Fixed point theorems for $\alpha$--contractive mappings of Meir--Keeler type and applications

TL;DR

This paper introduces a new class of $oldsymbol{ ext{α}}$--contractive mappings of Meir--Keeler type in complete metric spaces, proving fixed point theorems that extend existing results and applying them to boundary value problems.

Contribution

It defines $oldsymbol{ ext{α}}$--contractive mappings of Meir--Keeler type and establishes new fixed point theorems that generalize and improve prior results, with applications to differential equations.

Findings

Proved existence and uniqueness of fixed points for the new mappings.

Extended classical fixed point theorems to a broader class of contractions.

Applied the theorems to solve third order two-point boundary value problems.

Abstract

In this paper, we introduce the notion of $\alpha$--contractive mapping of Meir--Keeler type in complete metric spaces and prove new theorems which assure the existence, uniqueness and iterative approximation of the fixed point for this type of contraction. The presented theorems extend, generalize and improve several existing results in literature. To validate our results, we establish the existence and uniqueness of solution to a class of third order two point boundary value problems.