Existence Result for Difference Equations on Non-Uniform Grids via Upper and Lower Solution Method

2025/07/07 by Bandyopadhyay, Shalmali, Lor, Kimser
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2508.04706

Abstract

This paper establishes an existence theory for discrete second-order boundary value problems on non-uniform time grids using the upper and lower solution method. We consider difference equations of the form u^ΔΔ(t_i-1) + f(t_i, u(t_i), u^Δ(t_i-1)) = 0 on a non-uniform time grid t₀, t₁, …, t_n+2 with mixed boundary conditions u^Δ(t₀) = 0 and u(t_n+2) = g(t_n+2). This extends previous work on homogeneous boundary conditions to the non-homogeneous case, requiring a sophisticated functional analytic framework to handle the resulting affine function spaces. Our approach employs a decomposition strategy that separates boundary effects from the differential structure, enabling the application of Brouwer's Fixed Point Theorem to establish existence with solutions bounded between upper and lower functions.

Related

• Upper and Lower Solution Method for Regular Discrete Second-Order Single-Variable BVPs
• Existence Result for Singular Second Order Dynamic Equations with Mixed Boundary Conditions
• A boundary value problem of a generalised linear discrete time system\n with no solutions and infinitely many solutions
• Study of Solutions of Certain Kind of Non-Linear Differential Equations
• Convergence of the finite difference scheme for a general class of the\n spatial segregation of reaction-diffusion systems