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:semilinear evolution differential inclusions; mild solutions; measure of noncompactness; upper semicontinuous
In this paper, we discuss the nonlocal initial value problem
The study of nonlocal evolution equations was initiated by Byszewski . 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  when g and are compact. In [9-12] the semilinear evolution differential inclusions (1.1) were discussed when A generates a compact semigroup. Xue and Song  established the existence of mild solutions to the differential inclusions (1.1) when A generates an equicontinuous semigroup and is l.s.c. for a.e. . In  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  Zhu and Li studied the differential inclusions (1.1) when F admits a strongly measurable selector. In  the differential inclusions (1.1) were discussed when is a family of linear (not necessarily bounded) operators. In  local and global existence results for differential inclusions with infinite delay in a Banach space were considered. Benchohra and Ntouyas  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 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 : g is either compact or Lipschitz continuous and 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.
Let be a real Banach space. Let . A multivalued map is convex (closed) valued if is convex (closed) for all . We say that G is bounded on bounded sets if 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 the set is a nonempty, closed subset of X, and if for each open set N of X containing , there exists an open neighborhood M of such that . Also, G is said to be completely continuous if is relatively compact for every bounded subset . 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., , , imply ). Moreover, the following conclusions hold. Let and be closed for all , if G is u.s.c. and D is closed, then is closed. If is compact and D is closed, then G is u.s.c. if and only if is closed. Finally, we say that G has a fixed point if there exists such that .
A -semigroup is said to be compact if is compact for any . If the semigroup is compact, then is equicontinuous at all with respect to x in all bounded subsets of X; i.e., the semigroup is equicontinuous. If A is the generator of an analytic semigroup or a differentiable semigroup , then is an equicontinuous -semigroup (see ). In this paper, we suppose that A generates an equicontinuous semigroup on X. Since no confusion may occur, we denote by α the Hausdorff measure of noncompactness on both X and .
To prove the existence results in this paper, we need the following lemmas.
Lemma 2.10 (Fixed point theorem)
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.
The following lemma plays a crucial role in the proof of the main theorem.
Now we give the existence results under the above hypotheses.
Theorem 3.3If (1)-(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constantRwith
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).
Then there exists a positive integer K such that
Define , where means the closure of the convex hull in . Then is nonempty bounded closed convex on with . Let for all . Similarly to the above discussions, we know that for as and are both nonempty, closed, bounded and convex. Thus is a decreasing sequence consisting of subsets of . Moreover, set
(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.
From hypothesis (5), we obtain
From hypothesis (4), we know
Further, we have
(4) Γ is u.s.c. on W.
Since is compact, W is closed and is closed, we can come to the conclusion that Γ is u.s.c. (see ).
Finally, due to fixed point Lemma 2.9, Γ has at least one point , 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 is compact. In  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 , and we only require to be an equicontinuous semigroup. So, Theorem 3.3 generalizes and improves the related results in [8-12,14].
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. □
So, Theorem 3.3 is better than Theorem 3.5.
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:
Theorem 3.8If (1) and (3)-(6) are satisfied and
then there is at least one mild solution for (1.1) provided that there exists a constantRsatisfying
From hypothesis (5), we obtain
From hypothesis (4), we know
Therefore, we obtain
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:
We suppose that
(a) The differential operator is strongly elliptic .
Let and , then A generates an analytic semigroup on (). We suppose
From , we obtain g satisfies hypothesis (2).
Then equation (4.1) can be regarded as the following nonlocal semilinear evolution equation:
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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).