Periodic solutions of nonlinear second-order difference equations

Abstract

We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t+2)+by(t+1)+cy(t)=f(y(t)), where c≠0 and f:ℝ→ℝ is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant β>0 such that uf(u)>0 whenever |u|≥β. For such an equation we prove that if N is an odd integer larger than one, then there exists at least one N-periodic solution unless all of the following conditions are simultaneously satisfied: c=1, |b|<2, and N across-1(−b/2) is an even multiple of π.