A characterization of completeness of Menger PM-spaces

Pant, Neeraj; Pant, Abhijit; Nikolić, Rale M.; Ješić, Siniša N.

doi:10.1007/s11784-019-0732-9

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…In the next result, we show that Theorem 3.4 characterizes metric completeness of X. Various workers have proved fixed point theorems that characterize metric completeness [4,17,19,25,26]. In the next theorem, we show that completeness of the space is equivalent to fixed point property for a large class of mappings including both continuous and discontinuous mappings.…”

Section: Thenmentioning

confidence: 77%

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On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

Bisht

Rakočević

2021

Appl. Gen. Topol.

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<p>A Meir-Keeler type fixed point theorem for a family of mappings is proved in Menger probabilistic metric space (Menger PM-space). We establish that completeness of the space is equivalent to fixed point property for a larger class of mappings that includes continuous as well as discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ) type non-expansive mappings is established.</p>

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Section: Thenmentioning

confidence: 77%

“…Pant et al [17] (see also Bisht [2]) proved the following theorem where the Meir-Keeler [9] type operator ensures the convergence of sequence of iterates but does not ensure the existence of a fixed point.…”

Section: Introductionmentioning

confidence: 99%

On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

Bisht

Rakočević

2021

Appl. Gen. Topol.

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“…The study of contractive conditions which admit discontinuity at the fixed point and applications of such results in neural networks with discontinuous activation functions is presently a very active area of research e.g. Bisht [15] demonstrated that the problem of continuity of contractive maps at the fixed point has an affirmative answer in Menger PM spaces also. Bisht and Rakocević [3] obtained an interesting theorem which not only provides a new answer to the problem of continuity at the fixed point but, as shown below, also characterizes completeness of the metric space under the assumption of k-continuity.…”

Section: Remark 33 For a Single Valued Mapmentioning

confidence: 99%

Fixed point under set-valued relation-theoretic nonlinear contractions and application

Tomar

Joshi²,

Padaliya³

et al. 2019

Filomat

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We establish a relation theoretic version of the main result of Aydi et al. [H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler's fixed point theorem on partial metric space, Topol. Appl. (159), 2012, 3234-3242] and extend the results of Alam and Imdad [A. Alam, M. Imdad, Relation-theoretic contraction priciple, J. Fixed Point Theory Appl., 17(4), 2015, 693-702.] for a set-valued map in a partial Pompeiu-Hausdorff metric space. Numerical examples are presented to validate the theoretical finding and to demonstrate that our results generalize, improve and extend the recent results in different spaces equipped with binary relations to their set-valued variant exploiting weaker conditions. Our results provide a new answer, in the setting of relation theoretic contractions, to the open question posed by Rhoades on continuity at fixed point. We also show that, under the assumption of k-continuity, the solution to the Rhoades' problem given by Bisht and Rakocević characterizes completeness of the metric space. As an application of our main result, we solve an integral inclusion of Fredholm type.

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“…The problem of characterizing complete fuzzy metric spaces with the help of fixed point results has been recently discussed in [1,16,18,19,21,22] as a natural prolongation of the classical problem of obtaining necessary and sufficient conditions for the metric completeness via fixed point theorems (see e.g. [11,13,20,23,24,25]).…”

Section: Introductionmentioning

confidence: 99%

Contractions of Kannan-type and of Chatterjea-type on fuzzy quasi-metric spaces

Romaguera

2022

Results in Nonlinear Analysis

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We characterize the completeness of fuzzy quasi-metric spaces by means of a fixed point theorem of Kannan-type. Thus, we extend the classical characterization of metric completeness due to Subrahmanyam as well as recent results in the literature on the characterization of quasi-metric completeness and fuzzy metric completeness, respectively. We also introduce and discuss contractions of Chatterjea-type in this asymmetric context.

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A characterization of completeness of Menger PM-spaces

Cited by 5 publications

References 15 publications

On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

Fixed point under set-valued relation-theoretic nonlinear contractions and application

Contractions of Kannan-type and of Chatterjea-type on fuzzy quasi-metric spaces

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