Reverse-Order Lower and Upper Functions for Periodic Problems of Second-Order Singular Difference Equations

Abstract

We present sufficient conditions ensuring the lower and upper functions on the reversed-order for the periodic difference equations. This enables us to obtain the existence of positive periodic solutions of the second-order difference equation ${\mathrm{\Delta }}^{\mathrm{2}}u(t-\mathrm{1})=g(t)/u^{\mu }(t)-h(t)/u^{\lambda }(t)+f(t),\mathrm{}\mathrm{}\mathrm{}\mathrm{}t\in \mathbb{Z},$ where $g,\mathrm{}h:\mathbb{Z}\to [\mathrm{0},\mathrm{\infty })$, and $f:\mathbb{Z}\to ℝ$ are $T$-periodic functions, $\lambda ,\mathrm{}\mathrm{}\mu >\mathrm{0}$.