Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign

Abstract

In this article, we are concerned with the periodic solutions of first-order difference equation Δu(t − 1) = f (t, u(t)) − s, t ∈ Z, (P) where s ∈ R, f : Z × R → R is continuous with respect to u ∈ R, f (t, u) = f (t + T, u), T > 1 is an integer, Δu(t − 1) = u(t) − u(t − 1). We prove a result of Ambrosetti-Prodi-type for (P) by using the method of lower and upper solutions and topological degree. We relax the coercivity assumption on f in Bereanu and Mawhin [1] and obtain Ambrosetti-Prodi-type results.