Periodic solutions of asymptotically linear dynamical systems (Q1346268)

The equation (1) \(\ddot x + V_ x' (x,t) = 0\), \(x = x(t)\) \(kT_ 0\)- periodic curve \((k \in\mathbb{N})\) in \(\mathbb{R}^ n\), where \(V_ x' (x,t)\) denote the gradient of \(V\) with respect to \(x\), is considered. It is assumed that the potential function \(V\) is asymptotically quadratic, i.e. \(V(x,t) = (1/2) (A_ \infty x | x) + U(x,t)\) where (.1.) denotes the standard inner product in \(\mathbb{R}^ n\), \(A_ \infty = A_ \infty (t)\) is a symmetric real \(T_ 0\)-periodic \(n \times n\) matrix and \(U(x,t)\) is a function which is bounded having bounded gradient and whose Hessian matrix \(U_{xx}{''} (x,t)\) tends to zero (uniformly in \(t)\) as \(| x |\) goes to infinity. The paper deals with existence of periodic solutions of the equation (1) without any assumptions which have been used till now for the same problem.