Decaying positive global solutions of second order difference equations with mean curvature operator

TL;DR

This paper investigates positive solutions that decay at infinity for second order difference equations involving the mean curvature operator, using Sturm comparison theorems and fixed point methods.

Contribution

It introduces new Sturm comparison theorems for linear difference equations and applies a fixed point approach to establish existence of solutions.

Findings

Existence of positive decaying solutions on unbounded domains.

Development of new Sturm comparison theorems for difference equations.

Analysis of discretization effects from continuous to discrete problems.

Abstract

A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is examined by proving new Sturm comparison theorems for linear difference equations and using a fixed point approach based on a linearization device. %The process from the continuous problem to discrete one is examined, too. The process of discretization of the boundary value problem on the unbounded domain is examined, and some discrepancies between the discrete and the continuous case are pointed out, too.