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Existence results for abstract semilinear evolution differential inclusions with nonlocal conditions
Boundary Value Problems volume 2013, Article number: 153 (2013)
Abstract
In this paper, we use a new method to study semilinear evolution differential inclusions with nonlocal conditions in Banach spaces. We derive conditions for F and g for the existence of mild solutions. The results obtained here improve and generalize many known results.
MSC:34A60, 34G20.
1 Introduction
In this paper, we discuss the nonlocal initial value problem
where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e., ${C}_{0}$semigroup) $T(t)$ in a Banach space X, and $F:[0,1]\times X\to {P}_{c}(X)$, $g:C([0,1];X)\to X$ are given Xvalued functions.
The study of nonlocal evolution equations was initiated by Byszewski [1]. Since these represent mathematical models of various phenomena in physics, Byszewski’s work was followed by many others [2–7]. Subsequently, many authors have contributed to the study of the differential inclusions (1.1). Differential inclusions (1.1) were first considered by Aizicovici and Gao [8] when g and $T(t)$ are compact. In [9–12] the semilinear evolution differential inclusions (1.1) were discussed when A generates a compact semigroup. Xue and Song [13] established the existence of mild solutions to the differential inclusions (1.1) when A generates an equicontinuous semigroup and $F(t,\cdot )$ is l.s.c. for a.e. $t\in [0,1]$. In [14] the author proved the existence of mild solutions of the differential inclusions (1.1) when A generates an equicontinuous semigroup and a Banach space X which is separable and uniformly smooth. In [15] Zhu and Li studied the differential inclusions (1.1) when F admits a strongly measurable selector. In [16] the differential inclusions (1.1) were discussed when $\{A(t)\}$ is a family of linear (not necessarily bounded) operators. In [17] local and global existence results for differential inclusions with infinite delay in a Banach space were considered. Benchohra and Ntouyas [18] studied the secondorder initial value problems for delay integrodifferential inclusions. In [19, 20] the impulsive multivalued semilinear neutral functional differential inclusions were discussed in the case that the linear semigroup $T(t)$ is compact. The purpose of this paper is to continue the study of these authors. By using a new method, we prove the existence results of mild solutions for (1.1) under the following conditions of g and $T(t)$: g is either compact or Lipschitz continuous and $T(t)$ is an equicontinuous semigroup. So, our work extends and improves many main results such as those in [8–12, 14, 15].
The organization of this work is as follows. In Section 2, we recall some definitions and facts about setvalued analysis and the measure of noncompactness. In Section 3, we give the existence of mild solutions of the nonlocal initial value problem (1.1). In Section 4, an example is given to show the applications of our results.
2 Preliminaries
Let $(X,\parallel \cdot \parallel )$ be a real Banach space. Let ${P}_{c}(X)=\{A\subseteq X:\text{nonempty, closed, convex}\}$. A multivalued map $G:X\to X$ is convex (closed) valued if $G(x)$ is convex (closed) for all $x\in X$. We say that G is bounded on bounded sets if $G(B)$ is bounded in X for each bounded set B of X. The map G is called upper semicontinuous (u.s.c.) on X if for each ${x}_{0}\in X$ the set $G({x}_{0})$ is a nonempty, closed subset of X, and if for each open set N of X containing $G({x}_{0})$, there exists an open neighborhood M of ${x}_{0}$ such that $G(M)\subseteq N$. Also, G is said to be completely continuous if $G(B)$ is relatively compact for every bounded subset $B\subseteq X$. If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., ${x}_{n}\to {x}_{0}$, ${y}_{n}\to {y}_{0}$, ${y}_{n}\in G({x}_{n})$ imply ${y}_{0}\in G({x}_{0})$). Moreover, the following conclusions hold. Let $D\subset X$ and $G(x)$ be closed for all $x\in D$, if G is u.s.c. and D is closed, then $graph(G)$ is closed. If $\overline{G(D)}$ is compact and D is closed, then G is u.s.c. if and only if $graph(G)$ is closed. Finally, we say that G has a fixed point if there exists $x\in X$ such that $x\in G(x)$.
We denote by $C([0,1];X)$ the space of Xvalued continuous functions on $[0,1]$ with the norm $\parallel x\parallel =sup\{\parallel x(t)\parallel ;t\in [0,1]\}$, and by ${L}^{1}(0,1;X)$ the space of Xvalued Bochner functions on $[0,1]$ with the norm $\parallel x\parallel ={\int}_{0}^{1}\parallel x(s)\parallel \phantom{\rule{0.2em}{0ex}}ds$.
A ${C}_{0}$semigroup $T(t)$ is said to be compact if $T(t)$ is compact for any $t>0$. If the semigroup $T(t)$ is compact, then $t\to T(t)x$ is equicontinuous at all $t>0$ with respect to x in all bounded subsets of X; i.e., the semigroup $T(t)$ is equicontinuous. If A is the generator of an analytic semigroup $T(t)$ or a differentiable semigroup $T(t)$, then $T(t)$ is an equicontinuous ${C}_{0}$semigroup (see [21]). In this paper, we suppose that A generates an equicontinuous semigroup $T(t)$ on X. Since no confusion may occur, we denote by α the Hausdorff measure of noncompactness on both X and $C([0,1];X)$.
Definition 2.1 A function $x\in C([0,1];X)$ is a mild solution of (1.1) if

(1)
$x(t)=T(t)g(x)+{\int}_{0}^{t}T(ts)v(s)\phantom{\rule{0.2em}{0ex}}ds$,

(2)
$x(0)=g(x)$, where $v\in {S}_{F,x}=\{v\in {L}^{1}(I,X):v(t)\in F(t,x(t))\}$.
To prove the existence results in this paper, we need the following lemmas.
Lemma 2.2 [22]
If $W\subseteq C([0,1];X)$ is bounded, then $\alpha (W(t))\le \alpha (W)$ for all $t\in [0,1]$, where $W(t)=\{x(t);x\in W\}\subseteq X$. Furthermore, if W is equicontinuous on $[0,1]$, then $\alpha (W(t))$ is continuous on $[0,1]$, and $\alpha (W)=sup\{\alpha (W(t));t\in [0,1]\}$.
Lemma 2.3 [22]
If ${\{{W}_{n}\}}_{n=1}^{+\mathrm{\infty}}$ is a decreasing sequence of bounded closed nonempty subsets of X and ${lim}_{n\to +\mathrm{\infty}}\alpha ({W}_{n})=0$, then ${\bigcap}_{n=1}^{+\mathrm{\infty}}{W}_{n}$ is nonempty and compact in X.
Lemma 2.4 [23]
If ${\{{u}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset {L}^{1}(0,1;X)$ is uniformly integrable, then $\alpha ({\{{u}_{n}(t)\}}_{n=1}^{\mathrm{\infty}})$ is measurable and
Lemma 2.5 [24]
If the semigroup $T(t)$ is equicontinuous and $\eta \in {L}^{1}(0,1;{\mathrm{\Re}}^{+})$, then the set $\{t\to {\int}_{0}^{t}T(ts)x(s)\phantom{\rule{0.2em}{0ex}}ds;x\in {L}^{1}(0,1;{\mathrm{\Re}}^{+}),\parallel x(s)\parallel \le \eta (s),\phantom{\rule{0.1em}{0ex}}\mathit{\text{for a.e.}}\phantom{\rule{0.1em}{0ex}}s\in [0,1]\}$ is equicontinuous on $[0,1]$.
Lemma 2.6 [25]
If W is bounded, then for each $\epsilon >0$, there is a sequence ${\{{u}_{n}\}}_{n=1}^{\mathrm{\infty}}\subseteq W$ such that
A countable set ${\{{f}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset {L}^{1}(0,1;X)$ is said to be semicompact if

(a)
it is integrably bounded: $\parallel {f}_{n}(t)\parallel \le \omega (t)$ for a.e. $t\in [0,1]$ and every $n\ge 1$, where $\omega (\cdot )\in {L}^{1}(0,1;{\mathrm{\Re}}^{+})$;

(b)
the set ${\{{f}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is relatively compact for a.e. $t\in [0,1]$.
Lemma 2.7 [26]
Every semicompact set is weakly compact in the space ${L}^{1}(0,1;X)$.
If ${\{{f}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset {L}^{1}(0,1;X)$ is semicompact, then ${\{{\int}_{0}^{t}T(ts){f}_{n}(s)\phantom{\rule{0.2em}{0ex}}ds\}}_{n=1}^{\mathrm{\infty}}$ is relatively compact in $C([0,1];X)$ and, moreover, if ${f}_{n}\rightharpoonup {f}_{0}$, then
as $n\to \mathrm{\infty}$.
The map $F:W\subseteq X\to X$ is said to be α contraction if there exists a positive constant $k<1$ such that
for any bounded closed subset $Q\subseteq W$.
Lemma 2.9 [27–30] (Fixed point theorem)
If $W\subseteq X$ is a nonempty, bounded, closed, convex and compact subset, the map $F:W\to {2}^{W}$ is upper semicontinuous with $F(x)$ a nonempty, closed, convex subset of W for each $x\in W$, then F has at least one fixed point in W.
Lemma 2.10 [26] (Fixed point theorem)
If $W\subseteq X$ is nonempty, bounded, closed and convex, the map $F:W\to {2}^{W}$ is a closed α contraction map with $F(x)$ a nonempty, convex and compact subset of W for each $x\in W$, then F has at least one fixed point in W.
3 Main results
In this section, by using the measure of noncompactness and fixed point theorems, we give the existence results of the nonlocal initial value problem (1.1). Here we list the following hypotheses.

(1)
The ${C}_{0}$ semigroup $T(t)$ generated by A is equicontinuous. We denote $N=sup\{\parallel T(t)\parallel ;t\in [0,1]\}$.

(2)
$g:C([0,1];X)\to X$ is continuous and compact, there exist positive constants c and d such that $\parallel g(x)\parallel \le c\parallel x\parallel +d$, $\mathrm{\forall}x\in C([0,1];X)$.

(3)
The multivalued operator $F:[0,1]\times X\to {P}_{c}(X)$ satisfies the hypotheses: the set ${S}_{F,x}=\{v\in {L}^{1}(I,X):v(t)\in F(t,x(t));\text{for a.e.}t\in [0,1]\}$ is nonempty.
$t\to F(t,x)$ is measurable for every $x\in X$;
$x\to F(t,x)$ is u.s.c. for a.e. $t\in [0,1]$;

(4)
There exists $L\in {L}^{1}(0,1;{\mathrm{\Re}}^{+})$ such that for any bounded $D\subset X$,
$$\alpha (F(t,D))\le L(t)\alpha (D)$$
for a.e. $t\in [0,1]$.

(5)
There exist a function $m\in {L}^{1}(0,1;{\mathrm{\Re}}^{+})$ and a nondecreasing continuous function $\mathrm{\Omega}:{\mathrm{\Re}}^{+}\to {\mathrm{\Re}}^{+}$ such that
$$\parallel F(t,x)\parallel \le m(t)\mathrm{\Omega}(\parallel x\parallel )$$
for all $x\in X$, and a.e. $t\in [0,1]$.
Remark 3.1 If $dimX<\mathrm{\infty}$, then ${S}_{F,x}\ne \mathrm{\varnothing}$ for each $x\in C([0,1];X)$ (see Lasota and Opial [31]). If $dimX=\mathrm{\infty}$ and $x\in C([0,1];X)$, then the set ${S}_{F,x}$ is nonempty if and only if the function $Y:[0,1]\to \mathrm{\Re}$ defined by $Y(t)=inf\{\parallel v\parallel :v\in F(t,x(t))\}$ belongs to ${L}^{1}(0,1;{\mathrm{\Re}}^{+})$ (see Hu and Papageorgiou [32]).
The following lemma plays a crucial role in the proof of the main theorem.
Lemma 3.2 [26]
Under assumptions (3)(5), if we consider sequences ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset C([0,1];X)$ and ${\{{v}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset {L}^{1}(0,1;X)$, where ${v}_{n}\in {S}_{F,{x}_{n}}$, such that ${x}_{n}\to x$, ${v}_{n}\rightharpoonup v$, then $v\in {S}_{F,x}$.
Now we give the existence results under the above hypotheses.
Theorem 3.3 If (1)(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constant R with
Proof Define the operator $\mathrm{\Gamma}:C([0,1];X)\to C([0,1];X)$ by
We shall show that the multivalued Γ has at least one fixed point. The fixed point is then a mild solution of the problem (1.1).

(1)
We contract a bounded, convex, closed and compact set $W\subset C([0,1];X)$ such that Γ maps W into itself.
In view of (3.1), we know there exists a constant $\eta >0$ such that
where ${T}_{0}=N(cR+d)$.
Then there exists a positive integer K such that
Hence, we get $0={t}_{0}<{t}_{1}<{t}_{2}<\cdots <{t}_{K1}<{t}_{K}=1$ such that
We denote ${W}_{0}=\{x\in C([0,1];X),{\parallel x(t)\parallel}_{i}=sup\{\parallel x(t)\parallel :t\in [{t}_{i1},{t}_{i}]\}\le {T}_{0}+i\eta ,i=1,2,\dots ,K\}$, then ${W}_{0}\subseteq C([0,1];X)$ is nonempty, bounded, closed and convex.
For any $x\in {W}_{0}$, we have
Therefore
and
which implies $\mathrm{\Gamma}:{W}_{0}\to {2}^{{W}_{0}}$ is a bounded operator.
Define ${W}_{1}=\overline{\mathit{conv}}\mathrm{\Gamma}({W}_{0})$, where $\overline{\mathit{conv}}$ means the closure of the convex hull in $C([0,1];X)$. Then ${W}_{1}\subset C([0,1];X)$ is nonempty bounded closed convex on $[0,1]$ with ${W}_{1}\subseteq {W}_{0}$. Let ${W}_{n+1}=\overline{\mathit{conv}}\mathrm{\Gamma}({W}_{n})$ for all $n\ge 1$. Similarly to the above discussions, we know that ${W}_{n+1}\subseteq {W}_{n}$ for $n=1,2,\dots $ as ${W}_{1}\subseteq {W}_{0}$ and ${W}_{1},{W}_{2},\dots $ are both nonempty, closed, bounded and convex. Thus ${\{{W}_{n}\}}_{n=1}^{+\mathrm{\infty}}$ is a decreasing sequence consisting of subsets of $C([0,1];X)$. Moreover, set
then W is a convex, closed and bounded subset of $C([0,1];X)$ and $\mathrm{\Gamma}(W)\subseteq W$.
Now, we claim that W is nonempty and compact in $C([0,1];X)$. To do so, from Lemma 2.6, we know for arbitrary given $\u03f5>0$, there exist sequences ${\{{v}_{n}\}}_{n=1}^{+\mathrm{\infty}}\subset {S}_{F,{W}_{n}}$ such that
Since this is true for arbitrary $\u03f5>0$, we have
Because ${W}_{n}$ is decreasing for n, we can define
Let $n\to +\mathrm{\infty}$, we have
It implies that $\mu (t)=0$ for all $t\in [0,1]$. By Lemma 2.2, we know that ${lim}_{n\to +\mathrm{\infty}}\alpha ({W}_{n})=0$. Using Lemma 2.3, we obtain $W={\bigcap}_{n=1}^{+\mathrm{\infty}}{W}_{n}$ is nonempty and compact in $C([0,1];X)$.

(2)
We shall show that Γ is closed on W with closed convex values. It is very easy to see that Γ has convex values.
Let us now verity that $graph(\mathrm{\Gamma})$ is closed. Let ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset W$ with ${x}_{n}\to x$ in $C([0,1];X)$, and ${y}_{n}\in \mathrm{\Gamma}{x}_{n}$ with ${y}_{n}\to y$ in $C([0,1];X)$. Moreover, let ${\{{v}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset {L}^{1}(0,1;X)$ be a sequence such that ${v}_{n}\in {S}_{F,{x}_{n}}$ for any $n\ge 1$, and
As ${x}_{n}\to x$ in $C([0,1];X)$, we know that ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a bounded set of $C([0,1];X)$, we denote ${R}_{x}=sup\{\parallel {x}_{n}\parallel :n=1,2,\dots \}$.
From hypothesis (5), we obtain
Then we have the set ${\{{v}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is integrably bounded for a.e. $t\in [0,1]$.
From hypothesis (4), we know
for a.e. $t\in [0,1]$. Then the set ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is relatively compact for a.e. $t\in [0,1]$.
So, the set ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is semicompact. By applying Lemma 2.7, it yields that ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is weakly compact in ${L}^{1}(0,1;X)$. We get that there exists $v\in {L}^{1}(0,1;X)$ such that ${v}_{n}\rightharpoonup v$. Therefore, we infer that
Further, we have
and hence
By Lemma 3.2, it implies that $v\in {S}_{F,x}$, i.e., $y\in \mathrm{\Gamma}(x)$. Therefore $graph(\mathrm{\Gamma})$ is closed. And hence Γ has closed values on W.

(4)
Γ is u.s.c. on W.
Since $\overline{\mathrm{\Gamma}W}\subseteq W$ is compact, W is closed and $graph(\mathrm{\Gamma})$ is closed, we can come to the conclusion that Γ is u.s.c. (see [30]).
Finally, due to fixed point Lemma 2.9, Γ has at least one point $x\in \mathrm{\Gamma}(x)$, and x is a mild solution to the semilinear evolution differential inclusions with the nonlocal conditions (1.1). Thus the proof is complete. □
Remark 3.4 In [8–12] the authors discuss the nonlocal initial value problem (1.1) when $T(t)$ is compact. In [14] the existence of mild solutions of the differential inclusions (1.1) is proved when A generates an equicontinuous semigroup and Banach space X is separable and uniformly smooth. In this paper, by using a new method, we prove the operator Γ maps compact set W into itself. We do not impose any restriction on the coefficient $L(t)$, and we only require $T(t)$ to be an equicontinuous semigroup. So, Theorem 3.3 generalizes and improves the related results in [8–12, 14].
Theorem 3.5 [15]
If (1)(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constant $R>0$ such that
Proof In view of (3.4), we get
From Theorem 3.3, the nonlocal initial value problem (1.1) has at least one mild solution. □
Remark 3.6 If $N=1$, $c=\frac{1}{3}$, $d=0$, $\mathrm{\Omega}(x)=x$ and ${\int}_{0}^{1}m(s)\phantom{\rule{0.2em}{0ex}}ds=1$. We cannot obtain a constant R such that
By using Theorem 3.5, we do not know whether or not equation (1.1) has a mild solution. But we know there exists a constant $R=1$ such that
So, Theorem 3.3 is better than Theorem 3.5.
Theorem 3.7 [12]
If (1)(5) are satisfied and $\parallel g(x)\parallel \le d$, then there is at least one mild solution for (1.1) provided that
Proof In view of (3.5), we get there exists a constant R such that
By Theorem 3.3, we complete the proof of this theorem. □
Next, we give the existence result for (1.1) when g is Lipschitz continuous.
We suppose that:

(6)
There exists a constant $c\in {\mathrm{\Re}}^{+}$ such that $\parallel g(u)g(v)\parallel \le c\parallel uv\parallel $ for all $u,v\in C([0,1];X)$. Therefore, $\parallel g(x)\parallel \le c\parallel x\parallel +d$, where $d=\parallel g(0)\parallel $.
Theorem 3.8 If (1) and (3)(6) are satisfied and
then there is at least one mild solution for (1.1) provided that there exists a constant R satisfying
Proof With the same arguments as given in the first portion of the proof of Theorem 3.3, we know $\mathrm{\Gamma}:{W}_{0}\to {2}^{{W}_{0}}$ is a bounded map with convex values and is closed on ${W}_{0}$.
Now, we prove the values of Γ are compact in $C([0,1];X)$.
Let $x\in C([0,1];X)$ and ${y}_{n}\in \mathrm{\Gamma}(x)$. To prove that $\mathrm{\Gamma}(x)$ is compact, we have to show that ${y}_{n}$ has a subsequence converging to a point $y\in \mathrm{\Gamma}(x)$. We have ${v}_{n}\in {S}_{F,x}$ such that
From hypothesis (5), we obtain
Then we have the set ${\{{v}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is integrably bounded for a.e. $t\in [0,1]$.
From hypothesis (4), we know
for a.e. $t\in [0,1]$. Then the set ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is relatively compact for a.e. $t\in [0,1]$.
So, the set ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is semicompact. By applying Lemma 2.7, it yields that ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is weakly compact in ${L}^{1}(0,1;X)$. We get that there exists $v\in {L}^{1}(0,1;X)$ such that ${v}_{n}\rightharpoonup v$. Therefore, we infer that
and
By Lemma 3.2, it implies that $v\in {S}_{F,x}$, i.e.,$y\in \mathrm{\Gamma}(x)$. Therefore Γ has compact values.
Next, we prove Γ is an α contraction map. For any $B\subseteq {W}_{0}$, we have
From Lemma 2.6, we know for arbitrary given $\u03f5>0$, there exist sequences ${\{{v}_{n}\}}_{n=1}^{+\mathrm{\infty}}\subset {S}_{F,B}$ such that
Since this is true for arbitrary $\u03f5>0$, we have
Therefore, we obtain
Noting $Nc+4N{\int}_{0}^{t}L(s)\phantom{\rule{0.2em}{0ex}}ds<1$, therefore Γ is an α contraction map.
Finally, due to Lemma 2.10, Γ has at least one fixed point. This completes the proof. □
4 An example
In this section, as an application of our main results, an example is presented. We consider the following partial differential equation:
where Ω is a bounded domain in ${\mathrm{\Re}}^{n}$ with a smooth boundary ∂ Ω, ${a}_{\alpha}(\zeta )$ is a smooth real function on $\overline{\mathrm{\Omega}}$.
We suppose that

(a)
The differential operator ${\mathrm{\Sigma}}_{\alpha \le 2m}{a}_{\alpha}(\zeta ){D}^{\alpha}$ is strongly elliptic [21].

(b)
The function $k:[0,1]\times \mathrm{\Omega}\times \mathrm{\Omega}\times \mathrm{\Re}\to \mathrm{\Re}$ satisfies the following conditions:
(b_{1}) $k(t,\zeta ,\eta ,r)$ is a continuous function about r for a.e. $(t,\zeta ,\eta )\in [0,1]\times \mathrm{\Omega}\times \mathrm{\Omega}$.
(b_{2}) $k(t,\zeta ,\eta ,r)$ is measurable about $(t,\zeta ,\eta )$ for each fixed $r\in \mathrm{\Re}$.
(b_{3}) For any $R>0$, there is ${\beta}_{R}\in {L}^{1}([0,1]\times \mathrm{\Omega}\times \mathrm{\Omega}\times \mathrm{\Re};{\mathrm{\Re}}^{+})$ such that
for all $(t,\zeta ,\eta ,r),(t,{\zeta}^{\mathrm{\prime}},\eta ,r)\in ([0,1]\times \mathrm{\Omega}\times \mathrm{\Omega}\times \mathrm{\Re})$ with $r\le R$, and
uniformly for $\zeta \in \mathrm{\Omega}$.
(b_{4}) There exist $a(\cdot )\in L(0,1)$ and $d(\cdot )\in {L}^{2}([0,1]\times \mathrm{\Omega}\times \mathrm{\Omega},{\mathrm{\Re}}^{+})$ such that
for all $(t,\zeta ,\eta ,r)\in ([0,1]\times \mathrm{\Omega}\times \mathrm{\Omega}\times \mathrm{\Re})$.
Let $D(A)={H}^{2m}\cap {H}_{0}^{m}(\mathrm{\Omega})$ and $\mathit{Au}(\zeta )={\mathrm{\Sigma}}_{\alpha \le 2m}{a}_{\alpha}(\zeta ){D}^{\alpha}u(\zeta ,\cdot )$, then A generates an analytic semigroup on $X={L}^{2}(\mathrm{\Omega})$ ([21]). We suppose
From [33], we obtain g satisfies hypothesis (2).
Then equation (4.1) can be regarded as the following nonlocal semilinear evolution equation:
By using Theorem 3.3, the problem (4.1) has at least one mild solution $u\in C([0,1];{L}^{2}(\mathrm{\Omega}))$ provided that hypotheses (3)(5) and (3.1) hold.
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Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. The research was supported by the Scientific Research Foundation of Nanjing Institute of Technology (No: QKJA2011009).
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Keywords
 semilinear evolution differential inclusions
 mild solutions
 measure of noncompactness
 upper semicontinuous