On the existence of homoclinic type solutions of a class of inhomogenous second order Hamiltonian systems

We study the existence of homoclinic type solutions for second order Lagrangian systems of the type \(q̈(t)-q(t)+a(t)\nabla G(q(t))=f(t)\), where \(t\) \(\varepsilon\) \(\Bbb R\), \(q\) \(\varepsilon\) \(\Bbb R^n\), \(a:\Bbb R \longrightarrow \Bbb R^n\) is a continuous positive bounded function, \(G: \Bbb R^n \longrightarrow \Bbb R\) is a \(C^1\)-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and \(f:\Bbb R \longrightarrow \Bbb R^n\) is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of \(2k\)-periodic solutions of an approximative sequence of second order differential equations.