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29 pages, 416 KiB

Open AccessArticle

New Studies for Dynamic Programming and Fractional Differential Equations in Partial Modular b-Metric Spaces

by Abdurrahman Büyükkaya, Dilek Kesik, Ülkü Yeşil and Mahpeyker Öztürk

Fractal Fract. 2024, 8(12), 724; https://doi.org/10.3390/fractalfract8120724 - 9 Dec 2024

Cited by 1 | Viewed by 805

This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation

R

, proposing a novel definition for a generalized

ℷ_{C}

-type [...] Read more.

This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation

R

, proposing a novel definition for a generalized

ℷ_{C}

-type Suzuki

R

-contraction specific to these spaces. By doing so, we pave the way for a range of relation-theoretical common fixed-point theorems, highlighting the versatility of our approach. To illustrate the practical relevance of our findings, we present a compelling example. Ultimately, this work aims to enrich the existing academic discourse and stimulate further research and practical applications within the field. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)

27 pages, 333 KiB

Open AccessArticle

Fixed-Point Results for Multi-Valued Mappings in Topological Vector Space-Valued Cone Metric Spaces with Applications

by Hala Alzumi and Jamshaid Ahmad

Axioms 2024, 13(12), 841; https://doi.org/10.3390/axioms13120841 - 29 Nov 2024

Viewed by 660

Abstract

The objective of this research article is to introduce Kikkawa and Suzuki-type contractions in the setting of topological vector space-valued cone metric space with a solid cone and establish some new fixed point results for multi-valued mappings. The problem of finding fixed points [...] Read more.

The objective of this research article is to introduce Kikkawa and Suzuki-type contractions in the setting of topological vector space-valued cone metric space with a solid cone and establish some new fixed point results for multi-valued mappings. The problem of finding fixed points for multi-valued mappings satisfying locally contractive conditions on a closed ball is also addressed. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, we present an example. As a demonstration of the applicability of our principal theorem, we prove a result in homotopy theory. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

12 pages, 275 KiB

Open AccessArticle

Suzuki–Ćirić-Type Nonlinear Contractions Employing a Locally ζ-Transitive Binary Relation with Applications to Boundary Value Problems

by Doaa Filali and Faizan Ahmad Khan

Mathematics 2024, 12(13), 2058; https://doi.org/10.3390/math12132058 - 30 Jun 2024

Viewed by 1084

Abstract

This article is devoted to enhancing a class of generalized Suzuki-type nonlinear contractions following Pant to a class of Suzuki–Ćirić-type nonlinear contractions via comparison functions via a locally

ζ

-transitive relation and implemented the same to ascertain certain fixed-point results. The outcomes presented [...] Read more.

This article is devoted to enhancing a class of generalized Suzuki-type nonlinear contractions following Pant to a class of Suzuki–Ćirić-type nonlinear contractions via comparison functions via a locally

ζ

-transitive relation and implemented the same to ascertain certain fixed-point results. The outcomes presented herewith unify and generalize a few existing findings. An illustrative examples is offered to explain our findings. Our outcomes assist us in figuring out the unique solution to a boundary value problem. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)

14 pages, 297 KiB

Open AccessArticle

A Unified Approach and Related Fixed-Point Theorems for Suzuki Contractions

by Kastriot Zoto, Vesna Šešum-Čavić, Mirjana Pantović, Vesna Todorčević, Marsela Zoto and Stojan Radenović

Symmetry 2024, 16(6), 739; https://doi.org/10.3390/sym16060739 - 13 Jun 2024

Cited by 1 | Viewed by 962

Abstract

This paper aims to give an extended class of contractive mappings combining types of Suzuki contractions α-admissible mapping and Wardowski F-contractions in b-metric-like spaces. Our results cover and generalize many of the recent advanced results on the existence and uniqueness [...] Read more.

This paper aims to give an extended class of contractive mappings combining types of Suzuki contractions α-admissible mapping and Wardowski F-contractions in b-metric-like spaces. Our results cover and generalize many of the recent advanced results on the existence and uniqueness of fixed points and fulfill the Suzuki-type nonlinear hybrid contractions on various generalized metrics. Full article

(This article belongs to the Special Issue Symmetry in Metric Spaces and Topology)

9 pages, 281 KiB

Open AccessArticle

Pre-Symmetric w-Cone Distances and Characterization of TVS-Cone Metric Completeness

by Seyedeh Sara Karimizad and Ghasem Soleimani Rad

Mathematics 2024, 12(12), 1833; https://doi.org/10.3390/math12121833 - 12 Jun 2024

Cited by 2 | Viewed by 1051

Abstract

Motivated by two definitions of distance, “pre-symmetric w-distance” and “w-cone distance”, we define the concept of a pre-symmetric w-cone distance in a TVS-CMS and introduce its properties and examples. Also, we discuss the TVS-cone version of the recent results [...] Read more.

Motivated by two definitions of distance, “pre-symmetric w-distance” and “w-cone distance”, we define the concept of a pre-symmetric w-cone distance in a TVS-CMS and introduce its properties and examples. Also, we discuss the TVS-cone version of the recent results obtained by Romaguera and Tirado. Meanwhile, using Minkowski functionals, we show the equivalency between some consequences concerning a pre-symmetric w-distance in a usual metric space and a pre-symmetric w-cone distance in a TVS-CMS. Then, some types of various w-cone-contractions and the relations among them are investigated. Finally, as an application, a characterization of the completeness of TVS-cone metric regarding pre-symmetric concept is performed, which differentiates our results from former characterizations. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

16 pages, 306 KiB

Open AccessFeature PaperArticle

On Protected Quasi-Metrics

by Salvador Romaguera

Axioms 2024, 13(3), 158; https://doi.org/10.3390/axioms13030158 - 28 Feb 2024

Cited by 1 | Viewed by 1865

Abstract

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, [...] Read more.

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations. Full article

(This article belongs to the Section Geometry and Topology)

11 pages, 277 KiB

Open AccessArticle

Presymmetric w-Distances on Metric Spaces

by Salvador Romaguera and Pedro Tirado

Mathematics 2024, 12(2), 305; https://doi.org/10.3390/math12020305 - 17 Jan 2024

Cited by 4 | Viewed by 1151

Abstract

In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show [...] Read more.

In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show by means of a simple example that the problem of achieving, in an obvious way, its full extension to the framework of w-distances does not have an emphatic response. Motivated by this fact, we introduce the concept of presymmetric w-distance on metric spaces. We also give some properties and examples of this new structure and show that it provides a reasonable setting to obtain a real and hardly forced w-distance generalization of Suzuki’s theorem. This is realized in our main result, which consists of a fixed point theorem that involves presymmetric w-distances and certain contractions of Suzuki-type. We also discuss the relationship between our main result and the well-known w-distance full generalization of the Banach contraction principle, due to Suzuki and Takahashi. Connected to this approach, we prove another fixed point result that compares with our main result through some examples. Finally, we state a characterization of metric completeness by using our fixed point results. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

13 pages, 298 KiB

Open AccessArticle

Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces

by Basit Ali, Hammad Ali, Talat Nazir and Zakaria Ali

Mathematics 2023, 11(21), 4445; https://doi.org/10.3390/math11214445 - 26 Oct 2023

Cited by 2 | Viewed by 1456

Abstract

In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces, we introduce Suzuki-type contractions of quasi-metric spaces and prove some fixed point results. Further, we suggest a correction in the definition of another class of quasi-metrics known as

Δ

[...] Read more.

In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces, we introduce Suzuki-type contractions of quasi-metric spaces and prove some fixed point results. Further, we suggest a correction in the definition of another class of quasi-metrics known as

Δ

-symmetric quasi-metrics satisfying a weighted symmetry property. We discuss equivalence of various types of completeness of

Δ

-symmetric quasi-metric spaces. At the end, we consider the existence of fixed points of generalized Suzuki-type contractions of

Δ

-symmetric quasi-metric spaces. Some examples have been furnished to make sure that generalizations we obtain are the proper ones. Full article

(This article belongs to the Special Issue New Advances in Mathematical Analysis and Functional Analysis)

29 pages, 358 KiB

Open AccessArticle

Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

by Naeem Saleem, Maneesha Tur Raazzia, Nawab Hussain and Asim Asiri

Symmetry 2023, 15(8), 1572; https://doi.org/10.3390/sym15081572 - 11 Aug 2023

Cited by 6 | Viewed by 1015

Abstract

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context [...] Read more.

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an

α - θ -

Geraghty–Pata–Suzuki-type proximal contraction; an

α - θ -

generalized Geraghty–Pata–Suzuki-type proximal contraction; and an

α - θ -

modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

21 pages, 619 KiB

Open AccessArticle

Stability Results and Reckoning Fixed Point Approaches by a Faster Iterative Method with an Application

by Hasanen A. Hammad and Doha A. Kattan

Axioms 2023, 12(7), 715; https://doi.org/10.3390/axioms12070715 - 23 Jul 2023

Viewed by 1396

Abstract

In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive [...] Read more.

In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Our method opens the door to many expansions in the problems of monotone variational inequalities, image restoration, convex optimization, and split convex feasibility. Moreover, some experimental examples were conducted to gauge the usefulness and efficiency of the technique compared with the iterative methods in the literature. Finally, the proposed approach is applied to solve the nonlinear Volterra integral equation with a delay. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)

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22 pages, 452 KiB

Open AccessArticle

Fixed-Point Estimation by Iterative Strategies and Stability Analysis with Applications

by Hasanen A. Hammad and Doha A. Kattan

Symmetry 2023, 15(7), 1400; https://doi.org/10.3390/sym15071400 - 11 Jul 2023

Cited by 1 | Viewed by 2646

Abstract

In this study, we developed a new faster iterative scheme for approximate fixed points. This technique was applied to discuss some convergence and stability results for almost contraction mapping in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex [...] Read more.

In this study, we developed a new faster iterative scheme for approximate fixed points. This technique was applied to discuss some convergence and stability results for almost contraction mapping in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Moreover, some numerical experiments were investigated to illustrate the behavior and efficacy of our iterative scheme. The proposed method converges faster than symmetrical iterations of the S algorithm, Thakur algorithm and

K^{*}

algorithm. Eventually, as an application, the nonlinear Volterra integral equation with delay was solved using the suggested method. Full article

(This article belongs to the Special Issue Symmetry in Fixed Point Approaches and Nonlinear Functional Mathematical Equations)

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16 pages, 811 KiB

Open AccessArticle

Fractional Differential Boundary Value Equation Utilizing the Convex Interpolation for Symmetry of Variables

by Aftab Hussain

Symmetry 2023, 15(6), 1189; https://doi.org/10.3390/sym15061189 - 2 Jun 2023

Cited by 2 | Viewed by 1392

Abstract

In this paper, we introduce a novel form of interpolative convex contraction and develop some new theorems by utilizing the progressive method of interpolative convex contractions. We also obtain some fixed point results for a Suzuki convex contraction in orbitally S-complete F [...] Read more.

In this paper, we introduce a novel form of interpolative convex contraction and develop some new theorems by utilizing the progressive method of interpolative convex contractions. We also obtain some fixed point results for a Suzuki convex contraction in orbitally S-complete F-metric spaces. The second purpose of this research is to evaluate the effectiveness of the fixed point approach in solving fractional differential equations with boundary conditions. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

15 pages, 304 KiB

Open AccessArticle

On General Class of Z-Contractions with Applications to Spring Mass Problem

by Monairah Alansari and Mohammed Shehu Shagari

Symmetry 2023, 15(2), 302; https://doi.org/10.3390/sym15020302 - 21 Jan 2023

Viewed by 1309

Abstract

One of the latest techniques in metric fixed point theory is the interpolation approach. This notion has so far been examined using standard functional equations. A hybrid form of this concept is yet to be uncovered by observing the available literature. With this [...] Read more.

One of the latest techniques in metric fixed point theory is the interpolation approach. This notion has so far been examined using standard functional equations. A hybrid form of this concept is yet to be uncovered by observing the available literature. With this background information, and based on the symmetry and rectangular properties of generalized metric spaces, this paper introduces a novel and unified hybrid concept under the name interpolative Y-Hardy–Rogers–Suzuki-type Z-contraction and establishes sufficient conditions for the existence of fixed points for such contractions. As an application, one of the obtained results was employed to examine new criteria for the existence of a solution to a boundary valued problem arising in the oscillation of a spring. The ideas proposed herein advance some recently announced important results in the corresponding literature. A comparative example was constructed to justify the abstractions and pre-eminence of our obtained results. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)

21 pages, 362 KiB

Open AccessArticle

Some Characterizations of Complete Hausdorff KM-Fuzzy Quasi-Metric Spaces

by Salvador Romaguera

Mathematics 2023, 11(2), 381; https://doi.org/10.3390/math11020381 - 11 Jan 2023

Cited by 2 | Viewed by 1580

Abstract

Gregori and Romaguera introduced, in 2004, the notion of a KM-fuzzy quasi-metric space as a natural asymmetric generalization of the concept of fuzzy metric space in the sense of Kramosil and Michalek. Ever since, various authors have discussed several aspects of such spaces, [...] Read more.

Gregori and Romaguera introduced, in 2004, the notion of a KM-fuzzy quasi-metric space as a natural asymmetric generalization of the concept of fuzzy metric space in the sense of Kramosil and Michalek. Ever since, various authors have discussed several aspects of such spaces, including their topological and (quasi-)metric properties as well as their connections with domain theory and their relationship with other fuzzy structures. In particular, the development of the fixed point theory for these spaces and other related ones, such as fuzzy partial metric spaces, has received remarkable attention in the last 15 years. Continuing this line of research, we here establish general fixed point theorems for left and right complete Hausdorff KM-fuzzy quasi-metric spaces, which are applied to deduce characterizations of these distinguished kinds of fuzzy quasi-metric completeness. Our approach, which mixes conditions of Suzuki-type with contractions of

α - ϕ

-type in the well-known proposal of Samet et al., allows us to extend and improve some recent theorems on complete fuzzy metric spaces. The obtained results are accompanied by illustrative and clarifying examples. Full article

(This article belongs to the Special Issue Aggregation Functions and Indistinguishability Operators: Applications)

13 pages, 324 KiB

Open AccessArticle

Nonlinear Relation-Theoretic Suzuki-Generalized Ćirić-Type Contractions and Application to Fractal Spaces

by Asik Hossain, Mohammad Arif, Salvatore Sessa and Qamrul Haque Khan

Fractal Fract. 2022, 6(12), 711; https://doi.org/10.3390/fractalfract6120711 - 30 Nov 2022

Cited by 5 | Viewed by 1391

Abstract

In this article, we introduce the idea of relation-theoretic Suzuki-generalized nonlinear contractions and utilized the same to prove some fixed point results in an ℜ-complete partial metric space. Our newly established results are sharpened versions of earlier existing results in the literature. [...] Read more.

In this article, we introduce the idea of relation-theoretic Suzuki-generalized nonlinear contractions and utilized the same to prove some fixed point results in an ℜ-complete partial metric space. Our newly established results are sharpened versions of earlier existing results in the literature. Indeed, we give an application to construct multivalued fractals using a newly introduced contraction in the iterated function space. Full article

(This article belongs to the Section General Mathematics, Analysis)

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