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Anti-periodic solutions for nonlinear evolution equations
© Cheng et al.; licensee Springer 2012
Received: 26 April 2012
Accepted: 4 September 2012
Published: 19 September 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].
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.
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.
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 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 . The following lemmas are still needed in the proof of our main theorems.
Lemma 2.1 (see )
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
becomes a separable Banach space. The following is our main result of this part.
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 ;
- (iv)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 .
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.
Hence, a.e. .
the solution of the problem (3.2) is an anti-periodic solution of the problem (3.1). This completes the proof. □
where . Obviously, is hemicontinuous.
where the constants . The claim is proved.
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.
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 .
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:
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 graph measurable;
for almost all , is LSC;
- (iii)there exists an nonnegative function and a constant such that
for all , , where or with (c in ).
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 .
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.
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: .
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.
the last inclusion being a consequence of hypothesis (ii). So . Thus we prove the nonemptiness of .
then , i.e., is closed in . This proves the upper semicontinuity of from into .
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).
then , hence S is weakly compact in . □
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.
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 .
where B is a positive definite matrix. The hypotheses on the data (4.3) are as follows:
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 3.2 on the problem (4.3) and obtain:
Theorem 4.3 If hypotheses and hold, then the problem (4.3) has a solution .
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.
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