Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces

Abdolrahman Razani,Mozhgan Bagherboum

doi:10.1186/1687-1812-2013-331

Abdolrahman Razani, Mozhgan Bagherboum

Open Access

https://doi.org/10.1186/1687-1812-2013-331

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Abstract

The purpose of this paper is to investigate some strong convergence as well as stability results of some iterative procedures for a special class of mappings. First, this class of mappings called weak Jungck -contractive mappings, which is a generalization of some known classes of Jungck-type contractive mappings, is introduced. Then, using an iterative procedure, we prove the existence of coincidence points for such mappings. Finally, we investigate the strong convergence of some iterative Jungck-type procedures and study stability and almost stability of these procedures. Our results improve and extend many known results in other spaces. MSC:47H06, 47H10, 54H25, 65D15.

Highlights

Czerwik [ ] initiated the study of multivalued contractions in b-metric spaces.Definition
Using a Jungck-Picard iterative procedure, we investigate the existence of coincidence points and the uniqueness of the coincidence value of weak Jungck (φ, ψ)contractive mappings
This shows that the class of φquasinonexpansive mappings properly includes the class of weak Jungck (φ, ψ)-contractive mappings

Summary

Introduction

Czerwik [ ] initiated the study of multivalued contractions in b-metric spaces. Definition. A mapping T is said to be a weak Jungck (φ, ψ)-contractive mapping with respect to S if there exist an s-comparison function φ : R+ → R+ and a monotone increasing function ψ : R+ → R+ with upper semicontinuity from the right at ψ( ) = such that for all x, y ∈ Y , d(Tx, Ty) ≤ φ d(Sx, Sy) + ψ min d(Sx, Tx), d(Sx, Ty). If there exist a -comparison function φ and a monotone increasing function ψ with upper semicontinuity from the right at ψ( ) = such that for all x, y ∈ [ , ], d(Tx, Ty) ≤ φ d(x, y) + ψ min d(x, Tx), d(x, Ty) , taking x= This shows that the class of φquasinonexpansive mappings properly includes the class of weak Jungck (φ, ψ)-contractive mappings.

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