Fast homoclinic solutions for a class of damped vibration problems

In this paper, we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problemsu¨(t)+q(t)u̇(t)-L(t)u(t)+∇W(t,u(t))=0,where L(t)L(t) and W(t,x)W(t,x) are neither autonomous nor periodic in t. Our approach is variational and it is based on the critical point theory. We prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function. Moreover, some open problems proposed by Zhang and Ruan are resolved. Our theorems appear to be the first such result.