Iterating Fixed Point via Generalized Mann’s Iteration in Convex b-Metric Spaces with Application

A Asif,N Hussain,A Ali,M Arshad,M Alansari,Danilo Comminiello

doi:10.1155/2021/8534239

A Asif, N Hussain + Show 4 more

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https://doi.org/10.1155/2021/8534239

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Abstract

This manuscript investigates fixed point of single-valued Hardy-Roger’s typeF-contraction globally as well as locally in a convexb-metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom (F3) of theF-contraction is removed, and thus the mappingFis relaxed. An important approach used in the article is, though a subset closed ball of a complete convexb-metric space is not necessarily complete, the convergence of the Cauchy sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations.

Highlights

Introduction and PreliminariesVarious authors generalized metric space into many interesting generalizations
First we prove that the iteration 􏼈an􏼉 belongs to the closed ball
Conclusion is paper has introduced F-Hardy-Roger’s contraction on closed ball in convex b-metric space, and generalized Mann’s iteration theorem algorithm is used to find the fixed point. e existence of the limit point inside the closed ball is established despite of the fact that the subset closed ball in a complete b-metric space is not complete. e convergence of the chosen sequence is insured inside the closed ball without completeness of the ball

Summary

Introduction

Introduction and PreliminariesVarious authors generalized metric space into many interesting generalizations (see [1,2,3,4,5,6,7,8]). Assume that F ∈ Fb, (E, bk, η) be a convex bmetric space with k > 1, δn− 1 ∈ [0, 1) for n ∈ N, and Sε[a0] be a closed ball in E. en f: E ⟶ E is known as F-HardyRoger’s contraction on closed ball if for α, β, c ∈ [0, ∞) with k(k2δn− 1 + αk) + (k2 + 1)(β + 2ck) < 1, the following hold: τ + F kbk(fa, fb)􏼁 ≤F􏼂αbk(a, b) + β􏼈bk(a, fa) + bk(b, fb)􏼉

Results

Conclusion