Homoclinic solutions for second order differential equations generated by impulses (Q2918419)

The purpose of the paper is to study the existence of homoclinic solutions of the following system of second-order impulsive differential equations NEWLINE\[NEWLINE \begin{cases}\ddot u(t)+\nabla V(t,u(t))=f(t),\quad t\in(s_{k-1},s_{k}),\\ \Delta \dot u(s_{k})=g_{k}(u(s_{k})), \end{cases}\tag{1}NEWLINE\]NEWLINE where \(k\in\mathbb Z\), \(u\in\mathbb R^n\), \(\Delta\dot u(s_k)=\dot u(s_k^+)-\dot u(s_k^-)\) with \(\dot u(s_k^{\pm})=\lim_{t\to s_k^{\pm}}\dot u(t)\), \(f\in C(\mathbb R,\mathbb R^n)\), \(\nabla V(t,u)=\text{grad}_uV(t,u)\), \(g_k(u)=\text{grad}_uG_k(u)\), \(G_k\in C^1(\mathbb R^n,\mathbb R^n)\). It is assumed that there exist an \(m\in\mathbb N\) and \(T\in\mathbb R^+\) such that \(0=s_0<s_1<\dots <s_m=T\), \(s_{k+m}=s_k+T\) and \(g_{k+m}=g_k\) for all \(k\in\mathbb Z\) (that is, \(g_k\) is \(m\)-periodic in \(k\)). As usual, a solution \(u(t)\) of (1) is called homoclinic (to 0) if \(\lim_{t\to\pm\infty}u(t)=0\) and \(\lim_{t\to\pm\infty}\dot u(t^{\pm})=0\).NEWLINENEWLINEIn [\textit{X.~Han} and \textit{H. Zhang}, J. Comput. Appl. Math. 235, No. 5, 1531--1541 (2011; Zbl 1211.34008); \textit{H.~Zhang} and \textit{Z.~Li}, Nonlinear Anal., Real World Appl. 12, No. 1, 39--51 (2011; Zbl 1225.34019)], it was shown that under appropriate conditions, the system (1) possesses at least one nontrivial homoclinic solution generated by impulses when \(f\equiv 0\). In the present paper, the existence of homoclinic solutions of (1) via critical point theory is estsblished under weaker assumptions on \(V\) and \(g\). The work not only generalizes the known results, but also provides certain new methods and techniques.