Existence of Homoclinic Solutions for Nonlinear Second-Order Problems

In this work, we consider the second-order discontinuous equation in the real line,

$$u^{\prime \prime}(t)-ku(t) = f( t, u(t), u^{\prime}(t)), \quad a.e.t \in \mathbb {R},$$

with \({k > 0}\) and \({f : \mathbb{R}^{3} \rightarrow \mathbb{R}}\) an \({L^{1}}\)-Carathéodory function. The existence of homoclinic solutions in presence of not necessarily ordered lower and upper solutions is proved, without periodicity assumptions or asymptotic conditions. Some applications to Duffing-like equations are presented in last section.