Relation-theoretic metrical coincidence theorems

Alam, Aftab; Imdad, Mohammad

doi:10.2298/fil1714421a

Cited by 73 publications

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“…Our aim in this work is to proved some coincidence and common fixed point theorems for nonlinear contraction on metric space endowed with amorphous relation. The results proved herein generalize and unify main results of Berzig [9], Alam and Imdad [5] and several others. To demonstrate the validity of the hypotheses and degree of generality of our results, we also furnish some examples.…”

Section: Introductionsupporting

confidence: 90%

“…Remark 5. Indeed, Theorem 6 is more general as compared to Corollary 5.1 of Berzig [9] and Corollary 3 due to Alam and Imdad [5].…”

Section: Resultsmentioning

confidence: 92%

“…Otherwise, gw n gw n+1 for all n ∈ N 0 . Suppose that d(gw n 1 −1 , gw n 1 ) ≤ d(gw n 1 , gw n 1 +1 ), for some n 1 ∈ N. On using (5) and Lemma 1, we get…”

Section: Resultsmentioning

confidence: 98%

“…which shows that contraction condition of Theorem 2 (due to Alam and Imdad [5]) is not satisfied. Further, Theorem 1 is not applicable to the present example as underlying metric space (X, d) is not complete and R is not symmetric closure of any binary relation.…”

Section: Consequencesmentioning

confidence: 94%

“…Berzig [9] established the common fixed point theorem for nonlinear contraction under symmetric closure of a arbitrary relation. Most recently, Alam and Imdad [5] proved a relationtheoretic version of Banach contraction principle employing amorphous relation which in turn unify the several well known relevant order-theoretic fixed point theorems. Moreover, for further details one can consults [1, 2, 4-6, 9-11, 14, 20-22, 25, 26].…”

Section: Introductionmentioning

confidence: 94%

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

2018

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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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Section: Introductionsupporting

confidence: 90%

“…Remark 5. Indeed, Theorem 6 is more general as compared to Corollary 5.1 of Berzig [9] and Corollary 3 due to Alam and Imdad [5].…”

Section: Resultsmentioning

confidence: 92%

“…Otherwise, gw n gw n+1 for all n ∈ N 0 . Suppose that d(gw n 1 −1 , gw n 1 ) ≤ d(gw n 1 , gw n 1 +1 ), for some n 1 ∈ N. On using (5) and Lemma 1, we get…”

Section: Resultsmentioning

confidence: 98%

Section: Consequencesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 94%

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