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)\Delta u\left(t-1)=f\left(t,u\left(t))-s,\hspace{1em}t\in {\mathbb{Z}},\hspace{1.0em}\hspace{1.0em}\left(P) where s∈Rs\in {\mathbb{R}}, f:Z×R→Rf:{\mathbb{Z}}\times {\mathbb{R}}\to {\mathbb{R}} is continuous with respect to u∈Ru\in {\mathbb{R}}, f(t,u)=f(t+T,u)f\left(t,u)=f\left(t+T,u), T>1T\gt 1 is an integer, Δu(t−1)=u(t)−u(t−1)\Delta u\left(t-1)=u\left(t)-u\left(t-1). We prove a result of Ambrosetti-Prodi-type for (P)\left(P) by using the method of lower and upper solutions and topological degree. We relax the coercivity assumption on ff in Bereanu and Mawhin [1] and obtain Ambrosetti-Prodi-type results.

Keywords

• periodic solutions
• ambrosetti-prodi-type results
• lower and upper solutions
• topological degree
• 39a12
• 39a23