Caristi–Kirk and Oettli–Théra ball spaces and applications

Błaszkiewicz, Piotr; Ćmiel, Hanna; Linzi, Alessandro; Szewczyk, Piotr

doi:10.1007/s11784-019-0729-4

Cited by 3 publications

(12 citation statements)

References 13 publications

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“…Now, we return from this slight detour to give a quick series of generalizations of Karisti-Kirk theorem in various settings. Reasonings in proofs are similar to the ones presented in [4]. However, obtained results are more general.…”

Section: Generalizations Of Caristi-kirk Fixed Point Theoremsupporting

confidence: 78%

“…It was used to prove some fixed point theorems in metric, ultrametric and topological spaces. This idea was continued in [4,7,11]. In our paper we will generalize some results obtained in mentioned articles but for semimetric spaces.…”

Section: Introductionmentioning

confidence: 64%

“…We start by introducing some additional, necessary notions and then move to the new results. The proofs in this section are based on the methods used in [11] and [4].…”

Section: Generalizations Of Caristi-kirk Fixed Point Theoremmentioning

confidence: 99%

“…Of course, for K = 1 we obtain a metric space. Similarly as in [11] or [4] we will define the Caristi-Kirk and Oettli-Théra ball spaces.…”

Section: Generalizations Of Caristi-kirk Fixed Point Theoremmentioning

confidence: 99%

“…Theorem 4.1. [4] Let (X, B) be a spherically complete, strongly contractive ball space. Then for any…”

Section: Generalizations Of Caristi-kirk Fixed Point Theoremmentioning

confidence: 99%

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Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

Nowakowski¹,

Turoboś²

2021

Preprint

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In the paper we apply some of the results from the theory of ball spaces in semimetric spaces. This allowed us to obtain some fixed point theorems which we believe to be unknown to this day. We also show the limitations of the ball space approach to this topic. As a byproduct, we obtain the equivalence of some different notions of completness in semimetric spaces where the distance function is 1-continuous. In the second part of the article, we generalize Caristi-Kirk results for for b-metric spaces. Additionally, we obtain characterization of semicompleteness for 1-continuous b-metric space via fixed point theorem analogous to the result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion.

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Section: Generalizations Of Caristi-kirk Fixed Point Theoremsupporting

confidence: 78%

Section: Introductionmentioning

confidence: 64%

“…We start by introducing some additional, necessary notions and then move to the new results. The proofs in this section are based on the methods used in [11] and [4].…”

Section: Generalizations Of Caristi-kirk Fixed Point Theoremmentioning

confidence: 99%

“…Of course, for K = 1 we obtain a metric space. Similarly as in [11] or [4] we will define the Caristi-Kirk and Oettli-Théra ball spaces.…”

Section: Generalizations Of Caristi-kirk Fixed Point Theoremmentioning

confidence: 99%

“…Theorem 4.1. [4] Let (X, B) be a spherically complete, strongly contractive ball space. Then for any…”

Section: Generalizations Of Caristi-kirk Fixed Point Theoremmentioning

confidence: 99%

See 3 more Smart Citations

Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

Nowakowski¹,

Turoboś²

2021

Preprint

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Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

Nowakowski

Turoboś

2022

J. Fixed Point Theory Appl.

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In the paper, we apply some of the results from the theory of ball spaces in semimetric setting. This allows us to obtain fixed point theorems which we believe to be unknown to this day. As a byproduct, we obtain the equivalence of some different notions of completeness in semimetric spaces where the distance function is 1-continuous. In the second part of the article, we generalize the Caristi–Kirk results for b-metric spaces. Additionally, we obtain a characterization of semicompleteness for 1-continuous b-metric spaces via a fixed point theorem analogous to a result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion.

show abstract

A generic approach to measuring the strength of completeness/compactness of various types of spaces and ordered structures

Ćmiel

Kuhlmann

2021

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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We study spherical completeness of ball spaces and its stability under expansions. We give some criteria for ball spaces that guarantee that spherical completeness is preserved when the ball space is closed under unions of chains. This applies in particular to the spaces of closed ultrametric balls in ultrametric spaces with linearly ordered value sets, or more generally, with countable narrow value sets. Finally, we show that in general, chain union closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness.

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Caristi–Kirk and Oettli–Théra ball spaces and applications

Cited by 3 publications

References 13 publications

Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

A generic approach to measuring the strength of completeness/compactness of various types of spaces and ordered structures

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