Discussion of coupled and tripled coincidence point theorems for φ-contractive mappings without the mixed g-monotone property

Karapınar, Erdal; Roldán, Antonio; Shahzad, Naseer; Sintunavarat, Wutiphol

doi:10.1186/1687-1812-2014-92

Cited by 29 publications

(40 citation statements)

References 16 publications

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“…In proving our results, we use some relation-theoretic notions such as: R-completeness, R-closedness, R-continuity, ( , R)-continuity, R-compatibility, R-connected sets etc. In this course, we also observe that our results combine the idea contained in Karapinar et al [22] as the set M (utilized by Karapinar et al [22]) being subset of X 2 is, in fact, a binary relation on X. As consequences of our newly proved results, we deduce several other established metrical coincidence point theorems.…”

Section: Introductionsupporting

confidence: 82%

Relation-theoretic metrical coincidence theorems

Alam

Imdad

2017

Filomat

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In this article, we generalize some frequently used metrical notions such as: completeness, closedness, continuity, -continuity and compatibility to relation-theoretic setting and utilize these relatively weaker notions to prove our results on the existence and uniqueness of coincidence points involving a pair of mappings defined on a metric space endowed with an arbitrary binary relation. Particularly, under universal relation our results deduce the classical coincidence point theorems of Goebel, Jungck and others. Furthermore, our results generalize, modify, unify and extend several well-known results of the existing literature.

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Section: Introductionsupporting

confidence: 82%

Relation-theoretic metrical coincidence theorems

Alam

Imdad

2017

Filomat

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“…The relationships between these scientific areas are currently being studied. In this context, one of the most attractive research subjects in fixed point theory is the investigation of the existence and uniqueness of coincidence points of various operators in the setting of metric spaces (see [4][5][6][7][8][9][10]). …”

Section: Introductionmentioning

confidence: 99%

Coincidence point theorems on metric spaces via simulation functions

Hierro

Karapınar

Roldán-López-de-Hierro

et al. 2015

Journal of Computational and Applied Mathematics

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129

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a b s t r a c tDue to its possible applications, Fixed Point Theory has become one of the most useful branches of Nonlinear Analysis. In a very recent paper, Khojasteh et al. introduced the notion of simulation function in order to express different contractivity conditions in a unified way, and they obtained some fixed point results. In this paper, we slightly modify their notion of simulation function and we investigate the existence and uniqueness of coincidence points of two nonlinear operators using this kind of control functions.

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“…Then a subset D of X is said to be (g, R)-directed if for every pair of points x, y in D, there is z in X such that (x, gz) ∈ R and (y, gz) ∈ R. Definition 2.14. [25] Let (X, d) be a metric space equipped with a binary relation R and T, g two self-mappings on X. Then T and g are said to be R-compatible if lim n→∞ d(g(T x n ), T (gx n )) = 0, whenever lim n→∞ g(x n ) = lim n→∞ T (x n ), for any sequence {x n } ⊂ X such that the sequences {T x n } and {gx n } are R-preserving.…”

Section: Relevant Relation-theoretic Notionsmentioning

confidence: 99%

Common fixed point theorems under an implicit contractive condition on metric spaces endowed with an arbitrary binary relation and an application

Ahmadullah

Imdad

Mohammad

2019

Asian-European J. Math.

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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

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Discussion of coupled and tripled coincidence point theorems for φ-contractive mappings without the mixed g-monotone property

Cited by 29 publications

References 16 publications

Relation-theoretic metrical coincidence theorems

Relation-theoretic metrical coincidence theorems

Coincidence point theorems on metric spaces via simulation functions

Common fixed point theorems under an implicit contractive condition on metric spaces endowed with an arbitrary binary relation and an application

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