Fixed Points for a Pair of F-Dominated Contractive Mappings in Rectangular b-Metric Spaces with Graph

Tahair Rasham,Abdullah Shoaib,Giuseppe Marino

doi:10.3390/math7100884

Abstract

Recently, George et al. (in Georgea, R.; Radenovicb, S.; Reshmac, K.P.; Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005–1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. In this paper, we achieved fixed-point results for a pair of F-dominated mappings fulfilling a generalized rational F-dominated contractive condition in the better framework of complete rectangular b-metric spaces complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. Some examples are given to illustrate our conclusions. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results.

Highlights

Introduction and PreliminariesFixed-point theory is a basic tool in functional analysis
In the sequel George et al [2] furnished the notion of rectangular b-metric space (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space
Here, we investigate our results in a better framework of rectangular b-metric space

Summary

Introduction and Preliminaries

Fixed-point theory is a basic tool in functional analysis. Banach [1] has shown significant result for contraction mappings. Further recent results on rectangular b-metric spaces can be seen in [10,11]. We achieved fixed-point results for a pair of α-dominated mappings fulfilling a generalized rational F-dominated contractive condition in complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. The pair ( Z, dl ) is said a rectangular b-metric space (in short R.B.M.S) with coefficient b. Let ( Z, dl ) be a metric space, S : Z → P( Z ) be a multivalued mapping and α : Z × Z →. Let A ⊆ Z, the mapping S is said semi α∗ -admissible on A, if α( x, y) ≥ 1 implies α? It is clear that S and T are α-dominated but not α-admissible

Main Result

Fixed Points for Graphic Contractions

Conclusions