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

  • Jingrui Zhang1Email author,

    Affiliated with

    • Yi Cheng2, 3,

      Affiliated with

      • Changqin Yuan2 and

        Affiliated with

        • Fuzhong Cong2

          Affiliated with

          Boundary Value Problems20132013:15

          DOI: 10.1186/1687-2770-2013-15

          Received: 23 October 2012

          Accepted: 18 January 2013

          Published: 5 February 2013

          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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equa_HTML.gif

          where A : I × V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq1_HTML.gif is a nonlinear map, B : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq2_HTML.gif is a bounded linear map, φ : H H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq3_HTML.gif is a continuous map and F : I × H 2 V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq4_HTML.gif 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq5_HTML.gif;

          • φ ( x ) = x ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq6_HTML.gif;

          • φ ( x ) = 1 2 π 0 2 π x ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq7_HTML.gif;

          • φ ( x ) = i = 1 n β i x ( t i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq8_HTML.gif, where 0 < t 1 < t 2 < < t n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq9_HTML.gif are arbitrary, but fixed and i = 1 n | β i | 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq10_HTML.gif.

          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 [1418] 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq11_HTML.gif). We assume the nonlinear time invariant operator A to be monotone and the perturbation term to be multivalued, defined on I × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq12_HTML.gif with values in V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq13_HTML.gif (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 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq14_HTML.gif, ( I , Σ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq15_HTML.gif 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 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equb_HTML.gif
          Let A P f ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq16_HTML.gif, x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq17_HTML.gif, then the distance from x to A is given by d ( x , A ) = inf { | x a | : a A } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq18_HTML.gif. A multifunction F : I P f ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq19_HTML.gif is said to be measurable if and only if, for every z X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq20_HTML.gif, the function t d ( z , F ( t ) ) = inf { z x : x F ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq21_HTML.gif is measurable. A multifunction F : I 2 X { } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq22_HTML.gif is said to be graph measurable if Gr F = { ( t , x ) : x F ( t ) } Σ × B ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq23_HTML.gif with B ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq24_HTML.gif being the Borel σ-field of X. On P f ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq25_HTML.gif 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 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equc_HTML.gif

          for all A , B P f ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq26_HTML.gif.

          It is well known that ( P f ( X ) , h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq27_HTML.gif is a complete metric space and P f c ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq28_HTML.gif is a closed subset of it. When Z is a Hausdorff topological space, a multifunction G : Z P f ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq29_HTML.gif is said to be h-continuous if it is continuous as a function from Z into ( P f ( X ) , h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq27_HTML.gif.

          Let Y, Z be Hausdorff topological spaces and G : Y 2 Z { ϕ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq30_HTML.gif. We say that G ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq31_HTML.gif is ‘upper semicontinuous (USC)’ (resp., ‘lower semicontinuous (LSC)’) if for all C Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq32_HTML.gif nonempty closed, G ( C ) = { y Y : G ( y ) C ϕ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq33_HTML.gif (resp., G + ( C ) = { y Y : G ( y ) C } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq34_HTML.gif) is closed in Y. A USC multifunction has a closed graph in Y × Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq35_HTML.gif, while the converse is true if G is locally compact (i.e., for every y Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq36_HTML.gif, there exists a neighborhood U of y such that F ( U ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq37_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq38_HTML.gif, y d Z ( z , G ( y ) ) = inf { d Z ( z , v ) : v G ( y ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq39_HTML.gif is upper semicontinuous as R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq40_HTML.gif-valued function. Also, lower semicontinuity is equivalent to saying that if y n y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq41_HTML.gif in Y as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif, 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 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equd_HTML.gif

          Let I = [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq14_HTML.gif. By L 1 ( I , X ) w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq43_HTML.gif, we denote the Lebesgue-Bochner space L 1 ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq44_HTML.gif equipped with the norm g w = sup { t t g ( s ) d s : 0 t t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq45_HTML.gif, g L 1 ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq46_HTML.gif. A set D L p ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq47_HTML.gif is said to be ‘decomposable’ if for every g 1 , g 2 D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq48_HTML.gif and for every J I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq49_HTML.gif measurable, we have χ J g 1 + χ J c g 2 D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq50_HTML.gif.

          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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq51_HTML.gif, where V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq13_HTML.gif is the topological dual space of V. The system model considered here is based on this evolution triple. Let the embedding be compact. Let , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq52_HTML.gif denote the pairing of an element x V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq53_HTML.gif and an element y V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq54_HTML.gif. If x , y H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq55_HTML.gif, then , = ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq56_HTML.gif, where ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq57_HTML.gif is the inner product on H. The norm in any Banach space X will be denoted by X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq58_HTML.gif. Let 1 < q p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq59_HTML.gif be such that 1 p + 1 q = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq60_HTML.gif. We denote L p ( I , V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq61_HTML.gif by X. Then the dual space of X is L q ( I , V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq62_HTML.gif and is denoted by  X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif. For p, q satisfying the above conditions, from reflexivity of V that both X and X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif are reflexive Banach spaces (see Zeidler [23], p.411]).

          Define W p q ( I ) = { x : x X , x ˙ X } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq64_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq65_HTML.gif, the space ( W p q ( I ) , x W p q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq66_HTML.gif becomes a Banach space which is clearly reflexive and separable. Moreover, W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif embeds into C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif continuously (see Proposition 23.23 of [23]). So, every element in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif has a representative in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif. Since the embedding V H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq69_HTML.gif is compact, the embedding W p q ( I ) L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq70_HTML.gif is also compact (see Problem 23.13 of [23]). The pairing between X and X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif is denoted by , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq71_HTML.gif. By ‘⇀’ we denote the weakly convergence. The following lemmas are still needed in the proof of our main theorems.

          Lemma 2.1 (see [24])

          If X is a Banach space, C X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq72_HTML.gif is nonempty, closed and convex with 0 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq73_HTML.gif and G : C P k c ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq74_HTML.gif is an upper semicontinuous multifunction which maps bounded sets into relatively compact sets, then one of the following statements are true:
          1. (i)

            the set Γ = { x C : x λ G ( x ) , λ ( 0 , 1 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq75_HTML.gif is unbounded;

             
          2. (ii)

            the G ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq31_HTML.gif has a fixed point, i.e., there exists x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq76_HTML.gif such that x G ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq77_HTML.gif.

             

          Let X be a Banach space and let L 2 ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq78_HTML.gif be the Banach space of all functions u : I X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq79_HTML.gif which are Bochner integrable. D ( L 2 ( I , X ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq80_HTML.gif denotes the collection of nonempty decomposable subsets of L 2 ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq78_HTML.gif. Now, let us state the Bressan-Colombo continuous selection theorem.

          Lemma 2.2 (see [25])

          Let X be a separable metric space and let F : X D ( L 2 ( I , X ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq81_HTML.gif be a lower semicontinuous multifunction with closed decomposable values. Then F has a continuous selection.

          Let X be a separable Banach space and C ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq82_HTML.gif be the Banach space of all continuous functions. A multifunction F : I × X P w k c ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq83_HTML.gif is said to be Carathéodory type if for every x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq84_HTML.gif, F ( , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq85_HTML.gif is measurable, and for almost all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, F ( t , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq87_HTML.gif is h-continuous (i.e., it is continuous from X to the metric space ( P f ( X ) , h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq27_HTML.gif, where h is a Hausdorff metric).

          Let M C ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq88_HTML.gif, a multifunction F : I × X P w k c ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq83_HTML.gif is called integrably bounded on M if there exists a function λ : I R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq89_HTML.gif such that for almost all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, sup { y : y F ( t , x ( t ) ) , x ( ) M } λ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq90_HTML.gif. A nonempty subset M 0 C ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq91_HTML.gif is called σ-compact if there is a sequence { M k } k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq92_HTML.gif of compact subsets M k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq93_HTML.gif such that M 0 = k 1 M k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq94_HTML.gif. Let M 0 M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq95_HTML.gif be such that M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq96_HTML.gif 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq97_HTML.gif be Carathéodory type and integrably bounded. Then there exists a continuous function g : M L p ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq98_HTML.gif such that for almost all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq99_HTML.gif, if x ( ) M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq100_HTML.gif , then g ( x ) ( t ) ext F ( t , x ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq101_HTML.gif, and if x ( ) M M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq102_HTML.gif, then g ( x ) ( t ) ext ¯ F ( t , x ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq103_HTML.gif.

          3 Main results

          Let I = [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq14_HTML.gif, consider the following evolution inclusions:
          x ˙ + A ( t , x ) + B x F ( t , x ) , a.e.  I , x ( 0 ) = φ ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ1_HTML.gif
          (3.1)

          where A : I × V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq1_HTML.gif is a nonlinear map, B : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq2_HTML.gif is a bounded linear map, φ : H H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq3_HTML.gif is a continuous map and F : I × H 2 V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq4_HTML.gif is a multifunction satisfying some conditions mentioned later.

          Definition 3.1 A function x W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq104_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Eque_HTML.gif

          where x ( 0 ) = φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq105_HTML.gif, f ( t ) F ( t , x ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq106_HTML.gif for all v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq107_HTML.gif and almost all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif.

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

          (H1) A : I × V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq108_HTML.gif is an operator such that
          1. (i)

            t A ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq109_HTML.gif is measurable;

             
          2. (ii)
            for each t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, the operator A ( t , ) : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq110_HTML.gif is uniformly monotone and hemicontinuous, that is, there exists a constant C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq111_HTML.gif (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equf_HTML.gif
             
          for all x 1 , x 2 V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq112_HTML.gif, and the map s A ( t , x + s z ) , y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq113_HTML.gif is continuous on [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq114_HTML.gif for all x , y , z V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq115_HTML.gif;
          1. (iii)

            there exist a constant C 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq116_HTML.gif, a nonnegative function a ( ) L q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq117_HTML.gif and a nondecreasing continuous function η ( ) L q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq118_HTML.gif such that A ( t , x ) V a ( t ) + C 2 η ( x V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq119_HTML.gif for all x V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq120_HTML.gif, a.e. on I;

             
          2. (iv)
            there exist C 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq121_HTML.gif, C 4 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq122_HTML.gif, b ( ) L 1 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq123_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equg_HTML.gif
             
          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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equh_HTML.gif
          (H2) F : I × H P k ( V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq124_HTML.gif is a multifunction such that
          1. (i)

            ( t , x ) F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq125_HTML.gif is graph measurable;

             
          2. (ii)

            for almost all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, x F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq126_HTML.gif is LSC;

             
          3. (iii)
            there exist a nonnegative function b 1 ( ) L q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq127_HTML.gif and a constant C 5 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq128_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equi_HTML.gif
             

          where 1 k < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq129_HTML.gif.

          (H3)
          1. (i)

            B : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq130_HTML.gif is a bounded linear self-adjoint operator such that ( B x , x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq131_HTML.gif for all x V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq120_HTML.gif, a.e. on I;

             
          2. (ii)
            there exists a continuous function φ : L p ( I , H ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq132_HTML.gif such that
            φ ( u ) φ ( v ) u v C ( I , H ) u , v C ( I , H ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equj_HTML.gif
             

          and φ ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq133_HTML.gif.

          It is convenient to rewrite the system (3.1) as an operator equation in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. For x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq84_HTML.gif, we get
          A ( x ) ( t ) = A ( t , x ) , B ( x ) ( t ) = B x ( t ) , F ( x ) ( t ) = F ( t , x ( t ) ) , t I . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equk_HTML.gif
          It follows from Theorem 30.A of Zeidler [23] that the operator A : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq134_HTML.gif is bounded, monotone, hemicontinuous and coercive. By using the same technique, one can show that the operator F : L p ( I , H ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq135_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equl_HTML.gif

          for some constants M 1 ˆ , M 2 ˆ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq136_HTML.gif and all x L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq137_HTML.gif.

          We define
          L u = u ˙ , D ( L ) = { u W p q ( I ) : u ( 0 ) = ξ H } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ2_HTML.gif
          (3.2)
          where u ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq138_HTML.gif 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equm_HTML.gif

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

          Lemma 3.1 Let V H V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq139_HTML.gif be an evolution triple and let X = L p ( I , V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq140_HTML.gif, where 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq141_HTML.gif and 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq142_HTML.gif. Then the linear operator L : D ( L ) X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq143_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq144_HTML.gif and
          0 w L u , v u u D ( L ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equn_HTML.gif
          We have to show that v D ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq145_HTML.gif and w = L v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq146_HTML.gif, i.e., w = v ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq147_HTML.gif. Due to the arbitrariness of u, we choose u = ϕ z + ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq148_HTML.gif, where ϕ C 0 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq149_HTML.gif, ξ = u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq150_HTML.gif and z V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq151_HTML.gif. Then u ˙ = ϕ ˙ z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq152_HTML.gif, so L u , u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq153_HTML.gif. From w L u , v u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq154_HTML.gif, we obtain that
          0 w , v ξ 0 T φ ˙ v + φ w , z d t z V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equo_HTML.gif
          By the arbitrariness of z, one has that
          0 T ( φ ˙ v + φ w ) d t = 0 φ C 0 ( I ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equp_HTML.gif
          Hence, w = v ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq155_HTML.gif. Since v W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq156_HTML.gif, then w X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq157_HTML.gif. It remains to show that v D ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq145_HTML.gif. Using the integration by parts formula for functions in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif (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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ3_HTML.gif
          (3.3)

          Choose a set of functions ( a n ) n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq158_HTML.gif in H such that T a n v ( T ) ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq159_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif. For ξ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq160_HTML.gif, let u ( t ) = t a n + ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq161_HTML.gif, then u D ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq162_HTML.gif. By (3.3), we have v ( 0 ) = u ( 0 ) = ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq163_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif. Hence, v D ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq145_HTML.gif. This completes the proof. □

          Theorem 3.1 If 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ4_HTML.gif
          (3.4)

          has only one solution.

          Firstly, for every ϵ ( 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq164_HTML.gif, y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq165_HTML.gif and f X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq166_HTML.gif, we claim that the equation
          x ˙ + A ( t , x ) + B x = f ( t ) a.e.  I , x ( 0 ) = ( 1 ϵ ) φ ( y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ5_HTML.gif
          (3.5)
          has only one solution. By (H1) and (H3), it is easy to check that ( A + B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq167_HTML.gif is bounded, monotone, hemicontinuous and coercive. Moreover, by Lemma 3.1, L is a linear maximal monotone operator. Therefore, R ( L + A + B ) = V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq168_HTML.gif, i.e., L + ( A + B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq169_HTML.gif 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq170_HTML.gif. By W p q ( I ) C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq171_HTML.gif, then the operator P : W p q ( I ) W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq172_HTML.gif is defined as follows:
          P ( y ) = 0 t x ˙ ( s ) d s + ( 1 ϵ ) φ ( y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equq_HTML.gif
          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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equr_HTML.gif
          for all y 1 , y 2 W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq173_HTML.gif. Take an inner product over (3.5) with x 1 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq174_HTML.gif, 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equs_HTML.gif
          By (H1)(ii), we have
          P ( y 1 ) P ( y 2 ) 2 ( 1 ϵ ) φ ( y 2 ) φ ( y 1 ) x 1 x 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equt_HTML.gif
          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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ6_HTML.gif
          (3.6)

          Invoking the Banach fixed point theorem, the operator P has only one fixed point x ϵ = P ( x ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq175_HTML.gif, i.e., x ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq176_HTML.gif is the uniform solution of (3.4).

          Therefore, we define L ϵ : W p q ( I ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq177_HTML.gif as L ϵ x = x ˙ + A ( t , x ) + B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq178_HTML.gif and x ( 0 ) = ( 1 ϵ ) φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq179_HTML.gif. By Step 1, we have L ϵ : W p q ( I ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq180_HTML.gif is one-to-one and surjective, and so L ϵ 1 : X W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq181_HTML.gif is well defined.

          Step 2. L ϵ 1 : X L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq182_HTML.gif is completely continuous.

          We only need to show that L ϵ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq183_HTML.gif is continuous and maps a bounded set into a relatively compact set. We claim that L ϵ : W p q ( I ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq184_HTML.gif is continuous. In fact, let { x n } n 1 W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq185_HTML.gif such that x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq187_HTML.gif. From (H1)(ii) and (H3), we infer that x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif, A ( x n ) A ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq188_HTML.gif, B x n B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq189_HTML.gif, a.e. I as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif. Obviously, φ ( x n ) φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq190_HTML.gif. Therefore, L ϵ : W p q ( I ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq191_HTML.gif is continuous and L ϵ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq192_HTML.gif is continuous.

          Let K X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq193_HTML.gif be a bound set, for any f K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq194_HTML.gif, there is a priori bound in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif for the possible solution x ( t ) = L ϵ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq195_HTML.gif of (3.4). Then
          x ˙ + A ( t , x ) + B x = f ( t ) a.e.  I . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ7_HTML.gif
          (3.7)
          It follows that
          x ˙ , x + A x , x + B x , x = f ( t ) , x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ8_HTML.gif
          (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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equu_HTML.gif
          or
          A u , u C 3 u X p C 4 u X p 1 b L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equv_HTML.gif
          with p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq196_HTML.gif. But
          u ˙ , u = u ( T ) 2 u ( 0 ) 2 , f , u f X u X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equw_HTML.gif
          Therefore,
          C 3 x X p C 4 x X p 1 + f X x X + b L 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equx_HTML.gif
          or
          C 3 x X p C 4 x X p 1 + f X x X + b L 1 + x X 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equy_HTML.gif

          with p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq196_HTML.gif. So, there exists an M 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq197_HTML.gif such that x X M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq198_HTML.gif. Because of the boundedness of operators A, B, we obtain that there exists an M 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq199_HTML.gif such that x ˙ X M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq200_HTML.gif. Hence, x W p q M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq201_HTML.gif for some constant M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq202_HTML.gif. Therefore, we have L ϵ 1 ( K ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq203_HTML.gif is bounded in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. But W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif is compactly embedded in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. Therefore, L ϵ 1 ( K ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq203_HTML.gif is relatively compact in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif.

          Let N ˆ : L p ( I , H ) 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq205_HTML.gif be a multivalued Nemitsky operator corresponding to F and N ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq206_HTML.gif was defined by N ˆ ( x ) = { v X : v ( t ) F ( t , x ( t ) ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq207_HTML.gif a.e. on I.

          Step 3. N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif has nonempty, closed, decomposable values and is LSC.

          The closedness and decomposability of the values of N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif are easy to check. For the nonemptiness, note that if x L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq137_HTML.gif, by the hypothesis (H2)(i), ( t , x ) F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq125_HTML.gif is graph measurable, so we apply Aumann’s selection theorem and obtain a measurable map v : I V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq209_HTML.gif such that v ( t ) F ( t , x ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq210_HTML.gif a.e. on I. By the hypothesis (H2)(iii), v X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq211_HTML.gif. Thus, for every x L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq137_HTML.gif, N ˆ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq212_HTML.gif. To prove the lower semicontinuity of N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif, we only need to show that every u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq213_HTML.gif, x d ( u , N ˆ ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq214_HTML.gif is a USC R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq40_HTML.gif-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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equz_HTML.gif
          (see Hiai and Umegaki [27] Theorem 2.2). We will show that for every λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq215_HTML.gif, the superlevel set U λ = { x L p ( I , H ) : d ( u , N ˆ ( x ) ) λ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq216_HTML.gif is closed in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. Let { x n } n 1 U λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq217_HTML.gif and assume that x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. By passing to a subsequence if necessary, we may assume that x n ( t ) x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq218_HTML.gif a.e. on I as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif. By (H2)(ii), x d ( u ( t ) , F ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq219_HTML.gif is an upper semicontinuous R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq40_HTML.gif-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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equaa_HTML.gif

          Therefore, x U λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq220_HTML.gif and this proves the LSC of N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif. By Lemma 2.2, we obtain a continuous map f : L p ( I , H ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq221_HTML.gif such that f ( x ) N ˆ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq222_HTML.gif. To finish our proof, we need to solve the fixed point problem: x = L ϵ 1 f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq223_HTML.gif.

          Since the embedding V H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq224_HTML.gif is compact, the embedding W p q ( I ) L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq225_HTML.gif is compact. That is, x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq226_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif whenever x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq227_HTML.gif in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. By using the above relation and the continuity of f, we have f ( x n ) f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq228_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif whenever x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq227_HTML.gif in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. So, L ϵ 1 f : L p ( I , H ) L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq229_HTML.gif 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 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq230_HTML.gif is bounded.

          Let x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq231_HTML.gif, then we have L ϵ ( x σ ) = f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq232_HTML.gif. Note that
          x σ ˙ , x σ + A ( x σ ) , x σ + B x σ , x σ = f ( x ) , x σ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equab_HTML.gif
          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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ9_HTML.gif
          (3.9)
          or
          A u , u + B x , x C 3 u X p C 4 u X p 1 b L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ10_HTML.gif
          (3.10)
          with p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq196_HTML.gif. 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ11_HTML.gif
          (3.11)
          where γ 1 , γ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq233_HTML.gif. By (3.9), (3.10) and (3.11), if 1 k < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq129_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ12_HTML.gif
          (3.12)
          If 1 k < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq129_HTML.gif, p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq196_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ13_HTML.gif
          (3.13)
          Thus, by virtue of the inequalities (3.12) and (3.13), we can find a constant M 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq197_HTML.gif such that x X M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq198_HTML.gif for all x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq234_HTML.gif. From the boundedness of operators A, B and f, and the continuous embedding X L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq235_HTML.gif, we obtain A ( x ) X M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq236_HTML.gif, B x X M 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq237_HTML.gif and f ( x ) X M 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq238_HTML.gif for some constants M 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq199_HTML.gif, M 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq239_HTML.gif, M 4 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq240_HTML.gif and all x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq231_HTML.gif. Therefore,
          x ˙ X A ( x ) X + B x X + f ( x ) X M 2 + M 3 + M 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ14_HTML.gif
          (3.14)

          for all x Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq231_HTML.gif.

          It follows from (3.14) that x W p q x X + x ˙ X M ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq241_HTML.gif for some constant M ˆ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq242_HTML.gif. Hence, Γ is a bounded subset of W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. So, Γ is a bounded subset of L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif since the embedding W p q ( I ) L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq225_HTML.gif is compact.

          Invoking the Leray-Schauder theorem, one has that there exists an x ϵ W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq243_HTML.gif such that x ϵ = L ϵ 1 f ( x ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq244_HTML.gif, i.e., x ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq176_HTML.gif 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ15_HTML.gif
          (3.15)
          Let ( ϵ n ) n 1 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq245_HTML.gif and ϵ n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq246_HTML.gif. For every n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq247_HTML.gif, there exists an x n W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq248_HTML.gif 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ16_HTML.gif
          (3.16)
          By Step 3, we have that { x n } n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq249_HTML.gif is uniformly bounded. By the boundedness of the sequence { x n } n 1 W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq250_HTML.gif, it follows that the sequence { x ˙ n } n 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq251_HTML.gif is uniformly bounded and passing to subsequence if necessary, we may assume that x ˙ n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq252_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif. Evidently, u = x ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq253_HTML.gif and x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq227_HTML.gif in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. Since the embedding W p q ( I ) L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq254_HTML.gif is compact, then x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. Hence, from the hypothesis (H2)(ii), we obtain f ( x n ) f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq228_HTML.gif and f ( x ) F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq255_HTML.gif. Since the operator A is hemicontinuous and monotone and B is a continuous linear operator, thus A ( x n ) A ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq256_HTML.gif, B x n B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq257_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif. Therefore, we obtain x ˙ + A ( x ) + B x = f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq258_HTML.gif, f F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq259_HTML.gif a.e. on I. Since x n ( t ) x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq218_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif and φ : L p ( I , H ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq132_HTML.gif is continuous, then we have
          x n ( 0 ) = ( 1 ϵ n ) φ ( x n ) φ ( x ) = x ( 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equac_HTML.gif

          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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq260_HTML.gif is a multifunction such that
          1. (i)

            ( t , x ) F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq125_HTML.gif is graph measurable;

             
          2. (ii)

            for almost all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, x F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq126_HTML.gif has a closed graph; and (H2)(iii) hold.

             

          Theorem 3.2 If 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif.

          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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq261_HTML.gif has nonempty closed, convex values in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif and is USC from L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif into X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif furnished with the weak topology (denoted by X w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq262_HTML.gif). The closedness and convexity of the values of N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif are clear. To prove the nonemptiness, let x L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq137_HTML.gif and { s n } n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq263_HTML.gif be a sequence of step functions such that s n ( t ) x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq264_HTML.gif in H and s n ( t ) H x ( t ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq265_HTML.gif a.e. on I. Then by virtue of the hypothesis (H4)(i), for every n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq266_HTML.gif, t F ( t , s n ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq267_HTML.gif admits a measurable selector v n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq268_HTML.gif. From the hypothesis (H4)(iii), we have that v n X M 1 ˆ + M 2 ˆ x L p ( I , H ) k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq269_HTML.gif, so { v n ( t ) } n 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq270_HTML.gif is uniformly integrable. So, by the Dunford-Pettis theorem, and by passing to a subsequence if necessary, we may assume that v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq271_HTML.gif weakly in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif. 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equad_HTML.gif

          the last inclusion being a consequence of the hypothesis (H4)(ii). So, v N ˆ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq272_HTML.gif, which means that N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif is nonempty.

          Next, we show that N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif is USC from L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif into X w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq262_HTML.gif. Let Ξ be a nonempty and weakly closed subset of X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif. Obviously, it is sufficient to show that the set
          N ˆ 1 ( Ξ ) = { x L p ( I , H ) : N ˆ ( x ) Ξ ϕ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equae_HTML.gif
          is closed. Let { x n } n 1 N ˆ 1 ( Ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq273_HTML.gif and assume x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. Passing to a subsequence, we can get that x n ( t ) x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq218_HTML.gif a.e. on I. Let f n N ˆ ( x n ) Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq274_HTML.gif, n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq266_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equaf_HTML.gif
          So, by the Dunford-Pettis theorem, we may assume that f n f Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq275_HTML.gif in X w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq262_HTML.gif. 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equag_HTML.gif

          then f N ˆ ( x ) Ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq276_HTML.gif, i.e., N ˆ 1 ( Ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq277_HTML.gif is closed in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. This proves the upper semicontinuity of N ˆ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq208_HTML.gif from L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif into X w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq262_HTML.gif.

          We consider the following fixed point problem:
          x L ϵ 1 N ˆ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equah_HTML.gif
          Recalling that L ϵ 1 : X L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq278_HTML.gif is completely continuous, we see that L ϵ 1 N ˆ : L p ( I , H ) P k c ( L p ( I , H ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq279_HTML.gif 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 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equai_HTML.gif
          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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq243_HTML.gif such that x ϵ L ϵ 1 N ˆ ( x ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq280_HTML.gif, i.e., x ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq176_HTML.gif 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ17_HTML.gif
          (3.17)
          Let ( ϵ n ) n 1 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq245_HTML.gif and ϵ n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq246_HTML.gif. For every n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq247_HTML.gif, there exists an x n W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq248_HTML.gif 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ18_HTML.gif
          (3.18)
          By Step 3, { x n } n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq249_HTML.gif is uniformly bounded. By the boundedness of the sequence { x n } n 1 W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq250_HTML.gif, it follows that the sequence { x ˙ n } n 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq251_HTML.gif is uniformly bounded and, passing to subsequence if necessary, we may assume that x ˙ n x ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq281_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif. Thus, A ( x n ) A ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq256_HTML.gif, B x n B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq257_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif. Evidently, there exists f n N ( x n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq282_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq283_HTML.gif, so { f n ( t ) } n 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq284_HTML.gif is uniformly integrable. So, by the Dunford-Pettis theorem and by passing to a subsequence if necessary, we may assume that f n f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq285_HTML.gif weakly in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif. Therefore, we obtain x ˙ + A ( x ) + B x = f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq258_HTML.gif, f F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq259_HTML.gif a.e. on I. Since x n ( t ) x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq218_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif and φ : L p ( I , H ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq132_HTML.gif is continuous, then we have
          x n ( 0 ) = ( 1 ϵ n ) φ ( x n ) φ ( x ) = x ( 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equaj_HTML.gif
          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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq286_HTML.gif, for some M ˆ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq242_HTML.gif. So, S W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq287_HTML.gif is uniformly bounded. So, by the Dunford-Pettis theorem and by passing to a subsequence if necessary, we may assume that x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif weakly in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equak_HTML.gif

          Clearly, x ( 0 ) = φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq105_HTML.gif, then x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq288_HTML.gif. Thus, S is weakly compact in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. □

          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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ19_HTML.gif
          (4.1)

          where ext F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq289_HTML.gif denotes the extremal point set of F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq290_HTML.gif. We need the following hypothesis:

          (H5) F : I × H P w k c ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq291_HTML.gif is a multifunction such that
          1. (i)

            ( t , x ) F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq125_HTML.gif is graph measurable;

             
          2. (ii)

            for almost all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, x F ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq126_HTML.gif is h-continuous; and (H2)(iii) holds.

             

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

          Proof Since S e S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq292_HTML.gif, as in the proof of Theorem 3.1, we obtain a priori bound for S e http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq293_HTML.gif. We know that there exists M i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq294_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq295_HTML.gif such that x W p q < M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq296_HTML.gif and x C ( I , H ) < M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq297_HTML.gif for all x S e http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq298_HTML.gif. Let ψ ( t ) = b 2 ( t ) + C 5 M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq299_HTML.gif, ψ ( t ) L q + ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq300_HTML.gif. We may assume that | F ( t , x ) | ψ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq301_HTML.gif a.e. on I for all x H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq302_HTML.gif. By Theorem 3.1, let L 0 = x ˙ + A ( x ) + B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq303_HTML.gif, x ( 0 ) = φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq105_HTML.gif, then L 0 1 : W p q ( I ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq304_HTML.gif is well defined. So, let
          W = { v L q ( I , H ) : v ( t ) H ψ ( t )  a.e. on  I } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equal_HTML.gif
          then K ˆ = L 0 1 ( W ) W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq305_HTML.gif is a compact convex subset in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif. Obviously, K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq306_HTML.gif is convex. We only need to show the compactness. Let { x n } n 1 K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq307_HTML.gif, then there exists h n W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq308_HTML.gif such that L 0 ( x n ) = h n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq309_HTML.gif, i.e., x ˙ n = h n A ( x n ) B x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq310_HTML.gif. By the definition of W, W is uniformly bounded in L q ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq311_HTML.gif. By the Dunford-Pettis theorem, passing to a subsequence if necessary, we may assume that h n h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq312_HTML.gif in L q ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq311_HTML.gif for some h W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq313_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equam_HTML.gif
          Therefore, the sequence { x n } n 1 W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq250_HTML.gif is bounded. Because of the compactness of the embedding W p q ( I ) L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq314_HTML.gif, we have that the sequence { x n } n 1 L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq315_HTML.gif is relatively compact. So, by passing to a subsequence if necessary, we may assume that x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. Moreover, by the boundedness of the sequence { x n } n 1 W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq250_HTML.gif, it follows that the sequence { x ˙ n } n 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq251_HTML.gif is uniformly bounded and, passing to subsequence if necessary, we may assume that x ˙ n x ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq281_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif. Since the embedding W p q ( I ) C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq171_HTML.gif is continuous and W p q ( I ) L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq314_HTML.gif is compact, it follows that x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq227_HTML.gif in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif and x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif. Hence, x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq186_HTML.gif in H for all t I Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq316_HTML.gif, m ( Λ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq317_HTML.gif (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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq256_HTML.gif, B x n B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq257_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq63_HTML.gif and as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq42_HTML.gif, we obtain x ˙ + A ( x ) + B x = h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq318_HTML.gif a.e. on I and x ( 0 ) = φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq105_HTML.gif. Note that
          x ˙ n x ˙ + ( A ( t , x n ) A ( t , x ) ) = h n h ( B x n B x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ20_HTML.gif
          (4.2)
          Taking the inner product above with x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq319_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ21_HTML.gif
          (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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ22_HTML.gif
          (4.4)
          So, we can find τ I Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq320_HTML.gif such that
          x n ( τ ) x ( τ ) H 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equan_HTML.gif
          Using the integration by parts formula for functions in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif (see Zeidler [26], Proposition 23.23), for any t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ23_HTML.gif
          (4.5)
          By (4.5), we see that
          max t I x n ( t ) x ( t ) H 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equao_HTML.gif

          So, x n ( t ) x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq218_HTML.gif in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif. Since x = L 0 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq321_HTML.gif with h W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq313_HTML.gif, we conclude that L 0 1 ( W ) C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq322_HTML.gif is compact. From Lemma 2.3, we can find a continuous map f : K ˆ L q ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq323_HTML.gif such that f ( x ) ( t ) ext F ( t , x ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq324_HTML.gif a.e. on I for all x K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq325_HTML.gif. Then L 0 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq326_HTML.gif is a compact operator. On applying the Schauder fixed point theorem, there exists an x K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq325_HTML.gif such that x = L 0 1 f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq327_HTML.gif. This is a solution of (4.1), and so S e http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq328_HTML.gif in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. □

          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 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq329_HTML.gif is said to be a Kamke function if it is integrally bounded on the bounded sets, μ ( t , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq330_HTML.gif and the unique solution of the differential equation s ˙ ( t ) = μ ( t , s ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq331_HTML.gif, s ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq332_HTML.gif is s ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq333_HTML.gif.

          (H6) For each t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, there exists a Kamke function μ : I × R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq334_HTML.gif such that
          h ( F ( , x 1 ) , F ( , x 2 ) ) μ ( t , x 1 x 2 H 2 ) for all  x 1 , x 2 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equap_HTML.gif

          and (H5) hold.

          Theorem 4.2 If hypotheses (H1), (H3) and (H6) hold, then S e ¯ = S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq335_HTML.gif, where the closure is taken in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif.

          Proof Let x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq288_HTML.gif, then there exist f L q ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq336_HTML.gif and f ( x ) ( t ) F ( t , x ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq337_HTML.gif a.e. on I such that
          x ˙ ( t ) + A ( t , x ( t ) ) + B x = f ( t , x ) , x ( 0 ) = φ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ24_HTML.gif
          (4.6)
          As before, let W = { v L q ( I , H ) : v H ψ ( t )  a.e. on  I } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq338_HTML.gif, then K ˆ = L 0 1 ( W ) W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq305_HTML.gif is a compact convex subset in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif. For every y K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq339_HTML.gif, we define the multifunction
          Q ϵ ( t ) = { v F ( t , y ) : ( f v , x y ) 1 2 μ ( t , x y H 2 ) + ϵ } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equaq_HTML.gif
          Clearly, for every t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq86_HTML.gif, Q ϵ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq340_HTML.gif, and it is graph measurable. On applying Aumann’s selection theorem, we get a measurable function v : I H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq341_HTML.gif such that v ( t ) Q ϵ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq342_HTML.gif 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 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equar_HTML.gif
          We see that R ϵ : K ˆ 2 L q ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq343_HTML.gif has nonempty and decomposable values. It follows from Theorem 3 of [29] that R ϵ ( ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq344_HTML.gif is LSC. Therefore, y R ϵ ( y ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq345_HTML.gif 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq346_HTML.gif such that f ϵ ( y ) R ϵ ( y ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq347_HTML.gif for all y K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq339_HTML.gif. 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq348_HTML.gif such that g ϵ ( y ) ( t ) ext F ( t , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq349_HTML.gif almost everywhere on I, and f ϵ ( y ) g ϵ ( y ) w ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq350_HTML.gif for all y K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq351_HTML.gif. Now, let ϵ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq352_HTML.gif and set f ϵ n = f ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq353_HTML.gif, g ϵ n = g ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq354_HTML.gif. Note that g ϵ n ( y ) H ψ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq355_HTML.gif a.e. on I with ψ L q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq356_HTML.gif, so we have g ϵ n f ϵ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq357_HTML.gif in L q ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq311_HTML.gif. We consider the following problem:
          x ˙ ( t ) + A ( x ) ( t ) + B x = g ϵ n ( x ) ( t ) , x ( 0 ) = φ ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ25_HTML.gif
          (4.7)
          where g ϵ n ( x ) ext R ϵ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq358_HTML.gif. We see that L 0 1 g ϵ n : K ˆ K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq359_HTML.gif is a compact operator and by the Schauder fixed point theorem, we obtain a solution x ϵ n S e W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq360_HTML.gif of (4.1). We see that the sequence { x ϵ n } n 1 K ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq361_HTML.gif is uniformly bounded. So, by passing to a subsequence if necessary, we may assume that x ϵ n x ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq362_HTML.gif in W p q ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq67_HTML.gif. From the proof of Theorem 4.1, we know that x ϵ n x ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq363_HTML.gif in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif and x ˆ ( 0 ) = φ ( x ˆ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq364_HTML.gif. Note that L 0 x ϵ n L 0 x = g ϵ n ( x ϵ n ) f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq365_HTML.gif. 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 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equas_HTML.gif
          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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ26_HTML.gif
          (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 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ27_HTML.gif
          (4.9)
          By g ϵ n f ϵ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq357_HTML.gif in L q ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq311_HTML.gif and x ϵ n x ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq363_HTML.gif in L p ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq204_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ28_HTML.gif
          (4.10)
          Hence, there exists a constant N 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq366_HTML.gif, one has that
          | ( g ϵ n ( x ϵ n ) ( t ) f ϵ n ( x ϵ n ) ( t ) , x ϵ n ( t ) x ( t ) ) | < ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equat_HTML.gif
          as n > N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq367_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ29_HTML.gif
          (4.11)

          Hence, x ϵ ( t ) x ( t ) H 2 Q ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq368_HTML.gif, where Q ( 0 ) = x ϵ ( 0 ) x ( 0 ) H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq369_HTML.gif and Q ˙ ( t ) = μ ( t , Q ( t ) ) + 4 ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq370_HTML.gif. By (4.7), then Q ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq371_HTML.gif. Let ϵ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq352_HTML.gif, we have x ϵ ( t ) x ( t ) H 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq372_HTML.gif. Therefore, x = x ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq373_HTML.gif, i.e., x ϵ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq374_HTML.gif and x ϵ S e http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq375_HTML.gif, and so S S e ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq376_HTML.gif. Also, S is closed in C ( I , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq68_HTML.gif (see the proof of Theorem 3.2), thus S = S e ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq377_HTML.gif. □

          5 Examples

          As an application of the previous results, we introduce two examples. Let Ω be a bounded domain in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq378_HTML.gif with smooth boundary Ω, T = [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq379_HTML.gif, 0 < b < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq380_HTML.gif. Firstly, consider the following nonlinear evolution equation with a discontinuous right-hand side:
          u ( t , x ) t div ( | u | p 2 u ) + | u | p 2 u = f ( t , x , u ( t , x ) ) on  T × Ω , u ( t , x ) = 0 on  T × Ω , u ( 0 , x ) = 1 2 b 0 b u ( s , x ) d s + 1 2 u ( b , x ) on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ30_HTML.gif
          (5.1)

          The p-Laplacian div ( | u | p 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq381_HTML.gif arises in many applications such as Finsler geometry and non-Newtonian fluids. In [30], Liu showed the existence of anti-periodic solutions to the problem (5.1) where f ( t , x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq382_HTML.gif is continuous.

          Since f ( t , x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq382_HTML.gif is not continuous, the problem (5.1) need not have solutions. To obtain an existence theorem for (5.1), we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points of f ( t , x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq382_HTML.gif. So, we introduce the functions f 1 ( t , x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq383_HTML.gif and f 2 ( t , x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq384_HTML.gif defined by
          f 1 ( t , x , u ) = lim ̲ ξ u f ( t , x , ξ ) = sup ϵ > 0 inf | ξ u | < ϵ f ( t , x , ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equau_HTML.gif
          and
          f 2 ( t , x , u ) = lim ¯ ξ u f ( t , x , ξ ) = inf ϵ > 0 inf | ξ u | < ϵ f ( t , x , ξ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equav_HTML.gif
          Set
          f ˆ ( t , x , u ) = [ f 1 ( t , x , u ) , f 2 ( t , x , u ) ] = { v R : f 1 ( t , x , u ) v f 1 ( t , x , u ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equaw_HTML.gif
          Then, instead of (5.1), we study the following multivalued nonlinear evolution inclusion:
          u ( t , x ) t div ( | u | p 2 u ) + | u | p 2 u f ˆ ( t , x , u ( t , x ) ) on  T × Ω , u ( t , x ) = 0 on  T × Ω , u ( 0 , x ) = 1 2 b 0 b u ( s , x ) d s + 1 2 u ( b , x ) on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ31_HTML.gif
          (5.2)

          The hypotheses on the data of this problem (5.1) are the following:

          (H7)
          1. (i)

            f i ( t , x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq385_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq295_HTML.gif) are Nemitsky-measurable, i.e., u : T × Ω R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq386_HTML.gif for all measurable, u f i ( t , x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq387_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq295_HTML.gif) is measurable;

             
          2. (ii)
            there exists a 2 ( t ) L q ( t ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq388_HTML.gif, C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq389_HTML.gif, such that
            | f i ( t , x , u ) | a 2 ( t ) + C u k 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equax_HTML.gif
             

          where 1 k < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq129_HTML.gif.

          In this case, the evolution triple is V = W 0 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq390_HTML.gif, H = L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq391_HTML.gif and V = W 1 , q ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq392_HTML.gif. From the Sobolev embedding theorem, we see that all embeddings are compact. Let us define the following operator on V:
          A ( u ) ( t ) , v = Ω ( | u | p 2 u v + | u | p 2 u v ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equay_HTML.gif
          By the monotone property of p-Laplacian, it is easy to verify that A satisfies our hypothesis (H1). Let F : T × H P k c ( H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq393_HTML.gif be defined by
          F ( t , u ) = { g L 2 ( Ω ) : f 1 ( t , x , u ) g ( x ) f 2 ( t , x , u ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equaz_HTML.gif

          The hypothesis (H7) implies that (H4) is satisfied. Note that f 1 ( t , x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq394_HTML.gif is lower semicontinuous, f 2 ( t , x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq395_HTML.gif is upper semicontinuous, and so f ˆ ( t , x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq396_HTML.gif is USC (see [22], Example 2.8, p.371]). Let φ ( u ) = 1 2 b 0 b u ( s , x ) d s + 1 2 u ( b , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq397_HTML.gif, it is easy to check that φ satisfies our hypothesis (H3)(ii). Then, we rewrite equivalently (5.1) as (3.1) , with A and F as above. Finally, we can apply Theorem 3.2 to the problem (5.1) and obtain the following.

          Theorem 5.1 If the hypothesis (H7) holds, then the problem (5.1) has a nonempty set of solutions u L p ( T , W 0 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq398_HTML.gif such that u t L q ( T , W 1 , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq399_HTML.gif.

          Secondly, we present an example of a quasilinear distributed parameter control system, with a priori feedback (i.e., state dependent control constraint set). So, let T = [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq400_HTML.gif and Z R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq401_HTML.gif be a bounded domain with C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq402_HTML.gif-boundary Γ. Let D k = z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq403_HTML.gif, k { 1 , 2 , , N } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq404_HTML.gif, = k = 1 N 2 z k 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq405_HTML.gif. We consider the following control system:
          x t k = 1 N D k A k ( t , z , x , D x ) x = g ( t , z , x ( t , z ) ) u ( t , z ) a.e. on  T × Z , x | T × Γ = 0 , x ( 0 , z ) = 1 b 0 b x ( s , z ) d s a.e. on  Z , u ( t , z ) ext U ( t , z , x ( t , z ) ) a.e. on  T × Z . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equ32_HTML.gif
          (5.3)

          The hypotheses on the data (5.3) are the following:

          (H8) A k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq406_HTML.gif ( k = 1 , 2 , , N ) : T × Z × R × R N R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq407_HTML.gif are functions such that
          1. (i)

            ( t , z ) A k ( t , z , u , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq408_HTML.gif is measurable on T × Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq409_HTML.gif for every ( u , η ) R × R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq410_HTML.gif, ( u , η ) A k ( t , z , u , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq411_HTML.gif is continuous on R × R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq412_HTML.gif for all almost all ( t , z ) T × Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq413_HTML.gif;

             
          2. (ii)

            | A k ( t , z , u , η ) | α ˆ 1 ( t , z ) + c ˆ 1 ( z ) ( | u | + | η | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq414_HTML.gif with a nonnegative function α ˆ 1 L 2 ( I × Z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq415_HTML.gif and c ˆ 1 ( z ) L ( Z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq416_HTML.gif for almost all t T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq417_HTML.gif;

             
          3. (iii)

            k = 1 N ( A k ( t , z , u , η ) A k ( t , z , u , η ) ) ( η k η k ) | η η | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq418_HTML.gif for almost all t T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq417_HTML.gif;

             
          4. (iv)

            A k ( t , z , 0 , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq419_HTML.gif for all ( t , z ) T × Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq413_HTML.gif.

             
          (H9) The function g : T × Z × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq420_HTML.gif satisfies the following:
          1. (i)

            for all x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq421_HTML.gif, ( t , z ) g ( t , z , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq422_HTML.gif is measurable;

             
          2. (ii)

            for all ( t , z ) T × Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq423_HTML.gif, x g ( t , z , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq424_HTML.gif is continuous;

             
          3. (iii)
            for almost all ( t , z ) T × Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq423_HTML.gif and all x R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq421_HTML.gif, we have
            | g ( t , z , x ) | η 1 ( t , z ) + η 2 ( t ) | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_Equba_HTML.gif
             

          with η 1 L 2 ( T , L 2 ( Z ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-15/MediaObjects/13661_2012_Article_286_IEq425_HTML.gif, η 2 L