A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces

Karapınar, Erdal; Khojasteh, Farshid; Mitrović, Zoran D.

doi:10.3390/math7040308

Cited by 27 publications

(21 citation statements)

References 12 publications

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“…It is evident that the notions of b-metric and standard metric coincide in case of s = 1. For more details on b-metric spaces, see, e.g., [8][9][10][11] and corresponding references therein.…”

Section: Introductionmentioning

confidence: 99%

Admissible Hybrid Z-Contractions in b-Metric Spaces

Chifu

Karapınar

2019

Axioms

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In this manuscript, we introduce a new notion, admissible hybrid Z-contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the context of complete b-metric space. The given result not only unifies the several existing results in the literature, but also extends and improves them. We express some consequences of our main theorem by using variant examples of simulation functions. As applications, the well-posedness and the Ulam-Hyers stability of the fixed point problem are also studied.Axioms 2020, 9, 2 2 of 17As it was mentioned before, the theory has been advanced by reporting several new fixed point results that are obtained by changing the conditions on the given mappings. As a result, in the literature, there are so many different types of metric fixed point results that cause a disturbance, conflict, and disorder. For overcoming this problem, it needs to consider new theorems that cover several different results. One of the successful results in directions was given in [4] where admissible mappings were introduced to combine different structures. Other interesting results were given in [5] in which the notion of the simulation function was defined to combine many distinct contractions. The notion of the hybrid contractions can also be considered as a result of this trend: in two recent papers [6,7], the authors introduce a new type of contraction, namely admissible hybrid contraction, in order to unify several linear, nonlinear and interpolative contractions in the set-up of a complete metric and b-metric spaces.One of the main aims of this paper is to unify the several existing results in the literature by combining the interesting notions: admissible mappings, simulation functions, and hybrid contractions. Besides unifying the results, we express our results in the most generalized form: in the setting of a complete b-metric space. Next, we shall consider applications for our obtained results. In particular, we shall consider the well-posedness and the Ulam-Hyers stability of the fixed point problem. We shall give some consequences and we shall indicate how one can get more consequences from the main theorem of the paper. In the next section, we shall give some basic notions and results to provide a self-contained, easily readable paper. PreliminariesIn this section, we shall collect the necessary notations, notions, and results for the sake of the completeness of the paper. We first express the definition of the b-metric, as follows.Definition 1 (See, e.g., Bourbaki [1], Bakhtin [2], and Czerwik [3]). Let X be a nonempty set and let s ≥ 1 be a given real number. A functional d : X × X → [0, ∞) is said to be a b-metric with constant s, if

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Section: Introductionmentioning

confidence: 99%

Admissible Hybrid Z-Contractions in b-Metric Spaces

Chifu

Karapınar

2019

Axioms

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“…In this short note, we aimed to merge these two significant fixed-point theorems and extend them. This note can be thought as a continuation of Reference [10].…”

Section: Introductionmentioning

confidence: 82%

“…Since then, this result has been extended in several aspects (see, e.g., References [2][3][4][5][6][7][8][9][10] and the references therein).…”

Section: Introductionmentioning

confidence: 96%

Revisiting Ćirić-Type Contraction with Caristi’s Approach

2019

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“…After proving that these contractions possess fixed points, we express illustrative examples. We note that this work can be thought as a continuation of [6].…”

Section: Introductionmentioning

confidence: 98%

On Bilateral Contractions

et al. 2019

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In this manuscript, we introduce a new type of contraction, bilateral contraction which merges two significant approaches in the fixed point theory: Caristi type contractions and Jaggi type contractions. The principal aim of the main result is to enrich the literature by combining the techniques of the mentioned two celebrated results that belong to Jaggi and Caristi. We consider an example to indicate the validity and genuine nature of the main result.

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A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces

Abstract: In this paper, we revisit the renowned fixed point theorems belongs to Caristi and Banach. We propose a new fixed point theorem which is inspired from both Caristi and Banach. We also consider an example to illustrate our result.

Cited by 27 publications

References 12 publications

Admissible Hybrid Z-Contractions in b-Metric Spaces

Admissible Hybrid Z-Contractions in b-Metric Spaces

Revisiting Ćirić-Type Contraction with Caristi’s Approach

On Bilateral Contractions

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