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19 pages, 285 KiB

Open AccessArticle

Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces

by Moosa Gabeleh and Maggie Aphane

Axioms 2025, 14(6), 400; https://doi.org/10.3390/axioms14060400 - 23 May 2025

Viewed by 363

Let nonempty subsets E and F of a Banach space X be given, along with a mapping

S : E \cup F \to E \cup F

defined as noncyclic when

S (E) \subseteq E

and

S (F) \subseteq F

[...] Read more.

Let nonempty subsets E and F of a Banach space X be given, along with a mapping

S : E \cup F \to E \cup F

defined as noncyclic when

S (E) \subseteq E

and

S (F) \subseteq F

. In this case, an optimal pair of fixed points is defined as a point

(p, q) \in E \times F

where p and q are fixed points of

S

that estimate the distance between E and F. This article explores an extended version of Göhde’s fixed point problem to identify optimal fixed point pairs for noncyclic relatively nonexpansive maps in strictly convex Banach spaces, while introducing new classes of noncyclic Kannan contractions, noncyclic relatively Kannan nonexpansive contractions using the proximal projection mapping defined on union of proximal pairs, and proving additional existence results with supporting examples. Full article

20 pages, 386 KiB

Open AccessArticle

Some Fixed Point Results for Novel Contractions with Applications in Fractional Differential Equations for Market Equilibrium and Economic Growth

by Min Wang, Muhammad Din and Mi Zhou

Fractal Fract. 2025, 9(5), 324; https://doi.org/10.3390/fractalfract9050324 - 19 May 2025

Viewed by 389

Abstract

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction [...] Read more.

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction types, including

χ

-contractions, Banach contractions, enriched contractions, Kannan contractions, Bianchini contractions, Zamfirescu contractions, non-expansive mappings, and

(ρ, χ)

-enriched contractions. We establish related fixed point results for the novel contractions in normed spaces endowed with the binary relations preserving key symmetric properties, ensuring consistency and applicability. The Krasnoselskij iteration method is refined to incorporate symmetric constraints, facilitating fixed point identification within these spaces. By appropriately selecting constants in the definition of enriched

(I, ρ, χ)

-contractions, employing a suitable binary relation, or control function

χ \in Θ

, our framework generalizes and extends classical fixed point theorems. Illustrative examples highlight the significance of our findings in reinforcing fixed point conditions and demonstrating their broader applicability. Additionally, this paper explores how these ideas guarantee the stability of the production–consumption markets equilibrium and the economic growth model. Full article

(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)

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17 pages, 345 KiB

Open AccessArticle

Numerical Algorithm for Coupled Fixed Points in Normed Spaces with Applications to Fractional Differential Equations and Economics

by Lifang Guo, Salha Alshaikey, Abeer Alshejari, Muhammad Din and Umar Ishtiaq

Fractal Fract. 2025, 9(1), 37; https://doi.org/10.3390/fractalfract9010037 - 14 Jan 2025

Viewed by 807

Abstract

This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This [...] Read more.

This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This class of operators expands on existing cyclic contractions, including interpolative Kannan mappings, interpolative Reich–Rus–Ćirić contractions, and other known contractions in the literature. We demonstrate the existence and uniqueness of fixed points for these operators and provide an example to illustrate our findings. Moreover, we discuss the applications of our results in solving nonlinear integral equations. Furthermore, we introduce the idea of a coupled interpolative enriched cyclic Reich–Rus–Ćirić operator and establish the existence of a strongly coupled fixed-point theorem for this contraction. Finally, we provide an application to fractional differential equations to show the validity of the main result. Full article

(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)

13 pages, 267 KiB

Open AccessArticle

New Fixed Point Theorems for Generalized Meir–Keeler Type Nonlinear Mappings with Applications to Fixed Point Theory

by Shin-Yi Huang and Wei-Shih Du

Symmetry 2024, 16(8), 1088; https://doi.org/10.3390/sym16081088 - 22 Aug 2024

Cited by 2 | Viewed by 1361

Abstract

In this paper, we investigate new fixed point theorems for generalized Meir–Keeler type nonlinear mappings satisfying the condition (DH). As applications, we obtain many new fixed point theorems which generalize and improve several results available in the corresponding literature. An example is [...] Read more.

In this paper, we investigate new fixed point theorems for generalized Meir–Keeler type nonlinear mappings satisfying the condition (DH). As applications, we obtain many new fixed point theorems which generalize and improve several results available in the corresponding literature. An example is provided to illustrate and support our main results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

24 pages, 329 KiB

Open AccessArticle

Exploring Fixed-Point Theorems in k-Fuzzy Metric Spaces: A Comprehensive Study

by Muhammad Nazam, Seemab Attique, Aftab Hussain and Hamed H. Alsulami

Axioms 2024, 13(8), 558; https://doi.org/10.3390/axioms13080558 - 15 Aug 2024

Cited by 1 | Viewed by 1724

Abstract

Recently, k -fuzzy metric spaces were introduced by connecting the degree of nearness of two points with k parameters

(t_{1}, t_{2}, t_{3}, \dots, t_{k})

and the authors presented an analogue of Grabiec’s fixed-point [...] Read more.

Recently, k -fuzzy metric spaces were introduced by connecting the degree of nearness of two points with k parameters

(t_{1}, t_{2}, t_{3}, \dots, t_{k})

and the authors presented an analogue of Grabiec’s fixed-point result in k-fuzzy metric spaces along with other necessary notions. The results presented only addressed continuous mappings. For discontinuous mappings, there is no result in k-fuzzy metric spaces. In this paper, we obtain some fixed-point results stating necessary conditions for the existence of fixed points of mappings eliminating the continuity requirement in k-fuzzy metric spaces. We illustrate the hypothesis of our findings with examples. We provide a common fixed-point theorem and fixed-point theorems for single-valued k-fuzzy Kannan type contractions. As an application, we use a fixed-point result to ensure the existence of solution of fractional differential equations. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications)

11 pages, 277 KiB

Open AccessArticle

Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces

by Safeer Hussain Khan, Abdullah Eqal Al-Mazrooei and Abdul Latif

Axioms 2023, 12(6), 549; https://doi.org/10.3390/axioms12060549 - 2 Jun 2023

Cited by 3 | Viewed by 2000

Abstract

The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched

ρ

-contractions and enriched

ρ

-Kannan mappings in modular [...] Read more.

The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched

ρ

-contractions and enriched

ρ

-Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched

ρ

-contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched

ρ

-contractions. We make a first ever attempt to prove existence results for enriched

ρ

-Kannan mappings and deduce the result for

ρ

-Kannan mappings. Note that even

ρ

-Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples. Full article

(This article belongs to the Special Issue Research on Fixed Point Theory and Application)

10 pages, 264 KiB

Open AccessArticle

On Cyclic Contractive Mappings of Kannan and Chatterjea Type in Generalized Metric Spaces

by Mohammad Al-Khaleel, Sharifa Al-Sharif and Rami AlAhmad

Mathematics 2023, 11(4), 890; https://doi.org/10.3390/math11040890 - 9 Feb 2023

Cited by 10 | Viewed by 1992

Abstract

Novel cyclic contractions of the Kannan and Chatterjea type are presented in this study. With the aid of these brand-new contractions, new results for the existence and uniqueness of fixed points in the setting of complete generalized metric space have been established. Importantly, [...] Read more.

Novel cyclic contractions of the Kannan and Chatterjea type are presented in this study. With the aid of these brand-new contractions, new results for the existence and uniqueness of fixed points in the setting of complete generalized metric space have been established. Importantly, the results are generalizations and extensions of fixed point theorems by Chatterjea and Kannan and their cyclical expansions that are found in the literature. Additionally, several of the existing results on fixed points in generalized metric space will be generalized by the results presented in this work. Interestingly, the findings have a variety of applications in engineering and sciences. Examples have been given at the end to show the reliability of the demonstrated results. Full article

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

11 pages, 273 KiB

Open AccessArticle

Some Common Fixed-Circle Results on Metric Spaces

by Nabil Mlaiki, Nihal Taş, Elif Kaplan, Suhad Subhi Aiadi and Asma Karoui Souayah

Axioms 2022, 11(9), 454; https://doi.org/10.3390/axioms11090454 - 4 Sep 2022

Cited by 5 | Viewed by 1782

Abstract

Recently, the fixed-circle problems have been studied with different approaches as an interesting and geometric generalization. In this paper, we present some solutions to an open problem

C C

: what is (are) the condition(s) to make any circle [...] Read more.

Recently, the fixed-circle problems have been studied with different approaches as an interesting and geometric generalization. In this paper, we present some solutions to an open problem

C C

: what is (are) the condition(s) to make any circle

C_{ϖ_{0}, σ}

as the common fixed circle for two (or more than two) self-mappings? To do this, we modify some known contractions which are used in fixed-point theorems such as the Hardy–Rogers-type contraction, Kannan-type contraction, etc. Full article

(This article belongs to the Special Issue Special Issue in Honor of the 60th Birthday of Professor Hong-Kun Xu)

14 pages, 356 KiB

Open AccessEditor’s ChoiceArticle

Iteration of Operators with Contractive Mutual Relations of Kannan Type

by Ram N. Mohapatra, María A. Navascués, María V. Sebastián and Saurabh Verma

Mathematics 2022, 10(15), 2632; https://doi.org/10.3390/math10152632 - 27 Jul 2022

Cited by 9 | Viewed by 1962

Abstract

Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the [...] Read more.

Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the properties of the operators satisfying this type of condition; in particular, we study the case where the space is normed, and the maps are linear. Then we generalize some theorems proposed by this author on the existence of a fixed point of one operator or a common fixed point for two operators. Our results first prove the existence of a common fixed point of a set of self-maps of any cardinal number (countable or uncountable) satisfying the conditions of Kannan type in metric spaces. The same is proved for a set of maps satisfying the mutual relations of classical contractivity. We prove in both cases the convergence of iterative schemes involving operators with mutual relations of contractivity, proposing sufficient conditions for the iteration of the operators on any element of the space to converge to the common fixed point when a different operator is taken in each step. The results obtained are applied to operators acting on real functions, coming from the fractal convolution with the null function. Full article

(This article belongs to the Special Issue Fractal and Computational Geometry)

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13 pages, 288 KiB

Open AccessArticle

Extensions of Orthogonal p-Contraction on Orthogonal Metric Spaces

by Nurcan Bilgili Gungor

Symmetry 2022, 14(4), 746; https://doi.org/10.3390/sym14040746 - 5 Apr 2022

Cited by 10 | Viewed by 1776

Abstract

Fixed-point theory and symmetry are major and vigorous tools to working nonlinear analysis and applications, specially nonlinear operator theory and applications. The subject of examining the presence and inimitableness of fixed points of self-mappings defined on orthogonal metric spaces has become very popular [...] Read more.

Fixed-point theory and symmetry are major and vigorous tools to working nonlinear analysis and applications, specially nonlinear operator theory and applications. The subject of examining the presence and inimitableness of fixed points of self-mappings defined on orthogonal metric spaces has become very popular in the latest decade. As a result, many researchers reached more relevant conclusions. In this study, the notion of

ϕ

-Kannan orthogonal p-contractive conditions in orthogonal complete metric spaces is presented. W-distance mappings do not need to satisfy the symmetry condition, that is, such mappings can be symmetrical or asymmetrical. Self-distance does not need to be zero in w-distance mappings. The intent of this study is to enhance the recent development of fixed-point theory in orthogonal metric spaces and related nonlinear problems by using w-distance. On this basis, some fixed-point results are debated. Some explanatory examples are shown that indicate the currency of the hypotheses and grade of benefit of the suggested conclusions. Lastly, sufficient cases for the presence of a solution to nonlinear Fredholm integral equations are investigated through the main results in this study. Full article

11 pages, 258 KiB

Open AccessArticle

Generalizations of Kannan and Reich Fixed Point Theorems, Using Sequentially Convergent Mappings and Subadditive Altering Distance Functions

by Alireza Pourmoslemi, Shayesteh Rezaei, Tahereh Nazari and Mehdi Salimi

Mathematics 2020, 8(9), 1432; https://doi.org/10.3390/math8091432 - 26 Aug 2020

Cited by 2 | Viewed by 2362

Abstract

In this paper, first, using interpolative Kannan type contractions, a new fixed point theorem has been proved. Then, by applying sequentially convergent mappings and using subadditive altering distance functions, we generalize contractions in complete metric spaces. Finally, we investigate some fixed point theorems [...] Read more.

In this paper, first, using interpolative Kannan type contractions, a new fixed point theorem has been proved. Then, by applying sequentially convergent mappings and using subadditive altering distance functions, we generalize contractions in complete metric spaces. Finally, we investigate some fixed point theorems which are generalizations of Kannan and Reich fixed points. Full article

9 pages, 274 KiB

Open AccessArticle

New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations

by Pradip Debnath and Hari Mohan Srivastava

Symmetry 2020, 12(7), 1090; https://doi.org/10.3390/sym12071090 - 1 Jul 2020

Cited by 38 | Viewed by 2933

Abstract

The metric function generalizes the concept of distance between two points and hence includes the symmetric property. The aim of this article is to introduce a new and proper extension of Kannan’s fixed point theorem to the case of multivalued maps using Wardowski’s [...] Read more.

The metric function generalizes the concept of distance between two points and hence includes the symmetric property. The aim of this article is to introduce a new and proper extension of Kannan’s fixed point theorem to the case of multivalued maps using Wardowski’s F-contraction. We show that our result is applicable to a class of mappings where neither the multivalued version of Kannan’s theorem nor that of Wardowski’s can be applied to determine the existence of fixed points. Application of our result to the solution of integral equations has been provided. A multivalued Reich type generalized version of the result is also established. Full article

15 pages, 323 KiB

Open AccessArticle

On Fixed-Point Results in Controlled Partial Metric Type Spaces with a Graph

by Nizar Souayah and Mehdi Mrad

Mathematics 2020, 8(1), 33; https://doi.org/10.3390/math8010033 - 26 Dec 2019

Cited by 12 | Viewed by 2744

Abstract

Recently, Mlaiki et al. introduced the notion of a controlled metric type space which is a generalization of the b-metric space. In this work, we define the controlled partial metric type space and give some fixed-point theorems for extensions of Kannan contraction [...] Read more.

Recently, Mlaiki et al. introduced the notion of a controlled metric type space which is a generalization of the b-metric space. In this work, we define the controlled partial metric type space and give some fixed-point theorems for extensions of Kannan contraction in this space with suitable conditions. Moreover, as an application, we derive a fixed-point theorem for graphic contraction on the considered metric space endowed with a graph. Full article

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15 pages, 788 KiB

Open AccessArticle

Fundamental Questions and New Counterexamples for b-Metric Spaces and Fatou Property

by Ning Lu, Fei He and Wei-Shih Du

Mathematics 2019, 7(11), 1107; https://doi.org/10.3390/math7111107 - 14 Nov 2019

Cited by 11 | Viewed by 3966

Abstract

In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction [...] Read more.

In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction with contraction constant

λ \in [\frac{1}{s}, 1)

in a complete b-metric space with

s > 1

have a unique fixed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

19 pages, 288 KiB

Open AccessArticle

Solutions to Fredholm Integral Inclusions via Generalized Fuzzy Contractions

by Hamed H Al-Sulami, Jamshaid Ahmad, Nawab Hussain and Abdul Latif

Mathematics 2019, 7(9), 808; https://doi.org/10.3390/math7090808 - 2 Sep 2019

Cited by 5 | Viewed by 1849

Abstract

The aim of this study is to investigate the existence of solutions for the following Fredholm integral inclusion [...] Read more.

The aim of this study is to investigate the existence of solutions for the following Fredholm integral inclusion

φ (t) \in [f (t) + \int_{0}^{1} K (t, s, φ (s)) ϱ s]

for

t \in [0, 1],

where

f \in C [0, 1]

is a given real-valued function and

K : [0, 1]

\times [0, 1]

\times R \to K_{c v} (R)

a given multivalued operator, where

K_{c v}

represents the family of non-empty compact and convex subsets of

R

,

φ \in C [0, 1]

is the unknown function and

ϱ

is a metric defined on

C [0, 1]

. To attain this target, we take advantage of fixed point theorems for

α

-fuzzy mappings satisfying a new class of contractive conditions in the context of complete metric spaces. We derive new fixed point results which extend and improve the well-known results of Banach, Kannan, Chatterjea, Reich, Hardy-Rogers, Berinde and Ćirić by means of this new class of contractions. We also give a significantly non-trivial example to support our new results. Full article

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