A Novel Application of the Classical Banach Fixed Point Theorem

Abstract

Using the classical Banach fixed point theorem, we propose a novel method to obtain existence and uniqueness result pertaining to the solutions of semilinear elliptic partial differential equation of the type \(\Delta u+f(x,u,Du)=0\), in \(\Omega \subset {\mathbb {R}}^n \) and \(u|_{\partial \Omega }=0\), in a suitable Sobolev space. Here \(f{:}\;\Omega \times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is either a linear or a non-linear Lipshitz continuous function. The approach attempted here can be used as an algorithm by the numerical analysts to determine a solution to a partial differential equation of the above type.