In this paper, the problems of existence, non-existence, and uniqueness of homoclinic solutions are studied for relativistic Liénard equations:

where \(\nu >0\) is a constant, \(f\in C({\mathbb {R}},{\mathbb {R}}), g\in C({\mathbb {R}}\times {\mathbb {R}},{\mathbb {R}})\) with \(T-\)periodic in the variable t and \(p\in C({\mathbb {R}},{\mathbb {R}})\). Under some conditions, the authors obtain that the equation has at least one nontrivial homoclinic solution for \(p(t)\not \equiv 0\), and the equation has no nontrivial homoclinic solution for \(p(t)\equiv 0\). The arguments are based upon the method of upper and lower solutions.

本文研究了相对论Liénard方程的同宿解的存在、不存在和唯一性问题：

其中\(\nu >0\)是一个常数，\(f\in C({\mathbb {R}},{\mathbb {R}}), g\in C({\mathbb {R}}\次 {\mathbb {R}},{\mathbb {R}})\)与\(T-\)在变量t和\(p\in C({\mathbb {R}},{\mathbb {R}})\)。在某些条件下，作者得到方程对于\(p(t)\not \equiv 0\)至少有一个非平凡同宿解，而方程对于\(p(t)\equiv 0没有非平凡同宿解\)。参数基于上下解法。