Nonlinear contractions in metric spaces under locally T-transitive binary relations

Alam, Aftab; Imdad, Mohammad

doi:10.24193/fpt-ro.2018.1.02

Cited by 60 publications

(45 citation statements)

References 20 publications

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“…Most recently, Alam and Imdad [3,4] established a relation-theoretic version of Banach contraction principle employing amorphous relation which in turn generalizes several well known relevant order-theoretic fixed point theorems. For more details, one can consult ( [1,2,3,4,7,10,11,16,17,18,19,20,21,22,23] and references cited therein).…”

Section: Introductionmentioning

confidence: 99%

Relation-theoretic metrical fixed point theorems under nonlinear contractions

Ahmadullah¹,

Imdad²,

Gubran³

2019

FPT-RO

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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.

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Section: Introductionmentioning

confidence: 99%

Relation-theoretic metrical fixed point theorems under nonlinear contractions

Ahmadullah¹,

Imdad²,

Gubran³

2019

FPT-RO

Self Cite

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“…Definition 10. T-Transitive relation: [26] Given a map T : X → X, we say a relation R on X to be T -transitive if (Tu, Tw) ∈ R for all u, v, w ∈ X such that (Tu, Tv), (Tv, Tw) ∈ R.…”

Section: Definitionmentioning

confidence: 99%

“…Locally T-Transitive relation: [26] Let X be a nonempty set with a self-mapping T on it. A binary relation R on X is said to be locally T-transitive if for every (effectively) R-preserving sequence {u n } ⊆ T(X) (with range E := {u n : n ∈ N}), the binary relation R| E is transitive.…”

Section: Definition 12mentioning

confidence: 99%

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Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings

2019

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In recent times there have been two prominent trends in metric fixed point theory. One is the use of weak contractive inequalities and the other is the use of binary relations. Combining the two trends, in this paper we establish a relation-theoretic fixed point result for a mapping which is defined on a metric space with an arbitrary binary relation and satisfies a weak contractive inequality for any pair of points whenever the pair of points is related by a given relation. The uniqueness is obtained by assuming some extra conditions. The metric space is assumed to be R-complete. We use R-continuity of functions. The property of local T-transitivity of the relation R is used in the main theorem. There is an illustrative example. An existing fixed point result is generalized through the present work. We use a method in the proof of our main theorem which is a blending of relation-theoretic and analytic approaches.

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“…More recently, there have been some attempts to generalize the fixed point results for metric spaces endowed with binary relation. For more details, we refer the reader to [11][12][13][14][15][16][17][18][19]. In this paper, some coincidence point results in complete metric spaces endowed with transitive binary relations are first highlighted.…”

Section: Introductionmentioning

confidence: 99%