Non-spurious solutions to second order BVP by monotonicity methods

TL;DR

This paper demonstrates the convergence of solutions from discretized second-order boundary value problems to continuous solutions using monotonicity methods, ensuring non-spurious solutions and analyzing parameter dependence.

Contribution

It introduces a novel application of monotonicity methods to prove convergence and existence of non-spurious solutions for second-order BVPs with discretization.

Findings

Solutions of discrete problems converge to continuous solutions.

Existence of non-spurious solutions is established.

Continuous dependence on parameters is analyzed.

Abstract

We consider the following BVP $\ddot{x}\left( t\right) =f\left( t,\dot{x}\left( t\right) ,x\left( t\right) \right) -h\left( t\right) $, $% x\left( 0\right) =x\left( 1\right) =0$, where $f$ is continuous and satisfies some other conditions, $h\in H_{0}^{1}\left( 0,1\right) $ together with its discretization $$-\Delta^{2}x(k-1)+\frac{1}{n^{2}}f\left(\frac{k}{n}, n\Delta x\left(k-1\right), x\left(k\right)\right)=\frac{1}{n^{2}}h\left(\frac{k}{n}\right), k\in \left\{1, 2, \ldots,n \right\}.$$ Using monotonicity methods we obtain the convergence of a solutions to a family of discrete problems to the solution of a continuous one, i.e. the existence of non-spurious solutions to the above problems is considered. Continuous dependence on parameters for the continuous problem is also investigated.