On Convex F-Contraction in b-Metric Spaces

Huaping Huang,Kastriot Zoto,Stojan Radenović,Zoran D Mitrović

doi:10.3390/axioms10020071

Abstract

In this paper, we introduce a notion of convex F-contraction and establish some fixed point results for such contractions in b-metric spaces. Moreover, we give a supportive example to show that our convex F-contraction is quite different from the F-contraction used in the existing literature since our convex F-contraction does not necessarily contain the continuous mapping but the F-contraction contains such mapping. In addition, via some facts, we claim that our results indeed generalize and improve some previous results in the literature.

Highlights

We introduce the concept of convex F-contraction and give some sufficient conditions when the Picard sequence of convex F-contraction on b-metric space satisfies the Cauchy condition
Let T be an F-contraction of Reich type, i.e., there exist τ > 0 and α, β, γ ∈ [0, 1], α + β + γ = 1 such that τ + F (d( Tx, Ty)) ≤ F (αd( x, y) + βd( x, Tx ) + γd(y, Ty)), (5)
We give a supportive example to verify that the mapping T with regard to convex F-contraction is not necessarily continuous

Summary

Introduction

Let ( X, d) be a complete metric space and T : X → X be an F-contraction. Several authors proved fixed point results for F-contractions (see [2,3,4,5,6,7,8,9,10,11,12,13]). We introduce the concept of convex F-contraction and give some sufficient conditions when the Picard sequence of convex F-contraction on b-metric space satisfies the Cauchy condition. We present a fixed point result for such contraction.

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Results

Conclusion