Anti-periodic solutions for nonlinear evolution equations

*Correspondence: congfz67@126.com 1Fundamental Department, Aviation University of Air Force, Changchun, 130022, People’s Republic of China Full list of author information is available at the end of the article Abstract In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem { ẋ + A(t, x) + Bx = f (t) a.e. t ∈ R, x(t + T ) = –x(t),


Introduction
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, antiperiodic boundary conditions have been considered for the Schrödinger and Hill differential operator [, ]. Also, anti-periodic boundary conditions appear in the study of difference equations [, ]. Moreover, anti-periodic boundary conditions appear in physics in a variety of situations, see [-].
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 [-] and the references therein. In [], Chen studied the anti-periodic solution for the following first-order semilinear evolution equation: where A : R N → R N is a matrix, f : R × R N → R N is a continuous function satisfying f (t + T, u) = -f (t, -u) for all (t, u) ∈ R × R N . Here they assume that f (t, u) is a uniform bound with respect to u and T   A  < . We do not need these assumptions and consider the following semilinear anti-periodic problem: where A : R N → R N is a hemicontinuous function satisfying A(t + T, x) = -A(t, -x), f : R → R N is a measurable function satisfying f (t +T) = -f (t) for all t ∈ R and B is a bounded linear operator from R N to R N . We will establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions of Eq. (.) by the theory of topological degree.
In addition, we also consider the following nonlinear evolution inclusion problem: where I = [, T]. We refer the reader to the work of [, ]. These works focused on the problem in which the multivalued term F(t, x) 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 [-] 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 [, ]. 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 parameterṡ where the control sets U(·, ·) may also depend on the state variable x. Let F(t, x) = f (t, x, U(t, x)). Then Eq. (.) is reduced toẋ ∈ F(t, x), which is a particular case of the http://www.advancesindifferenceequations.com/content/2012/1/165 inclusion relation in Eq. (.). 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 , we state some basic knowledge from multivalued analysis. In Section , 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 .

Preliminaries
For convenience, we introduce some notations as follows. In Euclidean space, (·, ·) expresses an inner product, while | · | expresses the Euclidean norm. Let Let A ⊂ P f (X), x ∈ X, then the distance form x to A is given by d(x, A) = inf{|x -a| : a ∈ A}. A multifunction F : I → P f (X) is said to be measurable if and only if, for every z ∈ X, the function t → d(z, for all A, B ∈ P f (X). It is well known that (P f (X), h) is a complete metric space and P fc (X) is a closed subset of it. When Z is a Hausdorff topological space, a multifunction G : Z → P f (X) is said to be h-continuous if it is continuous as a function from Z into (P f (X), h).
Let Y , Z be Hausdorff topological spaces and G : Y →  Z \{φ}. We say that G(·) is 'upper semicontinuous (USC)' (resp. 'lower semicontinuous (LSC)'), if for all C ⊆ Z nonempty closed, A USC multifunction has a closed graph in Y × Z, while the converse is true if G is locally compact (i.e., for every y ∈ Y , there exists a neighborhood U of y such that F(U) 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 z ∈ Z, Also, lower semicontinuity is equivalent to saying that if y n → y in Y as n → ∞, then G(y) ⊆ limG(y n ) = z ∈ Z : lim d Z z, G(y n ) =  = z ∈ Z : z = lim z n , z n ∈ G(y n ), n ≥  . http://www.advancesindifferenceequations.com/content/2012/1/165 A set D ⊆ L  (I, X) is said to be 'decomposable' , if for every g  , g  ∈ D and for every J ⊆ I measurable, we have χ J g  + χ J c g  ∈ D. The following lemmas are still needed in the proof of our main theorems.
Lemma . (see []) If X is a Banach space, C ⊂ X is nonempty, closed and convex with  ∈ C, and G : C → P kc (C) is an upper semicontinuous multifunction which maps bounded sets into relatively compact sets, then one of the following statements is true: Let X be a Banach space and let L  (I, X) be the Banach space of all functions u : I → X which are Bochner integrable. D(L  (I, X)) denotes the collection of nonempty decomposable subsets of L  (I, X). Now, let us state the Bressan-Colombo continuous selection theorem.
Lemma . (see []) Let X be a separable metric space and let F : X → D(L  (I, X)) be a lower semicontinuous multifunction with closed decomposable values. Then F has a continuous selection.

The evolution equation
In this section, let whereẋ is the weak derivative of x. C T is a Banach space under the norm x c = max t∈R |x|. Equipped with the norm W , becomes a separable Banach space. The following is our main result of this part.
Theorem . Assume the following hold: for all x, y, z ∈ R N ; (iv) B : R N → R N is a bounded linear operator and there exists c ∈ R + such that In order to complete the proof of Theorem ., we need the following lemmas.

Lemma . (see []) Suppose is a bounded open set of a normal space X, f is compact in and p ∈ X\f (∂ ). Then the equation f (x) = p has at least one solution in , provided
. Then the problem (.) has a unique T-anti-periodic solution.
Proof Let x be a solution of (.) satisfying the boundary value condition By using integration from  to T and the relation x() = -x(T), one can see that Since B is a linear bounded operator, then Next, we claim that L : W , → L  ([, T]; R N ) is surjective. For this purpose, consider the Cauchy problem (.) http://www.advancesindifferenceequations.com/content/2012/1/165 It is well known that the above problem has a unique solution which can be written as follows: Since -x() = x(T), then we have that By hypothesis (iv), one has that (-Ie BT ) - exists; therefore, when we take the solution of the problem (.) is an anti-periodic solution of the problem (.). This completes the proof.
Proof of Theorem . Consider the homotopic systems of (.), where λ ∈ [, ]. Obviously, λf (t) -λA(t, x) is hemicontinuous. First, a priori bound of the solution set is derived. We claim that there is a priori bound in W , for the possible solution x(t) of (.). Take the inner product with x(t), and then integrate from  to T. It follows that Without loss of generality, we assume A(t, ) = . Since T  (ẋ, x) dt = , and then By hypothesis (iii), we deduce that which implies Integrating above from τ to t, we have that By (.) and (.), we obtain that there is some constant M  >  (independent of λ) such that Thus, Since the operator A is hemicontinuous, and B is a bounded linear operator, we show that where id is the identity. Consequently, N has a fixed point in by Lemma .. Namely, Eq. (.) has an anti-periodic solution.
Next, we prove the uniqueness. Suppose that x  , x  are two solutions of Eq. (.). Theṅ Take an inner product above with x x  and note that By using integration from  to T and the relation x() = -x(T), one can see that Hence, x  = x  a.e. t ∈ R. This ends the proof. Consider the following anti-periodic problem:

The evolution inclusions
where A : R N → R N is a hemicontinuous function, B is a bounded linear operator from R N to R N , and F : R × R N →  R N is a multifunction. By a solution x of the problem (.), we mean a function x ∈ W , (I, R N ), and there exists a function f (t) ∈ F(t, x(t)) such that for all v ∈ R N and almost all t ∈ I. http://www.advancesindifferenceequations.com/content/2012/1/165 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 (.) are as follows: (ii) for each t ∈ I, the operator A(t, ·) : R N → R N is uniformly monotone and hemicontinuous, that is, there exists a constant p >  such that (iii) there exists an nonnegative function b(·) ∈ L  + (I) and a constant c  >  such that H(B)).
x) is graph measurable; (ii) for almost all t ∈ I, x → F(t, x) has a closed graph; and H(F)  (iii) holds. We claim that N(·) has nonempty, closed, decomposable values and is LSC. The closedness and decomposability of the values of N(·) are easy to check. For the nonemptiness, note that if x ∈ L  ([, T]; R N ), by hypothesis H(F)  (i), (t, x) → F(t, x) is graph measurable, so we apply Aumann's selection theorem and obtain a measurable map v : F(t, x(t)) a.e. on I. By hypothesis H(F)  (iii), v ∈ L  ([, T]; R N ). Thus for every x ∈ R N , N(x) = ∅. To prove the lower semicontinuity of N(·), we only show that every u ∈ L  ([, T]; R N ), x → d(u, N(x)) is a USC R + -valued function. Note that

Theorem . If hypotheses H(A), H(B) and H(F)
(see Hiai and Umegaki [] Th. .). We shall show that for every λ ≥ , the superlevel set . By passing to a subsequence, if necessary, we may assume that x n (t) → x(t) a.e. on I as n → ∞. By hypothesis F(t, x)) is an upper semicontinuous R + -valued function. So, via Fatou's lemma, we have Therefore, x ∈ U λ and this proves the LSC of N(·). We apply Lemma . and obtain a continuous map f : N(x). To finish our proof, we only need to solve the fixed point problem: We claim that the set = {x ∈ L  ([, T]; R N ) :  http://www.advancesindifferenceequations.com/content/2012/1/165 with α < . By (.), we get that for some constant c  > . So, we have that Thus, we can find a constant c  >  such that x  ≤ c  . If α = , we can also find a constant Similar to the estimation of (.), we have that Then by virtue of hypothesis H(F)  (i), for every n ≥ , t → F(t, s n ) admits a measurable selector f n (t). From hypothesis H(F)  (iii), we have that there exists a constant c  >  such that So {f n } n≥ is uniformly integrable. By the Dunford-Pettis theorem, and by passing to a subsequence if necessary, we may assume that f is closed. Let {x n } n≥ ⊆ N - (C) and assume x n → x in W , (I, R N ). Passing to a subsequence, we can get that x n (t) → x(t) a.e. on I. Let f n ∈ N(x n ) ∩ C, n ≥ . Then by virtue of hypothesis H(F)  (iii) and the Dunford-Pettis theorem, we may assume that f n → f ∈ C weakly in L  ([, T]; R N ). As before, we have F(t, x) a.e. on I, then f ∈ N(x)∩C, i.e., N -(C) is closed in W , (I, R N ). This proves the upper semicontinuity of N(·) from W , (I, We consider the following fixed point problem: Recalling is bounded, as in the proof of Theorem .. Invoking Lemma ., there exists u ∈ W , (I, R N ) such that u ∈ L - N(u). Evidently, this is a solution of the problem (.). Let S denote the solution set of the problem (.). As in the proof of Theorem ., we have that |S| = sup{ u , : u ∈ S} ≤ M, where M > . By virtue of hypothesis H(F)  (iii) and the Dunford-Pettis theorem, we may assume that u n → u weakly in W , (I, R N ). As before, we have Lu ∈ conv lim {Lu n } n≥ ⊆ conv lim F(t, u n ) ⊆ F(t, u) a.e. on I, then u ∈ S, hence S is weakly compact in W , (I, R N ).

Examples
As an application of the previous results, we introduce two examples. Consider a class of neural networks described by the system of differential equationṡ Theorem . If for any x,x ∈ R N , there exists a constant α ∈ R + such that (g(x)g(x), xx) ≥ α|x -x|  , and g(-x) = -g(x), I(t + T) = -I(t) for all t ∈ R, x ∈ R N , then the problem (.) 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 I i ∈ , for any i = , , . . . , N , where denotes the class of functions from R N 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 H(C), then any I i , i = , , . . . , N , possesses only isolated jump discontinuities where I i is not necessary defined. Hence for all x ∈ R N , we have where I i (xi ) = lim ε→x i I i (ε), I i (x + i ) = lim ε→x i I i (ε). Thus the differential equations (.) become the following differential inclusions: x ∈ Ag(x) -Bx(t) + I(x) .
(  .  ) The existence and the stability of the equilibrium point of (.) were first discussed in [] (I(t) is constant). In [], the authors proved the existence of periodic solutions of (.) when I(t) is the continuous periodic input and g(x) is discontinuous.
We set F(t, x) = [I(x)], it is easy to check F(t, x) satisfies H(F)  . Thus, by Theorem ., we obtain the following theorem.
Theorem . If for any x,x ∈ R N , there exists a constant α ∈ R + such that (g(x)g(x), xx) ≥ α|x -x|  , and H(C) hold, then the problem (.) has a nonempty set of solutions x ∈ W , (I, R N ).
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 T = [, b], x = (ẋ  ,ẋ  , . . . ,ẋ N ). We consider the following control system: U(t, x(t)) a.e. t ∈ T, where B is a positive definite matrix. The hypotheses on the data (.) are as follows: H(a): a : T × R N → R + , g : T × R N → R are Carathéodory functions such that, for almost all t ∈ T,  < θ  ≤ a(t, x) ≤ θ  , g(t, x) ≤ η  (t) + η  (t)|x| α ,