Discussion on Generalized-(αψ,βφ)-Contractive Mappings via Generalized Altering Distance Function and Related Fixed Point Theorems

Maher Berzig,Antonio-Francisco Roldán-López-De-Hierro,Erdal Karapınar

doi:10.1155/2014/259768

Maher Berzig, Antonio-Francisco Roldán-López-De-Hierro + Show 1 more

Open Access

https://doi.org/10.1155/2014/259768

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Abstract

We extend the notion of (αψ,βφ)-contractive mapping, a very recent concept by Berzig and Karapinar. This allows us to consider contractive conditions that generalize a wide range of nonexpansive mappings in the setting of metric spaces provided with binary relations that are not necessarily neither partial orders nor preorders. Thus, using this kind of contractive mappings, we show some related fixed point theorems that improve some well known recent results and can be applied in a variety of contexts.

Highlights

Their results have been extended to contractivity conditions in which altering distance functions play an important role
Alghamdi and Karapınar [13] used a similar notion in G-metric spaces, and Berzig and Karapinar [14] considered a more general kind of contractivity conditions using a pair of generalized altering distance functions
Fixed Point Results on Metric Spaces Endowed with N

Summary

Introduction

Introduction and PreliminariesAfter the appearance of the pioneering Banach contractive mapping principle and due to its possible applications, fixed point theory has become one of the most useful branches of nonlinear analysis, with applications to very different settings, including, among others, resolution of all kind of equations (differential, integral, matrix, etc.), image recovery, convex minimization and split feasibility, and equilibrium problems.In the last decades, fixed point theorems in partially ordered metric spaces have attracted much attention, especially after the works of Ran and Reurings [1], Nieto and Rodrıguez-Lopez [2], Bhaskar and Lakshmikantham [3], Berinde and Borcut [4, 5], Karapınar [6, 7], Berzig and Samet [8], and Karapınar et al [9,10,11], among others. Let (X, d) be a complete metric space and N ∈ N \ {0} and T : X → X be an (αψ, βφ)-contractive mapping of type I satisfying the following conditions: (i) R1 and R2 are N-transitive; (ii) T is R1-preserving and R2-preserving; (iii) there exists x0 ∈ X such that x0R1Tx0 and x0R2Tx0; (iv) T is continuous. T has a fixed point; that is, there exists x∗ ∈ X such that Tx∗ = x∗.

Results

Conclusion