼ ff 2 L1 ðJ; EÞ : f ðtÞ 2 Fðt; xðtÞÞ; a:e: t 2 Jg: First we note F is superpositionally measurable by [58]. Next, we shall prove that N has a nonempty closed decomposable value and l.s.c. Let x 2PCðJ; EÞ. Since F has closed values, S1Fð;xðÞÞ is closed [30]. Because F is integrably bounded, S1Fð;xðÞÞ is nonempty (see Theorem 3.2 of [30]). It is readily verified, S1Fð;xðÞÞ is decomposable (see Theorem 3.1 of [30]). To check the lower semicontinuity of N, we need to show that, for every u 2 L1 ðJ; EÞ; x ! dðu; NðxÞÞ is upper semicontinuous (see Proposition 1.2.26 of [31]). To this end let u 2 L1 ðJ; EÞ be fixed. From Theorem 2.2 in [30] with /ðt; v Þ ¼ kuðtÞ  v k, we have that

dðu; NðxÞÞ ¼ inf ku  v kL1 ¼ v 2NðxÞ

v ðtÞ

Z

inf

2Fðt;xðtÞÞ

b

kuðtÞ  v ðtÞkdt ¼

Z

0

b

inf

0

zðtÞ2Fðt;xðtÞÞ

kuðtÞ  zðtÞkdt ¼

Z

b

dðuðtÞ; Fðt; xðtÞÞÞdt:

0

ð36Þ We shall show that for any k P 0, the set

uk ¼ fx 2 PCðJ; EÞ : dðu; NðxÞÞ P kg is closed. For this purpose, let fxn g # uk and assume that xn ! x in PCðJ; EÞ. Then, for all t 2 J; xn ðtÞ ! xðtÞ in E. By virtue of ðHF  Þ(i) the function z ! dðuðtÞ; Fðt; zÞÞ is upper semicontinuous. So, via the Fatou lemma, and (36) we have

k 6 lim sup dðu; Nðxn ÞÞ ¼ lim sup n!1

n!1

Z

b

dðuðtÞ; Fðt; xn ðtÞÞÞdt 6

0

Z 0

b

lim sup dðuðtÞ; Fðt; xn ðtÞÞÞdt 6

n!1

Z

b

dðuðtÞ; Fðt; xðtÞÞÞdt

0

¼ dðu; NðxÞÞ: Therefore x 2 uk and this proves the lower semicontinuity of N. This allows us to apply Theorem 3 of [15] and obtain a continuous map Z : PCðJ; EÞ ! L1 ðJ; EÞ such that ZðxÞ 2 NðxÞ, for every x 2 PCðJ; EÞ. Then, ZðxÞðsÞ 2 Fðs; xðsÞÞ a.e. s 2 J. Consider a map U : PCðJ; EÞ ! PCðJ; EÞ defined by

8 Rt a1 > > < Sa ðtÞðtÞðx0  gðx1 ÞÞ þ 0 ðt  sÞ T a ðtÞðt  sÞZðxÞðsÞds; t 2 J 0 ; k¼i X ðUxÞðtÞ ¼ Rt > Sa ðtÞðt  t k ÞIk ðx1 ðtk ÞÞ þ 0 ðt  sÞa1 T a ðtÞðt  sÞZðxÞðsÞds; > : Sa ðtÞðtÞðx0  gðx1 ÞÞ þ

t 2 J i ; 1 6 i 6 m:

k¼1

Arguing as in the proof of Theorem 3.3, we can show that there is a convex compact subset B of E such that the function

UjB : B ! B satisfies the conditions of Schauder fixed point theorem. Then there exists x 2 PCðJ; EÞ such that xðÞ ¼ ðUxÞðÞ. This means that x is a PC-mild solution for the problem (1). h Acknowledgments The authors thanks the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of the paper. References [1] E. Bajlekova, Fractional evolution equations in Banach spaces (Ph.D. thesis), Eindhoven University of Technology, 2001. [2] R.P. Agarwal, B. Ahmad, A. Alsaedi, N. Shahzad, On the dimension of the solution set for semilinear fractional differential inclusions, Abstr. Appl. Anal. 2012 (2012) 1–10 (Art. ID 305924). [3] R.P. Agarwal, B. Ahmad, A. Alsaedi, N. Shahzad, Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions, Adv. Differ. Equ. 74 (74) (2012) 1–10. [4] R.P. Agarwal, M. Benchohra, B.A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differ. Equ. Math. Phys. 44 (2008) 1–21. [5] R.P. Agarwal, M. Belmekki, M. Benchohra, A survey on semi-linear differential equations and inclusions involving Riemann–Liouville fractional derivative, Adv. Differ. Equ. 2009 (2009) 1–47 (Art. ID 981728). [6] Z. Agur, L. Cojocaru, G. Mazaur, R.M. Anderson, Y.L. Danon, Pulse mass measles vaccination across age shorts, Proc. Natl. Acad. Sci. USA 90 (1993) 11698–11702. [7] R.A. Al-Omair, A.G. Ibrahim, Existence of mild solutions of a semilinear evolution differential inclusions with nonlocal conditions, Electron. J. Differ. Equ. 2009 (42) (2009) 1–11. [8] J.P. Aubin, H. Frankoeska, Set-Valued Analysis, Birkhäuser, Boston, Basel, Berlin, 1990. [9] R. Bader, M. Kamenskii, V. Obukhowskii, On some class of operator inclusions with lower semicontinuous nonlinearity, Topol. Meth. Nonlinear Anal. 17 (2001) 143–156. [10] G. Ballinger, X. Liu, Boundedness for impulsive delay differential equations and applications in populations growth models, Nonlinear Anal.: TMA 53 (2003) 1041–1062. [11] J.M. Ball, Initial boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973) 16–90. [12] M. Belmekki, M. Benchohra, Existence results for fractional order semilinear functional differential equations with nondense domain, Nonlinear Anal.: TMA 72 (2010) 925–932.

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