# Existence results for abstract semilinear evolution differential inclusions with nonlocal conditions

- Tao Zhu
^{1}Email author, - Chao Song
^{1}and - Gang Li
^{2}

**2013**:153

https://doi.org/10.1186/1687-2770-2013-153

© Zhu et al.; licensee Springer. 2013

**Received: **12 April 2013

**Accepted: **10 June 2013

**Published: **1 July 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.

## Keywords

## 1 Introduction

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 *X*-valued 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 second-order 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 set-valued 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 *X*-valued 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 *X*-valued 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)
$x(t)=T(t)g(x)+{\int}_{0}^{t}T(t-s)v(s)\phantom{\rule{0.2em}{0ex}}ds$,

- (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* $\{t\to {\int}_{0}^{t}T(t-s)x(s)\phantom{\rule{0.2em}{0ex}}ds;x\in {L}^{1}(0,1;{\mathrm{\Re}}^{+}),\parallel x(s)\parallel \le \eta (s),\phantom{\rule{0.1em}{0ex}}\mathit{\text{for a.e.}}\phantom{\rule{0.1em}{0ex}}s\in [0,1]\}$ *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)
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*${\{{\int}_{0}^{t}T(t-s){f}_{n}(s)\phantom{\rule{0.2em}{0ex}}ds\}}_{n=1}^{\mathrm{\infty}}$

*is relatively compact in*$C([0,1];X)$

*and*,

*moreover*,

*if*${f}_{n}\rightharpoonup {f}_{0}$,

*then*

*as* $n\to \mathrm{\infty}$.

*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

- (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 $x\in X$;

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

- (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

- (1)
We contract a bounded, convex, closed and compact set $W\subset C([0,1];X)$ such that Γ maps

*W*into itself.

where ${T}_{0}=N(cR+d)$.

*K*such that

We denote ${W}_{0}=\{x\in C([0,1];X),{\parallel x(t)\parallel}_{i}=sup\{\parallel x(t)\parallel :t\in [{t}_{i-1},{t}_{i}]\}\le {T}_{0}+i\eta ,i=1,2,\dots ,K\}$, then ${W}_{0}\subseteq C([0,1];X)$ is nonempty, bounded, closed and convex.

which implies $\mathrm{\Gamma}:{W}_{0}\to {2}^{{W}_{0}}$ is a bounded operator.

then *W* is a convex, closed and bounded subset of $C([0,1];X)$ and $\mathrm{\Gamma}(W)\subseteq W$.

*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

*n*, we can define

- (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.

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 ${R}_{x}=sup\{\parallel {x}_{n}\parallel :n=1,2,\dots \}$.

Then we have the set ${\{{v}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is integrably bounded for a.e. $t\in [0,1]$.

for a.e. $t\in [0,1]$. Then the set ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is relatively compact for a.e. $t\in [0,1]$.

*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

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.

- (6)
There exists a constant $c\in {\mathrm{\Re}}^{+}$ such that $\parallel g(u)-g(v)\parallel \le c\parallel u-v\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 $C([0,1];X)$.

Then we have the set ${\{{v}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is integrably bounded for a.e. $t\in [0,1]$.

for a.e. $t\in [0,1]$. Then the set ${\{{v}_{n}(t)\}}_{n=1}^{\mathrm{\infty}}$ is relatively compact for a.e. $t\in [0,1]$.

By Lemma 3.2, it implies that $v\in {S}_{F,x}$, *i.e.*,$y\in \mathrm{\Gamma}(x)$. Therefore Γ has compact values.

*α*contraction map. For any $B\subseteq {W}_{0}$, we have

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

where Ω is a bounded domain in ${\mathrm{\Re}}^{n}$ with a smooth boundary *∂* Ω, ${a}_{\alpha}(\zeta )$ is a smooth real function on $\overline{\mathrm{\Omega}}$.

- (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 $r\in \mathrm{\Re}$.

_{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

uniformly for $\zeta \in \mathrm{\Omega}$.

_{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 $(t,\zeta ,\eta ,r)\in ([0,1]\times \mathrm{\Omega}\times \mathrm{\Omega}\times \mathrm{\Re})$.

*A*generates an analytic semigroup on $X={L}^{2}(\mathrm{\Omega})$ ([21]). We suppose

From [33], we obtain *g* satisfies hypothesis (2).

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.

## Declarations

### 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).

## Authors’ Affiliations

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