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Anti-periodic solutions for nonlinear evolution equations
Advances in Difference Equations volume 2012, Article number: 165 (2012)
In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem
where is a nonlinear map and B is a bounded linear operator from to . Sufficient conditions for the existence of the solution set are presented. Also, we consider the nonlinear evolution problems with a perturbation term which is multivalued. We show that, for this problem, the solution set is nonempty and weakly compact in for the case of convex valued perturbation and prove the existence theorems of anti-periodic solutions for the nonconvex case. All illustrative examples are provided.
Anti-periodic problems have important applications in auto-control, partial differential equations and engineering, and they have been studied extensively in the past ten years. For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems , and anti-periodic wavelets are discussed in . Recently, anti-periodic boundary conditions have been considered for the Schrödinger and Hill differential operator [3, 4]. Also, anti-periodic boundary conditions appear in the study of difference equations [5, 6]. Moreover, anti-periodic boundary conditions appear in physics in a variety of situations, see [7–10].
The study of anti-periodic solutions for nonlinear evolution equations was initiated by Okochi . Since then, many authors have been devoted to investigation of the existence of anti-periodic solutions to nonlinear evolution equations in Hilbert spaces. For the details, see [12–17] 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 . Here they assume that is a uniform bound with respect to u and . We do not need these assumptions and consider the following semilinear anti-periodic problem:
where is a hemicontinuous function satisfying , is a measurable function satisfying for all and B is a bounded linear operator from to . We will establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions of Eq. (1.2) by the theory of topological degree.
In addition, we also consider the following nonlinear evolution inclusion problem:
where . We refer the reader to the work of [18, 19]. These works focused on the problem in which the multivalued term is an even lower semi-continuous convex function with a compact assumption. But, in this paper, we prove the existence theorems of anti-periodic solutions for the cases of a convex and of a nonconvex valued perturbation term which is multivalued based on the techniques and results of the theory of set-valued analysis and the Leray-Schauder fixed point theorem. As far as we know, there are few papers which deal with this type of anti-periodic problems. For recent developments involving the existence of anti-periodic solutions of differential equations, inequalities and other interesting results on anti-periodic boundary value problems, the reader is referred to [20–27] and the references therein.
On the one hand, it is well known that the neural networks have been successfully applied to signal and image processing, pattern recognition and optimization. However, many neural networks with discontinuous neuron activation functions appear in the theoretical study on dynamics of neural networks, see [28, 29]. In order to solve some practical engineering problems, people also need to present new neural networks with discontinuous activation functions. Therefore, developing a new class of neural networks with discontinuous neuron activation functions and giving the conditions of the stability are very valuable in both theory and practice. Motivated by the above discussions, in this paper, we present a class of neural networks with discontinuous neuron activation functions. Based on our results, the existence and uniqueness of the equilibrium point is investigated.
On the other hand, it has been well recognized that differential inclusions, which are certainly of their own interest, provide a useful generalization of control systems governed by differential/evolution equations with control parameters
where the control sets may also depend on the state variable x. Let . Then Eq. (1.4) is reduced to , which is a particular case of the inclusion relation in Eq. (1.3). Hence, we present an example of a nonlinear anti-periodic distributed parameter control system with a priori feedback for our results.
This paper is organized as follows. In Section 2, we state some basic knowledge from multivalued analysis. In Section 3, we first establish the existence of anti-periodic solutions for an evolution equation by the theory of topological degree, and then, by applying the Leray-Schauder fixed point theorem, we prove the existence of anti-periodic solutions for convex and nonconvex cases. Finally, two examples for our results are presented in Section 4.
For convenience, we introduce some notations as follows. In Euclidean space, expresses an inner product, while expresses the Euclidean norm. Let denote the set of the map which satisfies , and the norm in is denoted by . We recall some basic definitions and facts from multivalued analysis which we shall need in what follows. For details, we refer to the book of Hu and Papageorgiou . Let , be the Lebesgue measurable space and X be a separable Banach space. Denote
Let , , then the distance form 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. A 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 , is upper semicontinuity as -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 . The following lemmas are still needed in the proof of our main theorems.
Lemma 2.1 (see )
If X is a Banach space, is nonempty, closed and convex with , and is an upper semicontinuous multifunction which maps bounded sets into relatively compact sets, then one of the following statements is true:
the set is unbounded;
the has a fixed point, i.e., there exists such that .
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 . Now, let us state the Bressan-Colombo continuous selection theorem.
Lemma 2.2 (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.
3 Main results
3.1 The evolution equation
In this section, let
where is the weak derivative of x. is a Banach space under the norm . Equipped with the norm
becomes a separable Banach space. The following is our main result of this part.
Theorem 3.1 Assume the following hold:
and for all ;
is measurable and ;
for each , the operator is uniformly monotone and hemicontinuous, that is, there exists a constant such that for all , and the map is continuous on for all ;
is a bounded linear operator and there exists such that
then the problem (1.2) has a unique T-anti-periodic solution.
In order to complete the proof of Theorem 3.1, we need the following lemmas.
Lemma 3.1 (see )
Suppose Γ is a bounded open set of a normal space X, f is compact in and . Then the equation has at least one solution in Γ, provided with .
Lemma 3.2 Consider the equation
where is a bounded linear operator, and there exists such that for all , and . Then the problem (3.1) has a unique T-anti-periodic solution.
Proof Let x be a solution of (3.1) satisfying the boundary value condition . Then x is a T-anti-periodic solution of (3.1). Denote for all , then is a linear operator.
Firstly, we show is one to one. Suppose , then a.e. , and so a.e. . Take an inner product above with and note that
By using integration from 0 to T and the relation , one can see that
Since B is a linear bounded operator, then
for some constant . It follows that
Hence, a.e. .
Next, we claim that is surjective. For this purpose, consider the Cauchy problem
It is well known that the above problem has a unique solution which can be written as follows:
Since , then we have that
By hypothesis (iv), one has that exists; therefore, when we take
the solution of the problem (3.2) is an anti-periodic solution of the problem (3.1). This completes the proof. □
Proof of Theorem 3.1 Consider the homotopic systems of (1.2),
where . Obviously, is hemicontinuous.
First, a priori bound of the solution set is derived. We claim that there is a priori bound in for the possible solution of (3.3). Take the inner product with , and then integrate from 0 to T. It follows that
Without loss of generality, we assume . Since , and then
By hypothesis (iii), we deduce that
for some constant . Hence, there is a constant such that
for some constant . By (3.3), one has that
Integrating above from τ to t, we have that
From , we know
By (3.4) and (3.5), we obtain that there is some constant (independent of λ) such that
for any . Thus,
Since the operator A is hemicontinuous, and B is a bounded linear operator, we show that
where the constants . The claim is proved.
Secondly, we can prove the existence of anti-periodic solutions for Eq. (1.2). Set
Then Γ is a bounded open set in . By Lemma 3.2, it is easy to see that
is well defined. We define the operator , . Obviously, N is compact. Hence, the fixed point of N in is the anti-periodic solutions of Eq. (1.2). Let
By (3.7), we obtain . So for each , then we have that
where id is the identity. Consequently, N has a fixed point in Γ by Lemma 3.1. Namely, Eq. (1.2) has an anti-periodic solution.
Next, we prove the uniqueness. Suppose that , are two solutions of Eq. (1.2). Then
Take an inner product above with and note that
By using integration from 0 to T and the relation , one can see that
Hence, a.e. . This ends the proof. □
3.2 The evolution inclusions
Let and be all the continuous functions from I to with the max norm. Let , and . is a separable Banach space under the norm .
Consider the following anti-periodic problem:
where is a hemicontinuous function, B is a bounded linear operator from to , and is a multifunction. By a solution x of the problem (3.8), we mean a function , and there exists a function such that
for all and almost all .
In this section, we prove two existence theorems under the hypothesis that the multivalued nonlinearity F is convex-valued (‘convex existence theorem’) or nonconvex-valued (‘nonconvex existence theorem’). The precise hypotheses on the data of the problem (3.8) are as follows:
: 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 , and the map is continuous on for all .
: is a bounded linear operator, and there exists such that
: is a multifunction such that
is graph measurable;
for almost all , is LSC;
there exists an nonnegative function and a constant such that
for all , , where or with (c in ).
: is a multifunction such that
is graph measurable;
for almost all , has a closed graph; and (iii) holds.
Theorem 3.2 If hypotheses , and hold, then the problem (3.8) has a solution .
Proof Let for all . By Theorem 3.1, we have is one to one and surjective, and so is well defined. So, we prove that
is completely continuous (i.e., it is continuous and maps bounded sets into relatively compact sets). To this end, let be bounded. We shall show that is relatively compact in . For this purpose, let , then with . By (3.7), we have and for some constant . From these bounds we infer that is bounded in . But is compactly embedded in . Therefore, is relatively compact in . Also, from the fact that is a compact operator, is continuous.
Next, let be the multivalued Nemitsky operator corresponding to F and N be defined by a.e. on I.
We claim that has nonempty, closed, decomposable values and is LSC. The closedness and decomposability of the values of are easy to check. For the nonemptiness, note that if , by hypothesis (i), is graph measurable, so we apply Aumann’s selection theorem and obtain a measurable map such that a.e. on I. By hypothesis (iii), . Thus for every , . To prove the lower semicontinuity of , we only show that every , is a USC -valued function. Note that
(see Hiai and Umegaki  Th. 2.2). We shall show that for every , the superlevel set is closed in . Let and assume that in . By passing to a subsequence, if necessary, we may assume that a.e. on I as . By hypothesis (ii), is an upper semicontinuous -valued function. So, via Fatou’s lemma, we have
Therefore, and this proves the LSC of .
We apply Lemma 2.2 and obtain a continuous map such that . To finish our proof, we only need to solve the fixed point problem: .
We claim that the set is bounded. Let , then . By hypothesis (iii), we can derive
with . By (3.6), we get that
for some constant . So, we have that
Thus, we can find a constant such that . If , we can also find a constant such that . Similar to the estimation of (3.7), we have that
for some constant . So, Γ is bounded in . Invoking Leray-Schauder’s alternative theorem, we obtain there exists such that , x is a solution of the problem (3.8). This ends the proof. □
Theorem 3.3 If hypotheses , and hold, then the problem (3.8) has a solution . Moreover, the solution set is weakly compact in .
Proof The proof is as that of Theorem 3.2. So, we only present those particular points where the two proofs differ.
In this case, the multivalued Nemistsky operator has nonempty closed, convex values in and is USC. The closedness and convexity of the values of are clear. To prove the nonemptiness, let and be a sequence of step functions such that
Then by virtue of hypothesis (i), for every , admits a measurable selector . From hypothesis (iii), we have that there exists a constant such that
So is uniformly integrable. By the Dunford-Pettis theorem, and by passing to a subsequence if necessary, we may assume that weakly in . Then from Theorem 3.1 in , we have
the last inclusion being a consequence of hypothesis (ii). So . Thus we prove the nonemptiness of .
Next, we show that is USC from into . Let C be a nonempty and weakly closed subset of . We need to show that the set
is closed. Let and assume in . Passing to a subsequence, we can get that a.e. on I. Let , . Then by virtue of hypothesis (iii) and the Dunford-Pettis theorem, we may assume that weakly in . As before, we have
then , i.e., is closed in . This proves the upper semicontinuity of from into .
We consider the following fixed point problem:
Recalling that is completely continuous, we see that is USC and maps bounded sets into relatively compact sets. We easily check that the set
is bounded, as in the proof of Theorem 3.2. Invoking Lemma 2.1, there exists such that . Evidently, this is a solution of the problem (3.8).
Let S denote the solution set of the problem (3.8). As in the proof of Theorem 3.2, we have that , where . By virtue of hypothesis (iii) and the Dunford-Pettis theorem, we may assume that weakly in . As before, we have
then , hence S is weakly compact in . □
As an application of the previous results, we introduce two examples. Consider a class of neural networks described by the system of differential equations
where is the vector of neuron state, is an diagonal matrix, where , , are the neuron self-inhibitions; is an positive definite matrix, which represents the neuron interconnection matrix. Moreover, is a mapping where , represents the neuron input-output activation and is the mapping of neuron inputs.
We set . It is easy to check satisfies the condition of Theorem 3.1. Moreover, is bounded and B is a positive definite matrix. Thus, by Theorem 3.1 we easily obtain the following theorem.
Theorem 4.1 If for any , there exists a constant such that , and , for all , , then the problem (4.1) has a unique anti-periodic solution.
Discontinuous dynamical systems, particularly neural networks with discontinuous activation functions, arise in a number of applications. Further, we need the following assumptions.
H(C): We have , for any , where Ω denotes the class of functions from to R which are monotone nondecreasing bounded and have at most a finite number of jump discontinuities in every compact interval.
We note that if I satisfies , then any , , possesses only isolated jump discontinuities where is not necessary defined. Hence for all , we have
where , . Thus the differential equations (4.1) become the following differential inclusions:
The existence and the stability of the equilibrium point of (4.1) were first discussed in  ( is constant). In , the authors proved the existence of periodic solutions of (4.1) when is the continuous periodic input and is discontinuous.
We set , it is easy to check satisfies . Thus, by Theorem 3.3, we obtain the following theorem.
Theorem 4.2 If for any , there exists a constant such that , and hold, then the problem (4.1) has a nonempty set of solutions .
Next, 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 , . We consider the following control system:
where B is a positive definite matrix. The hypotheses on the data (4.3) are as follows:
: , 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 , B satisfies hypothesis . Also, let be defined by
Using hypotheses and , it is straightforward to check that F satisfies hypothesis .
Rewrite the problem (4.3) in the following equivalent evolution inclusion form:
We can apply Theorem 3.2 on the problem (4.3) and obtain:
Theorem 4.3 If hypotheses and hold, then the problem (4.3) has a solution .
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The authors would like to express their sincere appreciation to the reviewer for his/her helpful comments in improving the presentation and quality of the paper. This work is partially supported by NSFC Grants 11171350 and Natural Science Foundation of Jilin Province Grants 201115133. The second author was partially supported by NSFC Grant (11171350) and Natural Science Foundation of Jilin Province (201115133) of China.
The authors declare that they have no competing interests.
YC carried out the main part of this manuscript. FC participated in the discussion and corrected the main theorem. HH provided all examples for our results. All authors read and approved the final manuscript.
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Cheng, Y., Cong, F. & Hua, H. Anti-periodic solutions for nonlinear evolution equations. Adv Differ Equ 2012, 165 (2012). https://doi.org/10.1186/1687-1847-2012-165
- anti-periodic solution
- evolution equation
- Leray-Schauder alternative theorem
- measurable selection
- continuous selection