An application of Darbo’s fixed point theorem in the investigation of periodicity of solutions of difference equations

Using the technique of Darbo’s fixed point theorem we investigate a nonlinear second order difference equation of the form Δ(rnΔxn)=anf(xn+1) where x:N0→Rx:N0→R, a,r:N0→Ra,r:N0→R and f:R→Rf:R→R is a continuous function. Additionally, the Sturm–Liouville difference equation is considered as a special case of the above equation. Sufficient conditions for the existence of an asymptotically periodic solution of this equation are obtained. An example illustrates the result. Next, the conditions under which this equation has no asymptotically periodic solutions are presented.