The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects

Abstract

In this paper, we study the existence of multiple solutions for a class of second-order impulsive Hamiltonian systems. We give some new criteria for guaranteeing that the impulsive Hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of B. Ricceri. We extend and improve on some recent results. Finally, some examples are presented to illustrate our main results.

Introduction

In this paper, we consider the following problem:

$$\{\begin{array}{c}-\ddot{u}+A\left(t\right)u=\lambda \nabla F(t,u)+\mu \nabla G(t,u)\text{,}\text{a.e. }t\in [0,T]\text{,}\hfill \\ \Delta \left({\dot{u}}^i\left(t_j\right)\right)={\dot{u}}^i\left(t_j^{+}\right)-{\dot{u}}^i\left(t_j^{-}\right)=I_{ij}\left(u^i\left(t_j\right)\right)\text{,}i=1,2,\dots ,N,j=1,2,\dots ,l\text{,}\hfill \\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0\text{,}\hfill \end{array}$$

where $A:[0,T]\to {\mathbb{R}}^{N\times N}$ is a continuous map from the interval $[0,T]$ to the set of $N$-order symmetric matrices, $\lambda ,\mu \in \mathbb{R},T$ is a real positive number, $u\left(t\right)=(u^1\left(t\right),u^2\left(t\right),\dots ,u^N\left(t\right)),t_j,j=1,2,\dots ,l$, are the instants where the impulses occur and $0=t_0<t_1<t_2<\cdots <t_l<t_{l+1}=T$, $I_{ij}:\mathbb{R}\to \mathbb{R}(i=1,2,\dots ,N,j=1,2,\dots ,l)$ are continuous and $F,G:[0,T]\times {\mathbb{R}}^N\to \mathbb{R}$ are measurable with respect to $t$, for every $u\in {\mathbb{R}}^N$, continuously differentiable in $u$, for almost every $t\in [0,T]$ and satisfy the following standard summability condition:

$$\underset{\left|u\right|\le b}{sup}(max\{\left|F(\cdot ,u)\right|,\left|G(\cdot ,u)\right|,\left|\nabla F(\cdot ,u)\right|,\left|\nabla G(\cdot ,u)\right|\})\in L^1\left([0,T]\right)$$

for all $b>0$.

Note that the above condition is satisfied by, for instance, simply assuming $\nabla F$ and $\nabla G$ continuous in $[0,T]\times {\mathbb{R}}^N$.

Moreover, without loss of generality, it is supposed that $F(t,0)=G(t,0)=0$ (for a.e. $t\in [0,T]$).

Denote by $\mathcal{C}$ the class of all functions $H:[0,T]\times {\mathbb{R}}^N\to \mathbb{R}$ that are measurable with respect to $t$, for every $u\in {\mathbb{R}}^N$, continuously differentiable in $u$, for almost every $t\in [0,T]$ and satisfy the condition (1.2).

Inspired by the excellent monograph of Mawhin and Willem [1], already, several authors have widely studied second-order Hamiltonian systems of the following type:

$$\{\begin{array}{c}\ddot{u}\left(t\right)=\nabla F(t,u\left(t\right))\text{,}\text{a.e. }t\in [0,T]\text{,}\hfill \\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0\text{,}\hfill \end{array}$$

where $F:[0,T]\times {\mathbb{R}}^N\to \mathbb{R}$ and $\nabla F$ is the gradient of $F$ with respect to $u$, obtaining the existence and multiplicity of periodic solutions for problem (1.3) using various techniques and different conditions on the nonlinearities, such as the coercive type potential condition, the even type potential condition, and the superquadratic potential condition. We refer the reader to [2], [3], [4], [5], [6]. More precisely, recently, in [5], Bonanno and Livrea studied the problem (1.3) with $F(t,u)=\frac{1}{2}A\left(t\right)u\cdot u-\lambda b\left(t\right)G\left(u\right)$:

$$\{\begin{array}{c}-\ddot{u}\left(t\right)+A\left(t\right)u=\lambda b\left(t\right)\nabla G\left(u\right)\text{,}\text{a.e. }t\in [0,T]\text{,}\hfill \\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0\text{,}\hfill \end{array}$$

where $A:[0,T]\to {\mathbb{R}}^{N\times N}$ is a matrix-valued function; $b\left(t\right)\in L^1\left([0,T]\right)\setminus \left\{0\right\}$ is nonnegative and $G:{\mathbb{R}}^N\to \mathbb{R}$. They proved the existence of infinitely many periodic solutions for the Hamiltonian systems (1.4) under an appropriate oscillating behavior of the nonlinear term. Moreover, they obtained the multiplicity of periodic solutions for problem (1.4) with a coercive potential and also in the noncoercive case.

On the other hand, impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians. For some recent works on the theory of impulsive differential equations we refer the reader to [7], [8], [9], [10]. Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory and so on (see [11], [12], [13], [14], [15], [16], [17], [18], [19]).

Furthermore, many classical approaches and tools have been used to study boundary value problems for impulsive differential equations, such as the coincidence degree theory of Mawhin, the method of upper and lower solutions with the monotone iterative technique, and some fixed point theorems in cones. However, the study of solutions for impulsive differential equations using the variational method has received considerably less attention (see, for example, [20], [21], [22], [23], [24]). The variational method is, to the best of our knowledge, novel and it may open a new approach for dealing with nonlinear problems with some kinds of discontinuities such as impulses.

Motivated by the above facts, in this paper, our aim is to study the multiplicity of solutions for impulsive Hamiltonian systems of the class (1.1) by using the variational method. It is worth stressing that we assume $G$ to satisfy a general growth condition permitting us to use a variational method. In addition, we note that we get the multiplicity results for two cases: when the nonlinearity $F$ is asymptotically quadratic and also when it is subquadratic, as $\left|u\right|\to \infty$.

This paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we discuss the existence of three weak solutions for problem (1.1). Some examples are presented in the last section.