Some fixed-circle theorems and discontinuity at fixed circle

Özgür, Nihal Yılmaz; Taş, Ni̇hal

doi:10.1063/1.5020497

Cited by 49 publications

(36 citation statements)

References 10 publications

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“…Fixed point theorems for contractive mappings which admit discontinuity at the fixed point and their applications to neural networks with discontinuous activation functions have emerged as a very active area of research (e.g. Bisht and Rakocevic [4,5], Ozgur and Tas [27,28], Rashid et al [33], Tas and Ozgur [38], Tas et al [39], Zheng and Wang [44]). The question of the existence of contractive mappings which admit discontinuity at the fixed point arose with the publication of two papers by Kannan [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…Recently some more solutions to the problem of continuity at fixed point and applications of such results to neural networks with discontinuous activation functions have been reported (e.g. Bisht and Pant [2,3], Bisht and Rakocevic [4,5], Ozgur and Tas [27,28], Rashid et al [33], Tas and Ozgur [38], Tas et al [39], Zheng and Wang [44]). All the known solutions of the Rhoades' problem (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…All the known solutions of the Rhoades' problem (e.g. [2,3], [4,5], [27,28], [30], [31], [44]) employ condition (ii) or some generalized form of (ii) together with a condition similar to condition (iii) or some weaker form of continuity, e.g., orbital continuity, continuity of f k or k-continuity for some k > 1.…”

Section: Introductionmentioning

confidence: 99%

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Caristi type and meir-keeler type fixed point theorems

Pant

Joshi

2019

Filomat

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We generalize the Caristi fixed point theorem by employing a weaker form of continuity and show that contractive type mappings that satisfy the conditions of our theorem provide new solutions to the Rhoades' problem on continuity at fixed point. We also obtain a Meir-Keeler type fixed point theorem which gives a new solution to the Rhoades' problem on the existence of contractive mappings that admit discontinuity at the fixed point. We prove that our theorems characterize completeness of the metric space as well as Cantor's intersection property.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Caristi type and meir-keeler type fixed point theorems

Pant

Joshi

2019

Filomat

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“…Various fixed-circle theorems have been obtained using different approaches on metric and some generalized metric spaces (see [5][6][7][8][9] for more details). For example, in [5], fixed-circle results were proved using the Caristi's inequality on metric spaces.…”

Section: Introductionmentioning

confidence: 99%

“…For example, in [5], fixed-circle results were proved using the Caristi's inequality on metric spaces. In [8], it was given a fixed-circle theorem for a self-mapping that maps a given circle onto itself. In [9], it was extended known fixed-circle results in many directions and introduced a new notion called as an F c -contraction.…”

Section: Introductionmentioning

confidence: 99%

New Types of Fc-Contractions and the Fixed-Circle Problem

2018

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In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of F c -contractions such as Ćirić type, Hardy-Rogers type, Reich type and Chatterjea type. Two illustrative examples are presented to show the effectiveness of our results. Also, it is given an application of a Ćirić type F c -contraction to discontinuous self-mappings which have fixed circles.

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Fixed ellipse theorems in metric spaces and an application of discontinuous activation function in neural networks

Ermiş,

Zaim Erçınar,

Gelişgen

2023

Math Methods in App Sciences

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In this article, we establish new fixed ellipse theorems that will provide a geometrical perspective to the classical metric fixed point theory. Also, we provide novel fixed ellipse consequences as the expanding of some existing and known fixed circle theorems in literature to ellipse case, and give non‐trivial illuminating examples, which show the impressiveness of our results. On the other hand, using the given fixed ellipse, we give some solutions for the open question which is posed by Rhoades with respect to the presence of contractive mappings that permissive discontinuity at the fixed point. Furthermore, we give an application to usage respecting activation functions handled at neural networks via the discontinuity and the obtained results of fixed ellipse.

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Some fixed-circle theorems and discontinuity at fixed circle

Cited by 49 publications

References 10 publications

Caristi type and meir-keeler type fixed point theorems

Caristi type and meir-keeler type fixed point theorems

New Types of Fc-Contractions and the Fixed-Circle Problem

Fixed ellipse theorems in metric spaces and an application of discontinuous activation function in neural networks

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