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Subharmonic solutions for a class of second-order impulsive Lagrangian systems with damped term

Xingyong Zhang

Author Affiliations

Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P.R. China

Boundary Value Problems 2013, 2013:218  doi:10.1186/1687-2770-2013-218

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/218


Received:16 July 2013
Accepted:27 August 2013
Published:7 November 2013

© 2013 Zhang; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, by using the mountain pass theorem, we investigate the existence of subharmonic weak solutions for a class of second-order impulsive Lagrangian systems with damped term under asymptotically quadratic conditions. Some new existence criteria are established. Finally, an example is presented to verify our results.

MSC: 37J45, 34C25, 70H05.

Keywords:
impulsive Lagrangian systems; damped term; subharmonic weak solutions; mountain pass theorem

1 Introduction and main results

In this paper, we investigate the existence of subharmonic weak solutions for the following second-order impulsive Lagrangian system with damped term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M7">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M9">View MathML</a>, B is a skew-symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10">View MathML</a> constant matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M12">View MathML</a> are symmetric and continuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10">View MathML</a> matrix-value functions on ℝ satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M15">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M16">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M17">View MathML</a>, where K, W are T-periodic in their first variable, and the following assumption:

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M18">View MathML</a>is measurable intfor every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M19">View MathML</a>and continuously differentiable inxfor a.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M20">View MathML</a>, and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M21">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M22">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M23">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M24">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M25">View MathML</a>and a.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M20">View MathML</a>.

Lagrangian systems are applied extensively to study the fluid mechanics, nuclear physics and relativistic mechanics. Especially, as a special case of Lagrangian systems, the following second-order Hamiltonian systems are considered by many authors:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M27">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M16">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M29">View MathML</a>, and the existence and multiplicity of periodic solutions, subharmonic solutions and homoclinic solutions are obtained via variational methods. We refer readers to [1-14]. Especially, in 2010, under the asymptotically quadratic conditions, Tang and Jiang [10] obtained the following interesting result.

Theorem A (see [10], Theorem 1.1)

Assume thatFsatisfies

(F) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M17">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M31">View MathML</a>areT-periodic in their first variable with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M2">View MathML</a>, and thatKandWsatisfy the following assumptions:

(H1) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M33">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M34">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M35">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M36">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M37">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M38">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M39">View MathML</a>;

(H4) There exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M40">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M41">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M42">View MathML</a>

(H5) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M44">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M45">View MathML</a>

(H6) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M46">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M47">View MathML</a>

Then system (1.2) has a nontrivialT-periodic solution.

In recent years, variational methods have been applied to study the existence and multiplicity of solutions for impulsive differential equations and lots of interesting results have been obtained, see [15-20].

In [15], Nieto and O’Regan considered a one-dimensional Dirichlet boundary value problem with impulses. They obtained that the solutions of the impulsive problem minimize some (energy) functional and the critical points of the functional are indeed solutions of the impulsive problem.

In [16], Nieto introduced a variational formulation for the following one-dimensional damped nonlinear Dirichlet problem with impulses:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M48">View MathML</a>

(1.3)

and gave the concept of a weak solution for such a problem. They obtained that the weak solutions of problem (1.3) are indeed the critical points of the functional:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M49">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M51">View MathML</a>. In [17] and [18], the authors also dealt with some one-dimensional impulsive problems with damped term by variational methods.

For higher dimensional dynamical systems, some interesting results have also been obtained (see [21-23]). In [21], Zhou and Li investigated the second-order Hamiltonian system with impulsive effects:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M52">View MathML</a>

(1.5)

By using the least action principle and the saddle point theorem, they obtained some existence results of solutions under sublinear condition and some reasonable conditions. In [22], system (1.5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M53">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M54">View MathML</a>, was also investigated. By using variational methods, the authors obtained that system (1.5) has at least three weak solutions. In [23], the authors investigated system (1.5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M55">View MathML</a>. They obtained that system (1.5) has infinitely many solutions under the assumptions that nonlinear term is superquadratic, asymptotically quadratic and subquadratic, respectively.

In recent years, via variational methods, some authors have been interested in studying the existence and multiplicity of periodic solutions and homoclinic solutions for the following Lagrangian systems with damped term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M56">View MathML</a>

(1.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11">View MathML</a> is a symmetric and continuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10">View MathML</a> matrix-valued function, B is a skew-symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M10">View MathML</a> constant matrix and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M16">View MathML</a>. They obtained some interesting results. We refer readers to [24-27].

In 2010, Li et al.[28] investigated the following system, more general than system (1.6), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M61">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M62">View MathML</a>

(1.7)

Motivated by [28], in [29], we investigated the following system, more general than system (1.7):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M63">View MathML</a>

(1.8)

By variational methods, under superquadratic or subquadratic conditions, we obtained that system (1.8) has infinitely many solutions. One can see more details of our results and more research background of system (1.8) in [29].

In [32], Luo et al. investigated the existence of subharmonic solutions with prescribed minimal period for the following one-dimensional second-order impulsive differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M64">View MathML</a>

(1.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M71">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M72">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M73">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M74">View MathML</a>.

In this paper, motivated by [10,15,16,21,28,29] and [32], we focus on the existence of subharmonic weak solutions for system (1.1), which is of impulsive conditions, and we study the problem under asymptotically quadratic conditions. To the best of our knowledge, there are few papers that consider such a problem for system (1.1). We call a solution u subharmonic if u is kT-periodic for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M76">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M77">View MathML</a>

In this paper, we make the following assumptions:

(P) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M78">View MathML</a> such that the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M80">View MathML</a>

(K1) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M82">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M83">View MathML</a>

(K2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M36">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M19">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M86">View MathML</a>;

(K3) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M87">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M88">View MathML</a>

(W1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M89">View MathML</a> uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M39">View MathML</a>;

(W2) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M93">View MathML</a>

(W3) There exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M94">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M95">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M96">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M97">View MathML</a>;

(W4) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M46">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M99">View MathML</a>

(W5) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M100">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M101">View MathML</a>

(I1) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M102">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M103">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M104">View MathML</a>

(I2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M106">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M19">View MathML</a>;

(I3) There exists a constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M108">View MathML</a>

This paper is organized as follows. In Section 2, we present the definition of a subharmonic classical solution, a subharmonic weak solution and the variational structure for system (1.1) and make some preliminaries. In Section 3, we present our main theorems and their proofs. In Section 4, an example is given to verify our main theorems.

2 Preliminaries

In this section, we present the variational structure of system (1.1), which is motivated by [15-17,28] and [29].

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M109">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M110">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M111">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M112">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M113">View MathML</a> is a Hilbert space. It is well known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M114">View MathML</a>

is also a norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a>. Obviously, if the condition (P) holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M117">View MathML</a> are equivalent. Moreover, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M118">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M119">View MathML</a>

(see Proposition 1.1 in [1]). Hence, there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M121">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M122">View MathML</a>

(2.1)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M123">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M124">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M125">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M126">View MathML</a> may not hold, which leads to impulsive effects.

Definition 2.1 Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M127">View MathML</a> and the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M129">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M130">View MathML</a>) exist. If u satisfies system (1.1), then we say that u is a subharmonic classical solution of system (1.1).

Remark 2.1 In [32], impulsive effects may occur periodically in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M132">View MathML</a>. In order to obtain a sequence of distinct subharmonic weak solutions (see Theorem 3.2 below), different from [32], in Definition 2.1, we assume that the impulsive effects only occur in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M103">View MathML</a>, which belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M135">View MathML</a>. In other words, u is absolutely continuous on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M136">View MathML</a> is absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M137">View MathML</a>. Moreover, note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M138">View MathML</a>. Then it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M139">View MathML</a>.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M97">View MathML</a>. Then, by T-periodicity of q, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M141">View MathML</a>. Moreover, obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M142">View MathML</a> is continuous on ℝ. We transform system (1.1) into the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M143">View MathML</a>

(2.2)

Then system (2.2) is equivalent to system (1.1) and its solutions are the solutions of system (1.1).

By the idea in [15], we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144">View MathML</a> and multiply the two sides of the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M145">View MathML</a>

by v and integrate it from 0 to kT. Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M146">View MathML</a>

(2.3)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M11">View MathML</a> is continuous on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M149">View MathML</a>. By integration by parts and the continuity of v, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M150">View MathML</a>

(2.4)

Definition 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M125">View MathML</a> is called a subharmonic weak solution of system (1.1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M152">View MathML</a>

holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144">View MathML</a>.

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M125">View MathML</a>is a subharmonic weak solution of system (1.1), thenuis a subharmonic classical solution of system (1.1).

Proof Motivated by [15], for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M155">View MathML</a>, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M157">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M158">View MathML</a>. Then, by Definition 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M159">View MathML</a>

(2.5)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M144">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M157">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M39">View MathML</a>. Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M163">View MathML</a>

(2.6)

Equations (2.5) and (2.6) imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M164">View MathML</a> and u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M165">View MathML</a>

Multiplying the above equality by v and integrating between 0 and kT, combining the argument of (2.4) and Definition 2.2, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M166">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M167">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M168">View MathML</a>. This completes the proof. □

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75">View MathML</a>, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M170">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M171">View MathML</a>

It follows from assumption (A) and Theorem 1.4 in [1] that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172">View MathML</a> is continuously differentiable and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M173">View MathML</a>

(2.7)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M112">View MathML</a>. Obviously, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M175">View MathML</a> is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M177">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M178">View MathML</a> is a subharmonic weak solution of system (1.1).

We will use the following mountain pass theorem to prove our results.

Lemma 2.2 (see [30])

LetEbe a real Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M179">View MathML</a>satisfy the (PS) condition. Ifϕsatisfies the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M180">View MathML</a>;

(ii) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M181">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M182">View MathML</a>;

(iii) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M183">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M184">View MathML</a>, thenϕpossesses a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M185">View MathML</a>given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M186">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M187">View MathML</a>is an open ball inEof radiusρcentered at 0, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M188">View MathML</a>

Remark 2.2 As shown in [31], a deformation lemma can be proved by replacing the usual (PS)-condition with the condition (C), and it turns out that Lemma 2.2 holds true under the condition (C). We say that ϕ satisfies the condition (C), i.e., for every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M190">View MathML</a> has a convergent subsequence if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M191">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M192">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M193">View MathML</a>.

3 Main results

Theorem 3.1Assume that (P), (K1), (K2), (W1)-(W4) and (I1)-(I3) hold. Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75">View MathML</a>, system (1.1) has at least onekT-periodic weak solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a>.

Proof We use Lemma 2.2 to prove the theorem. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M196">View MathML</a>.

Step 1. We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172">View MathML</a> satisfies assumption (ii) of Lemma 2.2. It follows from (W1) and (W2) that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M200">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M201">View MathML</a>

(3.1)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M202">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M203">View MathML</a>. Then it follows from (K1), (I2), (3.1) and (2.1) that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M204">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M205">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M206">View MathML</a>

Step 2. We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172">View MathML</a> satisfies assumption (iii) of Lemma 2.2. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M208">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M209">View MathML</a>. By the argument in [10], we know that (W3) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M210">View MathML</a>

(3.2)

and (K2) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M211">View MathML</a>

(3.3)

It follows from (3.2), (3.3), (W3) and (I1) that for sufficiently large s,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M212">View MathML</a>

By (W4), we can choose sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M215">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M216">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172">View MathML</a> satisfies assumption (iii) of Lemma 2.2.

Step 3. We prove that φ satisfies the (C)-condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a>. The proof is motivated by [10]. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M219">View MathML</a>, assume that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M220">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M221">View MathML</a>

(3.4)

Then it follows from antisymmetry of B, (K2) and (I3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M222">View MathML</a>

(3.5)

Next we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M190">View MathML</a> is bounded. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M224">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M226">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M227">View MathML</a>, and so there exists a subsequence, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M228">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M229">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a>. Then, by Proposition 1.2 in [1], we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M231">View MathML</a>. Hence, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M233">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M234">View MathML</a>. Thus, by conditions (P), (W2) and (I2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M235">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M236">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225">View MathML</a>. Then, by (3.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M238">View MathML</a>

(3.6)

Then it follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M78">View MathML</a> and (3.6) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M240">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M241">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M243">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M244">View MathML</a> and

(3.7)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M246">View MathML</a>. Then (3.7) and T-periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M247">View MathML</a> in t imply that

(3.8)

It follows from (3.8) and Lemma 1 in [6] that there exists a subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M249">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M250">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M251">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M252">View MathML</a>

(3.9)

By (W3), we have

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225">View MathML</a>. Then by Fatou’s lemma and (3.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M255">View MathML</a>

which contradicts (3.5). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M190">View MathML</a> is bounded. Going if necessary to a subsequence, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M257">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a>. Then, by Proposition 1.2 in [1], we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M259">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M260">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225">View MathML</a>. Similar to the argument of Theorem 3.1 in [28], it is easy to obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M262">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M263">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M225">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172">View MathML</a> satisfies the (C)-condition.

Finally, (K1), (W1) and (I2) imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M266">View MathML</a>. Hence, combining Step 1-Step 3 with Lemma 2.2 and Remark 2.2, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M172">View MathML</a> has at least a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M270">View MathML</a>. Then system (1.1) has at least one kT-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a>. This completes the proof. □

Remark 3.1 It is easy to see that Theorem 3.1 generalizes Theorem A. To be precise, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M278">View MathML</a>, Theorem 3.1 reduces to Theorem A.

Theorem 3.2Assume (P), (K1)-(K3), (W1)-(W5) and (I1)-(I3) hold. Then system (1.1) has a sequence of distinct subharmonic weak solutions with period<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M279">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M280">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M281">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M282">View MathML</a>.

Proof By Theorem 3.1, we know that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75">View MathML</a>, system (1.1) has at least one kT-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M286">View MathML</a>. By Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M287">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M288">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M290">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M291">View MathML</a>. Hence, by (K3), (W5) and (I1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M292">View MathML</a>

(3.10)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M293">View MathML</a> is uniformly bounded for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75">View MathML</a>.

Obviously, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M295">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M296">View MathML</a>, then we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M268">View MathML</a> is distinct to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M298">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M299">View MathML</a>. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M300">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M299">View MathML</a>, it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M302">View MathML</a>

Then, by (3.10), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M303">View MathML</a>, a contradiction. We can also find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M304">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M305">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M306">View MathML</a>. Otherwise, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M307">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M299">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M309">View MathML</a>. Then by (3.10), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M310">View MathML</a>, a contradiction. In the same way, we can obtain that system (1.1) has a sequence of distinct periodic solutions with period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M279">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M280">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M281">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M282">View MathML</a>. This completes the proof. □

4 Example

The following example is inspired partially by Example 3.1 in [10]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M315">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M316">View MathML</a>. Consider the following impulsive Lagrangian system with damped term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M317">View MathML</a>

(4.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M318">View MathML</a>

Obviously, the condition (P) holds and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M320">View MathML</a> and (K1), (K2), (W1) and (W2) hold with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M322">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M323">View MathML</a>

Then (W3) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M324">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M325">View MathML</a>

Hence, it is easy to see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M46">View MathML</a> such that (W4) holds by the above inequality. Obviously, (I1) and (I2) hold. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M327">View MathML</a>

So (I3) holds. Hence, by Theorem 3.1, we obtain that system (4.1) has at least one kT-periodic solution for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M75">View MathML</a>.

Moreover, it is easy to see that (K3) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M329">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M330">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M331">View MathML</a>. Then (W5) holds. Hence, by Theorem 3.2, we obtain that system (4.1) has a sequence of distinct subharmonic weak solutions with period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M279">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M334">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/218/mathml/M282">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Acknowledgements

This work is supported by Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No. 11226135) and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY201207032).

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