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

Ćmiel, Hanna; Kuhlmann, Franz-Viktor; Kuhlmann, Katarzyna

doi:10.48550/arxiv.1905.09930

Cited by 1 publication

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“…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: 63%

“…A discussion of its potential range of applications follows. We start with a result which generalizes Theorem 5.6. from [7] which characterizes complete 1-continuous (G)-semimetric spaces.…”

Section: A New Fixed Point Theoremmentioning

confidence: 99%

“…The next result presented is the Theorem of Ćmiel and Kuhlemanns' [7] which we will use to obtain a fixed point theorem for semimetric spaces. Before proceeding, let us recall two useful definitions.…”

Section: A New Fixed Point Theoremmentioning

confidence: 99%

“…Theorem 3.2. [7] Let (X, B) be a spherically complete ball space and let f : X → X. Then (i) If any f -closed A ⊂ X contains f -contracting B ∈ B as its subset (i.e., B ⊆ A, then for any…”

Section: A New 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: Introductionmentioning

confidence: 63%

Section: A New Fixed Point Theoremmentioning

confidence: 99%

Section: A New Fixed Point Theoremmentioning

confidence: 99%

Section: A New Fixed Point Theoremmentioning

confidence: 99%

See 2 more Smart Citations