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Properties of the solutions set for a class of nonlinear evolution inclusions with nonlocal conditions

Jingrui Zhang1*, Yi Cheng23, Changqin Yuan2 and Fuzhong Cong2

Author Affiliations

1 School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

2 Fundamental Department, Aviation University of Air Force, Changchun, 130022, People’s Republic of China

3 Institute of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

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Boundary Value Problems 2013, 2013:15  doi:10.1186/1687-2770-2013-15


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/15


Received:23 October 2012
Accepted:18 January 2013
Published:5 February 2013

© 2013 Zhang et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the nonlocal problems for nonlinear first-order evolution inclusions in an evolution triple of spaces. Using techniques from multivalued analysis and fixed point theorems, we prove existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term with nonlocal conditions. Also, we prove the existence of extremal solutions and a strong relaxation theorem. Some examples are presented to illustrate the results.

MSC: 34B15, 34B16, 37J40.

Keywords:
evolution inclusions; nonlocal conditions; Leray-Schauder alternative theorem; extremal solutions

1 Introduction

In this paper, we examine the following nonlinear nonlocal problem:

{ x ˙ + A ( t , x ) + B x F ( t , x ) , a.e.  I = [ 0 , T ] , x ( 0 ) = φ ( x ) ,

where A : I × V V is a nonlinear map, B : V V is a bounded linear map, φ : H H is a continuous map and F : I × H 2 V is a multifunction to be given later. Concerning the function φ, appearing in the nonlocal condition, we mention here four remarkable cases covered by our general framework, i.e.:

φ ( x ) = x ( T ) ;

φ ( x ) = x ( T ) ;

φ ( x ) = 1 2 π 0 2 π x ( s ) d s ;

φ ( x ) = i = 1 n β i x ( t i ) , where 0 < t 1 < t 2 < < t n are arbitrary, but fixed and i = 1 n | β i | 1 .

Many authors have studied the nonlocal Cauchy problem because it has a better effect in the applications than the classical initial condition. We begin by mentioning some of the previous work done in the literature. As far as we know, this study was first considered by Byszewski. Byszewski and Lakshmikantham [1,2] proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when F is a single-valued function satisfying Lipschitz-type conditions. The fully nonlinear case was considered by Aizicovici and Lee [3], Aizicovici and McKibben [4], Aizicovici and Staicu [5], García-Falset [6], García-Falset and Reich [7], and Paicu and Vrabie [8]. All these studies were motivated by the practical interests of such nonlocal Cauchy problems. For example, the diffusion of a gas through a thin transparent tube is described by a parabolic equation subjected to a nonlocal initial condition very close to the one mentioned above, see [9]. For the nonlocal problems of evolution equations, in [10], Ntouyas and Tsamatos studied the case with compactness conditions. Subsequently, Byszewski and Akca [11] established the existence of a solution to functional-differential equations when the semigroup is compact and φ is convex and compact on a given ball. In [12], Fu and Ezzinbi studied neutral functional-differential equations with nonlocal conditions. Benchohra and Ntouyas [13] discussed second-order differential equations under compact conditions. For more details on the nonlocal problem, we refer to the papers of [14-18] and the references therein.

It is worth mentioning that many of these documents assume that a nonlocal function meets certain conditions of compactness and A is a strongly continuous semigroup of operators or accretive operators in studying the evolution equations or inclusions with nonlocal conditions. However, one may ask whether there are similar results without the assumption on the compactness or equicontinuity of the semigroup. This article will give a positive answer to this question. The works mentioned above mainly establish the existence of mild solutions for evolution equations or inclusions with nonlocal conditions. However, in the present paper, we consider the cases of a convex and of a nonconvex valued perturbation term in the evolution triple of spaces ( V H V ). We assume the nonlinear time invariant operator A to be monotone and the perturbation term to be multivalued, defined on I × H with values in V (not in H). We will establish existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term, which is new for nonlocal problems. Our approach will be based on the techniques and results of the theory of monotone operators, set-valued analysis and the Leray-Schauder fixed point theorem.

We pay attention to the existence of extreme solutions [19] that are not only the solutions of a system with a convexified right-hand side, but also they are solutions of the original system. We prove that, under appropriate hypotheses, such a solution set is dense and codense in the solution set of a system with a convexified right-hand side (‘bang-bang’ principle). Our results extend those of [20] and are similar to those of [21] in an infinite dimensional space. Furthermore, the process of our proofs is much shorter, and our conditions are more general. Finally, some examples are also given to illustrate the effectiveness of our results.

The paper is divided into five parts. In Section 2, we introduce some notations, definitions and needed results. In Section 3, we present some basic assumptions and main results, the proofs of the main results are given based on the Leray-Schauder alternative theorem. In Section 4, the existence of extremal solutions and a relaxation theorem are established. Finally, two examples are presented for our results in Section 5.

2 Preliminaries

In this section we recall some basic definitions and facts from multivalued analysis which we will need in what follows. For details, we refer to the books of Hu and Papageorgiou [22] and Zeidler [23]. Let I = [ 0 , T ] , ( I , Σ ) be the Lebesgue measurable space and X be a separable Banach space. Denote

P w ( f ) k c ( X ) = { A X : nonempty, weakly (closed) compact and convex } .

Let A P f ( X ) , x X , then the distance from x to A is given by d ( x , A ) = inf { | x a | : a A } . A multifunction F : I P f ( X ) is said to be measurable if and only if, for every z X , the function t d ( z , F ( t ) ) = inf { z x : x F ( t ) } is measurable. A multifunction F : I 2 X { } is said to be graph measurable if Gr F = { ( t , x ) : x F ( t ) } Σ × B ( X ) with B ( X ) being the Borel σ-field of X. On P f ( X ) we can define a generalized metric, known in the literature as the ‘Hausdorff metric’, by setting

h ( A , B ) = max { sup a A d ( a , B ) , sup b B d ( b , A ) }

for all A , B P f ( X ) .

It is well known that ( P f ( X ) , h ) is a complete metric space and P f c ( X ) is a closed subset of it. When Z is a Hausdorff topological space, a multifunction G : Z P f ( X ) is said to be h-continuous if it is continuous as a function from Z into ( P f ( X ) , h ) .

Let Y, Z be Hausdorff topological spaces and G : Y 2 Z { ϕ } . We say that G ( ) is ‘upper semicontinuous (USC)’ (resp., ‘lower semicontinuous (LSC)’) if for all C Z nonempty closed, G ( C ) = { y Y : G ( y ) C ϕ } (resp., G + ( C ) = { y Y : G ( y ) C } ) is closed in Y. A USC multifunction has a closed graph in Y × Z , while the converse is true if G is locally compact (i.e., for every y Y , there exists a neighborhood U of y such that F ( U ) ¯ is compact in Z). A multifunction which is both USC and LSC is said to be ‘continuous’. If Y, Z are both metric spaces, then the above definition of LSC is equivalent to saying that for all z Z , y d Z ( z , G ( y ) ) = inf { d Z ( z , v ) : v G ( y ) } is upper semicontinuous as R + -valued function. Also, lower semicontinuity is equivalent to saying that if y n y in Y as n , then

G ( y ) lim ̲ G ( y n ) = { z Z : lim d Z ( z , G ( y n ) ) = 0 } = { z Z : z = lim z n , z n G ( y n ) , n 1 } .

Let I = [ 0 , T ] . By L 1 ( I , X ) w , we denote the Lebesgue-Bochner space L 1 ( I , X ) equipped with the norm g w = sup { t t g ( s ) d s : 0 t t T } , g L 1 ( I , X ) . A set D L p ( I , X ) is said to be ‘decomposable’ if for every g 1 , g 2 D and for every J I measurable, we have χ J g 1 + χ J c g 2 D .

Let H be a real separable Hilbert space, V be a dense subspace of H having structure of a reflexive Banach space, with the continuous embedding V H V , where V is the topological dual space of V. The system model considered here is based on this evolution triple. Let the embedding be compact. Let , denote the pairing of an element x V and an element y V . If x , y H , then , = ( , ) , where ( , ) is the inner product on H. The norm in any Banach space X will be denoted by X . Let 1 < q p < be such that 1 p + 1 q = 1 . We denote L p ( I , V ) by X. Then the dual space of X is L q ( I , V ) and is denoted by  X . For p, q satisfying the above conditions, from reflexivity of V that both X and X are reflexive Banach spaces (see Zeidler [23], p.411]).

Define W p q ( I ) = { x : x X , x ˙ X } , where the derivative in this definition should be understood in the sense of distribution. Furnished with the norm x W p q = x X + x ˙ X , the space ( W p q ( I ) , x W p q ) becomes a Banach space which is clearly reflexive and separable. Moreover, W p q ( I ) embeds into C ( I , H ) continuously (see Proposition 23.23 of [23]). So, every element in W p q ( I ) has a representative in C ( I , H ) . Since the embedding V H is compact, the embedding W p q ( I ) L p ( I , H ) is also compact (see Problem 23.13 of [23]). The pairing between X and X is denoted by , . By ‘⇀’ we denote the weakly convergence. The following lemmas are still needed in the proof of our main theorems.

Lemma 2.1 (see [24])

IfXis a Banach space, C X is nonempty, closed and convex with 0 C and G : C P k c ( C ) is an upper semicontinuous multifunction which maps bounded sets into relatively compact sets, then one of the following statements are true:

(i) the set Γ = { x C : x λ G ( x ) , λ ( 0 , 1 ) } is unbounded;

(ii) the G ( ) has a fixed point, i.e., there exists x C such that x G ( x ) .

Let X be a Banach space and let L 2 ( I , X ) be the Banach space of all functions u : I X which are Bochner integrable. D ( L 2 ( I , X ) ) denotes the collection of nonempty decomposable subsets of L 2 ( I , X ) . Now, let us state the Bressan-Colombo continuous selection theorem.

Lemma 2.2 (see [25])

LetXbe a separable metric space and let F : X D ( L 2 ( I , X ) ) be a lower semicontinuous multifunction with closed decomposable values. ThenFhas a continuous selection.

Let X be a separable Banach space and C ( I , X ) be the Banach space of all continuous functions. A multifunction F : I × X P w k c ( X ) is said to be Carathéodory type if for every x X , F ( , x ) is measurable, and for almost all t I , F ( t , ) is h-continuous (i.e., it is continuous from X to the metric space ( P f ( X ) , h ) , where h is a Hausdorff metric).

Let M C ( I , X ) , a multifunction F : I × X P w k c ( X ) is called integrably bounded on M if there exists a function λ : I R + such that for almost all t I , sup { y : y F ( t , x ( t ) ) , x ( ) M } λ ( t ) . A nonempty subset M 0 C ( I , X ) is called σ-compact if there is a sequence { M k } k 1 of compact subsets M k such that M 0 = k 1 M k . Let M 0 M be such that M 0 is dense in M and σ-compact. The following continuous selection theorem in the extreme point case is due to Tolstonogov [26].

Lemma 2.3 (see [26])

Let the multifunction F : I × X P w k c ( X ) be Carathéodory type and integrably bounded. Then there exists a continuous function g : M L p ( I , X ) such that for almost all t I , if x ( ) M 0 , then g ( x ) ( t ) ext F ( t , x ( t ) ) , and if x ( ) M M 0 , then g ( x ) ( t ) ext ¯ F ( t , x ( t ) ) .

3 Main results

Let I = [ 0 , T ] , consider the following evolution inclusions:

x ˙ + A ( t , x ) + B x F ( t , x ) , a.e.  I , x ( 0 ) = φ ( x ) , (3.1)

where A : I × V V is a nonlinear map, B : V V is a bounded linear map, φ : H H is a continuous map and F : I × H 2 V is a multifunction satisfying some conditions mentioned later.

Definition 3.1 A function x W p q ( I ) is called a solution to the problem (3.1) iff

x ˙ ( t ) , v + A ( t , x ( t ) ) , v + B x ( t ) , v = f ( t ) , v ,

where x ( 0 ) = φ ( x ) , f ( t ) F ( t , x ( t ) ) for all v V and almost all t I .

We will need the following hypotheses on the data problem (3.1).

(H1) A : I × V V is an operator such that

(i) t A ( t , x ) is measurable;

(ii) for each t I , the operator A ( t , ) : V V is uniformly monotone and hemicontinuous, that is, there exists a constant C 1 > 0 (independent of t) such that

A ( t , x 1 ) A ( t , x 2 ) , x 1 x 2 C 1 x 1 x 2 H p

for all x 1 , x 2 V , and the map s A ( t , x + s z ) , y is continuous on [ 0 , 1 ] for all x , y , z V ;

(iii) there exist a constant C 2 > 0 , a nonnegative function a ( ) L q ( I ) and a nondecreasing continuous function η ( ) L q ( I ) such that A ( t , x ) V a ( t ) + C 2 η ( x V ) for all x V , a.e. on I;

(iv) there exist C 3 > 0 , C 4 > 0 , b ( ) L 1 ( I ) such that

A ( t , x ) , x C 3 x V p C 4 x V p 1 + 1 2 T x ( 0 ) 2 b ( t ) a.e.  I , x V ,

or

A ( t , x ) , x C 3 x V p C 4 x V p 1 b ( t ) a.e.  I , x V , p > 2 .

(H2) F : I × H P k ( V ) is a multifunction such that

(i) ( t , x ) F ( t , x ) is graph measurable;

(ii) for almost all t I , x F ( t , x ) is LSC;

(iii) there exist a nonnegative function b 1 ( ) L q ( I ) and a constant C 5 > 0 such that

| F ( t , x ) | = sup { f V : f F ( t , x ) } b 1 ( t ) + C 5 x H k 1 x V  a.e.  I ,

where 1 k < p .

(H3)

(i) B : V V is a bounded linear self-adjoint operator such that ( B x , x ) 0 for all x V , a.e. on I;

(ii) there exists a continuous function φ : L p ( I , H ) H such that

φ ( u ) φ ( v ) u v C ( I , H ) u , v C ( I , H ) ,

and φ ( 0 ) = 0 .

It is convenient to rewrite the system (3.1) as an operator equation in W p q ( I ) . For x X , we get

A ( x ) ( t ) = A ( t , x ) , B ( x ) ( t ) = B x ( t ) , F ( x ) ( t ) = F ( t , x ( t ) ) , t I .

It follows from Theorem 30.A of Zeidler [23] that the operator A : X X is bounded, monotone, hemicontinuous and coercive. By using the same technique, one can show that the operator F : L p ( I , H ) X is bounded and satisfies

| F ( t , x ) | = sup { f X : f F ( t , x ) } M 1 ˆ + M 2 ˆ x L p ( I , H ) k 1

for some constants M 1 ˆ , M 2 ˆ > 0 and all x L p ( I , H ) .

We define

L u = u ˙ , D ( L ) = { u W p q ( I ) : u ( 0 ) = ξ H } , (3.2)

where u ˙ stands for the generalized derivative of u, i.e.,

0 T u ˙ ( t ) v ( t ) d t = 0 T u ( t ) v ˙ ( t ) d t v ( ) C 0 ( I ) .

For the proofs of main results, we need the following lemma.

Lemma 3.1Let V H V be an evolution triple and let X = L p ( I , V ) , where 1 < p < and 0 < T < . Then the linear operator L : D ( L ) X X defined by (3.2) is maximal monotone.

Proof In the sequel we will show that L is maximal monotone. To prove this, suppose that ( v , w ) X × X and

0 w L u , v u u D ( L ) .

We have to show that v D ( L ) and w = L v , i.e., w = v ˙ . Due to the arbitrariness of u, we choose u = ϕ z + ξ , where ϕ C 0 ( I ) , ξ = u 0 and z V . Then u ˙ = ϕ ˙ z , so L u , u = 0 . From w L u , v u 0 , we obtain that

0 w , v ξ 0 T φ ˙ v + φ w , z d t z V .

By the arbitrariness of z, one has that

0 T ( φ ˙ v + φ w ) d t = 0 φ C 0 ( I ) .

Hence, w = v ˙ . Since v W p q ( I ) , then w X . It remains to show that v D ( L ) . Using the integration by parts formula for functions in W p q ( I ) (see Zeidler [23], Proposition 23.23), we obtain from (3.2) that

0 v ˙ u ˙ , v u = 1 2 ( v ( T ) u ( T ) 2 v ( 0 ) u ( 0 ) 2 ) u D ( L ) . (3.3)

Choose a set of functions ( a n ) n 1 in H such that T a n v ( T ) ξ as n . For ξ H , let u ( t ) = t a n + ξ , then u D ( L ) . By (3.3), we have v ( 0 ) = u ( 0 ) = ξ as n . Hence, v D ( L ) . This completes the proof. □

Theorem 3.1If hypotheses (H1), (H2) and (H3) hold, the problem (3.1) has at least one solution.

Proof The process of proof is divided into four parts.

Step 1. We claim that the equation

x ˙ + A ( t , x ) + B x = f ( t ) a.e.  I , x ( 0 ) = ( 1 ϵ ) φ ( x ) (3.4)

has only one solution.

Firstly, for every ϵ ( 0 , 1 ] , y X and f X , we claim that the equation

x ˙ + A ( t , x ) + B x = f ( t ) a.e.  I , x ( 0 ) = ( 1 ϵ ) φ ( y ) (3.5)

has only one solution. By (H1) and (H3), it is easy to check that ( A + B ) is bounded, monotone, hemicontinuous and coercive. Moreover, by Lemma 3.1, L is a linear maximal monotone operator. Therefore, R ( L + A + B ) = V , i.e., L + ( A + B ) is surjective (see [23], p.868]). The uniqueness is clear. Hence, for the Cauchy problem (3.5) has a unique x y ( t ) W p q ( I ) . By W p q ( I ) C ( I , H ) , then the operator P : W p q ( I ) W p q ( I ) is defined as follows:

P ( y ) = 0 t x ˙ ( s ) d s + ( 1 ϵ ) φ ( y ) .

By (3.5), we have

P ( y 1 ) P ( y 2 ) + 0 t A ( s , x 1 ) A ( s , x 2 ) d s = ( 1 ϵ ) φ ( y 2 ) ( 1 ϵ ) φ ( y 1 )

for all y 1 , y 2 W p q ( I ) . Take an inner product over (3.5) with x 1 x 2 , then

( P ( y 1 ) P ( y 2 ) , x 1 x 2 ) + 0 t ( A ( s , x 1 ) A ( s , x 2 ) , x 1 x 2 ) d s = ( 1 ϵ ) ( φ ( y 2 ) φ ( y 1 ) , x 1 x 2 ) .

By (H1)(ii), we have

P ( y 1 ) P ( y 2 ) 2 ( 1 ϵ ) φ ( y 2 ) φ ( y 1 ) x 1 x 2 .

Hence,

P ( y 1 ) P ( y 2 ) C ( I , H ) ( 1 ϵ ) φ ( y 2 ) φ ( y 1 ) C ( I , H ) ( 1 ϵ ) y 2 y 1 C ( I , H ) . (3.6)

Invoking the Banach fixed point theorem, the operator P has only one fixed point x ϵ = P ( x ϵ ) , i.e., x ϵ is the uniform solution of (3.4).

Therefore, we define L ϵ : W p q ( I ) X as L ϵ x = x ˙ + A ( t , x ) + B x and x ( 0 ) = ( 1 ϵ ) φ ( x ) . By Step 1, we have L ϵ : W p q ( I ) X is one-to-one and surjective, and so L ϵ 1 : X W p q ( I ) is well defined.

Step 2. L ϵ 1 : X L p ( I , H ) is completely continuous.

We only need to show that L ϵ 1 is continuous and maps a bounded set into a relatively compact set. We claim that L ϵ : W p q ( I ) X is continuous. In fact, let { x n } n 1 W p q ( I ) such that x n x as n . From (H1)(ii) and (H3), we infer that x n x , A ( x n ) A ( x ) , B x n B x , a.e. I as n . Obviously, φ ( x n ) φ ( x ) . Therefore, L ϵ : W p q ( I ) X is continuous and L ϵ 1 is continuous.

Let K X be a bound set, for any f K , there is a priori bound in W p q ( I ) for the possible solution x ( t ) = L ϵ 1 f of (3.4). Then

x ˙ + A ( t , x ) + B x = f ( t ) a.e.  I . (3.7)

It follows that

x ˙ , x + A x , x + B x , x = f ( t ) , x . (3.8)

By (H1)(iv),

A u , u C 3 u X p C 4 u X p 1 + 1 2 u ( 0 ) 2 b L 1 ,

or

A u , u C 3 u X p C 4 u X p 1 b L 1

with p > 2 . But

u ˙ , u = u ( T ) 2 u ( 0 ) 2 , f , u f X u X .

Therefore,

C 3 x X p C 4 x X p 1 + f X x X + b L 1 ,

or

C 3 x X p C 4 x X p 1 + f X x X + b L 1 + x X 2 ,

with p > 2 . So, there exists an M 1 > 0 such that x X M 1 . Because of the boundedness of operators A, B, we obtain that there exists an M 2 > 0 such that x ˙ X M 2 . Hence, x W p q M for some constant M > 0 . Therefore, we have L ϵ 1 ( K ) is bounded in W p q ( I ) . But W p q ( I ) is compactly embedded in L p ( I , H ) . Therefore, L ϵ 1 ( K ) is relatively compact in L p ( I , H ) .

Let N ˆ : L p ( I , H ) 2 X be a multivalued Nemitsky operator corresponding to F and N ˆ was defined by N ˆ ( x ) = { v X : v ( t ) F ( t , x ( t ) ) } a.e. on I.

Step 3. N ˆ ( ) has nonempty, closed, decomposable values and is LSC.

The closedness and decomposability of the values of N ˆ ( ) are easy to check. For the nonemptiness, note that if x L p ( I , H ) , by the hypothesis (H2)(i), ( t , x ) F ( t , x ) is graph measurable, so we apply Aumann’s selection theorem and obtain a measurable map v : I V such that v ( t ) F ( t , x ( t ) ) a.e. on I. By the hypothesis (H2)(iii), v X . Thus, for every x L p ( I , H ) , N ˆ ( x ) . To prove the lower semicontinuity of N ˆ ( ) , we only need to show that every u X , x d ( u , N ˆ ( x ) ) is a USC R + -valued function. Note that

d ( u , N ˆ ( x ) ) = inf { u v X : v N ˆ ( x ) } = inf { [ 0 T u ( t ) v ( t ) V q d t ] 1 / q : v N ˆ ( x ) } = { 0 T inf { u ( t ) v ( t ) V q : v N ˆ ( x ) } d t } 1 / q = { 0 T [ d ( u ( t ) , F ( t , x ( t ) ) ) ] q d t } 1 / q

(see Hiai and Umegaki [27] Theorem 2.2). We will show that for every λ 0 , the superlevel set U λ = { x L p ( I , H ) : d ( u , N ˆ ( x ) ) λ } is closed in L p ( I , H ) . Let { x n } n 1 U λ and assume that x n x in L p ( I , H ) . By passing to a subsequence if necessary, we may assume that x n ( t ) x ( t ) a.e. on I as n . By (H2)(ii), x d ( u ( t ) , F ( t , x ) ) is an upper semicontinuous R + -valued function. So, via Fatou’s lemma, we have

λ q lim ¯ [ d ( u , N ˆ ( x n ) ) ] q = lim ¯ 0 T [ d ( u ( t ) , F ( t , x n ( t ) ) ) ] q d t 0 T lim ¯ [ d ( u ( t ) , F ( t , x n ( t ) ) ) ] q d t 0 T [ d ( u ( t ) , F ( t , x ( t ) ) ) ] q d t = [ d ( u , N ˆ ( x ) ) ] q .

Therefore, x U λ and this proves the LSC of N ˆ ( ) . By Lemma 2.2, we obtain a continuous map f : L p ( I , H ) X such that f ( x ) N ˆ ( x ) . To finish our proof, we need to solve the fixed point problem: x = L ϵ 1 f ( x ) .

Since the embedding V H is compact, the embedding W p q ( I ) L p ( I , H ) is compact. That is, x n x in L p ( I , H ) whenever x n x in W p q ( I ) . By using the above relation and the continuity of f, we have f ( x n ) f ( x ) in X whenever x n x in W p q ( I ) . So, L ϵ 1 f : L p ( I , H ) L p ( I , H ) is compact.

Step 4. We claim that the set Γ = { x L p ( I , H ) : x = σ L ϵ 1 f ( x ) , σ ( 0 , 1 ) } is bounded.

Let x Γ , then we have L ϵ ( x σ ) = f ( x ) . Note that

x σ ˙ , x σ + A ( x σ ) , x σ + B x σ , x σ = f ( x ) , x σ .

By (H1)(iv) and (H3)(i), one has that

A u , u + B x , x C 3 u X p C 4 u X p 1 + 1 2 u ( 0 ) 2 b L 1 , (3.9)

or

A u , u + B x , x C 3 u X p C 4 u X p 1 b L 1 (3.10)

with p > 2 . By using the integration by parts formula, we have

A ( x σ ) , x σ + B x σ , x σ = f ( x ) , x σ x σ ˙ , x σ 1 σ { 0 T f ( t , x ) V q d t } 1 / q { 0 T x V p d t } 1 / p + 1 2 σ 2 φ ( x ) 2 1 σ { 0 T ( h ( t ) + x H k 1 ) q d t } 1 / q x X + 1 2 σ 2 φ ( x ) 2 2 σ { 0 T | h ( t ) | q + x H q ( k 1 ) d t } 1 / q x X + 1 2 σ 2 φ ( x ) 2 2 σ { ( 0 T | h ( t ) | q ) 1 / q + ( 0 T x H q ( k 1 ) d t ) 1 / q } x X + 1 2 σ 2 φ ( x ) 2 γ 1 x X + γ 2 x X k + 1 2 σ 2 φ ( x ) 2 , (3.11)

where γ 1 , γ 2 > 0 . By (3.9), (3.10) and (3.11), if 1 k < p , then we have

x X p β 1 x X p 1 + β 2 x X k + β 3 x X + β 4 b L 1 . (3.12)

If 1 k < p , p > 2 , then we have

x X p β 1 x X p 1 + β 2 x X k + β 3 x X + β 4 b L 1 + β 5 x X 2 . (3.13)

Thus, by virtue of the inequalities (3.12) and (3.13), we can find a constant M 1 > 0 such that x X M 1 for all x Γ . From the boundedness of operators A, B and f, and the continuous embedding X L p ( I , H ) , we obtain A ( x ) X M 2 , B x X M 3 and f ( x ) X M 4 for some constants M 2 > 0 , M 3 > 0 , M 4 > 0 and all x Γ . Therefore,

x ˙ X A ( x ) X + B x X + f ( x ) X M 2 + M 3 + M 4 (3.14)

for all x Γ .

It follows from (3.14) that x W p q x X + x ˙ X M ˆ for some constant M ˆ > 0 . Hence, Γ is a bounded subset of W p q ( I ) . So, Γ is a bounded subset of L p ( I , H ) since the embedding W p q ( I ) L p ( I , H ) is compact.

Invoking the Leray-Schauder theorem, one has that there exists an x ϵ W p q ( I ) such that x ϵ = L ϵ 1 f ( x ϵ ) , i.e., x ϵ is a solution of the following problem:

x ˙ ϵ + A ( t , x ϵ ) + B x ϵ = f ( x ϵ ) , f ( x ϵ ) F ( t , x ϵ ) a.e.  I , x ϵ ( 0 ) = ( 1 ϵ ) φ ( x ϵ ) . (3.15)

Let ( ϵ n ) n 1 ( 0 , 1 ) and ϵ n 0 . For every n N , there exists an x n W p q ( I ) which is a solution of the following equations:

x ˙ n + A ( t , x n ) + B x n = f ( x n ) a.e.  I , f ( x n ) F ( t , x n ) , x n ( 0 ) = ( 1 ϵ n ) φ ( x n ) . (3.16)

By Step 3, we have that { x n } n 1 is uniformly bounded. By the boundedness of the sequence { x n } n 1 W p q ( I ) , it follows that the sequence { x ˙ n } n 1 X is uniformly bounded and passing to subsequence if necessary, we may assume that x ˙ n u in X . Evidently, u = x ˙ and x n x in W p q ( I ) . Since the embedding W p q ( I ) L p ( I , H ) is compact, then x n x in L p ( I , H ) . Hence, from the hypothesis (H2)(ii), we obtain f ( x n ) f ( x ) and f ( x ) F ( t , x ) . Since the operator A is hemicontinuous and monotone and B is a continuous linear operator, thus A ( x n ) A ( x ) , B x n B x in X as n . Therefore, we obtain x ˙ + A ( x ) + B x = f , f F ( t , x ) a.e. on I. Since x n ( t ) x ( t ) in L p ( I , H ) and φ : L p ( I , H ) H is continuous, then we have

x n ( 0 ) = ( 1 ϵ n ) φ ( x n ) φ ( x ) = x ( 0 ) .

Hence, x is a solution of (3.1). The proof is completed. □

Next, we consider the convex case, the assumption on F is as follows:

(H4) F : I × H P k c ( V ) is a multifunction such that

(i) ( t , x ) F ( t , x ) is graph measurable;

(ii) for almost all t I , x F ( t , x ) has a closed graph; and (H2)(iii) hold.

Theorem 3.2If hypotheses (H1), (H3) and (H4) hold, the problem (3.1) has at least one solution; moreover, the solution set is weakly compact in W p q ( I ) .

Proof The proof is as that of Theorem 3.1. So, we only present those particular points where the two proofs differ.

In this case, the multivalued Nemistsky operator N ˆ : L p ( I , H ) 2 X w has nonempty closed, convex values in X and is USC from L p ( I , H ) into X furnished with the weak topology (denoted by X w ). The closedness and convexity of the values of N ˆ ( ) are clear. To prove the nonemptiness, let x L p ( I , H ) and { s n } n 1 be a sequence of step functions such that s n ( t ) x ( t ) in H and s n ( t ) H x ( t ) H a.e. on I. Then by virtue of the hypothesis (H4)(i), for every n 1 , t F ( t , s n ( t ) ) admits a measurable selector v n ( t ) . From the hypothesis (H4)(iii), we have that v n X M 1 ˆ + M 2 ˆ x L p ( I , H ) k 1 , so { v n ( t ) } n 1 X is uniformly integrable. So, by the Dunford-Pettis theorem, and by passing to a subsequence if necessary, we may assume that v n v weakly in X . Then from Theorem 3.1 in [28], we have

v ( t ) conv ¯ lim ¯ { v n ( t ) } n 1 conv ¯ lim ¯ F ( t , s n ( t ) ) F ( t , x ( t ) ) a.e. on  I ,

the last inclusion being a consequence of the hypothesis (H4)(ii). So, v N ˆ ( x ) , which means that N ˆ ( ) is nonempty.

Next, we show that N ˆ ( ) is USC from L p ( I , H ) into X w . Let Ξ be a nonempty and weakly closed subset of X . Obviously, it is sufficient to show that the set

N ˆ 1 ( Ξ ) = { x L p ( I , H ) : N ˆ ( x ) Ξ ϕ }

is closed. Let { x n } n 1 N ˆ 1 ( Ξ ) and assume x n x in L p ( I , H ) . Passing to a subsequence, we can get that x n ( t ) x ( t ) a.e. on I. Let f n N ˆ ( x n ) Ξ , n 1 . Then by virtue of the hypothesis (H4)(iii), we have

f n X M 1 ˆ + M 2 ˆ x L p ( I , H ) k 1 .

So, by the Dunford-Pettis theorem, we may assume that f n f Ξ in X w . As before, we have

f ( t ) conv ¯ lim ¯ { f n ( t ) } n 1 conv ¯ lim ¯ F ( t , x n ( t ) ) F ( t , x ( t ) ) a.e. on  I ,

then f N ˆ ( x ) Ξ , i.e., N ˆ 1 ( Ξ ) is closed in L p ( I , H ) . This proves the upper semicontinuity of N ˆ ( ) from L p ( I , H ) into X w .

We consider the following fixed point problem:

x L ϵ 1 N ˆ ( x ) .

Recalling that L ϵ 1 : X L p ( I , H ) is completely continuous, we see that L ϵ 1 N ˆ : L p ( I , H ) P k c ( L p ( I , H ) ) is USC and maps bounded sets into relatively compact sets. We easily check that

Γ 1 = { x L p ( I , H ) : x λ L ϵ 1 N ˆ ( x ) , λ ( 0 , 1 ) }

is bounded, as a proof of Theorem 3.1. Invoking the Leray-Schauder fixed point theorem, one has that there exists an x ϵ W p q ( I ) such that x ϵ L ϵ 1 N ˆ ( x ϵ ) , i.e., x ϵ is a solution of the following problem:

x ˙ ϵ + A ( t , x ϵ ) + B x ϵ F ( t , x ϵ ) a.e.  I , x ϵ ( 0 ) = ( 1 ϵ ) φ ( x ϵ ) . (3.17)

Let ( ϵ n ) n 1 ( 0 , 1 ) and ϵ n 0 . For every n N , there exists an x n W p q ( I ) which is a solution of the following problem:

x ˙ n + A ( t , x n ) + B x n = f n ( t ) a.e.  I , f n ( t ) F ( t , x n ) , x n ( 0 ) = ( 1 ϵ n ) φ ( x n ) . (3.18)

By Step 3, { x n } n 1 is uniformly bounded. By the boundedness of the sequence { x n } n 1 W p q ( I ) , it follows that the sequence { x ˙ n } n 1 X is uniformly bounded and, passing to subsequence if necessary, we may assume that x ˙ n x ˙ in X . Thus, A ( x n ) A ( x ) , B x n B x in X as n . Evidently, there exists f n N ( x n ) , by virtue of the hypothesis (H4)(iv), we have that f n X M 1 ˆ + M 2 ˆ x L p ( I , H ) k 1 , so { f n ( t ) } n 1 X is uniformly integrable. So, by the Dunford-Pettis theorem and by passing to a subsequence if necessary, we may assume that f n f weakly in X . Therefore, we obtain x ˙ + A ( x ) + B x = f , f F ( t , x ) a.e. on I. Since x n ( t ) x ( t ) in L p ( I , H ) and φ : L p ( I , H ) H is continuous, then we have

x n ( 0 ) = ( 1 ϵ n ) φ ( x n ) φ ( x ) = x ( 0 ) .

Hence, evidently x is a solution of (3.1). As in the proof of Theorem 3.1, we have that | S | = sup { x W p q : x S } M ˆ , for some M ˆ > 0 . So, S W p q ( I ) is uniformly bounded. So, by the Dunford-Pettis theorem and by passing to a subsequence if necessary, we may assume that x n x weakly in W p q ( I ) . As before, we have

L ϵ ( x ) ( t ) conv ¯ lim ¯ { L ϵ x n ( t ) } n 1 conv ¯ lim ¯ F ( t , x n ( t ) ) F ( t , x ( t ) ) a.e. on  I .

Clearly, x ( 0 ) = φ ( x ) , then x S . Thus, S is weakly compact in W p q ( I ) . □

4 Relaxation theorem

Now, we prove the existence of extremal solutions and a strong relaxation theorem. Consider the extremal problem of the following evolution inclusion:

x ˙ + A ( t , x ) + B x ext F ( t , x ) a.e.  I , x ( 0 ) = φ ( x ) , (4.1)

where ext F ( t , x ) denotes the extremal point set of F ( t , x ) . We need the following hypothesis:

(H5) F : I × H P w k c ( H ) is a multifunction such that

(i) ( t , x ) F ( t , x ) is graph measurable;

(ii) for almost all t I , x F ( t , x ) is h-continuous; and (H2)(iii) holds.

Theorem 4.1If hypotheses (H1), (H3) and (H5) hold, then the problem (4.1) has at least one solution.

Proof Since S e S , as in the proof of Theorem 3.1, we obtain a priori bound for S e . We know that there exists M i > 0 , i = 1 , 2 such that x W p q < M 1 and x C ( I , H ) < M 2 for all x S e . Let ψ ( t ) = b 2 ( t ) + C 5 M 2 , ψ ( t ) L q + ( I ) . We may assume that | F ( t , x ) | ψ ( t ) a.e. on I for all x H . By Theorem 3.1, let L 0 = x ˙ + A ( x ) + B x , x ( 0 ) = φ ( x ) , then L 0 1 : W p q ( I ) X is well defined. So, let

W = { v L q ( I , H ) : v ( t ) H ψ ( t )  a.e. on  I } ,

then K ˆ = L 0 1 ( W ) W p q ( I ) is a compact convex subset in C ( I , H ) . Obviously, K ˆ is convex. We only need to show the compactness. Let { x n } n 1 K ˆ , then there exists h n W such that L 0 ( x n ) = h n , i.e., x ˙ n = h n A ( x n ) B x n . By the definition of W, W is uniformly bounded in L q ( I , H ) . By the Dunford-Pettis theorem, passing to a subsequence if necessary, we may assume that h n h in L q ( I , H ) for some h W . From the definition of W, we have

x n W p q = L 0 1 ( L 0 x n ) W p q = L 0 1 h n W p q M 1 .

Therefore, the sequence { x n } n 1 W p q ( I ) is bounded. Because of the compactness of the embedding W p q ( I ) L p ( I , H ) , we have that the sequence { x n } n 1 L p ( I , H ) is relatively compact. So, by passing to a subsequence if necessary, we may assume that x n x in L p ( I , H ) . Moreover, by the boundedness of the sequence { x n } n 1 W p q ( I ) , it follows that the sequence { x ˙ n } n 1 X is uniformly bounded and, passing to subsequence if necessary, we may assume that x ˙ n x ˙ in X . Since the embedding W p q ( I ) C ( I , H ) is continuous and W p q ( I ) L p ( I , H ) is compact, it follows that x n x in C ( I , H ) and x n x in L p ( I , H ) . Hence, x n x in H for all t I Λ , m ( Λ ) = 0 (m being the Lebesgue measure on R). Since A is hemicontinuous and monotone and B is a continuous linear operator, thus A ( x n ) A ( x ) , B x n B x in X and as n , we obtain x ˙ + A ( x ) + B x = h a.e. on I and x ( 0 ) = φ ( x ) . Note that

x ˙ n x ˙ + ( A ( t , x n ) A ( t , x ) ) = h n h ( B x n B x ) . (4.2)

Taking the inner product above with x n x and integrating from 0 to T, one can see that

0 T ( A ( t , x n ) A ( t , x ) , x n x ) d t = 0 T ( h n h , x n x ) d t 0 T ( B x n B x , x n x ) d t 0 T ( x ˙ n x ˙ , x n x ) d t 0 T ( h n h , x n x ) d t + x n ( 0 ) x ( 0 ) 2 = 0 T ( h n , x n x ) d t 0 T ( h , x n x ) d t + x n ( 0 ) x ( 0 ) 2 0 T φ H x n x H d t + 0 T h H x n x H d t + φ ( x n ) φ ( x ) 2 2 0 T φ H x n x H d t + φ ( x n ) φ ( x ) 2 0 as  n . (4.3)

By the hypothesis (H1)(iii), it follows that

0 T ( A ( t , x n ) A ( t , x ) , x n x ) d t C 1 0 T x n x H p d t 0 as  n . (4.4)

So, we can find τ I Λ such that

x n ( τ ) x ( τ ) H 0 as  n .

Using the integration by parts formula for functions in W p q ( I ) (see Zeidler [26], Proposition 23.23), for any t I , we have

x n ( t ) x ( t ) H 2 = x n ( τ ) x ( τ ) H 2 + 2 τ t ( x ˙ n ( s ) x ˙ ( s ) , x n ( s ) x ( s ) ) d s = x n ( τ ) x ( τ ) H 2 + 2 τ t ( h n ( s ) h ( s ) , x n ( s ) x ( s ) ) d s 2 τ t A ( x n ) ( s ) A ( x ) ( s ) , x n ( s ) x ( s ) d s 2 τ t B x n ( s ) B x ( s ) , x n ( s ) x ( s ) d s x n ( τ ) x ( τ ) H 2 + 2 0 T ( h n ( t ) h ( t ) , x n ( t ) x ( t ) ) d t x n ( τ ) x ( τ ) H 2 + 4 0 T | φ ( t ) | x n ( t ) x ( t ) H d t x n ( τ ) x ( τ ) H 2 + 4 φ ( t ) L q ( I ) x n ( t ) x ( t ) L p ( I , H ) . (4.5)

By (4.5), we see that

max t I x n ( t ) x ( t ) H 0 as  n .

So, x n ( t ) x ( t ) in C ( I , H ) . Since x = L 0 1 ( h ) with h W , we conclude that L 0 1 ( W ) C ( I , H ) is compact. From Lemma 2.3, we can find a continuous map f : K ˆ L q ( I , H ) such that f ( x ) ( t ) ext F ( t , x ( t ) ) a.e. on I for all x K ˆ . Then L 0 1 f is a compact operator. On applying the Schauder fixed point theorem, there exists an x K ˆ such that x = L 0 1 f ( x ) . This is a solution of (4.1), and so S e in W p q ( I ) . □

For the relation theorem of the problem (4.1), we need the following definition and hypotheses.

Definition 4.1 A Carathéodory function μ : I × R + R + is said to be a Kamke function if it is integrally bounded on the bounded sets, μ ( t , 0 ) 0 and the unique solution of the differential equation s ˙ ( t ) = μ ( t , s ( t ) ) , s ( 0 ) = 0 is s ( t ) 0 .

(H6) For each t I , there exists a Kamke function μ : I × R + R + such that

h ( F ( , x 1 ) , F ( , x 2 ) ) μ ( t , x 1 x 2 H 2 ) for all  x 1 , x 2 H

and (H5) hold.

Theorem 4.2If hypotheses (H1), (H3) and (H6) hold, then S e ¯ = S , where the closure is taken in C ( I , H ) .

Proof Let x S , then there exist f L q ( I , H ) and f ( x ) ( t ) F ( t , x ( t ) ) a.e. on I such that

x ˙ ( t ) + A ( t , x ( t ) ) + B x = f ( t , x ) , x ( 0 ) = φ ( x ) . (4.6)

As before, let W = { v L q ( I , H ) : v H ψ ( t )  a.e. on  I } , then K ˆ = L 0 1 ( W ) W p q ( I ) is a compact convex subset in C ( I , H ) . For every y K ˆ , we define the multifunction

Q ϵ ( t ) = { v F ( t , y ) : ( f v , x y ) 1 2 μ ( t , x y H 2 ) + ϵ } .

Clearly, for every t I , Q ϵ ( t ) , and it is graph measurable. On applying Aumann’s selection theorem, we get a measurable function v : I H such that v ( t ) Q ϵ ( t ) almost everywhere on I. So, we define the multifunction

R ϵ ( y ) = { v S F ( , y ) : ( f v , x y ) 1 2 μ ( t , x y H 2 ) + ϵ } .

We see that R ϵ : K ˆ 2 L q ( I , H ) has nonempty and decomposable values. It follows from Theorem 3 of [29] that R ϵ ( ) ¯ is LSC. Therefore, y R ϵ ( y ) ¯ is LSC and has closed and decomposable values. So, we apply Lemma 2.2 to get a continuous map f ϵ : K ˆ L q ( I , H ) such that f ϵ ( y ) R ϵ ( y ) ¯ for all y K ˆ . Invoking II-Theorem 8.31 of [22] (in [22], p.260]), we can find a continuous map g ϵ : K ˆ L q ( I , H ) such that g ϵ ( y ) ( t ) ext F ( t , y ) almost everywhere on I, and f ϵ ( y ) g ϵ ( y ) w ϵ for all y K ˆ . Now, let ϵ 0 and set f ϵ n = f ϵ , g ϵ n = g ϵ . Note that g ϵ n ( y ) H ψ ( t ) a.e. on I with ψ L q ( I ) , so we have g ϵ n f ϵ n in L q ( I , H ) . We consider the following problem:

x ˙ ( t ) + A ( x ) ( t ) + B x = g ϵ n ( x ) ( t ) , x ( 0 ) = φ ( x ) , (4.7)

where g ϵ n ( x ) ext R ϵ ( x ) . We see that L 0 1 g ϵ n : K ˆ K ˆ is a compact operator and by the Schauder fixed point theorem, we obtain a solution x ϵ n S e W p q ( I ) of (4.1). We see that the sequence { x ϵ n } n 1 K ˆ is uniformly bounded. So, by passing to a subsequence if necessary, we may assume that x ϵ n x ˆ in W p q ( I ) . From the proof of Theorem 4.1, we know that x ϵ n x ˆ in C ( I , H ) and x ˆ ( 0 ) = φ ( x ˆ ) . Note that L 0 x ϵ n L 0 x = g ϵ n ( x ϵ n ) f ( x ) . So, we have that

( x ˙ ϵ n ( t ) x ˙ ( t ) , x ϵ n ( t ) x ( t ) ) + ( A ( x ϵ n ) ( t ) A ( x ) ( t ) , x ϵ n ( t ) x ( t ) ) + ( B x ϵ n ( t ) B x ( t ) , x ϵ n ( t ) x ( t ) ) = ( g ϵ n ( x ϵ n ) ( t ) f ( x ) ( t ) , x ϵ n ( t ) x ( t ) ) .

However,

( A ( x ϵ n ) ( t ) A ( x ) ( t ) , x ϵ n ( t ) x ( t ) ) 0 , ( B x ϵ n ( t ) B x ( t ) , x ϵ n ( t ) x ( t ) ) 0 a.e.  I . (4.8)

Then

( x ˙ ϵ n ( t ) x ˙ ( t ) , x ϵ n ( t ) x ( t ) ) ( g ϵ n ( x ϵ n ) ( t ) f ( x ) ( t ) , x ϵ n ( t ) x ( t ) ) = ( g ϵ n ( x ϵ n ) ( t ) f ϵ n ( x ϵ n ) ( t ) , x ϵ n ( t ) x ( t ) ) + ( f ϵ n ( x ϵ n ) ( t ) f ( x ) ( t ) , x ϵ n ( t ) x ( t ) ) . (4.9)

By g ϵ n f ϵ n in L q ( I , H ) and x ϵ n x ˆ in L p ( I , H ) , we have that

( g ϵ n ( x ϵ n ) ( t ) f ϵ n ( x ϵ n ) ( t ) , x ϵ n ( t ) x ( t ) ) = ( g ϵ n ( x ϵ n ) ( t ) f ϵ n ( x ϵ n ) ( t ) , x ϵ n ( t ) x ˆ ( t ) ) + ( g ϵ n ( x ϵ n ) ( t ) f ϵ n ( x ϵ n ) ( t ) , x ˆ ( t ) x ( t ) ) 0 a.e.  I . (4.10)

Hence, there exists a constant N 0 > 0 , one has that

| ( g ϵ n ( x ϵ n ) ( t ) f ϵ n ( x ϵ n ) ( t ) , x ϵ n ( t ) x ( t ) ) | < ϵ

as n > N 0 . It follows that

1 2 d d t x ϵ n x H 2 = ( x ϵ n ˙ ( t ) x ˙ ( t ) , x ϵ n ( t ) x ( t ) ) ( g ϵ n ( x ϵ n ) ( t ) f ϵ n ( x ϵ n ) ( t ) , x ϵ n ( t ) x ( t ) ) + ( f ϵ n ( x ϵ n ) ( t ) f ( x ) ( t ) , x ϵ n ( t ) x ( t ) ) ( f ϵ n ( x ϵ n ) ( t ) f ( x ) ( t ) , x ϵ n ( t ) x ( t ) ) + ϵ 1 2 μ ( t , x ϵ n x H 2 ) + 2 ϵ .