Periodic Solutions of Some Impulsive Hamiltonian Systems with Convexity Potentials

and Applied Analysis 3 Theorem 1.1 2, Theorem 1.1 . Suppose that V and W are reflexive Banach spaces, φ ∈ C1 V × W,R , φ v, · is weakly upper semi-continuous for all v ∈ V , and φ ·, w : V → R is convex for all w ∈ W and φ′ is weakly continuous. Assume that φ 0, w −→ −∞ 1.9 as ‖w‖ → ∞ and for everyM > 0, φ v,w −→ ∞, 1.10 as ‖v‖ → ∞ uniformly for ‖w‖ ≤ M. Then φ has at least one critical point. 2. Main Results Theorem 2.1. Assume that assumption A holds. If further H1 F t, · is convex for a.e. t ∈ 0, T , and H2 there exist η, θ > 0 such that Gk x ≥ η|x| θ, for all x ∈ R, then 1.1 possesses at least one solution in H1 T . Remark 2.2. H1 implies there exists a point x for which ∫T 0 ∇F t, x dt 0. 2.1 Proof of Theorem 2.1. It follows Remark 2.2, 1.6 , and H2 that φ u 1 2 ∫T 0 |u̇ t |dt ∫T 0 F t, u t − F t, x dt ∫T 0 F t, x dt m ∑ k 1 Gk u tk 1 2 ∫T 0 |u̇ t |dt ∫T 0 F t, x dt ∫T 0 ( f t, x , u t − xdt m ∑ k 1 Gk u tk ≥ 1 2 ∫T 0 |u̇ t |dt ∫T 0 F t, x dt ∫T 0 ( f t, x , ũ ) dt m ∑ k 1 η|ũ u| mθ ≥ 1 2 ∫T 0 |u̇ t |dt − (∫T 0 ∣f t, x ∣dt ) ‖ũ‖∞ mη|u| −mη‖ũ‖∞ mθ ≥ 1 2 ∫T 0 |u̇ t |dt − C0 (∫T 0 |u̇ t |dt )1/2 mη|u| mθ, 2.2 for all u ∈ H1 T and some positive constant C0. As ‖u‖ → ∞ if and only if |u|2 ‖u̇‖2 1/2 → ∞, we have φ u → ∞ as ‖u‖ → ∞. By Theorem 1.1 and Corollary 1.1 in 1 , φ has a minimum point in H1 T , which is a critical point of φ. Hence, problem 1.1 has at least one weak solution. 4 Abstract and Applied Analysis Theorem 2.3. Assume that assumption A and H1 hold. If further H3 there exist η, θ > 0 and α ∈ 0, 2 such that Gk x ≤ η|x|α θ for all x ∈ R and H4 there exist some β > α and γ > 0 such that |x|−β ∫T 0 F t, x dt ≤ −γ, 2.3 for |x| ≥ M and t ∈ 0, T , whereM is a constant, then 1.1 possesses at least one solution in H1 T . Remark 2.4. We can find that our condition H4 is very different from condition vii in 7 since we prove this by the saddle point theorem substituted for the least action principle. Proof of Theorem 2.3. We prove φ satisfies the PS condition at first. Suppose {un} is such an sequence that {φ un } is bounded and limn→∞φ′ un 0. We will prove it has a convergent subsequence. By H3 and 1.6 , we have m ∑ k 1 Gk u tk ≤ m ∑ k 1 η|ũ tk u| mθ ≤ 4mη|ũ tk | |u| ) mθ ≤ C1‖u̇‖2 C2|u| C3, 2.4 for some positive constants C1, C2, C3. By Remark 2.2, 1.6 , and 2.4 , we have φ un 1 2 ∫T 0 |u̇n t |dt ∫T 0 F t, un t − F t, x dt ∫T 0 F t, x dt m ∑ k 1 Gk un tk 1 2 ∫T 0 |u̇n t |dt ∫T 0 F t, x dt ∫T 0 ( f t, x , un t − x ) dt m ∑ k 1 Gk un tk 1 2 ∫T 0 |u̇n t |dt ∫T 0 F t, x dt ∫T 0 ( f t, x , ũn ) dt m ∑ k 1 Gk un tk ≥ 1 2 ∫T 0 |u̇n t |dt − (∫T 0 ∣f t, x ∣dt ) ‖ũn‖∞ − C1‖u̇n‖2 − C2|un| − C4 ≥ 1 2 ∫T 0 |u̇n t |dt − C5 (∫T 0 |u̇n t |dt )1/2 − C1‖u̇n‖2 − C2|un| − C4, 2.5 for some positive constants C4, C5, which implies that C|un| ≥ (∫T 0 |u̇n t |dt )1/2 − C6, 2.6 Abstract and Applied Analysis 5 for some positive constants C, C6. By 1.6 , the above inequality implies that ‖ũn‖∞ ≤ C7 ( |un| 1 ) , 2.7and Applied Analysis 5 for some positive constants C, C6. By 1.6 , the above inequality implies that ‖ũn‖∞ ≤ C7 ( |un| 1 ) , 2.7 for the positive constant C7. The one has |un t | ≥ |un| − |ũn| ≥ |un| − ‖ũn‖∞ ≥ |un| − C7 ( |un| 1 ) , ∀t ∈ 0, T . 2.8 If {|un|} is unbounded, we may assume that, going to a subsequence if necessary, |un| −→ ∞ as n −→ ∞. 2.9 By 2.8 and 2.9 , we have |un t | ≥ 1 2 |un|, 2.10 for all large n and every t ∈ 0, T . By 2.10 and H4 , one has |un t | ≥ M for all large n. It follows from H4 , 2.4 , 2.6 , 2.7 , and above inequality that φ un ≤ ( C|un| C6 )2 − ∫T 0 γ |un t |dt C2‖ũ‖α∞ C2|u| C3 ≤ ( C|un| C6 )2 − 2−β|un|Tγ C8 ( |un| 1 )α C2|u| C3, 2.11 for large n and the positive constant C8, which contradicts the boundedness of φ un since β > α. Hence |un| is bounded. Furthermore, un is bounded by 2.6 . A similar calculation to Lemma 3.1 in 9 shows that φ satisfies the PS condition. We now prove that φ satisfies the other conditions of the saddle point theorem. Assume that H̃1 T {u ∈ H1 T : u 0}, then H1 T H̃ 1 T ⊕ R. From above calculation, one has φ u ≥ 1 2 ∫T 0 |u̇ t |dt − C5 (∫T 0 |u̇ t |dt )1/2 − C1‖u̇‖2 − C4, 2.12 for all u ∈ H̃1 T , which implies that φ u −→ ∞, 2.13 6 Abstract and Applied Analysis as ‖u‖ → ∞ in H̃1 T . Moreover, by H3 and H4 we have φ x ∫T 0 F t, x dt m ∑ k 1 Gk x ≤ − Tγ |x| mη|x| mθ, 2.14 for |x| > M, which implies that φ x −→ −∞, 2.15 as |x| → ∞ in R since β > α. Now Theorem 2.3 is proved by 2.13 , 2.15 , and the saddle point theorem. Theorem 2.5. Assume that assumption A holds. Suppose that F t, · , Gk x are concave and satisfy H5 Gk x ≤ −η|x| θ for some positive constant η, θ > 0, then 1.1 possesses at least one solution in HT. Proof of Theorem 2.5. Consider the corresponding functional φ on R × H̃1 T given by φ u − 2 ∫T 0 |u̇ t |dt − ∫T 0 F t, u t dt − m ∑ k 1 Gk u tk , 2.16 which is continuously differentiable, bounded, and weakly upper semi-continuous on H1 T . Similar to the proof of Lemma 3.1 in 2 , one has that φ x w is convex in x ∈ R for every w ∈ H̃1 T . By the condition, we have −Gk x w ≥ −2Gk 1/2 x Gk −w . Similar to the proof of Theorem 3.1, we have φ x w − 1 2 ∫T 0 |ẇ|dt − ∫T 0 F t, x w dt − m ∑ k 1 Gk x w ≥ − 1 2 ∫T 0 |ẇ|dt − (∫T 0 ∣f t, x ∣dt ) ‖w‖∞ − m ∑ k 1 Gk x w C9 ≥ − 1 2 ∫T 0 |ẇ|dt − C0 (∫T 0 |ẇ|dt )1/2 − 2Gk ( 1 2 x ) Gk −w C9, 2.17 Abstract and Applied Analysis 7 which means φ x w → ∞ as |x| → ∞, uniformly forw ∈ H̃1 T with ‖w‖ ≤ M by H5 and 1.6 . On the other hand,and Applied Analysis 7 which means φ x w → ∞ as |x| → ∞, uniformly forw ∈ H̃1 T with ‖w‖ ≤ M by H5 and 1.6 . On the other hand, φ w − 1 2 ∫T 0 |ẇ|dt − ∫T 0 F t,w dt − m ∑ k 1 Gk w ≤ − 1 2 ∫T 0 |ẇ|dt C0 (∫T 0 |ẇ|dt )1/2 mη‖w‖∞ C9, 2.18 which implies that φ w → −∞ as ‖w‖ → ∞ ∈ H̃1 T by H5 and 1.6 . We complete our proof by Theorem 1.1. Acknowledgment The first author was supported by the Postgraduate Research and Innovation Project of Hunan Province CX2011B078 . References 1 J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74, Springer, Berlin, Germany, 1989. 2 C.-L. Tang and X.-P. Wu, “Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems,” Journal of Differential Equations, vol. 248, no. 4, pp. 660–692, 2010. 3 J. Mawhin, “Semicoercive monotone variational problems,” Académie Royale de Belgique, Bulletin de la Classe des Sciences, vol. 73, no. 3-4, pp. 118–130, 1987. 4 C.-L. Tang, “An existence theorem of solutions of semilinear equations in reflexive Banach spaces and its applications,” Académie Royale de Belgique, Bulletin de la Classe des Sciences, vol. 4, no. 7–12, pp. 317–330, 1996. 5 J. J. Nieto, “Variational formulation of a damped Dirichlet impulsive problem,” Applied Mathematics Letters, vol. 23, no. 8, pp. 940–942, 2010. 6 J. J. Nieto and D. O’Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 680–690, 2009. 7 J. Zhou and Y. Li, “Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1594–1603, 2010. 8 H. Zhang and Z. Li, “Periodic solutions of second-order nonautonomous impulsive di erential equations,” International Journal of Qualitative Theory of Differential Equations and Applications, vol. 2, no. 1, pp. 112–124, 2008. 9 J. Zhou and Y. Li, “Existence andmultiplicity of solutions for some Dirichlet problems with impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2856–2865, 2009. 10 W. Ding and D. Qian, “Periodic solutions for sublinear systems via variational approach,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2603–2609, 2010. 11 H. Zhang and Z. Li, “Periodic and homoclinic solutions generated by impulses,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 39–51, 2011. 12 Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 155–162, 2010. 13 D. Zhang and B. Dai, “Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3153–3160, 2011. 14 L. Yang and H. Chen, “Existence and multiplicity of periodic solutions generated by impulses,” Abstract and Applied Analysis, vol. 2011, Article ID 310957, 15 pages, 2011. 8 Abstract and Applied Analysis 15 J. Sun, H. Chen, and J. J. Nieto, “Infinitely many solutions for second-order Hamiltonian system with impulsive effects,”Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 544–555, 2011. 16 J. Sun, H. Chen, J. J. Nieto, and M. Otero-Novoa, “The multiplicity of solutions for perturbed secondorder Hamiltonian systemswith impulsive effects,”Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4575–4586, 2010.


Introduction
Consider the following systems: ü t f t, u t , a.e.t ∈ 0, T , where 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 7 there is no result about convexity conditions with impulsive effects.By using different techniques, we obtain different results from 7 .
We recall some basic facts which will be used in the proofs of our main results.Let where •, • denotes the inner product in R n .The corresponding norm is defined by The space H 1 T has some important properties.For u ∈ H 1 T , let u 1/2T T 0 u t dt, and u u t − u.Then one has Sobolev's inequality see Proposition 1.3 in 1 : Consider the corresponding functional ϕ on H 1 T given by It follows from assumption A and the continuity of g k one has that ϕ is continuously differentiable and weakly lower semicontinuous on H 1 T .Moreover, we have for u, v ∈ H 1 T and ϕ is weakly continuous and the weak solutions of problem 1.1 correspond to the critical points of ϕ see 8 .
as w → ∞ and for every M > 0, as v → ∞ uniformly for w ≤ M. Then ϕ has at least one critical point.

Main Results
Theorem 2.1.Assume that assumption A holds.If further • is convex for a.e.t ∈ 0, T , and  Proof of Theorem 2.3.We prove ϕ satisfies the PS condition at first.Suppose {u n } is such an sequence that {ϕ u n } is bounded and lim n → ∞ ϕ u n 0. We will prove it has a convergent subsequence.By H 3 and 1.6 , we have for some positive constants C 1 , C 2 , C 3 .By Remark 2.2, 1.6 , and 2.4 , we have for some positive constants C 4 , C 5 , which implies that for some positive constants C, C 6 .By 1.6 , the above inequality implies that for the positive constant C 7 .The one has If {|u n |} is unbounded, we may assume that, going to a subsequence if necessary, By 2.8 and 2.9 , we have

2.11
for large n and the positive constant C 8 , which contradicts the boundedness of ϕ u n since β > α.Hence |u n | is bounded.Furthermore, u n is bounded by 2.6 .A similar calculation to Lemma 3.1 in 9 shows that ϕ satisfies the PS condition.We now prove that ϕ satisfies the other conditions of the saddle point theorem.Assume that H 1 From above calculation, one has which is continuously differentiable, bounded, and weakly upper semi-continuous on H 1 T .Similar to the proof of Lemma 3.1 in 2 , one has that ϕ x w is convex in x ∈ R n for every w ∈ H

Abstract and Applied Analysis 7 which
means ϕ x w → ∞ as |x| → ∞, uniformly for w ∈ H 1 T with w ≤ M by H 5 and 1.6 .On the other hand, ϕ w → −∞ as w → ∞ ∈ H1  T by H 5 and 1.6 .We complete our proof by Theorem 1.1.
Assume that assumption A and H 1 hold.If further H 3 there exist η, θ > 0 and α ∈ 0, 2 such that G k x ≤ η|x| α θ for all x ∈ R n and H 4 there exist some β > α and γ > 0 such that M and t ∈ 0, T , where M is a constant, then 1.1 possesses at least one solution in H 1 T .Remark 2.4.We can find that our condition H 4 is very different from condition vii in 7 since we prove this by the saddle point theorem substituted for the least action principle.
∞, we have ϕ u → ∞ as u → ∞.By Theorem 1.1 and Corollary 1.1 in 1 , ϕ has a minimum point in H 1 T , which is a critical point of ϕ.Hence, problem 1.1 has at least one weak solution.