Some Fixed Point Theorems in $(\alpha,\beta)$- Metric Spaces with applications to Fredholm integral and non-linear differential equations

TL;DR

This paper introduces a new class of metric spaces called $(\alpha,eta)$-metric spaces, establishes fixed point theorems for novel contraction mappings within these spaces, and applies these results to solve integral and differential equations.

Contribution

It defines $(\alpha,eta)$-metric spaces, introduces new contraction mappings, and extends fixed point theorems, including Kannan's and Reich's, with applications to integral and differential equations.

Findings

Established fixed point theorems for $(\alpha,eta)$-contraction mappings.

Extended classical fixed point results to the new $(\alpha,eta)$-metric space setting.

Applied the theoretical results to solve Fredholm integral and non-linear differential equations.

Abstract

In this paper, we presented a new type of metric space called $(\alpha,\beta)$-metric space along with some novel contraction mappings named $(\alpha,\beta)$-contraction and weak $(\alpha,\beta)$-contraction mapping. We established some fixed point theorem for these newly introduced contractive mappings. Our results extended some fixed point results in the existing literature. We also provide an example which holds for the weak $(\alpha,\beta)$-contraction. Furthermore, we proved Kannan's fixed point theorem and Reich's fixed point theorem in the setting of $(\alpha,\beta)$-metric spaces. At the end, as applications the Fredholm integral and non-linear differential equations are solved in order to validate the theoretically obtained conclusions