Existence of homoclinic solutions for a class of second order ordinary differential equations

In this paper, we study the existence of homoclinic solutions for the following class of second order ordinary differential equations (ODEs) equation(P){−(A(u(t))u′(t))′+u(t)=h(t,u(t))+g(t,u′(t)),t∈Ru(±∞)=u′(±∞)=0, where A,hA,h and gg are nonnegative continuous functions. Using Galerkin’s Method and assuming additional hypotheses on A,hA,h and gg, we show the existence of a homoclinic solution to problem (P).