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Applicable Analysis: An International Journal

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On a class of fractional order

differential inclusions with infinite delays

Tran Dinh Ke ^a , Valeri Obukhovskii ^b , Ngai-Ching Wong ^c & Jen- Chih Yao ^d

a Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

b Faculty of Mathematics, Voronezh State University, Universitetskays Pl. 1, 394006 Voronezh, Russia

c Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

d Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Version of record first published: 28 Jul 2011

To cite this article: Tran Dinh Ke, Valeri Obukhovskii, Ngai-Ching Wong & Jen-Chih Yao (2011):

On a class of fractional order differential inclusions with infinite delays, Applicable Analysis: An International Journal, DOI:10.1080/00036811.2011.601454

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2011, 1–23, iFirst

On a class of fractional order differential inclusions with infinite delays

Tran Dinh Ke^a, Valeri Obukhovskii^b, Ngai-Ching Wong^c* and Jen-Chih Yao^d

aDepartment of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam;^bFaculty of Mathematics, Voronezh State University, Universitetskays Pl. 1, 394006 Voronezh, Russia;^cDepartment of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan;^dCenter for General Education,

Kaohsiung Medical University, Kaohsiung 807, Taiwan Communicated by D.-C. Chang

(Received 6 December 2010; final version received 25 June 2011) Our aim is to study fractional order differential inclusions with infinite delays in Banach spaces. We impose the regularity condition on multivalued nonlinearity in terms of measures of noncompactness to get the existence result. Some properties of the solution map are proved.

Keywords: fractional order differential inclusion; fractional derivative;

functional differential inclusion; infinite delay; initial value problem;

continuous dependence; multivalued map; measure of noncompactness;

fixed point; condensing map

AMS Subject Classifications: 34K37; 26A33; 34A60; 34G25; 34K09;

34K30; 47H04; 47H08; 47H10; 47H11

1. Introduction

LetEbe a Banach space. We are concerned with the following problem:

CDuðtÞ 2Fðt,ut,r^NuÞ, t2J:¼ ½0,T, ð1:1Þ

r^Nuð0Þ ¼U0, ð1:2Þ

uðsÞ ¼’ðsÞ, s2 ð1, 0Þ, ð1:3Þ

whereN1 is an integer,2(N1,N],u: (1,T]!Eis the unknown function,

CD denotes the Caputo fractional derivative, r^Nu¼(u,u⁰,. . .,u^(N1)) and F: [0,T] B E^N! P(E) is a multivalued map with nonempty compact convex values. HereP(E) stands for the collection of all subsets ofE,Bis a phase space of delays and ut2 B is the history of the state function u up to the time t, that is

*Corresponding author. Email: wong@math.nsysu.edu.tw

ISSN 0003–6811 print/ISSN 1563–504X online ß2011 Taylor & Francis

DOI: 10.1080/00036811.2011.601454 http://www.informaworld.com

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ut(s)¼u(tþs) fors2(1, 0]. The initial dataU0¼ ðeu0,eu1,. . .,euN1Þare given inE^N and the initial function’2 Bis such that’ð0Þ ¼eu₀.

The subject of fractional differential equations has recently received much attention due to its important applications in modelling phenomena of science and engineering. The employment of differential equations with fractional order allows to deal with many problems in numerous areas including fluid flow, rheology, electrical networks, viscoelasticity, electrochemistry, etc. For complete references, we refer to some significant works, e.g., the monographs of Kilbas et al. [1], Kiryakova [2], Miller and Ross [3] and Podlubny [4]. In the past few years, there has been a great contribution in fractional differential equations. Let us refer to some relevant works in [5–18]. With initial or boundary conditions, some particular cases of (1.1) without delay were studied. The equation

CDuðtÞ ¼fðt,uðtÞÞ or the inclusion

CDuðtÞ 2Fðt,uðtÞÞ

in the cases 2(0, 1] or 2(1, 2] were considered in [6,8–11,13]. Similar problems with the Riemann–Liouville fractional derivatives were also investigated, for instance, in [12,17,18]. The readers can find more works in the survey of Argawal et al. [19].

In addition, inclusion (1.1) can be seen as a generalized model of high-order ordinary differential equations, an example of which is the equation considered in [20]:

u^ðnÞðtÞ ¼fðt,uðtÞ,u⁰ðtÞ,. . .,u^ðn1ÞðtÞ,ðTuÞðtÞÞ, whereT is an integral operator.

There are several approaches that can be used to get the solvability for local, global or extremal solutions of fractional differential equations or inclusions. One can use the method of upper and lower solutions to obtain the results as in [9].

Another approach develops a comparison principle and iteration schemes to receive the existence results as in [14–16]. The method that is widely used consists of transforming problems to corresponding fixed point equations or inclusions, followed by applying some known fixed point theorems.

Our approach is employing the fixed point theory technique for multivalued condensing maps under assumptions expressed in terms of the measure of noncompactness (MNC). The method used in this note allows us to solve the problems of differential inclusions in infinite-dimensional spaces with a general form of nonlinearity. Furthermore, by using this method, we need not impose the Lipschitz condition on the nonlinearity. Instead of Lipschitz assumptions, we suppose that the nonlinearityFsatisfies a regularity condition expressed in terms of the Hausdorff MNC. In the sequel, we define a suitable MNC, prove that the solution multioperator is condensing with respect to this new MNC and find solutions by using the fixed point theory of condensing multimaps presented in [21].

Besides [21], an employment of this approach can be found, e.g., in [17,22] and other works. For more applications of multivalued analysis to differential equations

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and inclusions, the readers are referred to [23–25]. The reader may also find a complete reference to MNCs in the monographs [26,27].

The rest of this article is organized as follows. In the next section we recall some basic facts related to fractional calculus, measures of noncompactness, multivalued maps and the phase space for delay differential equations. Section 3 is devoted to the formulations and proofs of the local and global existence results. In the last section, we study the continuous dependence of the solution set of the stated problem on initial data.

2. Preliminaries 2.1. Fractional calculus

We start this section with some notation and definitions in fractional calculus. For the motivations of these definitions, see, for example [1–4].

Definition 2.1 The fractional integral of order40 of a functionf2L¹(0,T;E) is defined by

I₀ fðtÞ ¼ 1 ðÞ

Zt

0

ðtsÞ¹fðsÞds, whereis the Gamma function.

We use Bochner integral in the foregoing definition and in the rest of this work.

Definition 2.2 For a functionf2C^N([0,T];E), the Caputo fractional derivative of order2(N1,N] is defined by

CD₀ fðtÞ ¼ 1 ðNÞ

Z t

0

ðtsÞ^N1f^ðNÞðsÞds:

It should be noted that there are some notions of fractional derivatives in which the Riemann–Liouville and Caputo definitions have been used widely. Many application problems, expressed by differential equations of fractional order, require initial conditions related to u(0), u⁰(0), etc., and the Caputo fractional derivative satisfies these demands. Foru2C^N([0,T];E), we have the following formulae:

CD₀I₀uðtÞ ¼uðtÞ, ð2:1Þ

I₀^CD₀uðtÞ ¼uðtÞ X^N1

k¼0

u^ðkÞð0Þ

k! t^k: ð2:2Þ

2.2. Phase space

LetBbe a linear space, with a seminormj jB, consisting of functions mapping (1, 0] intoE. The definition of the phase spaceB, introduced by Hale and Kato [28], can be given by the following axioms. Ifv: (1,T]!Eis such thatvj[0,T]2C([0,T];E) andv02 B, then

(B1)vt2 B for allt2[0,T];

(B2) the functiont°vtis continuous on [0,T];

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(B3) jvtjB4K(t)sup{kv(s)kE: 04s4t}þM(t)jv0jB, where K,M: [0,T]![0,1), K is continuous,Mis bounded and they are independent ofv.

We may consider the following examples of phase spaces satisfying all the above properties.

(1) For 40, let B ¼C be the space of continuous functions : (1; 0]!E having a limit lim!1e ðÞwith

j j_B¼ sup

150

ek ðÞk:

(2) (Spaces of ‘fading memory’). Let B ¼C be the space of functions : (1; 0]!Esuch that

(a) is continuous on [r; 0],r40;

(b) is Lebesgue measurable on (1;r) and there exists a nonnegative Lebesgue integrable function : (1;r)!R^þ such that is Lebesgue integrable on (1;r); moreover, there exists a locally bounded functionP: (1; 0]!R^þsuch that, for all0,(þ)P()() a.e.

2(1;r). Then, j jB¼ sup

r0

k ðÞk þ Z _r

1

ðÞk ðÞkd:

A simple example of such a space is given by ()¼e,2R. For more examples of phase spaces, see [28].

2.3. Measures of noncompactness and multivalued maps

Let us recall some basic facts from the multivalued analysis, which will be used in this article. LetEbe a Banach space. We denote

. P(E)¼{A E:A6¼ ;},

. Pv(E)¼{A2 P(E) :Ais convex}, . K(E)¼{A2 P(E) :Ais compact}, . Kv(E)¼Pv(E)\K(E).

We will use the following definition of the MNC (see, e.g. [21]).

Definition 2.3 Let (A,5) be a partially ordered set. A function :P(E)! A is called an MNC inEif

ðcoÞ ¼ðÞ for every2 PðEÞ,

wherecois the closure of the convex hull of. An MNC is called (i) monotone, if0,12 P(E),0 1implies(0)4(1);

(ii) nonsingular, if({a}[)¼() for anya2 E,2 P(E);

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(iii) invariant with respect to union with compact set, if (K[)¼() for every relatively compact setK Eand2 P(E).

IfAis a cone in a normed space, we say that is

(iv) algebraically semi-additive, if (0þ1)4(0)þ(1) for any 0, 12 P(E);

(v) regular, if()¼0 is equivalent to the relative compactness of.

An important example of MNC is the Hausdorff MNC, which satisfies all the above properties:

ðÞ ¼inff": has a finite"-netg:

It should be mentioned that the Hausdorff MNC has also the following properties:

. Semihomogeneity: (t)4jtj () for any2 P(E) andt2R;

. in a separable Banach spaceE, ðÞ ¼limm!1sup_x2dðx,E_mÞ, where {Em} is a sequence of finite-dimensional subspaces of E such that E_m E_mþ1, m¼1, 2,. . . andS1

m¼1E_m¼ E.

Let Xbe a metric space.

Definition 2.4 A multivalued map (multimap)F:X! P(E) is said to be:

(i) upper semicontinuous (u.s.c) if F¹(V)¼{x2X:F(x) V} is an open subset ofXfor every open setV E;

(ii) closed if its graphF¼{(x,y) :y2 F(x)} is a closed subset ofX E;

(iii) compact if its rangeF(X) is relatively compact inE;

(iv) quasicompact if its restriction to any compact subset A Xis compact.

Definition 2.5 A multimapF:X E !K(E) is said to be condensing with respect to an MNC (-condensing) if for every bounded set X that is not relatively compact, we have

ðF ðÞÞ 64

¼ ðÞ:

Suppose that D E is a nonempty closed convex subset of E and UD is a nonempty relatively open subset ofD. We denote byU_Dand@UD, the closure and the boundary ofUDin the relative topology ofD, respectively.

Letbe a monotone nonsingular MNC inE. The application of the topological degree theory for condensing multimaps (see, e.g. [21]) yields the following fixed point theorems.

THEOREM 2.1 [21, Corollary 3.3.1] Let M be a bounded convex closed subset ofE and F:M !Kv(M) an u.s.c. -condensing multimap. Then the fixed point set FixF:¼{x:x2 F(x)}is a nonempty compact set.

The following theorem presents a version for multimaps of the classical Leray–Schauder alternative.

THEOREM 2.2 [21, Corollary 3.3.3] LetU_Dbe a bounded open neighbourhood of a2D andF :U_D!KvðDÞan u.s.c-condensing multimap satisfying the boundary condition

xa 62ðF ðxÞ aÞ

for all x2@UDand0541.Then FixF is a nonempty compact set.

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Definition 2.6 Let G: [0,T]!K(E) be a multifunction and p51.Then G is said to be

. L^p-integrable, if it admits a Bochner L^p-integrable selection. That is there exists g: [0,T]!E, g(t)2G(t) for a.e. t2[0,T] such that RT

0 kgðsÞk^p_Eds51;

. L^p-integrably bounded, if there exists a function2L^p([0,T]) such that kGðtÞk:¼supfkgk_E:g2GðtÞg4ðtÞ for a.e.t2 ½0,T:

The set of allL^p-integrable selections ofGwill be denoted byS^p_G.

The multifunctionGis calledmeasurableifG¹(V) measurable (with respect to the Lebesgue measure onJ:¼[0,T]) for any open subsetVof E. We say thatGis strongly measurableif there exists a sequenceGn: [0,T]!K(E),n¼1, 2,. . .of step multifunctions such that

n!1lim HðG_nðtÞ,GðtÞÞ ¼0 for a.e.t2 ½0,T, whereHis the Hausdorff metric inK(E).

It is known that, whenEis a separable Banach space, the notion of measurable multifunctions has some equivalences. More precisely, for a multifunction G: [0,T]!K(E), the following conditions are equivalent to each other (see, e.g. [21]):

(1) Gis measurable;

(2) for every countable dense set {xn} of E, the functions ’n: [0,T]!R, defined by

’nðtÞ ¼dðxn,GðtÞÞ are measurable;

(3) Ghas a Castaing presentation: there is a countable family {gn} of measurable selections ofGsuch that

[¹

n¼1

g_nðtÞ ¼GðtÞ for a.e.t2[0,T];

(4) Gis strongly measurable.

Furthermore, if G is measurable and L^p-integrably bounded, then it is L^p-integrable. If G is L^p-integrable on [0,d] for some p51, then G is also L¹-integrable. In this case, we have a multifunctiont°Rt

0GðsÞdsdefined by Zt

0

GðsÞds:¼ Zt

0

gðsÞdx:g2S¹_G

8t2 ½0,d:

The following -estimate ( is the Hausdorff MNC), which is similar to [21, Theorem 4.2.3] will be used in the sequel.

LEMMA 2.3 Assume that E is a separable Banach space. Let G: [0,d]! P(E) be L^p-integrable,L^p-integrably bounded multifunction such that

ðGðtÞÞ4qðtÞ

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for a.e. t2[0,d].Here,q2L^p_þð½0,dÞ.Then Zt

0

GðsÞds

4 Zt

0

qðsÞds

for all t2[0,d].In particular,if the multifunction G: [0,d]!K(E)is measurable and L^p-integrably bounded then the function (G())is integrable and,moreover,

Zt

0

GðsÞds

4 Zt

0

ðGðsÞÞds for all t2[0,d].

Now we consider the multimap F: [0,T] B E^N!Kv(E) in our problem (1.1)–(1.3).

Definition 2.7 We say thatFsatisfies theupper Carathe´odory conditions if

(1) the multifunction F(,,U) : [0,T]!Kv(E) admits a strongly measurable selection for each (,U)2 B E^N, and

(2) the multimapF(t,,) :B E^N!Kv(E) is u.s.c for a.e.t2[0,T].

The multimapF is said to beL^p-locally integrably bounded if for eachr40, there exists a function!r2L^p([0,T]) such that

kFðt,,UÞk ¼supfkzk_E:z2Fðt,,UÞg4!_rðtÞ for all (,U)2 B E^Nsatisfyingjj_Bþ kUk_EN4r.

Let CE(1,T) denote the linear topological space of functionsu: (1,T]!E satisfying that

u02 B and uj_½0,T2C^N1ð½0,T;EÞ, endowed with the seminorm

kuk_C_E_ð1,_TÞ¼ ju0j_Bþ kuk_C^N1_ð½0,_T_;EÞ: Foru2 C_E(1,T), consider the superposition multifunction

F:½0,T !KvðEÞ, FðtÞ ¼Fðt,ut,r^NuðtÞÞ:

By the axioms of the phase space, we see thatt°ut2 Bis a continuous function.

Further, the functionr^Nu: [0,T]!E^Nis continuous, too. ThenFisL^p-integrable providedFsatisfies upper Carathe´odory conditions and isL^p-locally bounded. The proof can be made in the same way as in [21, Theorem 1.3.5].

As the consequence, we can define on CE(1,T) the superposition multi- operatorPFby

P_FðuÞ ¼ f2L^pð0,T;EÞ: ðtÞ 2Fðt,u_t,r^NuðtÞÞfor a.e.t2 ½0,Tg:

We have the following property of weak closedness of P_F, whose proof can be proceeded as in [21, Lemma 5.1.1].

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LEMMA 2.4 Let{un}be a sequence inCE(1,T)converging to u2 CE(1,T)and suppose that the sequence {n} L^p(0,T;E), n2 PF(un) weakly converges to a function.Then2 P_F(u).

3. Existence results

In the sequel, let Ebe a separable Banach space. We give the definition of a mild solution for (1.1)–(1.3), in accordance with formula (2.2), as follows.

Definition 3.1 For a given 2(0,T], a function u2 CE(1,) is called a mild solution to problem (1.1)–(1.3) on the interval (1, ] if it satisfies the integral equation

uðtÞ ¼

’ðtÞ, fort40, X

N1 k¼0

t^k

k!eu_kþ 1 ðÞ

Zt

0

ðtsÞ¹ðsÞds fort2 ½0,, 8>

<

>: where2 PF(u).

We assume that the multimap F in problem (1.1)–(1.3) satisfies the following hypotheses:

(F1)F: [0,T] B E^N!Kv(E) satisfies the upper Carathe´odory conditions;

(F2)FisL^p-locally integrably bounded forp4_Nþ1¹ ;

(F3) there exists a functionk2L^p(0,T;E) such that for any bounded subsetsQ B andj E,j¼0,. . .,N1, we have

F t,Q,^N1Y

j¼0

_j

!!

4kðtÞ ðQÞ þX^N1

j¼0

ð_jÞ

! ,

where

ðQÞ ¼sup

40

ðQðÞÞ, ð3:1Þ

QðÞ ¼ fqðÞ:q2 Qg:

Remark 3.1 In the case E¼R^m, condition (F3) follows from (F2). In fact, the L^p-locally integrably boundedness of F implies that the set Fðt,Q,QN1

j¼0 _jÞ is bounded inR^mfor a.e.t2[0,T], and hence it is a precompact set.

If dim(E)¼ þ1, then a particular case of fulfilling (F3) is the following condition:

Fðt, ,Þ:B E^N!KvðEÞ

is completely continuous for a.e. t2[0,T], i.e. F(t,,) maps each bounded set in B E^Ninto a precompact set inE.

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For a given2(0,T], we set

S:L^pð0,;EÞ !Cð½0,;EÞ, SðÞðtÞ ¼ 1

ðÞ Zt

0

ðtsÞ¹ðsÞds, ð3:2Þ

uðtÞ ¼

’ðtÞ, ift40, X

N1 k¼0

t^k

k!euk, if 04t4, 8>

><

>>

: and

GðuÞ ¼uþS P_FðuÞ: ð3:3Þ One sees that u2 CE(1,) is a mild solution of problem (1.1)–(1.3) on interval (1,] if and only if it is a fixed point ofG. From now on, we can restrictGon the subsetD C^N1([0,];E) defined as

D ¼ fv2C^N1ð½0,;EÞ,vð0Þ ¼eu0¼’ð0Þg, ð3:4Þ by settingG(v)¼ G(v[’]), where

v½’ðtÞ ¼ ’ðtÞ, ift40, vðtÞ, if 04t4:

(

We first verify some features of S which will be used to obtain some important properties of the multioperator G. Define the following collection of operators Sj:L^p(0,;E)!C([0,];E),j¼0, 1,. . .,N1:

S₀¼S: ð3:5Þ

IfN41 then SjðÞðtÞ ¼ d^j

dt^jSðÞðtÞ ¼ 1 ðjÞ

Z t

0

ðtsÞ^j1ðsÞds, j¼1,. . .,N1: ð3:6Þ PROPOSITION 3.1 The operators Sj,j¼0, 1,. . .,N1have the following properties:

(S1)There exist constants Cj40,j¼0, 1,. . .,N1,such that kS_jðÞðtÞ S_jðÞðtÞk^p_E4C^p_j

Zt

0

kðsÞ ðsÞk^p_Eds, ,2L^pð0,;EÞ:

(S2)For each compact set K E and sequence{n} L^p(0,;E)such that{n(t)} K for a.e. t2[0,],the weak convergencen* 0implies Sj(_n)!Sj(0)in C([0,];E).

Proof

(i) By using the Ho¨lder inequality we have kSjðÞðtÞ SjðÞðtÞk_E4 1

ðjÞ Zt

0

ðtsÞ^j1kðsÞ ðsÞÞk_Eds 4 1

ðjÞ h Z ^t

0

ðtsÞ^ðj1Þ^p=ð^p1Þdsi^p1_ph Z ^t

0

kðsÞ ðsÞk^p_Edsi¹_p :

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Then

kSjðÞðtÞ SjðÞðtÞk^p_E4C^p_j Z t

0

kðsÞ ðsÞk^p_Eds, where

Cj¼

h p1 ðjÞp1

i^p1_p T^j¹^p ðjÞ: Notice that (j)p140,j¼0,. . .,N1 sincep4_Nþ1¹ .

(ii) To prove (S2), notice that, without loss of generality, {n(t)} E⁰ for all t2[0,], where E⁰¼spK is the separable Banach space spanned by the compact setK. Moreover, it is clear also that {S_j(_n)(t)} E⁰for allt2[0,] andj¼0, 1,. . .,N1. Then, applying Lemma 2.3 we obtain

fS_jð Þ_n ð Þgt

1

ðjÞ Zt

0

ðtsÞ^j1 ðf_nðsÞgÞds¼0:

Hence, the sequencefSjðnÞðtÞg¹_n¼1 Eis relatively compact for everyt2[0,].

On the other hand, we have kSjðnÞðt2Þ SjðnÞðt1Þk_E

¼ 1

ðjÞ Z t2

0

ðt₂sÞ^j1_nðsÞds Zt1

0

ðt₁sÞ^j1_nðsÞds

E

4 1 ðjÞ

Zt2

t1

ðt2sÞ^j1nðsÞds

E

þ 1

ðjÞ Zt1

0

½ðt₂sÞ^j1 ðt₁sÞ^j1_nðsÞds

E

:

Since {n(s)} Kfor a.e. s2[0,], the right side of this inequality tends to zero as t2!t1uniformly with respect to n. So {Sj(n)} is an equicontinuous set. Thus from the Arzela–Ascoli theorem, we obtain that the sequence {Sj(n)} C([0,];E) is relatively compact.

Property (S1) ensures that eachSj:L^p(0,;E)!C([0,];E),j¼0,. . .,N1, is a bounded linear operator. Then it is continuous with respect to the topology of weak sequential convergence, that is the weak convergencen* 0ensuringSj(n)*Sj(0).

Taking into account that {Sj(n)} is relatively compact, we arrive at the conclusion thatSj(n)!Sj(0) strongly in C([0,];E). g We have the following technical result, whose proof constitutes of the arguments from [21, Theorem 4.2.1, Corollary 4.2.1, Remarks 4.2.1 and 4.2.2].

PROPOSITION 3.2 Let the sequence of functions {n} L^p(0,;E) be L^p-integrably bounded:

k_nðtÞk_E4ðtÞ, for all n¼1, 2,. . ., and a.e. t2 ½0,, where2L^p(0,).Assume that

ðfnðtÞgÞ4qðtÞ

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for a.e. t2[0,], where q2L^p(0,). Then for every 40 there exists a compact set K E,a set m [0,], meas(m)5and a sequence of functions G L^p(0,;E)with values in K,such that for every n51 there exists bn2Gfor which

knðtÞ bnðtÞk_E42qðtÞ þ, t2 ½0,nm:

In addition,one can choose the sequence{bn}so that bn¼0on mand this sequence is weakly compact.

Now using this result, we will prove the following proposition.

PROPOSITION 3.3 Let the sequence {n} L^p(0,;E) satisfy the conditions of Proposition3.2.Then we have

ðfS_jð_nÞðtÞgÞ42C_j Zt

0

jqðsÞj^pds

¹_p

, j¼1,. . .,N1, for any t2[0,].

Proof We follow the arguments in [21, Theorem 4.2.2] with some slight modifications. For any 40, choose 2(0,) such that for all m [0,], with meas(m)5, we have

Z

m

jðsÞj^pds5:

Taking m and {bn} corresponding to {n} from Proposition 3.2, we have by Proposition 3.1 that the sequences {Sj(bn)},j¼0,. . .,N1 are relatively compact in C([0,];X). Furthermore

kS_jð_nÞðtÞ S_jðb_nÞðtÞk^p_E 4C^p_j

Zt

0

knðsÞ bnðsÞk^p_Eds 4C^p_j

Z

½0,tnm

knðsÞ bnðsÞk^p_EdsþC^p_j Z

½0,t\m

knðsÞk^p_Eds 4C^p_j

Z

½0,tnm

½2qðsÞ þ^pdsþC^p_j Z

m

jðsÞj^pds 4C^p_j

Zt

0

j2qðsÞ þj^pdsþ

:

Therefore, the relatively compact setSjG(t) forms aCjðRt

0j2qðsÞ þj^pdsþÞ¹^p-net for the set {Sj(n)(t)}. This proves the proposition since40 is arbitrary. g Definition 3.2 A sequence {n} in L^p(0,;E) is called semicompact if it is L^p-integrably bounded and the set {n(t)} is relatively compact inEfor a.e.t2[0,].

PROPOSITION 3.4 Let {n} be a semicompact sequence in L^p(0,;E). Then {n} is weakly compact in L^p(0,;E).For each04j4N1, {Sj(n)}is relatively compact in C([0,];E).Moreover,ifn* 0then Sj(n)!Sj(0),j¼0,. . .,N1.

Proof The weak compactness of {_n} inL^p(0,;E) is the consequence of the results in [29, Corollary 3.4]. Since {n(t)} is relatively compact in E for a.e. t2[0,],

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from Proposition 3.3, we have that for eachj¼0,. . .,N1, the sequence {Sj(n)(t)}

is relatively compact inEfor a.e.t2[0,].

On the other hand, from the assumptions, there exists a function 2L^p(0,) such that

k_nðtÞk4ðtÞ, for alln¼1, 2,. . ., and t2 ½0,: Thanks to the Ho¨lder inequality, one thus has

kS_jð_nÞðt2Þ S_jð_nÞðt1Þk_E

¼ 1

ðjÞ Zt2

0

ðt₂sÞ^j1_nðsÞds Zt1

0

ðt₁sÞ^j1_nðsÞds

E

4 1 ðjÞ

Zt2

t1

ðt2sÞ^j1nðsÞds

E

þ 1

ðjÞ Zt1

0

½ðt₂sÞ^j1 ðt₁sÞ^j1_nðsÞds

E

4 1 ðjÞ

Zt2

t1

ðt2sÞ^ðj1Þ^p⁰ds _p¹0 Zt2

t1

jðsÞj^pds

¹_p

þ 1

ðjÞ Zt1

0

jðt2sÞ^j1 ðt1sÞ^j1j^p⁰ds

_p¹0 Zt1

0

jðsÞj^pds

¹_p

, where p⁰ is the conjugate of p. The last inequality implies that {S_j(_n)} is equicontinuous in C([0,];E), and then it is relatively compact in C([0,];E).

The last conclusion follows the same lines as in the proof of Proposition 3.1. g LEMMA 3.5 Suppose that F satisfies(F1)–(F3). Then the composition

G ¼uþS P_F is a closed multioperator with compact values.

Proof It suffices to prove the assertions forS PF. Let {vn} be such thatvn!vin D. Assume that _n2 P_F(v_n[’]) and z_n¼S(_n)2S(P_F(v_n[’])), z_n!z in C^N1([0,];E). We prove thatz2S(PF(v[’])). Since

fnðtÞg Fðt,ðvn½’Þ_t,r^NvnðtÞÞ,

we see that {n} is integrably bounded by (F2) and the following inequality holds by (F3)

ðf_nðtÞgÞ4kðtÞ ðfðv_n½’Þ_tgÞ þX^N1

j¼0

ðfv^ð_n^jÞðtÞgÞ

! :

For each j¼0,. . .,N1, the sequence fv^ð_n^j^Þg converges in C([0,];E). Then ðfv^ð_n^jÞðtÞgÞ ¼0 for a.e.t2[0,]. On the other hand,

ðfðvn½’Þ_tgÞ ¼sup

40

ðfvn½’ðtþÞgÞ 4sup

s2½0,t

ðfvnðsÞgÞ ¼0: ð3:7Þ

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Thus

ðfnðtÞgÞ ¼0, for a.e.t2 ½0,,

and then {n} is a semicompact sequence. By Proposition 3.4, we may assume, without loss of generality, that there exists2L^p(0,;E) such that

_n* and z_n¼Sð_nÞ !SðÞ ¼z:

Using Lemma 2.4, we obtain 2 PF(v[’]) and we deduce that z¼S()2S(PF(v[’])).

It remains to show that, for v2 D and {n} chosen in PF(v[’]), the sequence {S(n)} is relatively compact inC^N1([0,];E). Hypotheses (F2)–(F3) imply that {n} is semicompact. Using Proposition 3.4, we obtain that Sj(n), j¼0,. . .,N1, is relatively compact in C([0,];E). Therefore, {S(n)} is relatively compact in

C^N1([0,];E). The proof is completed. g

In order to prove the u.s.c. property ofS PF, we need the following result.

THEOREM 3.6 [21] Let X and Y be metric spaces and F:X!K(Y) a closed quasicompact multimap.Then F is u.s.c.

LEMMA 3.7 Let the conditions of Lemma3.5hold.Then the multioperatorGis u.s.c.

Proof In view of Theorem 3.6 and the result in Lemma 3.5, it suffices to check that Gis a quasicompact multimap. LetA Dbe a compact set. We prove thatG(A) is a relatively compact subset of C^N1([0,];E). Assume that {zn} G(A). Then zn¼uþS(n), where n2 PF(vn[’]), for a certain sequence {vn} A. Hypotheses (F2)–(F3) yield the fact that {n} is semicompact and then it is a weakly compact sequence inL^p(0,;E). Similar arguments as in the proof of Lemma 3.5 imply that {S(n)} is relatively compact in C^N1([0,];E). Thus we have the desired

conclusion. g

We are in a position to prove thatGis a condensing multioperator. We first need an MNC constructed suitably for our problem. Denote

:PðC^N1ð½0,;EÞÞ !R_þ, ðÞ ¼ sup

t2½0,

e^LtX^N1

j¼0

d^j dt^jðtÞ

: ð3:8Þ

Here,

d^j

dt^jðtÞ:¼ d^j

dt^jvðtÞ: v2

, and the constantLis chosen so that

‘:¼4X^N1

j¼0

Cj sup

t2 ½0,T

Z t

0

e^LpðtsÞk^pðsÞds

¹_p

51, ð3:9Þ

wherek() is the function from condition (F3).

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Further, consider

modC: PðC^N1ð½0,;EÞÞ !R_þ, modCðÞ ¼lim

!0sup

v2

04j4N1max max

jt1t2j5kv^ð^j^Þðt1Þ v^ð^j^Þðt2Þk, ð3:10Þ which is called the modulus of equicontinuity of in C^N1([0,];E). Consider the function

:PðC^N1ð½0,;EÞÞ !R²

þ, ðÞ ¼ max

D2DðÞððDÞ, modCðDÞÞ, ð3:11Þ whereD() is the collection of all countable subsets ofand the maximum is taken in the sense of the ordering in the coneR²

þ. By the same arguments as in [21], one can see thatis well-defined. That is, the maximum archives inD(), andis an MNC in the space C^N1([0,];E), which fulfils all properties in Definition 2.3 (see [21, Example 2.1.3] for details).

LEMMA 3.8 Under conditions of Lemma 3.5, the multioperator G:D!K(D) is -condensing.

Proof Let Dbe such that

ðGðÞÞ5ðÞ: ð3:12Þ We show that is relatively compact. Let (G()) be achieved on a sequence {zn} G(), i.e.

ðGðÞÞ ¼ ððfz_ngÞ, mod_Cðfz_ngÞÞ:

Then

zn¼uþSðnÞ, n2 P_Fðvn½’Þ wherefvng : Now inequality (3.12) implies

ðfzngÞ5ðfvngÞ: ð3:13Þ It follows from (F3) that

ðfnðsÞgÞ4kðsÞ ðfðvn½’Þ_sgÞ þX^N1

j¼0

ðfv^ð_n^j^ÞðsÞgÞ

!

for everys2[0,]. In view of (3.1), we have ðfðvn½’Þ_sgÞ ¼sup

40

ðfvn½’ðsþÞgÞ ¼ sup

2 ½0,s

ðfvnðÞgÞ:

Then

ðfnðsÞgÞ4kðsÞe^Ls sup

2 ½0,s

e^L ðfvnðÞgÞ þe^LsX^N1

j¼0

ðfv^ð_n^jÞðsÞgÞ

!

42e^LskðsÞðfvngÞ:

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Now the application of Proposition 3.3 forSj,j¼0,. . .,N1, yields ðfSjðnÞðtÞgÞ44Cj

Z t

0

e^pLsk^pðsÞds

¹_p

ðfvngÞ for anyt2 ½0,: Recalling that

S_jð_nÞðtÞ ¼ d^j

dt^jSð_nÞðtÞ ¼ d^j

dt^jz_nðtÞ d^j

dt^juðtÞ ¼z^ð_n^j^ÞðtÞ d^j

dt^juðtÞ, t50, we arrive at

e^Lt^N1X

j¼0

ðfz^ð_n^j^ÞðtÞgÞ ¼e^Lt^N1X

j¼0

z^ð_n^j^ÞðtÞ d^j dt^juðtÞ

¼e^Lt^N1X

j¼0

ðfSjðnÞðtÞgÞ44X^N1

j¼0

Cj

Zt

0

e^LpðtsÞk^pðsÞds

¹_p

ðfvngÞ:

Putting this relation together with (3.13), we obtain

ðfvngÞ4ðfzngÞ ¼ sup

t2 ½0,

e^LtX^N1

j¼0

ðfz^ð_n^j^ÞðtÞgÞ4‘ðfvngÞ:

Therefore({vn})¼0. This implies

ðfv^ð_n^j^ÞðtÞgÞ ¼0, j¼0,. . .,N1 for all t2 ½0,:

Using (F2)–(F3) again, one gets that {n} is a semicompact sequence. Then, Proposition 3.4 ensures that {Sj(n)} is relatively compact in C([0,];E) for j¼0,. . .,N1. This yields that {zn} is relatively compact in C^N1([0,];E). Hence mod_C({z_n})¼0. Finally,

ðÞ ¼ ð0, 0Þ:

g Now we go to the main results of this section. The following assertion is the local existence result for (1.1)–(1.3).

THEOREM 3.9 Suppose that conditions (F1)–(F3) are satisfied. Then there exists 2(0,T]such that problem(1.1)–(1.3)has at least one mild solution inCE(1,).

Proof We take an arbitrary number40 and denote 0¼ ðKþ1ÞþMj’j_B, K¼ max

t2 ½0,TKðtÞ, M¼ sup

t2 ½0,T

MðtÞ,

p⁰¼ p

p1, and Q¼ ZT

0

j!₀ðsÞj^pds

¹p

,

where!0is given in Definition 2.7. Taking into account thatp4_Nþ1¹ , it is obvious thatj¹_p40 for allj¼0,. . .,N1. Thus we can take2(0,T], which is small

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so that

1

ðjÞQ 1 ðj1Þp⁰þ1

_p¹0

^j¹^p4

N, ð3:14Þ

for allj¼0,. . .,N1.

Denote byBthe closed convex bounded subset of Ddefined by B¼ fu2 D :kuuk_CN1g:

Then foru2Bandvchosen from

GðuÞ ¼uþS PFðu½’Þ, we have

kvðtÞ uðtÞk_E4 1 ðÞ

Z t

0

ðtsÞ¹kFðs,u_s,r^NuðsÞÞkds 4 1

ðÞ Z t

0

ðtsÞ¹!0ðsÞds, ð3:15Þ for allt2(0,]. Here we use assumption (F2) with a note that

jusj_Bþ kr^NuðsÞk_E^N4KðsÞkuk_Cð½0,s;EÞþMðsÞj’jBþ kuk_C^N1_ð½0,_;EÞ

4ðKþ1Þkuk_CN1ð½0,;EÞþMj’j_B 40:

An application of the Ho¨lder inequality to (3.15) gives kvðtÞ uðtÞk_E4 1

ðÞ Zt

0

ðtsÞ^ð1Þ^p⁰ds _p¹0 Zt

0

j!₀ðsÞj^pds

¹_p

4 1 ðÞQ

h 1 ð1Þp⁰þ1

i_p¹0

t¹^p4 N, thanks to (3.14). Similarly, forj¼1,. . .,N1, we have

d^j

dt^jvðtÞ d^j dt^juðtÞ

E

4 1 ðjÞ

Z _t

0

ðtsÞ^j1!0ðsÞds 4 1

ðjÞ Z t

0

ðtsÞ^ðj1Þ^p⁰ds _p¹0 Zt

0

j!0ðsÞj^pds

¹_p

4 1

ðjÞQ 1 ðj1Þp⁰þ1

_p¹0

t^j¹^p 4

N: So, the following estimate holds

X

N1 j¼0

sup

t2 ½0,

d^j

dt^jðvðtÞ uðtÞÞ

4,

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or equivalently,

kvuk_CN1ð½0,;EÞ4:

Therefore, G maps B into B. Hence, the proof is completed by invoking the

conclusion of Theorem 2.1. g

In order to get the global existence result, we need to replace assumption (F2) with a stronger one. Precisely, we impose the assumption that

(F2⁰) There exists a function!2L^p([0,T]) such that

kFðt,,UÞk:¼supfkfk_E: f2Fðt,,UÞg4!ðtÞð1þ jjBþ kUk_ENÞ, for all2 BandU 2E^N.

In addition, we need the following version of generalized Bellman–Gronwall inequality (see, e.g. [30]).

LEMMA 3.10 Assume that f(t),g(t)and y(t)are non-negative,integrable functions on [0,T]satisfying the integral inequality

yðtÞ4gðtÞ þ Zt

0

fðsÞyðsÞds, t2 ½0,T: Then we have

yðtÞ4gðtÞ þ Zt

0

expn Z ^t

s

fðÞdo

fðsÞgðsÞds, t2 ½0,T:

THEOREM 3.11 Under assumptions (F1), (F2⁰) and (F3), the set of solutions for problem(1.1)–(1.3) on the interval[0,T]is nonempty and compact.

Proof We apply Theorem 2.2 to prove that FixG is a nonempty compact set.

Combining the results of Lemmas 3.5, 3.7 and 3.8, it remains to check that, ifu2 D_T is such that

uu2ðGðuÞ uÞ

for2(0, 1], thenumust belong to a bounded set inC^N1([0,T];E). Indeed, we have the inequality by using (F2⁰) forj¼0,. . .,N1

d^j

dt^juðtÞ d^j dt^juðtÞ

E

4 ðjÞ

Zt

0

ðtsÞ^j1kFðs,us,r^NuðsÞÞkds 4 1

ðjÞ Zt

0

ðtsÞ^j1!ðsÞð1þ ju_sj_Bþ kr^NuðsÞk_ENÞds:

Using the fact that

jusj_Bþ kr^NuðsÞk_E^N4Mj’jBþ ðKþ1Þkuk_C^N1_ð½0,_s;EÞ, we have

d^j dt^juðtÞ

E

4kU0k_EN

X

N1 k¼0

T^k

k! þMj’j_Bþ1 ðjÞ

Zt

0

ðtsÞ^j1!ðsÞds þ Kþ1

ðjÞ Zt

0

ðtsÞ^j1!ðsÞkuk_CN1ð½0,s;EÞÞds

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forj¼0,. . .,N1. By the Ho¨lder inequality, we get d^j

dt^juðtÞ

E

4kU0k_E^NX^N1

k¼0

T^k k!

þMj’j_Bþ1 ðjÞ

h 1

ðj1Þp⁰þ1 i_p¹0

T^j¹^p ZT

0

j!ðsÞj^pds

¹p

þ Kþ1 ðjÞ

h 1

T^j¹^p Zt

0

j!ðsÞj^pkuk^p_CN1ð½0,s;EÞds

¹_p

:

Let

Cj¼ 1 ðjÞ

h 1

T^j¹^p, C¼maxfC_j: j¼0,. . .,N1g,

g0 ¼NkU0k_E^NX^N1

k¼0

T^k

k!þNCðMj’j_Bþ1Þ ZT

0

j!ðsÞj^pds

¹_p

, fðsÞ ¼ ½NCðKþ1Þ¹^p!ðsÞ, s2 ½0,T:

Then

kuk_C^N1_ð½0,_t;E_Þ4g0þ Z t

0

jfðsÞj^pkuk^p_CN1ð½0,s;EÞds

¹_p

:

This implies

vðtÞ42^pg^p₀þ2^p Zt

0

jfðsÞj^pvðsÞds,

wherevðtÞ ¼ kuk^p_CN1ð½0,t;EÞ. In accordance with Lemma 3.10, we obtain the estimate

vðtÞ ¼ kuk^p_CN1ð½0,t;EÞ42^pg^p₀ 1þ ZT

0

exp 2^p Z T

s

jfðÞj^pd

jfðsÞj^pds

for allt2[0,T]. The last inequality leads to the fact that kuk_CN1ð½0,T;EÞ4R0, where

R0¼2g0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ

ZT

0

exp 2^p Z T

s

jfðÞj^pd

jfðsÞj^pds

p

s

:

Finally, takinga¼u2 D_Tand applying Theorem 2.2 for U_D_T¼ fu2 D_T:kuk_C^N1_ð½0,_T;E_ÞRg

withR4R0, we conclude that FixGis nonempty compact set. g

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4. Properties of solution map

In this section we will study the continuous dependence of the solution set F of problem (1.1)–(1.3) on the initial data (’,U)2 B E^N. To attain these ends we will need two additional hypotheses on the phase spaceB.

It will be supposed that

(B4) there existsl40 such thatk (0)k4lk kB for all 2 B;

(B5) there exists m, 04m4þ1, such that for every sequence { n} B with k n 0kB!0 the sequence { n()} is relatively compact inEfor every2[m, 0].

It is easy to see that both spaces presented as examples in Section 2.2 satisfy these properties. In fact,l¼1 in both cases, andm¼ þ1in case (1) andm¼rin case (2).

Further, we assume that the multimap F: [0,T] B E^N!Kv(E) satisfies conditions (F1), (F2⁰) and the following slightly modified condition of -regularity:

(F3⁰) there exists a function k2L^p(0,T;E) such that for any bounded subsets Q Bandj E,j¼0,. . .,N1, we have

Fðt,Q,^N1Y

j¼0

jÞ

!

4kðtÞ sup

m440

ðQðÞÞ þX^N1

j¼0

ðjÞ

! :

Now, we consider in the spaceB E^NC^N1([0,T];E) the subset

D¼ fð ;eu0,. . .,euN1;vÞ: ð0Þ ¼eu0¼vð0Þ,eu1¼v⁰ð0Þ,. . .,euN1¼v^ðN1Þð0Þg, which is closed by (B4). We define a family of multioperators

:D! PðC^N1ð½0,T;EÞÞ by

ð ;eu₀,. . .,eu_N1;vÞ ¼euþS P_Fðv½ Þ, where

euðtÞ ¼X^N1

k¼0

t^k

k!euk, 04t4T:

It is clear that v2ð ;eu0,. . .,euN1;vÞ implies that the function v[ ]2 C_E(1,T) belongs to the solution setF( ,U), whereU¼ ðeu0,. . .,euN1Þ:

LEMMA 4.1 The multioperator is closed, i.e.,for any sequences {( _n,Un,vn)} D and wn2( n,Un,vn), the conditions k n ₀kB!0, kUnU0k_EN!0, kvnv0k_CN1!0, andkwnw0k_CN1!0imply w02( 0,U0,v0).

Proof Consider a sequence {fn} L^p(0,T;E) such that fn2 P_F(vn[ n]) and wn¼eu_nþSðfnÞ: ð4:1Þ Here,

eu_nðtÞ ¼X^N1

k¼0

t^k

k!ðeukÞ_n, 04t4T, andðeukÞ_n, k¼0,. . .,N1 are the components ofUn.

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From hypothesis (B3) it follows that the sequence {vn[ n]t} is uniformly bounded with respect tot2[0,T]. Then (F2⁰) implies that the sequence {fn} isL^p-integrably bounded.

Further, from condition (F3⁰) we get for a.e.t2[0,T]:

ff_nðtÞg

ð Þ4 F t, fv_n½ _n_tg,fU_ng 4kðtÞ sup

m440

fvn½ n_tðÞg

þ^N1X

j¼0

fðeujÞ_ng

!

¼kðtÞ sup

m440

fv_n½ _n_tðÞg

¼

kðtÞmax n

sup_044t ðfvnðÞgÞ, sup_tm4⁰₄₀ ðf nð⁰ÞgÞ o

, if 04t5m, kðtÞ sup

tm44t

fvnðÞg

ð Þ, ifm4t:

8<

:

Applying hypothesis (B5), we conclude that both values are vanishing and so the sequence {fn} is semicompact. By Proposition 3.4 and Lemma 2.4, we may assume, with loss of generality, that fn*f0, where f02 P_F(v0[ 0]). Applying again Proposition 3.4, and passing to the limit in (4.1), we obtain

w0¼eu₀þSðf0Þ,

concluding the proof. g

Now we are in position to prove the main result of this section. Denote bythe closed subset of the spaceB E^Ndefined by

:¼ fð ,UÞ ¼ ð ;eu0,. . .,euN1Þ: ð0Þ ¼eu0g:

For (’,U)2, let F(’,U) CE(1,T) be the set of solutions of problem (1.1)–(1.3).

THEOREM 4.2 Under conditions (B1)–(B5), (F1), (F2⁰)and (F3⁰),the multimap _F: ! PðC_Eð1,TÞÞ

is upper semicontinuous.

Proof At first, we prove the upper semicontinuity of the multimap :!K(C^N1([0,T];E)) defined as

ð’,UÞ ¼ fv2ð’,U,vÞg:

Suppose to the contrary that there exist "0, and sequences {(’n,Un)} , with k’n’0kB!0,kU_nU₀k_EN!0, {vn} C^N1([0,T];E),vn2(’n,Un) such that

v_n62W_"₀ðð’₀,U₀ÞÞ, n1, ð4:2Þ whereW_"₀ denotes the "0-neighbourhood of a set. We have

v_n2ð’_n,U_n,v_nÞ, n1, ð4:3Þ i.e.,

vn¼eu_nþSðfnÞ,