Some New Results on Coincidence Points for Multivalued Suzuki-Type Mappings in Fairly Complete Spaces

Naeem Saleem,Iqra Habib,Manuel De La Sen

doi:10.3390/computation8010017

Abstract

In this paper, we introduce Suzuki-type ( α , β , γ g ) - generalized and modified proximal contractive mappings. We establish some coincidence and best proximity point results in fairly complete spaces. Also, we provide coincidence and best proximity point results in partially ordered complete metric spaces for Suzuki-type ( α , β , γ g ) - generalized and modified proximal contractive mappings. Furthermore, some examples are presented in each section to elaborate and explain the usability of the obtained results. As an application, we obtain fixed-point results in metric spaces and in partially ordered metric spaces. The results obtained in this article further extend, modify and generalize the various results in the literature.

Highlights

Due to wide range of applications, researchers around the globe are attracted towards this principle to generalize, modify and extend this pioneer result. These modifications are consisting upon three pillars (1) generalizing the contractive conditions, (2) generalizing the underlying space and (3) modifying the single valued mapping with multivalued mapping
In the first result we will prove that the pair (g, M) which satisfies the Suzuki-type (α, β, γg )−generalized proximal contraction has a coincidence best proximity point in the frame work of fairly complete spaces
If we choose q = (−1, 0) and r = (−1, 1) the inequality (13) does not holds. This shows that the pair (g, M) satisfy Suzuki-type (α, β, γg )−generalized proximal contractive condition; further remaining conditions of Theorem 3 holds, the pair (g, M) has two coincidence best proximity points (−1, 1) and (−1, −1)

Summary

Introduction and Preliminaries

([13]) Let (Y, d) be a complete metric space and mapping M : Y → Y satisfies d( Mq, Mr) ≤ αd(q, r) + βd( Mq, r), for all q, r ∈ Y where α ∈ [0, 1) and β ∈ [0, ∞) the mapping M has a "fixed point". ([14]) Let (Y, d) be a complete metric space and mapping M : Y → Y satisfies d(q, Mq) < d(q, r) implies d( Mq, Mr) < d(q, r), for all q, r ∈ Y mapping M has a unique fixed point in Y. In 2012, Samet et al ([18]) introduced α − ψ−contractive and α−admissible mappings and established some fixed-point theorems for such mappings in complete metric spaces.

Main Results

Some Results Related to Partially Ordered Metric Space

Application to Fixed-Point Theory

Conclusions