Decaying positive global solutions of second order difference equations with mean curvature operator
Zuzana Došlá, Serena Matucci, Pavel Řehák
TL;DR
The paper addresses the existence of globally positive, decaying solutions for a second-order nonlinear difference equation with the Euclidean mean curvature operator on the half-line. It combines new Sturm-type comparison results for linear difference equations with a Schauder fixed-point approach in a Fréchet space, using a linearization device and recessive solutions to obtain a priori bounds. A central contribution is a solvability criterion: if the associated linear majorant admits a positive decreasing solution, then a positive decaying solution to the original BVP exists; the method yields explicit conditions and corollaries that ensure solvability for a range of boundary values. The work clarifies discrete–continuous distinctions in decay behavior and provides tools (recessive solutions, majorants, and fixed-point arguments) that extend the discrete mean-curvature theory and inform discretizations of PDEs with mean curvature operators.
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.