- Open Access
Anti-periodic extremal problems for a class of nonlinear evolution inclusions in
© Zhang et al.; licensee Springer. 2014
- Received: 5 September 2013
- Accepted: 24 February 2014
- Published: 12 March 2014
The aim of this paper is to establish the existence of anti-periodic solutions to the following nonlinear anti-periodic problem: a.e. , , in where denotes the extremal point set of the multifunction , and is a nonlinear map from to . Sufficient conditions for the existence of extremal solutions are presented. Also, we prove that the extremal point set of this problem is compact in and dense in the solution set of nonlinear evolution problems with a convex valued perturbation which is multivalued. We apply our results on the control system with a priori feedback.
- anti-periodic solution
- evolution inclusion
- the Schauder fixed point theorem
- continuous selection
where is a matrix, is a continuous function satisfying for all . Chen et al.  also studied this problem in a real separable Hilbert space when A is a dense self-adjoint operator which only has a point spectrum. Recently, Q Liu , ZH Liu  and Wang  considered anti-periodic problem of nonlinear evolution equation in a real reflexive Banach space and obtained some existence results by applying the theory of pseudo-monotone perturbations of maximal monotone mappings and the theory of evolution operators. However, there are few results about the extremal anti-periodic problem, which is connected with the ‘Bang-Bang’ control problem in the above works. For the case of periodic problems, we refer to the work of Li-Xue , Xue-Yu  in Hilbert and Xue-Cheng  in Banach space. Inspired by , which considers the problem in Banach space, we continue to consider the existence of solutions for a nonlinear evolution inclusion in by relaxing the constraint conditions. Our approach will be based on techniques and results of the theory of the extremal continuous selection theorem and the Schauder fixed point theorem.
The paper is divided into four parts. In Section 2, we introduce some notation, and definitions we need for the results. In Section 3, we present some basic assumptions and main results, the proofs of the main results are given based on the Schauder fixed point theorem and the extremal continuous selection theorem. Finally, an example is presented for our results in Section 4.
for all . It is well known that is a complete metric space and is a closed subset of it. When Z is a Hausdorff topological space, a multifunction is said to be h-continuous if it is continuous as a function from Z into .
A set is said to be ‘decomposable’, if for every and for every measurable we have . Let X be a Banach space and let be the Banach space of all functions which are Bochner integrable. denotes the collection of nonempty decomposable subsets of . The following lemmas are still needed in the proof of our main theorems.
Lemma 2.1 (see )
Let X be a separable metric space and let be a lower semicontinuous multifunction with closed decomposable values. Then F has a continuous selection.
Let X be a separable Banach Space and be the Banach space of all continuous functions. A multifunction is said to be Carathéodory type, if for every , is measurable, and for almost all , is h-continuous. (i.e. it is continuous form X to the metric space where h is Hausdorff metric). Let . A multifunction is called integrably bounded on M if there exists a function such that for almost all , . A nonempty subset is called σ-compact if there is a sequence of compact subsets such that . Let , such that is dense in M and σ-compact. The following continuous selection theorem in the extreme point case is due to Tolstonogov .
Lemma 2.2 (see )
Let the multifunction be of Carathéodory type and integrably bounded. Then there exists a continuous function such that for almost all , if , then , and if , then .
Let and be all the continuous functions from I to with the max norm. We let , and . is a separable Banach space under the norm .
for all and almost all . The precise hypotheses on the data of problem (1) are the following:
for each , the operator is uniformly monotone and hemicontinuous, that is, there exists a constant such that for all .
is graph measurable;
for almost all , is h-continuous;
- (iii)there exist a nonnegative function and a constant such that
for all , , where .
We still need the following lemma.
Lemma 3.1 (see )
where . Then problem (2) has a unique T-anti-periodic solution.
Theorem 3.1 If hypotheses and hold, then problem (1) has at least one solution.
So, in . Since with , we conclude that is compact. From Lemma 2.2, we can find a continuous map such that a.e. on I for all . Then is a compact operator. On applying the Schauder fixed point theorem, there exists a such that . This is a solution of (1), and so in . □
For the relation theorem of problem (1), we need the following definition and hypotheses.
Definition 3.1 (see )
Under the above hypotheses, let S denote the solution set of the equation , .
Theorem 3.2 If hypotheses and hold and, moreover, satisfies the OSL condition, we have , where the closure is taken in .
From , we know that . Let , we have , i.e. for any . Therefore, , i.e. and , and so . Also S is closed in (see the proof of Theorem 3.1), thus . □
The hypotheses on the data (10) are the following.
with , , , .
for all , is measurable;
for all , is h-continuous;
for almost all and all , , with .
Using hypotheses and , it is straightforward to check that F satisfies hypothesis .
We can apply Theorem 3.1 on problem (11) and obtain the following.
Theorem 4.1 If the hypotheses and hold, then problem (10) has a solution .
The first author was also supported by NSFC Grant 11172036. The authors are also thankful to the referee for careful reading of the paper and valuable comments.
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