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 u+ f (x, u, Du) = 0, in ⊂ R n and u| ∂ = 0, in a suitable Sobolev space. Here f : ×R×R n → 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.

Expansions of real closed fields with the Banach fixed point property

“…In 1922, Banach introduced his fixed point theorem, which implies that if $$(X,d)$$ is complete and $$E$$ is closed then every contraction on $$E$$ has a fixed point. This theorem later became an important tool in many branches of mathematics, especially in analysis and studies on differential equations (cf., e.g., [3, 14]). There are many researches on converses of the Banach fixed point theorem (cf., e.g., [2, 6, 11, 13]).…”