Existence results for abstract semilinear evolution differential inclusions with nonlocal conditions
© Zhu et al.; licensee Springer. 2013
Received: 12 April 2013
Accepted: 10 June 2013
Published: 1 July 2013
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.
Keywordssemilinear evolution differential inclusions mild solutions measure of noncompactness upper semicontinuous
where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e., -semigroup) in a Banach space X, and , are given X-valued functions.
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 .
We denote by the space of X-valued continuous functions on with the norm , and by the space of X-valued Bochner functions on with the norm .
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 .
, where .
To prove the existence results in this paper, we need the following lemmas.
Lemma 2.2 
If is bounded, then for all , where . Furthermore, if W is equicontinuous on , then is continuous on , and .
Lemma 2.3 
If is a decreasing sequence of bounded closed nonempty subsets of X and , then is nonempty and compact in X.
Lemma 2.4 
Lemma 2.5 
If the semigroup is equicontinuous and , then the set is equicontinuous on .
Lemma 2.6 
it is integrably bounded: for a.e. and every , where ;
the set is relatively compact for a.e. .
Lemma 2.7 
Every semicompact set is weakly compact in the space .
for any bounded closed subset .
If is a nonempty, bounded, closed, convex and compact subset, the map is upper semicontinuous with a nonempty, closed, convex subset of W for each , then F has at least one fixed point in W.
Lemma 2.10  (Fixed point theorem)
If is nonempty, bounded, closed and convex, the map is a closed α contraction map with a nonempty, convex and compact subset of W for each , then F has at least one fixed point in W.
3 Main results
The semigroup generated by A is equicontinuous. We denote .
is continuous and compact, there exist positive constants c and d such that , .
The multivalued operator satisfies the hypotheses: the set is nonempty.
is measurable for every ;
- (4)There exists such that for any bounded ,
- (5)There exist a function and a nondecreasing continuous function such that
for all , and a.e. .
The following lemma plays a crucial role in the proof of the main theorem.
Lemma 3.2 
Under assumptions (3)-(5), if we consider sequences and , where , such that , , then .
Now we give the existence results under the above hypotheses.
We contract a bounded, convex, closed and compact set such that Γ maps W into itself.
We denote , then is nonempty, bounded, closed and convex.
which implies is a bounded operator.
then W is a convex, closed and bounded subset of and .
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 in , we know that is a bounded set of , we denote .
Then we have the set is integrably bounded for a.e. .
for a.e. . Then the set is relatively compact for a.e. .
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].
Theorem 3.5 
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.
Theorem 3.7 
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.
There exists a constant such that for all . Therefore, , where .
Proof With the same arguments as given in the first portion of the proof of Theorem 3.3, we know is a bounded map with convex values and is closed on .
Now, we prove the values of Γ are compact in .
Then we have the set is integrably bounded for a.e. .
for a.e. . Then the set is relatively compact for a.e. .
By Lemma 3.2, it implies that , i.e.,. Therefore Γ has compact values.
Noting , 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 with a smooth boundary ∂ Ω, is a smooth real function on .
The differential operator is strongly elliptic .
The function satisfies the following conditions:
(b1) is a continuous function about r for a.e. .
(b2) is measurable about for each fixed .
uniformly for .
for all .
From , we obtain g satisfies hypothesis (2).
By using Theorem 3.3, the problem (4.1) has at least one mild solution provided that hypotheses (3)-(5) and (3.1) hold.
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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