LAUSR

This article analyzes nonlinear, second-order difference equations subject to “left-focal” two-point boundary conditions. Our research questions are: RQ1: What are new, sufficient conditions under which solutions to our “discrete” problem will exist?; RQ2: What, if any, is the relationship between solutions to the discrete problem and solutions of the “continuous”, left-focal analogue involving second-order ordinary differential equations? Our approach involves obtaining new a priori bounds on solutions to the discrete problem, with the bounds being independent of the step size. We then apply these bounds, through the use of topological degree theory, to yield the existence of at least one solution to the discrete problem. Lastly, we show that solutions to the discrete problem will converge to solutions of the continuous problem.