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

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 $u^{\mathrm{\Delta }\mathrm{\Delta }}(t_{i-1})+f(t_i,u(t_i),u^{\mathrm{\Delta }}(t_{i-1}))=0$ on a non-uniform time grid $t_0,t_1,\dots ,t_{n+2}$ with mixed boundary conditions $u^{\mathrm{\Delta }}(t_0)=0$ 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.