Existence of Homoclinic Solutions for a Class of Nonlinear Second-order Problems

We are concerned with the existence of homoclinic solutions for the nonlinear problems P \(\begin{aligned} \left\{ \begin{array}{ll} u''+\omega u'-ku=f(t,u,u'),\ \ t\in \mathbb {R},\\ \lim \limits _{|t|\rightarrow +\infty }u(t)=0, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime \prime }+\omega u^{\prime }-ku=f(t,u,u^{\prime }),t\in \mathbb{R},\hfill \\ \underset{|t|\to +\infty }{lim}u(t)=0,\hfill \end{array}\right)\end{array}$$ where \(\omega \in \mathbb {R},~k>0\) $\omega \in \mathbb{R},k>0$ are real constants, and \(f: \mathbb {R}^{3}\rightarrow \mathbb {R}\) $f:{\mathbb{R}}^3\to \mathbb{R}$ is an \(L^{1}-\) $L^1-$ Carathéodory function. Under some suitable conditions, the existence of homoclinic solutions for problem (P) and the corresponding coupled systems are provided. The proofs of the main results are based on the method of upper and lower solutions.