Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation

Abstract

Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces endowed with binary relations and involving suitable comparison functions are presented. Our results generalize, improve, and extend several recent results. As an application, we study the existence of solutions for some class of integral equations.

Observations on relation-theoretic coincidence theorems under Boyd–Wong type nonlinear contractions

“…Several metrical fixed point theorems under arbitrary binary relations are proved by various authors such as Khan et al [10], Ayari et al [27], Roldán-López-de-Hierro [28], Roldán-López-de-Hierro and Shahzad [29], and Shahzad et al [30], which are generalizations of the relation-theoretic contraction principle due to Alam and Imdad [26]. Here we can point out that arbitrary binary relation is general enough and often does not work for certain contractions, so that various fixed/coincidence point theorems are proved in metric spaces equipped with different types of binary relations, for example, preorder (Roldán-López-de-Hierro and Shahzad [11]), transitive relations (Shahzad et al [31]), finitely transitive relations (Berzig and Karapinar [32], Berzig et al [33]), locally finitely transitive relations (Turinici [34,35]), locally finitely T-transitive relations (Alam et al [36]), and locally T-transitive relations (see Alam and Imdad [37]).…”

Relation-theoretic metrical coincidence theorems

“…In recent years, various fixed and coincidence point theorems are proved in metric spaces equipped with different types of binary relations, e. ., partial order (see Ran and Reurings [1], Nieto and Rodríguez-López [2] and Nieto and Rodríguez-López [3]), preorder (see Turinici [4], Roldán and Karapinar [5], Roldán-López-de-Hierro and Shahzad [6]), transitive relation (see Ben-El-Mechaiekh [7], Shahzad et al [8]), finitely transitive relation (see Berzig and Karapinar [9], Berzig et al [10]), tolerance (see Turinici [11,12]), strict order (see Ghods et al [13]), symmetric closure (see Samet and Turinici [14], Berzig [15]) and arbitrary binary relation (see Alam and Imdad [16], Roldán-López-de-Hierro [17], Roldán-López-de-Hierro and Shahzad [18], Shahzad et al [19], Khan et al [20], Ayari et al [21]). In the present context, the contraction condition remains relatively weaker than usual contraction as it is required to hold merely for those elements which are related in the underlying relation.…”

A best proximity point theorem for special generalized proximal β-quasi contractive mappings

“…A useful lemma concerning the comparison functions Φ β was performed in [10]. (1) ϕ β is nondecreasing;…”

“…[10]) Let β ∈ (0, +∞). A β-comparison function is a map ϕ : [0, +∞) → [0, +∞) satisfying the following properties:…”