 Research
 Open Access
Properties of the solutions set for a class of nonlinear evolution inclusions with nonlocal conditions
 Jingrui Zhang^{1}Email author,
 Yi Cheng^{2, 3},
 Changqin Yuan^{2} and
 Fuzhong Cong^{2}
https://doi.org/10.1186/16872770201315
© Zhang et al.; licensee Springer. 2013
 Received: 23 October 2012
 Accepted: 18 January 2013
 Published: 5 February 2013
Abstract
In this paper, we consider the nonlocal problems for nonlinear firstorder 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
 LeraySchauder alternative theorem
 extremal solutions
1 Introduction
where $A:I\times V\to {V}^{\ast}$ is a nonlinear map, $B:V\to {V}^{\ast}$ is a bounded linear map, $\phi :H\to H$ is a continuous map and $F:I\times H\to {2}^{{V}^{\ast}}$ 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.:

$\phi (x)=x(T)$;

$\phi (x)=x(T)$;

$\phi (x)=\frac{1}{2\pi}{\int}_{0}^{2\pi}x(s)\phantom{\rule{0.2em}{0ex}}ds$;

$\phi (x)={\sum}_{i=1}^{n}{\beta}_{i}x({t}_{i})$, where $0<{t}_{1}<{t}_{2}<\cdot \cdot \cdot <{t}_{n}$ are arbitrary, but fixed and ${\sum}_{i=1}^{n}{\beta}_{i}\le 1$.
Many authors have studied the nonlocal Cauchy problem because it has a better effect in the applications than the classical initial condition. We begin by mentioning some of the previous work done in the literature. As far as we know, this study was first considered by Byszewski. Byszewski and Lakshmikantham [1, 2] proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when F is a singlevalued function satisfying Lipschitztype conditions. The fully nonlinear case was considered by Aizicovici and Lee [3], Aizicovici and McKibben [4], Aizicovici and Staicu [5], GarcíaFalset [6], GarcíaFalset 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 functionaldifferential equations when the semigroup is compact and φ is convex and compact on a given ball. In [12], Fu and Ezzinbi studied neutral functionaldifferential equations with nonlocal conditions. Benchohra and Ntouyas [13] discussed secondorder differential equations under compact conditions. For more details on the nonlocal problem, we refer to the papers of [14–18] and the references therein.
It is worth mentioning that many of these documents assume that a nonlocal function meets certain conditions of compactness and A is a strongly continuous semigroup of operators or accretive operators in studying the evolution equations or inclusions with nonlocal conditions. However, one may ask whether there are similar results without the assumption on the compactness or equicontinuity of the semigroup. This article will give a positive answer to this question. The works mentioned above mainly establish the existence of mild solutions for evolution equations or inclusions with nonlocal conditions. However, in the present paper, we consider the cases of a convex and of a nonconvex valued perturbation term in the evolution triple of spaces ($V\subset H\subset {V}^{\ast}$). We assume the nonlinear time invariant operator A to be monotone and the perturbation term to be multivalued, defined on $I\times H$ with values in ${V}^{\ast}$ (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, setvalued analysis and the LeraySchauder 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 righthand 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 righthand side (‘bangbang’ 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 LeraySchauder 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
for all $A,B\in {P}_{f}(X)$.
It is well known that $({P}_{f}(X),h)$ is a complete metric space and ${P}_{fc}(X)$ is a closed subset of it. When Z is a Hausdorff topological space, a multifunction $G:Z\to {P}_{f}(X)$ is said to be hcontinuous if it is continuous as a function from Z into $({P}_{f}(X),h)$.
Let $I=[0,T]$. By ${L}_{1}{(I,X)}_{w}$, we denote the LebesgueBochner space ${L}_{1}(I,X)$ equipped with the norm ${\parallel g\parallel}_{w}=sup\{\parallel {\int}_{t}^{{t}^{\prime}}g(s)\phantom{\rule{0.2em}{0ex}}ds\parallel :0\le t\le {t}^{\prime}\le T\}$, $g\in {L}_{1}(I,X)$. A set $D\subseteq {L}_{p}(I,X)$ is said to be ‘decomposable’ if for every ${g}_{1},{g}_{2}\in D$ and for every $J\subseteq I$ measurable, we have ${\chi}_{J}{g}_{1}+{\chi}_{{J}^{c}}{g}_{2}\in D$.
Let H be a real separable Hilbert space, V be a dense subspace of H having structure of a reflexive Banach space, with the continuous embedding $V\to H\to {V}^{\ast}$, where ${V}^{\ast}$ is the topological dual space of V. The system model considered here is based on this evolution triple. Let the embedding be compact. Let $\u3008\cdot ,\cdot \u3009$ denote the pairing of an element $x\in {V}^{\ast}$ and an element $y\in V$. If $x,y\in H$, then $\u3008\cdot ,\cdot \u3009=(\cdot ,\cdot )$, where $(\cdot ,\cdot )$ is the inner product on H. The norm in any Banach space X will be denoted by ${\parallel \cdot \parallel}_{X}$. Let $1<q\le p<\mathrm{\infty}$ be such that $\frac{1}{p}+\frac{1}{q}=1$. We denote ${L}_{p}(I,V)$ by X. Then the dual space of X is ${L}_{q}(I,{V}^{\ast})$ and is denoted by ${X}^{\ast}$. For p, q satisfying the above conditions, from reflexivity of V that both X and ${X}^{\ast}$ are reflexive Banach spaces (see Zeidler [23], p.411]).
Define ${W}_{pq}(I)=\{x:x\in X,\dot{x}\in {X}^{\ast}\}$, where the derivative in this definition should be understood in the sense of distribution. Furnished with the norm ${\parallel x\parallel}_{{W}_{pq}}={\parallel x\parallel}_{X}+{\parallel \dot{x}\parallel}_{{X}^{\ast}}$, the space $({W}_{pq}(I),{\parallel x\parallel}_{{W}_{pq}})$ becomes a Banach space which is clearly reflexive and separable. Moreover, ${W}_{pq}(I)$ embeds into $C(I,H)$ continuously (see Proposition 23.23 of [23]). So, every element in ${W}_{pq}(I)$ has a representative in $C(I,H)$. Since the embedding $V\to H$ is compact, the embedding ${W}_{pq}(I)\to {L}_{p}(I,H)$ is also compact (see Problem 23.13 of [23]). The pairing between X and ${X}^{\ast}$ is denoted by $\u300a\cdot ,\cdot \u300b$. By ‘⇀’ we denote the weakly convergence. The following lemmas are still needed in the proof of our main theorems.
Lemma 2.1 (see [24])
 (i)
the set $\mathrm{\Gamma}=\{x\in C:x\in \lambda G(x),\lambda \in (0,1)\}$ is unbounded;
 (ii)
the $G(\cdot )$ has a fixed point, i.e., there exists $x\in C$ such that $x\in G(x)$.
Let X be a Banach space and let ${L}^{2}(I,X)$ be the Banach space of all functions $u:I\to X$ which are Bochner integrable. $D({L}^{2}(I,X))$ denotes the collection of nonempty decomposable subsets of ${L}^{2}(I,X)$. Now, let us state the BressanColombo continuous selection theorem.
Lemma 2.2 (see [25])
Let X be a separable metric space and let $F:X\to D({L}^{2}(I,X))$ be a lower semicontinuous multifunction with closed decomposable values. Then F has a continuous selection.
Let X be a separable Banach space and $C(I,X)$ be the Banach space of all continuous functions. A multifunction $F:I\times X\to {P}_{wkc}(X)$ is said to be Carathéodory type if for every $x\in X$, $F(\cdot ,x)$ is measurable, and for almost all $t\in I$, $F(t,\cdot )$ is hcontinuous (i.e., it is continuous from X to the metric space $({P}_{f}(X),h)$, where h is a Hausdorff metric).
Let $M\subset C(I,X)$, a multifunction $F:I\times X\to {P}_{wkc}(X)$ is called integrably bounded on M if there exists a function $\lambda :I\to {R}_{+}$ such that for almost all $t\in I$, $sup\{\parallel y\parallel :y\in F(t,x(t)),x(\cdot )\in M\}\le \lambda (t)$. A nonempty subset ${M}_{0}\subset C(I,X)$ is called σcompact if there is a sequence ${\{{M}_{k}\}}_{k\ge 1}$ of compact subsets ${M}_{k}$ such that ${M}_{0}={\bigcup}_{k\ge 1}{M}_{k}$. Let ${M}_{0}\subset M$ be such that ${M}_{0}$ is dense in M and σcompact. The following continuous selection theorem in the extreme point case is due to Tolstonogov [26].
Lemma 2.3 (see [26])
Let the multifunction $F:I\times X\to {P}_{wkc}(X)$ be Carathéodory type and integrably bounded. Then there exists a continuous function $g:M\to {L}_{p}(I,X)$ such that for almost all $t\in I$, if $x(\cdot )\in {M}_{0}$ , then $g(x)(t)\in extF(t,x(t))$, and if $x(\cdot )\in M\setminus {M}_{0}$, then $g(x)(t)\in \overline{ext}\phantom{\rule{0.2em}{0ex}}F(t,x(t))$.
3 Main results
where $A:I\times V\to {V}^{\ast}$ is a nonlinear map, $B:V\to {V}^{\ast}$ is a bounded linear map, $\phi :H\to H$ is a continuous map and $F:I\times H\to {2}^{{V}^{\ast}}$ is a multifunction satisfying some conditions mentioned later.
where $x(0)=\phi (x)$, $f(t)\in F(t,x(t))$ for all $v\in V$ and almost all $t\in I$.
We will need the following hypotheses on the data problem (3.1).
 (i)
$t\to A(t,x)$ is measurable;
 (ii)for each $t\in I$, the operator $A(t,\cdot ):V\to {V}^{\ast}$ is uniformly monotone and hemicontinuous, that is, there exists a constant ${C}_{1}>0$ (independent of t) such that$\u3008A(t,{x}_{1})A(t,{x}_{2}),{x}_{1}{x}_{2}\u3009\ge {C}_{1}{\parallel {x}_{1}{x}_{2}\parallel}_{H}^{p}$
 (iii)
there exist a constant ${C}_{2}>0$, a nonnegative function $a(\cdot )\in {L}_{q}(I)$ and a nondecreasing continuous function $\eta (\cdot )\in {L}_{q}(I)$ such that ${\parallel A(t,x)\parallel}_{{V}^{\ast}}\le a(t)+{C}_{2}\eta ({\parallel x\parallel}_{V})$ for all $x\in V$, a.e. on I;
 (iv)there exist ${C}_{3}>0$, ${C}_{4}>0$, $b(\cdot )\in {L}_{1}(I)$ such that$\u3008A(t,x),x\u3009\ge {C}_{3}{\parallel x\parallel}_{V}^{p}{C}_{4}{\parallel x\parallel}_{V}^{p1}+\frac{1}{2T}{\parallel x(0)\parallel}^{2}b(t)\phantom{\rule{1em}{0ex}}\text{a.e.}I,\mathrm{\forall}x\in V,$
 (i)
$(t,x)\to F(t,x)$ is graph measurable;
 (ii)
for almost all $t\in I$, $x\to F(t,x)$ is LSC;
 (iii)there exist a nonnegative function ${b}_{1}(\cdot )\in {L}_{q}(I)$ and a constant ${C}_{5}>0$ such that$F(t,x)=sup\{{\parallel f\parallel}_{{V}^{\ast}}:f\in F(t,x)\}\le {b}_{1}(t)+{C}_{5}{\parallel x\parallel}_{H}^{k1}\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in V\text{a.e.}I,$
where $1\le k<p$.
 (i)
$B:V\to {V}^{\ast}$ is a bounded linear selfadjoint operator such that $(Bx,x)\ge 0$ for all $x\in V$, a.e. on I;
 (ii)there exists a continuous function $\phi :{L}_{p}(I,H)\to H$ such that$\parallel \phi (u)\phi (v)\parallel \le {\parallel uv\parallel}_{C(I,H)}\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in C(I,H),$
and $\phi (0)=0$.
for some constants $\stackrel{\u02c6}{{M}_{1}},\stackrel{\u02c6}{{M}_{2}}>0$ and all $x\in {L}_{p}(I,H)$.
For the proofs of main results, we need the following lemma.
Lemma 3.1 Let $V\subseteq H\subseteq {V}^{\ast}$ be an evolution triple and let $X={L}_{p}(I,V)$, where $1<p<\mathrm{\infty}$ and $0<T<\mathrm{\infty}$. Then the linear operator $L:D(L)\subseteq X\to {X}^{\ast}$ defined by (3.2) is maximal monotone.
Choose a set of functions ${({a}_{n})}_{n\ge 1}$ in H such that $T{a}_{n}\to v(T)\xi $ as $n\to \mathrm{\infty}$. For $\xi \in H$, let $u(t)=t{a}_{n}+\xi $, then $u\in D(L)$. By (3.3), we have $v(0)=u(0)=\xi $ as $n\to \mathrm{\infty}$. Hence, $v\in D(L)$. 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.
has only one solution.
Invoking the Banach fixed point theorem, the operator P has only one fixed point ${x}_{\u03f5}=P({x}_{\u03f5})$, i.e., ${x}_{\u03f5}$ is the uniform solution of (3.4).
Therefore, we define ${L}_{\u03f5}:{W}_{pq}(I)\to {X}^{\ast}$ as ${L}_{\u03f5}x=\dot{x}+A(t,x)+Bx$ and $x(0)=(1\u03f5)\phi (x)$. By Step 1, we have ${L}_{\u03f5}:{W}_{pq}(I)\to {X}^{\ast}$ is onetoone and surjective, and so ${L}_{\u03f5}^{1}:{X}^{\ast}\to {W}_{pq}(I)$ is well defined.
Step 2. ${L}_{\u03f5}^{1}:{X}^{\ast}\to {L}_{p}(I,H)$ is completely continuous.
We only need to show that ${L}_{\u03f5}^{1}$ is continuous and maps a bounded set into a relatively compact set. We claim that ${L}_{\u03f5}:{W}_{pq}(I)\to {X}^{\ast}$ is continuous. In fact, let ${\{{x}_{n}\}}_{n\ge 1}\subset {W}_{pq}(I)$ such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$. From (H1)(ii) and (H3), we infer that ${x}_{n}\to x$, $A({x}_{n})\to A(x)$, $B{x}_{n}\to Bx$, a.e. I as $n\to \mathrm{\infty}$. Obviously, $\phi ({x}_{n})\to \phi (x)$. Therefore, ${L}_{\u03f5}:{W}_{pq}(I)\to {X}^{\ast}$ is continuous and ${L}_{\u03f5}^{1}$ is continuous.
with $p>2$. So, there exists an ${M}_{1}>0$ such that ${\parallel x\parallel}_{X}\le {M}_{1}$. Because of the boundedness of operators A, B, we obtain that there exists an ${M}_{2}>0$ such that ${\parallel \dot{x}\parallel}_{{X}^{\ast}}\le {M}_{2}$. Hence, ${\parallel x\parallel}_{{W}_{pq}}\le M$ for some constant $M>0$. Therefore, we have ${L}_{\u03f5}^{1}(K)$ is bounded in ${W}_{pq}(I)$. But ${W}_{pq}(I)$ is compactly embedded in ${L}_{p}(I,H)$. Therefore, ${L}_{\u03f5}^{1}(K)$ is relatively compact in ${L}_{p}(I,H)$.
Let $\stackrel{\u02c6}{N}:{L}_{p}(I,H)\to {2}^{{X}^{\ast}}$ be a multivalued Nemitsky operator corresponding to F and $\stackrel{\u02c6}{N}$ was defined by $\stackrel{\u02c6}{N}(x)=\{v\in {X}^{\ast}:v(t)\in F(t,x(t))\}$ a.e. on I.
Step 3. $\stackrel{\u02c6}{N}(\cdot )$ has nonempty, closed, decomposable values and is LSC.
Therefore, $x\in {U}_{\lambda}$ and this proves the LSC of $\stackrel{\u02c6}{N}(\cdot )$. By Lemma 2.2, we obtain a continuous map $f:{L}_{p}(I,H)\to {X}^{\ast}$ such that $f(x)\in \stackrel{\u02c6}{N}(x)$. To finish our proof, we need to solve the fixed point problem: $x={L}_{\u03f5}^{1}\circ f(x)$.
Since the embedding $V\to H$ is compact, the embedding ${W}_{pq}(I)\to {L}_{p}(I,H)$ is compact. That is, ${x}_{n}\to x$ in ${L}_{p}(I,H)$ whenever ${x}_{n}\rightharpoonup x$ in ${W}_{pq}(I)$. By using the above relation and the continuity of f, we have $f({x}_{n})\to f(x)$ in ${X}^{\ast}$ whenever ${x}_{n}\rightharpoonup x$ in ${W}_{pq}(I)$. So, ${L}_{\u03f5}^{1}\circ f:{L}_{p}(I,H)\to {L}_{p}(I,H)$ is compact.
Step 4. We claim that the set $\mathrm{\Gamma}=\{x\in {L}_{p}(I,H):x=\sigma {L}_{\u03f5}^{1}\circ f(x),\sigma \in (0,1)\}$ is bounded.
for all $x\in \mathrm{\Gamma}$.
It follows from (3.14) that ${\parallel x\parallel}_{{W}_{pq}}\le {\parallel x\parallel}_{X}+{\parallel \dot{x}\parallel}_{{X}^{\ast}}\le \stackrel{\u02c6}{M}$ for some constant $\stackrel{\u02c6}{M}>0$. Hence, Γ is a bounded subset of ${W}_{pq}(I)$. So, Γ is a bounded subset of ${L}_{p}(I,H)$ since the embedding ${W}_{pq}(I)\to {L}_{p}(I,H)$ is compact.
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:
 (i)
$(t,x)\to F(t,x)$ is graph measurable;
 (ii)
for almost all $t\in I$, $x\to F(t,x)$ 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 ${W}_{pq}(I)$.
Proof The proof is as that of Theorem 3.1. So, we only present those particular points where the two proofs differ.
the last inclusion being a consequence of the hypothesis (H4)(ii). So, $v\in \stackrel{\u02c6}{N}(x)$, which means that $\stackrel{\u02c6}{N}(\cdot )$ is nonempty.
then $f\in \stackrel{\u02c6}{N}(x)\cap \mathrm{\Xi}$, i.e., ${\stackrel{\u02c6}{N}}^{1}(\mathrm{\Xi})$ is closed in ${L}_{p}(I,H)$. This proves the upper semicontinuity of $\stackrel{\u02c6}{N}(\cdot )$ from ${L}_{p}(I,H)$ into ${X}_{w}^{\ast}$.
Clearly, $x(0)=\phi (x)$, then $x\in S$. Thus, S is weakly compact in ${W}_{pq}(I)$. □
4 Relaxation theorem
where $extF(t,x)$ denotes the extremal point set of $F(t,x)$. We need the following hypothesis:
 (i)
$(t,x)\to F(t,x)$ is graph measurable;
 (ii)
for almost all $t\in I$, $x\to F(t,x)$ is hcontinuous; and (H2)(iii) holds.
Theorem 4.1 If hypotheses (H1), (H3) and (H5) hold, then the problem (4.1) has at least one solution.
So, ${x}_{n}(t)\to x(t)$ in $C(I,H)$. Since $x={L}_{0}^{1}(h)$ with $h\in W$, we conclude that ${L}_{0}^{1}(W)\subseteq C(I,H)$ is compact. From Lemma 2.3, we can find a continuous map $f:\stackrel{\u02c6}{K}\to {L}_{q}(I,H)$ such that $f(x)(t)\in extF(t,x(t))$ a.e. on I for all $x\in \stackrel{\u02c6}{K}$. Then ${L}_{0}^{1}\circ f$ is a compact operator. On applying the Schauder fixed point theorem, there exists an $x\in \stackrel{\u02c6}{K}$ such that $x={L}_{0}^{1}\circ f(x)$. This is a solution of (4.1), and so ${S}_{e}\ne \mathrm{\varnothing}$ in ${W}_{pq}(I)$. □
For the relation theorem of the problem (4.1), we need the following definition and hypotheses.
Definition 4.1 A Carathéodory function $\mu :I\times {R}^{+}\to {R}^{+}$ is said to be a Kamke function if it is integrally bounded on the bounded sets, $\mu (t,0)\equiv 0$ and the unique solution of the differential equation $\dot{s}(t)=\mu (t,s(t))$, $s(0)=0$ is $s(t)\equiv 0$.
and (H5) hold.
Theorem 4.2 If hypotheses (H1), (H3) and (H6) hold, then $\overline{{S}_{e}}=S$, where the closure is taken in $C(I,H)$.
Hence, ${\parallel {x}_{\u03f5}(t)x(t)\parallel}_{H}^{2}\le Q(t)$, where $Q(0)={\parallel {x}_{\u03f5}(0)x(0)\parallel}_{H}^{2}$ and $\dot{Q}(t)=\mu (t,Q(t))+4\u03f5$. By (4.7), then $Q(0)=0$. Let $\u03f5\to 0$, we have ${\parallel {x}_{\u03f5}(t)x(t)\parallel}_{H}\to 0$. Therefore, $x=\stackrel{\u02c6}{x}$, i.e., ${x}_{\u03f5}\to x$ and ${x}_{\u03f5}\in {S}_{e}$, and so $S\subseteq \overline{{S}_{e}}$. Also, S is closed in $C(I,H)$ (see the proof of Theorem 3.2), thus $S=\overline{{S}_{e}}$. □
5 Examples
The pLaplacian $div({\mathrm{\nabla}u}^{p2}\mathrm{\nabla}u)$ arises in many applications such as Finsler geometry and nonNewtonian fluids. In [30], Liu showed the existence of antiperiodic solutions to the problem (5.1) where $f(t,x,\cdot )$ is continuous.
The hypotheses on the data of this problem (5.1) are the following:
 (i)
${f}_{i}(t,x,u)$ ($i=1,2$) are Nemitskymeasurable, i.e., $u:T\times \mathrm{\Omega}\to R$ for all measurable, $u\to {f}_{i}(t,x,u)$ ($i=1,2$) is measurable;
 (ii)there exists ${a}_{2}(t)\in {L}^{q}{(t)}_{+}$, $C>0$, such that${f}_{i}(t,x,u)\le {a}_{2}(t)+C{\parallel u\parallel}^{k1},$
where $1\le k<p$.
The hypothesis (H7) implies that (H4) is satisfied. Note that ${f}_{1}(t,x,\cdot )$ is lower semicontinuous, ${f}_{2}(t,x,\cdot )$ is upper semicontinuous, and so $\stackrel{\u02c6}{f}(t,x,\cdot )$ is USC (see [22], Example 2.8, p.371]). Let $\phi (u)=\frac{1}{2b}{\int}_{0}^{b}u(s,x)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{2}u(b,x)$, 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 $u\in {L}_{p}(T,{W}_{0}^{1,p}(\mathrm{\Omega}))$ such that $\frac{\partial u}{\partial t}\in {L}_{q}(T,{W}^{1,p}(\mathrm{\Omega}))$.
The hypotheses on the data (5.3) are the following:
 (i)
$(t,z)\to {A}_{k}(t,z,u,\eta )$ is measurable on $T\times Z$ for every $(u,\eta )\in R\times {R}^{N}$, $(u,\eta )\to {A}_{k}(t,z,u,\eta )$ is continuous on $R\times {R}^{N}$ for all almost all $(t,z)\in T\times Z$;
 (ii)
${A}_{k}(t,z,u,\eta )\le {\stackrel{\u02c6}{\alpha}}_{1}(t,z)+{\stackrel{\u02c6}{c}}_{1}(z)(u+\eta )$ with a nonnegative function ${\stackrel{\u02c6}{\alpha}}_{1}\in {L}^{2}(I\times Z)$ and ${\stackrel{\u02c6}{c}}_{1}(z)\in {L}^{\mathrm{\infty}}(Z)$ for almost all $t\in T$;
 (iii)
${\sum}_{k=1}^{N}({A}_{k}(t,z,u,\eta ){A}_{k}(t,z,u,{\eta}^{\prime}))({\eta}_{k}{\eta}_{k}^{\prime})\ge {\eta {\eta}^{\prime}}^{2}$ for almost all $t\in T$;
 (iv)
${A}_{k}(t,z,0,0)=0$ for all $(t,z)\in T\times Z$.
 (i)
for all $x\in R$, $(t,z)\to g(t,z,x)$ is measurable;
 (ii)
for all $(t,z)\in T\times Z$, $x\to g(t,z,x)$ is continuous;
 (iii)for almost all $(t,z)\in T\times Z$ and all $x\in R$, we have$g(t,z,x)\le {\eta}_{1}(t,z)+{\eta}_{2}(t)x$
with ${\eta}_{1}\in {L}_{2}(T,{L}^{2}(Z))$, ${\eta}_{2}\in {L}^{\mathrm{\infty}}(T)$.
 (i)
for all $x\in R$, $(t,z)\to U(t,z,x)$ is measurable;
 (ii)
for all $(t,z)\in T\times Z$, $x\to U(t,z,x)$ is hcontinuous;
 (iii)
for almost all $(t,z)\in T\times Z$ and all $x\in R$, $U(t,z,x)\le \gamma $, with $\gamma >0$.
for all $v\in {H}_{0}^{1}(Z)$. Let $\phi (u)=\frac{1}{b}{\int}_{0}^{b}u(s,z)\phantom{\rule{0.2em}{0ex}}ds$.
Using hypotheses (H9) and (H10), it is straightforward to check that F satisfies the hypothesis (H5).
It is easy to get the following theorem by applying Theorem 4.1 to the problem (5.3).
Theorem 5.2 If hypotheses (H8)(H10) hold, then the problem (5.3) has one solution $x\in {L}_{2}(T,{H}_{0}^{1}(Z))\cap C(T,{L}^{2}(Z))$ with $\frac{\partial x}{\partial t}\in {L}_{2}(T,{H}^{1}(Z))$.
Declarations
Acknowledgements
The authors are in debt to the anonymous referees whose comments helped them to improve the final version of this paper. This work is partially supported by the National Natural Science Foundation of China (No. 11172036, 11171350, 10902125) and the Natural Science Foundation of Jilin Province Grants 201115133.
Authors’ Affiliations
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