Variational methods on finite dimensional Banach spaces and discrete problems

In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at leastone positive solution without requiring any asymptotic condition at innity on thenonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach isbased on variational methods and, in particular, on a local minimum theorem andits variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.