Periodic and homoclinic solutions generated by impulses

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Abstract

In this paper we study the existence of periodic and homoclinic solutions for a class of second order differential equations of the form q̈+Vq(t,q)=f(t) with impulsive conditions Δq̇(sk)=gk(q(sk)) via variational methods. Our results show that under appropriate conditions such a system possesses at least one non-zero periodic solution and at least one non-zero homoclinic solution and these solutions are generated by impulses when f0. Furthermore, one of the results gives us a lower bound of the number of periodic solutions generated by impulses and this lower bound is determined by the number of impulses of the system in a period of the solution.

Introduction

In the present paper we are interested in the existence of periodic and homoclinic solutions of the following second order impulsive differential equations, q̈+Vq(t,q)=f(t),for t(sk1,sk),Δq̇(sk)=gk(q(sk)), where kZ, qRn, Δq̇(sk)=q̇(sk+)q̇(sk) with q̇(sk±)=limtsk±q̇(t), Vq(t,q)=gradqV(t,q), fC(R,Rn), gk(q)=gradqGk(q), GkC1(Rn,Rn) for each kZ, and there exist an mN and a TR+ such that 0=s0<s1<<sm=T, sk+m=sk+T and gk+mgk for all kZ (that is, gk is m-periodic in k).

Impulsive differential equations arise from real world and are used to describe the dynamics of processes which possess sudden, discontinuous jumps. Such processes naturally occur in control theory, biology, optimization theory and some physics and mechanics problems. Due to their significance, a lot of efforts have been made in studying the qualitative properties of such systems; see, for instance, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].

Most of the results in the literature concerning the existence of solutions of impulsive differential equations are obtained via fixed point theory, topological degree theory (including continuation method and coincidence degree theory) and comparison method (including upper and lower solutions method and monotone iterative method). See [12], [2], [5], [6], [8], [13], [14], [10] and the references therein.

Recently some authors studied the existence of solutions for impulsive systems via variational methods [7], [15], [16], [17], [18]. In [15], Tian and Ge studied the existence of solutions for a second order impulsive system of a more general form than (0.1)–(0.2) with Sturm–Liouville boundary conditions. In [7], [18], the authors studied the existence of weak solutions for a system similar to (0.1)–(0.2) with Dirichlet boundary conditions. In [16], [17], Zhang and Li studied the existence of periodic solutions for similar impulsive systems. All the results of [7], [15], [16], [17], [18] can be seen as generalizations of corresponding ones for second order ordinary differential equations. In other words, these results can be applied to impulsive systems when the impulses are absent and still give the existence of solutions in this situation. This, in some sense, means that the nonlinear term Vq plays a more important role than the impulsive terms gk do in guaranteeing the existence of solutions in these results.

However, in the present paper we are more interested in the existence of solutions of system (0.1)–(0.2) controlled by the impulses. More precisely, in this paper we will study the existence of periodic and homoclinic solutions of system (0.1)–(0.2) generated by impulses. Here, we say a solution of (0.1)–(0.2) is generated by impulses if this solution is non-trivial when gk0 for some 1km, but it is trivial when gk0 for all 1km. For example, if system (0.1)–(0.2) does not possess non-zero periodic solution when gk0 for all 1km, then a non-zero periodic solution q of system (0.1)–(0.2) with gk0 for some 1km is called a periodic solution generated by impulses.

The results of this paper show that under appropriate conditions, system (0.1)–(0.2) possesses at least one non-zero periodic solutions and that this periodic solution is generated by impulses when f0. In particular, one of the results (Theorem 2) gives a lower bound of the number of lT-periodic solutions generated by impulses. This lower bound is completely determined by the number of impulses occur in l periods of the system, that is, this bound is totally determined by the number of sk-s in the interval (0,lT].

For the existence of homoclinic solutions of impulsive systems, to the best of the authors’ knowledge, there is still no result in the literature, but there are many papers studying the existence of homoclinic solutions for ordinary differential equations, for instance, [19], [20], [21], [22], [23], [24], [25], [26], [27]. Since 1980s, some mathematicians have began studying the existence of homoclinic solutions for Hamiltonian systems via variational methods and fruitful results were proved. For example, in [24], [26], [27] the authors proved the multiple existence of homoclinic solutions for a class of Hamiltonian systems via variational methods; in [19], [20], [22], the authors took another approach: they first proved the existence of T-periodic solutions via variational methods, and then they showed that there exists a homoclinic solution as the limit of a sequence T-periodic solutions when T goes to .

Based on the facts mentioned above, in this paper we study the existence of homoclinic solutions of (0.1)–(0.2) via variational methods. Our results show that system (0.1)–(0.2) possesses at least one non-zero homoclinic solution and this homoclinic solution is generated by impulses when f0.

This paper is organized as follows: in Section 1, the main results of this paper are stated and the non-existence of non-zero periodic and homoclinic solutions is proved when f0 and gk0 for k=1,2,,m, and two examples are also given to illustrate the results; in Section 2, the variational structure of (0.1)–(0.2) is studied, and the equivalence of periodic solutions of (0.1)–(0.2) and critical points of the action functional is proved; in Sections 3 Existence of periodic solutions, 4 Existence of homoclinic solutions, Theorem 1–2 and Theorem 3 are proved respectively.

Section snippets

Main results

To state the main results we need the following assumptions.

  • (V1)

    V:R×RnR is continuous differentiable and T-periodic, and there exist positive constants b1, b2>0 such that b1|q|2V(t,q)b2|q|2for all (t,q)[0,T]×Rn;

  • (V2)

    V(t,q)Vq(t,q)q2V(t,q) for all (t,q)[0,T]×Rn;

  • (g1)

    lim|q|0gk(q)|q|=0 for k=1,,m;

  • (g2)

    there exists a μ>2 such that gk(q)qμGk(q)<0,for qRn{0} and k=1,2,,m;

  • (g3)

    for all k=1,2,,m, lim inf|q|Gk(q)|q|20.

Remark 1

By Fact 2.1 of [19], (g2) implies 0>Gk(q)Gk(q|q|)|q|μif 0<|q|1,0>Gk(q)Gk(q|q|)|q|μif |q|

Preliminaries

For each integer l1 set HlT={q:RRnq is absolutely continuous, q̇L2((0,lT),Rn) and q(t)=q(t+lT) for tR}. Obviously, equipped with the inner product q,p=0lT[q(t)p(t)+q̇(t)ṗ(t)]dt,q,pHlT,HlT is a Hilbert space. Let l denote the norm on HlT induced by , and denote the uniform norm on HlT. Set Kl={ksk(0,lT]}={1,2,,lm}. Consider the functional Il on HlT defined as follows, Il(q)=0lT[12|q̇|2V(t,q)+fq]dt+kKlGk(q(sk)). Define Jl:HlTR by Jl(q)=0lT|q(t)̇|22V(t,q(t))dt,

Existence of periodic solutions

In this section Theorem 1, Theorem 2 will be proved, and the main tool used is the well-known Mountain Pass Lemma:

Mountain Pass Lemma

Let E be a real Banach space and IC1(E,R) satisfying the Palais–Smale (P.S.) condition. Suppose I(0)=0 and

  • (I1)

    there are constants ρ,β>0 such that I|Bρβ, where Bρ={xEx<ρ}, and

  • (I2)

    there is an eEBρ such that I(e)0.

Then I possesses a critical value cβ . Moreover c can be characterized asc=infhΓmaxt[0,1]I(h(t)),whereΓ={hC([0,1],E)|h(0)=0,h(1)=e}.

Lemma 2

Under the hypotheses of Theorem 1,

Existence of homoclinic solutions

In this section we prove Theorem 3 through two lemmas. We assume that the hypotheses of Theorem 3 hold in this section.

Set fl be a 2lT-periodic extension of f|(lT,lT] on R for each integer l>0. Consider the system q̈+Vq(t,q)=fl(t),for t(sk1,sk),Δq̇(sk)=gk(q(sk)), with the periodic boundary conditions q(lT)q(lT)=q̇(lT±)q̇(lT±)=0. Set K̃l={kZsk(lT,lT]}={lm+1,,lm} and Ĩl(q)=lTlT[12|q̇|2V(t,q)+flq]dt+skK̃lGk(q(sk)).Ĩl is defined on the Hilbert space H2lT={q:RRnq

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and helpful comments.

References (29)

  • M. Benchohra et al.

    Impulsive Differential Equations and Inclusions

    (2006)
  • J. Chu et al.

    Impulsive periodic solutions of first-order singular differential equations

    Bull. Lond. Math. Soc.

    (2008)
  • W.M. Haddad et al.

    Impulsive and Hybrid Dynamical Systems. Stability, Dissipativity, and Control

    (2006)
  • V. Lakshmikantham et al.

    Theory of Impulsive Differential Equations

    (1989)
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