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36 pages, 3130 KB

Open AccessArticle

Rational (a, p)−Quasicontractions and Fractional Delayed Nonlocal Caputo Problems via Hammerstein Operators

by Mahpeyker Öztürk

Fractal Fract. 2026, 10(3), 148; https://doi.org/10.3390/fractalfract10030148 - 26 Feb 2026

Viewed by 389

We introduce and study a new class of nonlinear operators on metric spaces, called rational

(a, p) -

quasicontractions. Within this framework, we establish Greguš-type fixed-point theorems for closed, convex subsets of Banach spaces. The results establish the existence [...] Read more.

We introduce and study a new class of nonlinear operators on metric spaces, called rational

(a, p) -

quasicontractions. Within this framework, we establish Greguš-type fixed-point theorems for closed, convex subsets of Banach spaces. The results establish the existence and uniqueness of fixed points, as well as the convergence of the Picard iteration for every initial guess. We show that rational

(a, p) -

quasicontractions strictly extend several classical contractive classes, including Hardy-Rogers, Kannan, Chatterjea, and rational contractions, and we provide explicit examples exhibiting the properness of these inclusions. As an application, we consider a nonlocal boundary value problem for a Caputo fractional differential equation of order

α \in (1, 2)

with distributed delay and mixed nonlocal boundary conditions. By rewriting the problem as a Hammerstein-Volterra integral equation on a cone, and imposing natural growth and rational Lipschitz conditions on the delayed nonlinearity, we show that the associated Hammerstein operator is a rational

(a, p) -

quasicontraction. This yields the existence, uniqueness, and global attractivity of a positive solution. Two model fractional nonlinearities with delayed feedback are discussed in detail, along with a numerical scheme that illustrates the predicted geometric convergence of the discrete Picard iteration in the Caputo fractional setting. Full article

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

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10 pages, 244 KB

Open AccessArticle

On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results

by Zouaoui Bekri, Nicola Fabiano, Mohammed Ahmed Alomair and Abdulaziz Khalid Alsharidi

Mathematics 2025, 13(24), 4011; https://doi.org/10.3390/math13244011 - 16 Dec 2025

Viewed by 466

Abstract

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter

p \geq 1

, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated [...] Read more.

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter

p \geq 1

, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated mappings

ħ^{p}

. We establish fixed-point theorems, ensuring both existence and uniqueness of fixed points for continuous self-maps on complete metric spaces that satisfy these p-contractive conditions. The proofs are constructed via geometric estimates on the iterates and by transferring the fixed point from the p-th iterate

ħ^{p}

to the original mapping ħ. Our results unify and broaden several well-known fixed-point theorems reported in previous studies, including those of Banach, Hardy–Rogers, and Zamfirescu as special cases. Full article

(This article belongs to the Section C: Mathematical Analysis)

22 pages, 350 KB

Open AccessArticle

Generalized Common Best Proximity Point Results in Fuzzy Metric Spaces with Application

by Umar Ishtiaq, Fahad Jahangeer, Doha A. Kattan and Ioannis K. Argyros

Symmetry 2023, 15(8), 1501; https://doi.org/10.3390/sym15081501 - 28 Jul 2023

Cited by 5 | Viewed by 2133

Abstract

The symmetry of fuzzy metric spaces has benefits for flexibility, ambiguity tolerance, resilience, compatibility, and applicability. They provide a more comprehensive description of similarity and offer a solid framework for working with ambiguous and imprecise data. We give fuzzy versions of some celebrated [...] Read more.

The symmetry of fuzzy metric spaces has benefits for flexibility, ambiguity tolerance, resilience, compatibility, and applicability. They provide a more comprehensive description of similarity and offer a solid framework for working with ambiguous and imprecise data. We give fuzzy versions of some celebrated iterative mappings. Further, we provide different concrete conditions on the real valued functions

J, S : (0, 1] \to R

for the existence of the best proximity point of generalized fuzzy

(J, S)

-iterative mappings in the setting of fuzzy metric space. Furthermore, we utilize fuzzy versions of

(J, S)

-proximal contraction,

(J, S)

-interpolative Reich–Rus–Ciric-type proximal contractions,

(J, S)

-Kannan type proximal contraction and

(J, S)

-interpolative Hardy Roger’s type proximal contraction to examine the common best proximity points in fuzzy metric space. Also, we establish several non-trivial examples and an application to support our results. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications, 3rd Edition)

15 pages, 304 KB

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 1615

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)

11 pages, 273 KB

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 6 | Viewed by 2220

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)

11 pages, 302 KB

Open AccessArticle

Interpolative Reich–Rus–Ćirić and Hardy–Rogers Contraction on Quasi-Partial b-Metric Space and Related Fixed Point Results

by Vishnu Narayan Mishra, Luis Manuel Sánchez Ruiz, Pragati Gautam and Swapnil Verma

Mathematics 2020, 8(9), 1598; https://doi.org/10.3390/math8091598 - 17 Sep 2020

Cited by 33 | Viewed by 3183

Abstract

The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction [...] Read more.

The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction in the framework of quasi-partial b-metric space and proved the corresponding common fixed point theorem by adopting the notion of interpolation. The results are further validated with the application based on them. Full article

(This article belongs to the Special Issue Variational Inequality)

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18 pages, 352 KB

Open AccessArticle

Multiparametric Contractions and Related Hardy-Roger Type Fixed Point Theorems

by Antonio Francisco Roldán López de Hierro, Erdal Karapınar and Andreea Fulga

Mathematics 2020, 8(6), 957; https://doi.org/10.3390/math8060957 - 11 Jun 2020

Cited by 15 | Viewed by 2712

Abstract

In this paper we present some novel fixed point theorems for a family of contractions depending on two functions (that are not defined on

t = 0

) and on some parameters that we have called multiparametric contractions. We develop our study in [...] Read more.

In this paper we present some novel fixed point theorems for a family of contractions depending on two functions (that are not defined on

t = 0

) and on some parameters that we have called multiparametric contractions. We develop our study in the setting of b-metric spaces because they allow to consider some families of functions endowed with b-metrics deriving from similarity measures that are more general than norms. Taking into account that the contractivity condition we will employ is very general (of Hardy-Rogers type), we will discuss the validation and usage of this novel condition. After that, we introduce the main results of this paper and, finally, we deduce some consequences of them which illustrates the wide applicability of the main results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)

13 pages, 290 KB

Open AccessArticle

Fixed Point Problems on Generalized Metric Spaces in Perov’s Sense

by Liliana Guran, Monica-Felicia Bota and Asim Naseem

Symmetry 2020, 12(5), 856; https://doi.org/10.3390/sym12050856 - 22 May 2020

Cited by 4 | Viewed by 2879

Abstract

The aim of this paper is to give some fixed point results in generalized metric spaces in Perov’s sense. The generalized metric considered here is the w-distance with a symmetry condition. The operators satisfy a contractive weakly condition of Hardy–Rogers type. The [...] Read more.

The aim of this paper is to give some fixed point results in generalized metric spaces in Perov’s sense. The generalized metric considered here is the w-distance with a symmetry condition. The operators satisfy a contractive weakly condition of Hardy–Rogers type. The second part of the paper is devoted to the study of the data dependence, the well-posedness, and the Ulam–Hyers stability of the fixed point problem. An example is also given to sustain the presented results. Full article

12 pages, 281 KB

Open AccessArticle

On Some New Multivalued Results in the Metric Spaces of Perov’s Type

by Liliana Guran, Monica-Felicia Bota, Asim Naseem, Zoran D. Mitrović, Manuel de la Sen and Stojan Radenović

Mathematics 2020, 8(3), 438; https://doi.org/10.3390/math8030438 - 17 Mar 2020

Cited by 4 | Viewed by 3229

Abstract

The purpose of this paper is to present some new fixed point results in the generalized metric spaces of Perov’s sense under a contractive condition of Hardy–Rogers type. The data dependence of the fixed point set, the well-posedness of the fixed point problem [...] Read more.

The purpose of this paper is to present some new fixed point results in the generalized metric spaces of Perov’s sense under a contractive condition of Hardy–Rogers type. The data dependence of the fixed point set, the well-posedness of the fixed point problem and the Ulam–Hyers stability are also studied. Full article

(This article belongs to the Special Issue Abstract Metric Spaces: Usefulness in Statistics and Fixed Point Theory)

19 pages, 288 KB

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 2068

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

12 pages, 276 KB

Open AccessArticle

On Almost b-Metric Spaces and Related Fixed Point Results

by Nabil Mlaiki, Katarina Kukić, Milanka Gardašević-Filipović and Hassen Aydi

Axioms 2019, 8(2), 70; https://doi.org/10.3390/axioms8020070 - 1 Jun 2019

Cited by 10 | Viewed by 4355

Abstract

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy–Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b [...] Read more.

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy–Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b-metrics to the corresponding b-metrics. Later, we show that this approach can not work for all kinds of contractions. To confirm this, we present a proof in which the contraction condition is such that it cannot be reduced to corresponding b-metrics. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

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