Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential EquationsView this Special Issue
Periodic Solutions of Some Impulsive Hamiltonian Systems with Convexity Potentials
We study the existence of periodic solutions of some second-order Hamiltonian systems with impulses. We obtain some new existence theorems by variational methods.
Consider the following systems: where with for each , there exists an such that , and we suppose that satisfies the following assumption. is measurable in for and continuously differentiable in for a.e. , and there exist such that for all and .
Many solvability conditions for problem (1.1) without impulsive effect are obtained, such as, the coercivity condition, the convexity conditions (see [1–4] and their references), the sublinear nonlinearity conditions, and the superlinear potential conditions. Recently, by using variational methods, many authors studied the existence of solutions of some second-order differential equations with impulses. More precisely, Nieto in [5, 6] considers linear conditions, [7–10] the sublinear conditions, and [11–16] the sublinear conditions and the other conditions. But to the best of our knowledge, except  there is no result about convexity conditions with impulsive effects. By using different techniques, we obtain different results from .
We recall some basic facts which will be used in the proofs of our main results. Let with the inner product where denotes the inner product in . The corresponding norm is defined by
The space has some important properties. For , let , and . Then one has Sobolev's inequality (see Proposition in ):
Consider the corresponding functional on given by
It follows from assumption and the continuity of one has that is continuously differentiable and weakly lower semicontinuous on . Moreover, we have for and is weakly continuous and the weak solutions of problem (1.1) correspond to the critical points of (see ).
Theorem 1.1 ([2, Theorem 1.1]). Suppose that and are reflexive Banach spaces, , is weakly upper semi-continuous for all , and is convex for all and is weakly continuous. Assume that as and for every , as uniformly for . Then has at least one critical point.
2. Main Results
Theorem 2.1. Assume that assumption holds. If further is convex for a.e. , and there exist , such that , for all , then (1.1) possesses at least one solution in .
Remark 2.2. implies there exists a point for which
Proof of Theorem 2.1. It follows Remark 2.2, (1.6), and that for all and some positive constant . As if and only if , we have as . By Theorem 1.1 and Corollary 1.1 in , has a minimum point in , which is a critical point of . Hence, problem (1.1) has at least one weak solution.
Theorem 2.3. Assume that assumption and hold. If further there exist , and such that for all and there exist some and such that for and , where is a constant, then (1.1) possesses at least one solution in .
Remark 2.4. We can find that our condition is very different from condition (vii) in  since we prove this by the saddle point theorem substituted for the least action principle.
Proof of Theorem 2.3. We prove satisfies the (PS) condition at first. Suppose is such an sequence that is bounded and . We will prove it has a convergent subsequence. By and (1.6), we have for some positive constants , , . By Remark 2.2, (1.6), and (2.4), we have for some positive constants , , which implies that for some positive constants , . By (1.6), the above inequality implies that for the positive constant . The one has If is unbounded, we may assume that, going to a subsequence if necessary, By (2.8) and (2.9), we have for all large and every . By (2.10) and , one has for all large . It follows from , (2.4), (2.6), (2.7), and above inequality that for large and the positive constant , which contradicts the boundedness of since . Hence is bounded. Furthermore, is bounded by (2.6). A similar calculation to Lemma 3.1 in  shows that satisfies the (PS) condition. We now prove that satisfies the other conditions of the saddle point theorem. Assume that , then . From above calculation, one has for all , which implies that as in . Moreover, by and we have for , which implies that as in since . Now Theorem 2.3 is proved by (2.13), (2.15), and the saddle point theorem.
Theorem 2.5. Assume that assumption holds. Suppose that are concave and satisfy for some positive constant , then (1.1) possesses at least one solution in .
Proof of Theorem 2.5. Consider the corresponding functional on given by which is continuously differentiable, bounded, and weakly upper semi-continuous on . Similar to the proof of Lemma 3.1 in , one has that is convex in for every . By the condition, we have . Similar to the proof of Theorem 3.1, we have which means as , uniformly for with by and (1.6). On the other hand, which implies that as by and (1.6). We complete our proof by Theorem 1.1.
The first author was supported by the Postgraduate Research and Innovation Project of Hunan Province (CX2011B078).
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