Non-Iterative Set-Theoretic Approach to Fixed Point Theorems in Multiplicative Banach Spaces

Abstract

This paper introduces an alternative approach to proving fixed point theorems, by passing the conventional iterative method. Instead, we employ sets and intersections, inspired by Boyd and Wong's concept. Our focus is on multiplicative normed spaces, examining three types of contractions: Banach, Kannan, and Chatterjea. The key technique involves considering sets of "almost fixed points," which are points located within a certain distance from their image. As this distance approaches 1, these sets diminish and converge to a single point. Cantor's intersection theorem then directly yields the fixed point. We present new formulas that describe the rate at which these sets contract for each type of contraction. It is also noted that multiplicative normed spaces are essentially regular normed spaces under a different appearance, so the existence results are not novel but merely rephrased. What is novel are the speed estimates. Examples, including an application to a Volterra integral equation, demonstrate the findings.