Discrete Arzelà-Ascoli Theorem on the Half Line and Its Application

Abstract

The compactness criterion in $ l_{\diamond}=\left\{x(\,\cdot\,)\in l^{\infty}(0,\infty))\mid\lim\limits_{t\rightarrow\infty}x(t)=x(\infty)\right\} $ is given and as an application, we study the existence of positive solutions of second order boundary value problems of difference equation on the half line by the fixed-point theorem in cones.