Existence of homoclinic solutions for a class of nonlinear second-order problems (Q6594629)

The authors study the existence of homoclinic solutions for a particular class of second order nonlinear ordinary differential equations with an \(L^1\)-Carathéodory right hand side, for which a linear damping term prevents the direct application of previously known similar results. The main Theorem 3.1 states that, assuming the existence of upper and lower solutions to the problem, there also exists a homoclinic solution. It is proved through a fixed-point argument for a suitable operator equation, which is obtained using a Green's function approach. An explicit example is given to illustrate the statement, and, finally, the result is then extended to coupled systems of similar type.