Relation-theoretic metrical coincidence and common fixed point theorems under nonlinear contractions

Abstract

In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those are contained in Berzig [J. Fixed Point Theory Appl. 12, 221-238 (2012))] and Alam and Imdad [To appear in Filomat (arXiv:1603.09159 (2016))]. Interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to Berzig. Finally, we use examples to highlight the accomplished improvements in the results of this paper.Throughout this paper, R stands for a 'non-empty binary relation' (i.e., R ∅) instead of 'binary relation' while N 0 , Q and Q c stand the set of whole numbers (N 0 = N ∪ {0}), the set of rational numbers and the set of irrational numbers respectively. Definition 4. [19] A binary relation R defined on a non-empty set X is called complete if every pair of elements of X are comparable under that relation i.e., for all u, v in X, either (u, v) ∈ R or (v, u) ∈ R which is denoted by [u, v] ∈ R. Proposition 1. [4] Let R be a binary relation defined on a non-empty set X. Then (u, v) ∈ R s if and only if [u, v] ∈ R. Definition 5. [4] Let f be a self-mapping defined on a non-empty set X. Then a binary relation R onDefinition 6.[5] Let ( f, g) be a pair of self-mappings defined on a non-empty set X. Then a binary relation R on X isNotice that on setting g = I, (the identity mapping on X) Definition 6 reduces to Definition 5. Definition 7.[4] Let R be a binary relation defined on a non-empty set X. Then a sequence {u n } ⊂ X is said to be an R-preserving if (u n , u n+1 ) ∈ R, ∀ n ∈ N 0 . Definition 8. [5] Let (X, d) be a metric space equipped with a binary relation R. Then (X, d) is said to be an R-complete if every R-preserving Cauchy sequence in X converges to a point in X. Remark 1. [5] Every complete metric space is R-complete, where R denotes a binary relation. Moreover, if R is universal relation, then notions of completeness and R-completeness are same.Definition 9. [5] Let (X, d) be a metric space equipped with a binary relation R. Then a mappings f : X → X is said to be an R-continuous at u if u n d −→ u, for any R-preserving sequence {u n } ⊂ X, we have f u n d −→ f u. Moreover, f is said to be an R-continuous if it is R-continuous at every point of X. Definition 10. [5] Let ( f, g) be a pair of self-mappings defined on a metric space (X, d) equipped with a binary relation R. Then f is said to be a (g, R)-continuous at x if gu n d −→ gu, for any R-preserving sequence {u n } ⊂ X, we have f u n d −→ f u. Moreover, f is called a (g, R)-continuous if it is (g, R)-continuous at every point of X.Notice that on setting g = I (the identity mapping on X), Definition 10 reduces to Definition 9.Remark 2. Every continuous mapping is R-continuous, where R denotes a binary relation. Moreover, if R is universal relation, then notions of R-continuity and continuity are same.Definition 11.[4] Let ...

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