Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications

Berzig, Maher

doi:10.1007/s11784-013-0094-7

Cited by 28 publications

(27 citation statements)

References 5 publications

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“…Then we can construct the sequence {x n } in E such that the relation (1) holds. Owing to (9), (10), and (11) and since the continuity of T, then we have then we obtain the result.…”

Section: Corollary 3 (Ahmadullah Et Al 5 )mentioning

confidence: 65%

“…Samet and Turinici extended the concept of metric spaces and partial metric spaces to define and construct fixed point theorems in metric spaces with binary relation. After that, Berzig considered on the coincidence and common fixed point for contractive mappings, which investigated on metric spaces endowed with binary relation. Asgari and Mousavi reconsidered this space for coupled fixed point theorems while Khan et al were constructing the fixed point theorem in bimetric spaces.…”

Section: Introductionmentioning

confidence: 99%

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Some results on implicit contractive conditions in metric spaces endowed with arbitrary binary relations

Chuensupantharat

2018

Math Methods in App Sciences

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The main purpose of this study is to establish some coincidence and common fixed point theorems for 2 self‐mappings via the condition of implicit function on metric spaces provided with arbitrary binary relation. In our results, the main theorems are the generalized version of the literatures, and we also give some examples to guarantee our main results. Moreover, we show that the result to guarantee a coincidence point which are more useful than the previous results of Banach's contraction which cannot be applied. Furthermore, we include some consequences for proving the corollary, which satisfied several well‐known contractions that are obtained from the main results.

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“…Then we can construct the sequence {x n } in E such that the relation (1) holds. Owing to (9), (10), and (11) and since the continuity of T, then we have then we obtain the result.…”

Section: Corollary 3 (Ahmadullah Et Al 5 )mentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Some results on implicit contractive conditions in metric spaces endowed with arbitrary binary relations

Chuensupantharat

2018

Math Methods in App Sciences

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“…Remark 10. Corollary 7 is an improved version of Corollary 5.14 of Berzig [9] and Corollary 3.3 (corresponding to condition (19)) due to Ahmadullah et al [2]. Also under the universal relation (i.e., R = X × X), Theorems 4 and 6 unify to the following lone corollary: Corollary 8.…”

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confidence: 86%

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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“…Due to its importance and applications potential, the Banach Contraction Mapping Principle has been investigated heavily by many authors. Consequently, a number of generalizations of this celebrated principle have appeared in the literature (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). …”

Section: Introductionmentioning

confidence: 99%

“…They showed a theorem subsumes many known results in the literature. For further study about contractive mappings in a metric space endowed with binary relation, we refer the reader to [3][4][5][6][7], [13], [17] and [19].…”

Section: Introductionmentioning

confidence: 99%

Fixed point theorems in bimetric space endowed with binary relation and applications

Khan¹,

Berzig²,

Chandok

2015

Miskolc Math Notes

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Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications

Cited by 28 publications

References 5 publications

Some results on implicit contractive conditions in metric spaces endowed with arbitrary binary relations

Some results on implicit contractive conditions in metric spaces endowed with arbitrary binary relations

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

Fixed point theorems in bimetric space endowed with binary relation and applications

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