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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 271616, 14 pages
http://dx.doi.org/10.1155/2012/271616
Research Article

Variational Approach to Quasi-Periodic Solution of Nonautonomous Second-Order Hamiltonian Systems

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

Received 18 May 2011; Accepted 9 December 2011

Academic Editor: Stephan A. van Gils

Copyright © 2012 Juhong Kuang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal with the quasi-periodic solutions of the following second-order Hamiltonian systems , where , and we present a new approach via variational methods and Minmax method to obtain the existence of quasi-periodic solutions to the above equation.

1. Introduction

In this paper, we consider the quasi-periodic solutions of the following second-order Hamiltonian system: where , and .

A special class of the system (1.1) is the following autonomous second-order Hamilton system with convex potential : For the scalar case [1] and for the vectorial case [2], Berger and Chen have established the existence and uniqueness of almost periodic solution of (1.2). In [2], Berger and Chen assume that is almost periodic, and the potential is of the form where is a symmetric positive-definite matrix and is a convex function. They also need the growth condition.

In [3], Carminati states a local version of the results of Berger and Chen, assuming that is convex only near the minimum of . The above growth condition is not used by Carminati. To prove the existence and uniqueness of bounded or almost periodic solution of (1.2), Carminati assumes that is bounded or almost periodic and the potential is of form (1.3), where is a symmetric positive-definite matrix and is a convex function of class on the ball , where reaches its minimum in this ball at .

When is autonomous in the system (1.1), Padilla [4] states the existence of the quasi periodic solution by using critical point theory, but it assumes that the Diophantine condition is satisfied.

As to the system (1.1), using a variational method, Zakharin and Parasyuk [5] have studied the existence of almost (quasi)periodic solutions for the system where , is a compact convex subset of and is convex and differentiable on , for each . The authors use a variational method on a Hilbert space of Besicovitch almost periodic functions which looks like a Sobolev space. By this method, the authors establish the existence of generalized solutions and in the quasi-periodic case, they prove that these solutions are classical. To prove the existence of quasi-periodic solutions, Zakharin and Parasyuk [5] assume that is quasi-periodic in , and is strongly monotone on with positive modulus . They also assume that the boundary of is a differentiable manifold of class such that, for each , the gradient makes an acute angle with an external normal unit vector to at the point in the Theorem  4.3 of [5] or a similar condition using the projection operator on the closed convex in the Theorem  4.2 of [5].

More recently in [6], Ayachi and Blot provided new variational settings to study the almost periodic solutions of a class of nonlinear neutral delay equation where is a differentiable function, denotes the partial differential with respect to the th vector variable, and is fixed. When they consider the almost periodicity in the sense of Corduneau [7], they obtain some results on the structure of the set of Bohr almost periodic solutions in the case that is autonomous and convex. When they consider the almost periodicity in the sense of Besicovitch [8], they assume that satisfies a Lipschitiz condition and is convex and obtain the existence of Besicovitch almost periodic solution by the least action principle.

A special case of the above equation is the system (1.1); our main purpose is to apply Minmax method to study the existence of quasi periodic solutions to the system (1.1), and we do not assume that is Lipschitzian, but we assume that satisfies some growth conditions, then we obtain results of existence of quasi periodic solution to the system (1.1). Moreover, when we consider only a frequency , our results will cover the results of periodic solutions to the system (1.1).

The present paper is organized as follows. In Section 1 we review some notations and definitions of almost periodic functions. In Section 2, we state our main theorems. In Section 3, in order to prove our main results, we will state our basic lemmas. In Section 4, we prove our main results and give an example.

Now we give some notations and definitions of almost periodic functions.

Definition 1.1 (Fink [9]). A function is said to be Bohr almost periodic, if for any , there is a constant , such that in any interval of length , there exists such that the inequality is satisfied for all .

Definition 1.2 (He [10]). is so called almost periodic in uniformly for when, for each compact subset in , for each , there exists , and for each , there exists such that
is the space of the Bohr almost periodic functions from to , endowed with the norm and it is a Banach space.
; endowed with the norm , it is a Banach space.

A fundamental property of almost periodic functions is that such functions have convergent means, that is, the following limit exists:

The Fourier-Bohr coefficients of are the complex vectors

and and it is a countable set,

when , is the completion of with respect to the norm When is a Hilbert spaces and its norm is associated to the inner product the elements of these spaces are called Besicovitch almost periodic functions.

We use the generalized derivative of defined by , and we will identify the equivalence class and its continuous representant Then we define , endowed with the norm Its norm is associated to the inner product , and is a Hilbert space.

For convenience, we denote ( is rationally independent), is the set of all Fourier exponents of , which is called the spectrum of , it is easily obtained that is a linear subspace of and is a Hilbert space.

2. Main Theorems

In this section, we state our main results. First, we give the following list of assumptions on :(f1), and is almost periodic in uniformly for ,(f2) is almost periodic in uniformly for ,(f3) for any , ,

Theorem 2.1. Suppose satisfies (f1)–(f3), the functional , defined by is continuously differentiable on , and is defined by
Moreover, if is a critical point of in , then

Definition 2.2. When satisfies (2.4) in Theorem 2.1, we say that is a weak solution of (1.1).

Theorem 2.3. Suppose that satisfies (f1)–(f3), and(f4) there exists , for a.e. and all , such that(f5)There exists Then (1.1) has at least a quasi periodic solution.

Theorem 2.4. Suppose that satisfies (f1)–(f4), and(f6)One has
Then (1.1) has at least a quasi periodic solution by saddle point theorem.

Remark 2.5. When only contains a frequency , is periodic in with periodic , which means that (f3) is satisfied; our results cover some results in [11].

3. Basic Lemmas

To apply critical point theory to study the quasi periodic solution of (1.1), we will state our basic lemmas, which will be used in the proofs of our main results.

Lemma 3.1. If , then

Proof. For any , It is easily obtained that there exists a constant , such that So

Lemma 3.2. For any , if the sequence converges weakly to , then converges uniformly to on any compact subset of .

Proof. By Lemma 3.1, the injection of into , with its natural norm , is continuous. Since in , it follows that in . By Banach-Steinhaus theorem, is bounded in , and hence in , we need to show that the sequence is equiuniformly continuous, for any , let then
Denoting for , we have
By Arzela-Ascoli theorem, is relatively compact on any compact of . By the uniqueness of the weak limit, every uniformly convergent subsequence of converges to on any compact of .

Lemma 3.3. If and then there exists , such that

Proof. Since, by Lemma 3.1, has the Fourier expansion The Cauchy-Schwarz inequality and Parseval equality imply that

Lemma 3.4 (saddle point theorem). Let be a real Banach space, , where and is finite dimensional. Suppose that satisfies the PS condition and(I1) there exist constants , such that ;(I2) there exists a constant , such that .
Then possesses a critical value and where .

4. The Proof of Main Results

In this section, we prove the main results stated in Section 2.

Proof of Theorem 2.1. First step: we show that has at every point a directional derivative given by (2.3).
It follows easily from Lemma 3.1 and (), for any that we have
So is everywhere finite on . For fixed in , , let us define
There exists , such that
For are fixed in , there exists , such that . Since is almost periodic in uniformly for , we have that is uniformly continuous on , where is compact subset in , so
Moreover, by Lemma 3.1, So has, at , a Gâteaux derivative .
Second step: we show that the mapping is continuous.
For any , is fixed in and , let with , by Lemma 3.1, it is easily obtained and . Since is uniformly continuous on , is compact subset in , then there exists , such that and we have
We denote , then, for all and , such that , we have
The above inequality holds, which implies the continuity of so that is Fréchet differentiable on .
If is a critical point of in , for all , we have by (), then for all , we have
Since is dense in , we have , for all ; therefore, , and then we obtain (2.4) by using Blot [12]. The proof of Theorem 2.1 is completed.

Proof of Theorem 2.3. By Theorem 2.1, is continuously differentiable on . Next we will prove that is weakly lower semicontinuous on .
By Lemma 3.2, if converges weakly to , then converges uniformly to on any compact of .
Since , and is almost periodic in uniformly for , then is almost periodic, and converges uniformly to on any compact of .
Let Then it is easily obtained that moreover, so Moreover, is convex and continuous, so is weakly lower semi-continuous.
For , we have , where then,
As if and only if the above inequality and (f5) imply that
Since is a Hilbert space and is weakly lower semi-continuous, the proof of Theorem 2.3 is completed.

Proof of Theorem 2.4. Let , denote the subspace of functions with mean value zero in , and denote the subspace of constant functions in . By Theorem 2.1, we know the functional is continuously differentiable on .
For any ,
So we see that by (f6), there exists , such that where , so (I1) and (I2) of Lemma 3.4. are satisfied.
Finally, we show that condition holds, that is, each sequence in such that and contains a convergent subsequence.
Letting with , since , there exists some such that for all and ; we obtain, for , and hence because of Lemma 3.3. Now , hence there exists , such that by using (4.24), we obtain and then . By (4.24), thus is bounded in and hence contains a subsequence, relabeled which weakly converges to some . Now, the equality holds, and Lemma 3.2 implies that as , so , and the condition holds, then the proof of Theorem 2.4 is completed by saddle point theorem.
Example 4.1. Consider the scalar problem: where and . is a projection operator from to , and . and
In this case, , and hence, when , if . So it is easy to check that the conditions (f1)–(f5) are satisfied, then (4.28) has at least a quasi periodic solution by using Theorem 2.3.
When , if . So it is easy to check that the conditions (f1)–(f4) and (f6) are satisfied, then (4.28) has at least a quasi periodic solution by using Theorem 2.4.

Acknowledgment

This project is supported by National Natural Science Foundation of China (No. 10871053 and No. 11031002).

References

  1. M. S. Berger and Y. Y. Chen, “Forced quasiperiodic and almost periodic oscillations of nonlinear Duffing equations,” Nonlinear Analysis, vol. 19, no. 3, pp. 249–257, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. M. S. Berger and Y. Y. Chen, “Forced quasiperiodic and almost periodic solution for nonlinear systems,” Nonlinear Analysis, vol. 21, no. 12, pp. 949–965, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. C. Carminati, “Forced systems with almost periodic and quasiperiodic forcing term,” Nonlinear Analysis, vol. 32, no. 6, pp. 727–739, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. P. Padilla, “Existence of nonmimimal quasi periodic solutions for second order equations,” Topological Method in Nonlinear Analysis, vol. 20, pp. 217–227, 2001.
  5. S. F. Zakharin and I. O. Parasyuk, “Generalized and classical almost periodic solutions of Lagrangian systems,” Funkcialaj Ekvacioj. Serio Internacia, vol. 42, no. 3, pp. 325–338, 1999. View at Zentralblatt MATH
  6. M. Ayachi and J. Blot, “Variational methods for almost periodic solutions of a class of neutral delay equations,” Abstract and Applied Analysis, vol. 2008, Article ID 153285, 13 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. C. Corduneau, Almost Periodic Functions, Chelsea, New York, NY, USA, 2th edition, 1989.
  8. A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, Cambridge, UK, 1932.
  9. A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377, Springer, Berlin, Germany, 1974.
  10. C. Y. He, Almost Periodic Differential Equations, Higher Education Publishing House, Beijing, China, 1992.
  11. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  12. J. Blot, “Oscillations presque-périodiques forcées d'équations d'Euler-Lagrange,” Bulletin de la Société Mathématique de France, vol. 122, no. 2, pp. 285–304, 1994. View at Zentralblatt MATH