Anti-periodic extremal problems for a class of nonlinear evolution inclusions in
Journal of Inequalities and Applications volume 2014, Article number: 111 (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.
In this paper, we consider the following anti-periodic problems:
in where , satisfies some conditions mentioned later, is a nonlinear map from to . Anti-periodic problems of evolution inclusions have important applications in many fields, such as auto-control, partial differential equations and engineering, etc. The study of anti-periodic solutions for nonlinear evolution equations was initiated by Okochi . Since then, many authors devoted themselves to the investigation of the existence of anti-periodic solutions to nonlinear evolution equations in Hilbert spaces. For the details, see [2–7] and the references therein. In , Chen studied the anti-periodic solution for the following first order semilinear evolution equation:
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 convenience, we introduce some notation as follows. In Euclidean space, expresses inner product, while expresses the Euclidean norm. Let denote the set of the map which satisfies , and the norm in is denoted by . If , the . We recall some basic definitions and facts from multivalued analysis which we shall need in the sequel. For details we refer to the book of Zeidler . Let , be the Lebesgue measurable space and X be a separable Banach space. Denote
Let , , then the distance from x to A is given by . A multifunction is said to be measurable if and only if, for every , the function is measurable. A multifunction is said to be graph measurable, if with being the Borel σ-field of X. On we can define a generalized metric, known in the literature as the ‘Hausdorff metric’, by setting
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 .
Let Y, Z be Hausdorff topological spaces and . We say that is ‘upper semicontinuous (USC)’ (resp., ‘lower semicontinuous (LSC)’), if for all nonempty closed, (resp., ) is closed in Y. An USC multifunction has a closed graph in , while the converse is true if G is locally compact (i.e. for every there exists a neighborhood U of y such that is compact in Z). A multifunction which is both USC and LSC is said to be ‘continuous’. If Y, Z are both metric spaces, then the above definition of LSC is equivalent to saying that for all , has upper semicontinuity as an -valued function. Also, lower semicontinuity is equivalent to saying that if in Y as , then
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 .
3 Main results
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 .
Consider the following anti-periodic problem:
where is a hemicontinuous function, is a multifunction. By a solution x of problem (1), we mean a function and there exists a function such that
for all and almost all . The precise hypotheses on the data of problem (1) are the following:
: is a nonlinear function such that
for each , the operator is uniformly monotone and hemicontinuous, that is, there exists a constant such that for all .
: is a multifunction such that
is graph measurable;
for almost all , is h-continuous;
there exist a nonnegative function and a constant such that
for all , , where .
We still need the following lemma.
Lemma 3.1 (see )
If the hypothesis holds, consider the equation
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.
Proof We define as and . By Lemma 3.1, we have is one to one and surjective, and so is well defined. As in the proof of Theorem 3.1, we obtain a priori bound for which denotes the solution set of problem (1). We know that there exist , , such that and for all . Let , . We may assume that , a.e. on I for all . So let
then is compact convex subset in . Obviously is convex. We only need to show the compactness. Let , then there exists such that , i.e. . By the definition of W, so W is uniformly bounded in . By the Dunford-Pettis theorem, passing to a subsequence if necessary, we may assume that in for some . From the definition of W, we have
Therefore, the sequence is bounded. Because of the compactness of the embedding , we find that the sequence is relatively compact. So by passing to a subsequence if necessary, we may assume that in . Moreover, by the boundedness of the sequence , it follows that the sequence is uniformly bounded and passing to a subsequence if necessary, we may assume that in . Since the embedding is continuous and is compact, it follows that in and in . Hence, in for all , (m being the Lebesgue measure on R). Since A is hemicontinuous and monotone. Thus, in and as , we obtain a.e. on I and . Note that
Taking the inner product above with and integrating from 0 to T, one can see that
By hypothesis , it follows that
So, we can find such that
Using the integration by parts formula for functions in , for any we have
By (5), we see that
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 )
The multifunction mapping is called ‘one-sided Lipschitz (OSL)’ continuous if there is an integrable function such that for every , , and there exists such that
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 .
Proof Let , then there exist and a.e. on I, such that
As before let , then is compact convex subset in . For every , we define the multifunction
Clearly, for every , , and it is graph measurable. On applying the Aumann selection theorem, we get a measurable function such that almost everywhere on I. So we define the multifunction
We see that has nonempty and decomposable values. It follows from Theorem 3 of  that is LSC. Therefore is LSC and has closed and decomposable values. So we apply Lemma 2.1 to get a continuous map such that for all . Invoking II-Theorem 8.31 of  (in [, p.260]), we can find a continuous map such that almost everywhere on I, and for all . Now let and set , . Note that a.e. on I with , so we have in . We consider the following problem:
where . We see that is a compact operator and by the Schauder fixed point theorem, we obtain a solution of (7). We see that the sequence is uniformly bounded. So by passing to a subsequence if necessary, we may assume that in . From the proof of Theorem 3.1, we know that in and . Note that . So, we have
By in and in , we have
Hence, there exists a constant , and one has
as . It follows that
Let , then . Integrating over (9) from 0 to t, one has
By using the Gronwall inequality, we have
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 . □
4 An application
We present an example of a nonlinear anti-periodic distributed parameter control system, with a priori feedback (i.e. state dependent control constraint set). Let , . Consider the following control system:
The hypotheses on the data (10) are the following.
: , are Carathéodory functions such that for almost all
with , , , .
: is a multifunction such that
for all , is measurable;
for all , is h-continuous;
for almost all and all , , with .
Let be the operator defined by . Evidently, using hypothesis , it is straightforward to check that A satisfies hypothesis . Also, let be defined by
Using hypotheses and , it is straightforward to check that F satisfies hypothesis .
Rewrite problem (10) in the following equivalent evolution inclusion form:
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 .
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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.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final version.
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Zhang, J., Cheng, Y., Li, C. et al. Anti-periodic extremal problems for a class of nonlinear evolution inclusions in . J Inequal Appl 2014, 111 (2014). https://doi.org/10.1186/1029-242X-2014-111