Existence of Subharmonic Solutions for a Class of Second-Order -Laplacian Systems with Impulsive Effects
By using minimax methods in critical point theory, a new existence theorem of infinitely many periodic solutions is obtained for a class of second-order -Laplacian systems with impulsive effects. Our result generalizes many known works in the literature.
Consider the following -Laplacian system with impulsive effects: where , , , and are continuous and is -periodic in for all , is the gradient of with respect to . is a -periodic positive definite symmetric matrix.
Throughout this paper, we always assume the following condition holds.(A) is measurable in for all and continuously differentiable in for a.e. , and there exist , such that for all and a.e. .
For the sake of convenience, in the sequel, we define .
When , , , problem (1.1) becomes the following second-order Hamiltonian system:
For , , problem (1.1) involves impulsive effects. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is jumps) in their values. Since these processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the processes, it is natural to suppose that these perturbations act instantaneously, that is, in the form of impulse. Processes of this type are often investigated in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books [9, 10].
There are many methods for finding periodic solutions of impulsive differential equations, such as the monotone-iterative technique, a numerical-analytical method, the method of upper and lower solutions, and the method of bilateral approximations. For more information about periodic solutions of impulsive differential equations, one can refer to the papers [11–18]. However, there are few papers [19–25] concerning periodic solutions of impulsive differential equations by variational methods. So it is a novel method to employ variational methods to investigate the existence of periodic solutions for impulsive differential equations.
Motivated by the above papers, we study the existence of subharmonic solutions for problem (1.1) by applying minimax methods in critical point theory. Our result is new, which seems not to be found in the literature.
Throughout this paper, let satisfy .
In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we construct a variational structure. With this variational structure, we can reduce the problem of finding solutions of problem (1.1) to that of seeking the critical points of the corresponding functional.
Let be a positive integer and the Sobolev space defined by with the norm
Take and multiply the two sides of the equality by and integrate from 0 to ; we have Moreover, by , one has Together with (2.4), we get
Definition 2.1. We say that a function is a weak solution of problem (1.1) if the identity holds for any .
Define the functional on by where
It follows from assumption (A) that the functional is continuously differentiable on and for . By the continuity of , , one has that . Hence, . For any , we have
For , let and ; then it follows from Proposition 1.1 in  that where , and if , then where . Let ; then . We will use the following lemma to prove our main results.
Lemma 2.2 (see ). Let be a real Banach space with , where is finite dimensional. Suppose that satisfies the (PS) condition, and (a)there exist constants , such that , where , and denotes the boundary of ; (b)there exists an and such that if , then .
Then possesses a critical value which can be characterized as , where .
It is well known that a deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition. So Lemma 2.2 holds true under condition (C).
3. Main Result and Proof
Theorem 3.1. Assume that (A) holds and satisfy the following conditions: (I1)there exists such that
(I2)for any ,
(H1), for all ;(H2) uniformly for a.e. ;(H3) uniformly for a.e. ;(H4)there exists a positive constant such that uniformly for a.e. ;(H5)there exists such that uniformly for a.e. ,
where and . Then problem (1.1) has a sequence of distinct periodic solutions with period satisfying and as .
Remark 3.2. As far as we know, there is no paper considering subharmonic solutions of impulsive differential equations. Our result is new.
Proof. The proof is divided into three steps. In the following, denote different positive constants.Step 1. The functional satisfies condition (C). Let satisfying as and is bounded; then, there exists a constant such that From (H4), there exists such that By assumption (A), for , there exists such that which together with (3.4) implies that By (3.3) and (3.6), we have Since is continuous -periodic positive definite symmetric matrix on , there exist constants such that It follows from (3.8) and (I1) that By (3.7) and (3.9), we get From (H5), there exists such that By assumption (A), for , there exists such that Thus, from (3.11) and (3.12), we have which together with (3.3) and (I2) implies that Hence, is bounded. If , we have which together with (3.10) implies that is bounded. If , then from (2.12), we get Since , it follows from (3.10) that is bounded too. Therefore, is bounded in . Hence, there exists a subsequence, still denoted by , such that From (2.11), we have From (3.3) and (3.18), we have By (3.8), we know that , which together with the boundedness of and (3.19) implies that From the boundedness of , the continuity of , and (3.18), we have It follows from (A), (3.18) and the boundedness of that which together with (3.20), (3.21), (3.22), and (3.23) implies that It is easy to see from the boundedness of and (3.18) that Let . Then, we have It follows from (3.25) and (3.26) that From (3.17), we get By (3.27), (3.28), and Hölder's inequality, we have Hence, from (3.29) and (3.30), we obtain That is, as . Since has the Kadec-Klee property, we have in . Therefore, the functional satisfies condition (C). Step 2. From (H2), for any small , there exists small enough such that For and , it follows from (2.13) that which implies that . Then from (I1), (3.8), and (3.33), we have Let ; then from (3.24), we have for all and . This implies that condition (a) of Lemma 2.2 holds. Step 3. Let , . Choose ; then from (H3), there exists such that By assumption (A), for , there exists such that which together with (3.37) implies that Thus, from (H1), (I1), (3.8), and (3.39), we have From (H3), we can choose suitable large such that Let , where , . Since is finite dimensional, there exists a constant such that By (I1), we have From (3.39), (3.42), and (3.43), we obtain From (H3), we can choose suitable such that If , then we get It follows from (3.46) that where , . Notice that for any , we have Hence, (3.47) holds for all whenever . Set then , where By (3.41) and (3.47), we have Furthermore, for all , it follows from (H1), (3.8), and (3.43) that Then by Lemma 2.2, has at least a critical point whose critical value satisfies Similar to the proof of , let be a -periodic solution; we can prove that there exists a positive integer such that for all . Otherwise, as , which contradicts to (3.53). Repeating this process, we can obtain a sequence of distinct periodic solutions of problem (1.1). From (3.41), we know that is nonconstant. The proof is complete.
In this section, we give an example to illustrate our result.
Example 4.1. Let , , , and consider the following -Laplacian system with impulsive effects Let where , . It is easy to check that satisfies (A), (H1), and (H2). By a direct computation, we have which show that (H3), (H4), and (H5) hold. On the other hand, where , . It is easy to see that satisfies (I1) and (I2). Hence, from Theorem 3.1, problem (4.1) has a sequence of distinct nonconstant periodic solutions with period satisfying and as .
This paper is partially supported by the NNSF (no. 10771215) of China. W.-Z. Gong is supported by Guangxi Natural Science Foundation (2010GXNSFA013125) and X. H. Tang is supported by the NNSF (no. 10771215) of China.
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