The main purpose of this article is to establish relation-theoretic metrical fixed point theorems via an implicit contractive condition which is general enough to yield a multitude of corollaries corresponding to several well known contraction conditions (e.g. Banach (Fundam. Math. 3:133-181, 1922) Bull. 16:201-206, 1973),Ćirić (Proc. Am. Math. Soc. 45:267-273, 1974) and several others) wherein even such corollaries are new results on their own. As an example we utilize our main results, to prove a theorem on the existence and uniqueness of the solution of an integral equation besides providing an illustrative example. MSC: Primary 47H10; secondary 54H25

We establish fixed point theorems for nonlinear contractions on a metric space (not essentially complete) endowed with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those contained in Samet and Turinici [Commun. Math. Anal. 13, 82-97 (2012)] and Alam and Imdad [J. Fixed Point Theory Appl. 17(4), 693-702 (2015)]. Interestingly a corollary to one of our main results proved under symmetric closure of any binary relation remains a sharpened version of a theorem due to Samet and Turinici. Finally, we use examples to highlight the realized improvements in the results proved in this paper.

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 ...

We consider a relatively new hybrid generalized F -contraction involving a pair of mappings and utilize the same to prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying generalized (F, ϕ)-contraction condition under common limit range property in complete metric spaces. A similar result involving a hybrid pair of mappings satisfying a Rational type Hardy-Rogers (F, ϕ)-contractive condition is also proved. Our results generalize and improve several results of the existing literature. As applications of our results, we prove two theorems for the existence of solutions of certain system of functional equations arising in dynamic programming, and Volterra integral inclusion besides providing an illustrative example.2010 Mathematics Subject Classification. 47H09, 47H10, 45G99, 90C39. Key words and phrases. multi-valued mappings; hybrid pair of mappings; common limit range property; occasionally coincidentally idempotent mappings; dynamic programming; integral inclusion.

The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction conditions in one go besides yielding several new ones. We also provide an example to demonstrate the generality of our results over several well known corresponding results of the existing literature. Finally, we utilize our results to prove an existence theorem for ensuring the solution of an integral equation. Definition 2.4. [27]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 x, y in X, either (x, y) ∈ R or (y, x) ∈ R which is denoted by [x, y] ∈ R. Proposition 2.1. [4] Let R be a binary relation defined on a non-empty set X. ThenDefinition 2.6.[5] Let T and g be two 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 2.6 reduces to Definition 2.5.Definition 2.7.[4] Let R be a binary relation defined on a non-empty set X. Then a sequence {x n } ⊂ X is said to be R-preserving if (x n , x n+1 ) ∈ R, ∀ n ∈ N 0 .Definition 2.8.[5] Let (X, d) be a metric space equipped with a binary relation R. Then (X, d) is said to be R-complete if every R-preserving Cauchy sequence in X converges to a point in X. 4 Remark 2.1. [5] Every complete metric space is R-complete, where R denotes a binary relation. Particularly, if R is universal relation, then notions of completeness and R-completeness coincide. Definition 2.9. [5] Let (X, d) be a metric space equipped with a binary relation R. Then a self-mapping T on X is said to be R-continuous at x if T x n d −→ T x whenever x n d −→ x, for any R-preserving sequence {x n } ⊂ X. Moreover, T is said to be R-continuous if it is R-continuous at every point of X. Definition 2.10. [5] Let (X, d) be a metric space equipped with a binary relation R and g a self-mapping on X. Then a self-mapping T on X is said to be (g, R)-continuous at x if T x n d −→ T x, for any R-preserving sequence {x n } ⊂ X with gx n d −→ gx. Moreover, T is called (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 2.10 reduces to Definition 2.9.Remark 2.2. Every continuous mapping is R-continuous, where R denotes a binary relation. Particularly, if R is universal relation, then notions of R-continuity and continuity coincide.Definition 2.11. [4] Let (X, d) be a metric space. Then a binary relation R on X is said to be d-self-closed if for any