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

Zuzana Dosla,Pavel Řehák,Serena Matucci

doi:10.14232/ejqtde.2020.1.72

Zuzana Dosla, Pavel Řehák + Show 1 more

Open Access

https://doi.org/10.14232/ejqtde.2020.1.72

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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 of discretization of the boundary value problem on the unbounded domain is examined, and some discrepancies between the discrete and the continuous cases are pointed out, too.

Highlights

In this paper we study the boundary value problem (BVP) on the half-line for difference equation with the Euclidean mean curvature operator subject to the conditions
Our existence result is based on a fixed point theorem for operators defined in a Fréchet space by a Schauder’s linearization device. This method is originated in [10], later extended to the discrete case in [20], and recently developed in [15]. This tool does not require the explicit form of the fixed point operator T and simplifies the check of the topological properties of T on the unbounded domain, since these properties become an immediate consequence of a-priori bounds for an associated linear equation
We prove a new Sturm-type comparison theorem for linear difference equations

Summary

Introduction

In this paper we study the boundary value problem (BVP) on the half-line for difference equation with the Euclidean mean curvature operator subject to the conditions. This method is originated in [10], later extended to the discrete case in [20], and recently developed in [15] This tool does not require the explicit form of the fixed point operator T and simplifies the check of the topological properties of T on the unbounded domain, since these properties become an immediate consequence of a-priori bounds for an associated linear equation. These bounds are obtained in an implicit form by means of the concepts of recessive solutions for second order linear equations. Throughout the paper we emphasize some discrepancies, which arise between the continuous case and the discrete one

Discrete versus continuous decay

A Sturm-type comparison theorem for linear equations

Main result: solvability of BVP

Comments and open problems