On the Dirichlet problem for a Duffing type equation

Recently the classical variational problem for a Duffing type equation received again some attention. In [1], [2], [7], some variational approaches were used in order to receive the existence of solutions for both periodic and Dirichlet type boundary value problems. Mainly direct method is applied under various conditions pertaining to at most quadratic growth imposed on the nonlinear term given in [2] and further relaxed in [7]. Dirichlet problems for such equations could also be considered by some other methods, for example min-max theorem due to Manashevich, [8]. In [6] the author gives some historical results concerning the Dirichlet problem for Duffing type equations and discusses the methods which are used in reaching the existence results which are different from the ones which we use and comprise the classical variational approach, the topological method.