AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 616427 10.1155/2012/616427 616427 Research Article Periodic Solutions of Some Impulsive Hamiltonian Systems with Convexity Potentials Chen Dezhu Dai Binxiang Sun Juntao 1 School of Mathematical Sciences and Computing Technology, Central South University, Hunan Changsha 410083 China 2012 28 11 2012 2012 31 08 2012 18 11 2012 2012 Copyright © 2012 Dezhu Chen and Binxiang Dai. 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.

We study the existence of periodic solutions of some second-order Hamiltonian systems with impulses. We obtain some new existence theorems by variational methods.

1. Introduction

Consider the following systems: (1.1)u¨(t)=f(t,u(t)),a.e.t[0,T],Δu˙(tk)=gk(u(tk-)),k=1,2,,m,u(T)-u(0)=u˙(T)-u˙(0)=0, where k,un,Δu˙(tk)=u˙(tk+)-u˙(tk-) with u˙(tk±)=limttk±u˙(t),gk(u)=graduGk(u),GkC1(n,n) for each k, there exists an m such that 0=t0<t1<<tm<tm+1=T, and we suppose that f(t,u)=graduF(t,u) satisfies the following assumption.

(A)F(t,x) is measurable in t for xn and continuously differentiable in x for a.e. t[0,T], and there exist aC(+,+),  bL1(0,T;+) such that (1.2)|F(t,x)|+|f(t,x)|a(|x|)b(t),

for all xn and t[0,T].

Many solvability conditions for problem (1.1) without impulsive effect are obtained, such as, the coercivity condition, the convexity conditions (see  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,  the sublinear conditions, and  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 (1.3)HT1={u:[0,T]nabsolutelycontinuous;u(0)=u(T),u˙(t)L2(0,T;n)}, with the inner product (1.4)u,v=0T(u(t),v(t))dt+0T(u˙(t),v˙(t))dt,u,vHT1, where (·,·) denotes the inner product in n. The corresponding norm is defined by (1.5)u=(0T(u(t),u(t))dt+0T(u˙(t),u˙(t))dt)1/2,uHT1.

The space HT1 has some important properties. For uHT1, let u-=(1/2T)0Tu(t)dt, and u~=u(t)-u-. Then one has Sobolev's inequality (see Proposition 1.3 in ): (1.6)u~2T120T|u˙(t)|2dt.

Consider the corresponding functional φ on HT1 given by (1.7)φ(u)=120T|u˙(t)|2dt+0TF(t,u(t))dt+k=1mGk(u(tk)).

It follows from assumption (A) and the continuity of gk one has that φ is continuously differentiable and weakly lower semicontinuous on HT1. Moreover, we have (1.8)φ(u),v=0T(u˙(t),v˙(t))dt+0T(f(t,u(t)),v(t))dt+k=1m(gk(u(tk)),v(tk)), for u,vHT1 and φ is weakly continuous and the weak solutions of problem (1.1) correspond to the critical points of φ (see ).

Theorem 1.1 ([<xref ref-type="bibr" rid="B2">2</xref>, Theorem  1.1]).

Suppose that V and W are reflexive Banach spaces, φC1(V×W,R), φ(v,·) is weakly upper semi-continuous for all vV, and φ(·,w):VR is convex for all wW and φ is weakly continuous. Assume that (1.9)φ(0,w)- as w and for every M>0, (1.10)φ(v,w)+, as v uniformly for wM. Then φ has at least one critical point.

2. Main Results Theorem 2.1.

Assume that assumption (A) holds. If further

F(t,·) is convex for a.e. t[0,T], and

there exist η, θ>0 such that Gk(x)η|x|+θ, for all xn, then (1.1) possesses at least one solution in HT1.

Remark 2.2.

( H 1 ) implies there exists a point x- for which (2.1)0TF(t,x-)dt=0.

Proof of Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>.

It follows Remark 2.2, (1.6), and (H2) that (2.2)φ(u)=120T|u˙(t)|2dt+0T(F(t,u(t))-F(t,x-))dt+0TF(t,x-)dt+k=1mGk(u(tk))=120T|u˙(t)|2dt+0TF(t,x-)dt+0T(f(t,x-),u(t)-x-)dt+k=1mGk(u(tk))120T|u˙(t)|2dt+0TF(t,x-)dt+0T(f(t,x-),u~)dt+k=1mη|u~+u-|+mθ120T|u˙(t)|2dt-(0T|f(t,x-)|dt)u~+mη|u-|-mηu~+mθ120T|u˙(t)|2dt-C0(0T|u˙(t)|2dt)1/2+mη|u-|+mθ, for all uHT1 and some positive constant C0. As u if and only if (|u|2+u˙22)1/2, we have φ(u)+ as u. By Theorem 1.1 and Corollary 1.1 in , φ has a minimum point in HT1, which is a critical point of φ. Hence, problem (1.1) has at least one weak solution.

Theorem 2.3.

Assume that assumption (A) and (H1) hold. If further

there exist η, θ>0 and α(0,2) such that Gk(x)η|x|α+θ for all xn and

there exist some β>α and γ>0 such that (2.3)|x|-β0TF(t,x)dt-γ,

for |x|M and t[0,T], where M is a constant, then (1.1) possesses at least one solution in HT1.

Remark 2.4.

We can find that our condition (H4) 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 <xref ref-type="statement" rid="thm2.2">2.3</xref>.

We prove φ satisfies the (PS) condition at first. Suppose {un} is such an sequence that {φ(un)} is bounded and limnφ(un)=0. We will prove it has a convergent subsequence. By (H3) and (1.6), we have (2.4)k=1mGk(u(tk))k=1mη|u~(tk)+u-|α+mθ4mη(|u~(tk)|α+|u-|α)+mθC1u˙2α+C2|u-|α+C3, for some positive constants C1, C2, C3. By Remark 2.2, (1.6), and (2.4), we have (2.5)φ(un)=120T|u˙n(t)|2dt+0T(F(t,un(t))-F(t,x-))dt+0TF(t,x-)dt+k=1mGk(un(tk))=120T|u˙n(t)|2dt+0TF(t,x-)dt+0T(f(t,x-),un(t)-x-)dt+k=1mGk(un(tk))=120T|u˙n(t)|2dt+0TF(t,x-)dt+0T(f(t,x-),u~n)dt+k=1mGk(un(tk))120T|u˙n(t)|2dt-(0T|f(t,x-)|dt)u~n-C1u˙n2α-C2|u-n|α-C4120T|u˙n(t)|2dt-C5(0T|u˙n(t)|2dt)1/2-C1u˙n2α-C2|u-n|α-C4, for some positive constants C4, C5, which implies that (2.6)C|u-n|α/2(0T|u˙n(t)|2dt)1/2-C6, for some positive constants C, C6. By (1.6), the above inequality implies that (2.7)u~nC7(|u-n|α/2+1), for the positive constant C7. The one has (2.8)|un(t)||u-n|-|u~n||u-n|-u~n|u-n|-C7(|u-n|α/2+1),t[0,T]. If {|u-n|} is unbounded, we may assume that, going to a subsequence if necessary, (2.9)|u-n|as  n. By (2.8) and (2.9), we have (2.10)|un(t)|12|u-n|, 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 (2.11)φ(un)(C|u-n|α/2+C6)2-0Tγ|un(t)|βdt+C2u~α+C2|u-|α+C3(C|u-n|α/2+C6)2-2-β|u-n|βTγ+C8(|u-n|α/2+1)α+C2|u-|α+C3, for large n and the positive constant C8, which contradicts the boundedness of φ(un) since β>α. Hence (|u-n|) is bounded. Furthermore, (un) is bounded by (2.6). A similar calculation to Lemma 3.1 in  shows that φ satisfies the (PS) condition. We now prove that φ satisfies the other conditions of the saddle point theorem. Assume that H~T1={uHT1:u-=0}, then HT1=H~T1n. From above calculation, one has (2.12)φ(u)120T|u˙(t)|2dt-C5(0T|u˙(t)|2dt)1/2-C1u˙2α-C4, for all uH~T1, which implies that (2.13)φ(u)+, as u in H~T1. Moreover, by (H3) and (H4) we have (2.14)φ(x)=0TF(t,x)dt+k=1mGk(x)-Tγ|x|β+mη|x|α+mθ, for |x|>M, which implies that (2.15)φ(x)-, as |x| in n 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

Gk(x)-η|x|+θ for some positive constant η,θ>0, then (1.1) possesses at least one solution in HT1.

Proof of Theorem <xref ref-type="statement" rid="thm2.3">2.5</xref>.

Consider the corresponding functional φ on n×H~T1 given by (2.16)φ(u)=-120T|u˙(t)|2dt-0TF(t,u(t))dt-k=1mGk(u(tk)), which is continuously differentiable, bounded, and weakly upper semi-continuous on HT1. Similar to the proof of Lemma 3.1 in , one has that φ(x+w) is convex in xn for every wH~T1. By the condition, we have -Gk(x+w)-2Gk((1/2)x)+Gk(-w). Similar to the proof of Theorem 3.1, we have (2.17)φ(x+w)=-120T|w˙|2dt-0TF(t,x+w)dt-k=1mGk(x+w)-120T|w˙|2dt-(0T|f(t,x-)|dt)w-k=1mGk(x+w)+C9-120T|w˙|2dt-C0(0T|w˙|2dt)1/2-2Gk(12x)+Gk(-w)+C9, which means φ(x+w)+ as |x|, uniformly for wH~T1 with wM by (H5) and (1.6). On the other hand, (2.18)φ(w)=-120T|w˙|2dt-0TF(t,w)dt-k=1mGk(w)-120T|w˙|2dt+C0(0T|w˙|2dt)1/2+mηw+C9, which implies that φ(w)- as wH~T1 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).

Mawhin J. Willem M. Critical Point Theory and Hamiltonian Systems 1989 74 Berlin, Germany Springer 982267 ZBL0772.47033 Tang C.-L. Wu X.-P. Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems Journal of Differential Equations 2010 248 4 660 692 10.1016/j.jde.2009.11.007 2578444 ZBL1191.34053 Mawhin J. Semicoercive monotone variational problems Académie Royale de Belgique, Bulletin de la Classe des Sciences 1987 73 3-4 118 130 938142 Tang C.-L. 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 1996 4 7–12 317 330 1475760 ZBL1194.47071 Nieto J. J. Variational formulation of a damped Dirichlet impulsive problem Applied Mathematics Letters 2010 23 8 940 942 10.1016/j.aml.2010.04.015 2651478 ZBL1197.34041 Nieto J. J. O'Regan D. Variational approach to impulsive differential equations Nonlinear Analysis. Real World Applications 2009 10 2 680 690 10.1016/j.nonrwa.2007.10.022 2474254 ZBL1167.34318 Zhou J. Li Y. Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects Nonlinear Analysis: Theory, Methods & Applications 2010 72 3-4 1594 1603 10.1016/j.na.2009.08.041 2577560 ZBL1193.34057 Zhang H. Li Z. Periodic solutions of second-order nonautonomous impulsive di erential equations International Journal of Qualitative Theory of Differential Equations and Applications 2008 2 1 112 124 Zhou J. Li Y. Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects Nonlinear Analysis: Theory, Methods & Applications 2009 71 7-8 2856 2865 10.1016/j.na.2009.01.140 2532812 ZBL1175.34035 Ding W. Qian D. Periodic solutions for sublinear systems via variational approach Nonlinear Analysis: Real World Applications 2010 11 4 2603 2609 10.1016/j.nonrwa.2009.09.007 2661927 ZBL1209.34046 Zhang H. Li Z. Periodic and homoclinic solutions generated by impulses Nonlinear Analysis: Real World Applications 2011 12 1 39 51 10.1016/j.nonrwa.2010.05.034 2728662 ZBL1225.34019 Zhang Z. Yuan R. An application of variational methods to Dirichlet boundary value problem with impulses Nonlinear Analysis: Real World Applications 2010 11 1 155 162 10.1016/j.nonrwa.2008.10.044 2570535 ZBL1191.34039 Zhang D. Dai B. Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions Computers & Mathematics with Applications 2011 61 10 3153 3160 10.1016/j.camwa.2011.04.003 2799840 ZBL1222.34031 Yang L. Chen H. Existence and multiplicity of periodic solutions generated by impulses Abstract and Applied Analysis 2011 2011 15 310957 10.1155/2011/310957 2800077 Sun J. Chen H. Nieto J. J. Infinitely many solutions for second-order Hamiltonian system with impulsive effects Mathematical and Computer Modelling 2011 54 1-2 544 555 10.1016/j.mcm.2011.02.044 2801909 ZBL1225.37070 Sun J. Chen H. Nieto J. J. Otero-Novoa M. The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects Nonlinear Analysis: Theory, Methods & Applications 2010 72 12 4575 4586 10.1016/j.na.2010.02.034 2639205 ZBL1198.34036