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20 pages, 312 KB

Open AccessArticle

Existence and Uniqueness of a Solution of a Boundary Value Problem Used in Chemical Sciences via a Fixed Point Approach

by Umar Ishtiaq, Fahad Jahangeer, Mubariz Garayev and Ioan-Lucian Popa

Symmetry 2025, 17(1), 127; https://doi.org/10.3390/sym17010127 - 16 Jan 2025

Cited by 1 | Viewed by 770

In this paper, we present Proinov-type fixed point theorems in the setting of bi-polar metric spaces and fuzzy bi-polar metric spaces. Fuzzy bi-polar metric spaces with symmetric property extend classical metric spaces to address dual structures and uncertainty, ensuring consistency and balance. We [...] Read more.

Ω, Π : (0, \infty) \to R

for the existence of fixed points via the

(Ω, Π)

-contraction in bi-polar metric spaces. Further, we define real-valued functions

Ω, Π : (0, 1] \to R

to obtain fixed point theorems in fuzzy bi-polar metric spaces. We apply

(Ω, Π)

fuzzy bi-polar version of a Banach fixed point theorem to show the existence of solutions. Furthermore, we provide some non-trivial examples to show the validity of our results. In the end, we find the existence and uniqueness of a solution of integral equations and boundary value problem used in chemical sciences by applying main results. Full article

(This article belongs to the Section Mathematics)

31 pages, 399 KB

Open AccessArticle

A New Study on the Fixed Point Sets of Proinov-Type Contractions via Rational Forms

by Mi Zhou, Xiaolan Liu, Naeem Saleem, Andreea Fulga and Nihal Özgür

Symmetry 2022, 14(1), 93; https://doi.org/10.3390/sym14010093 - 6 Jan 2022

Cited by 6 | Viewed by 2034

Abstract

In this paper, we presented some new weaker conditions on the Proinov-type contractions which guarantees that a self-mapping T has a unique fixed point in terms of rational forms. Our main results improved the conclusions provided by Andreea Fulga (On [...] Read more.

(ψ, φ) -

Rational Contractions) in which the continuity assumption can either be reduced to orbital continuity,

k -

continuity, continuity of

T^{k}

, T-orbital lower semi-continuity or even it can be removed. Meanwhile, the assumption of monotonicity on auxiliary functions is also removed from our main results. Moreover, based on the obtained fixed point results and the property of symmetry, we propose several Proinov-type contractions for a pair of self-mappings

(P, Q)

which will ensure the existence of the unique common fixed point of a pair of self-mappings

(P, Q)

. Finally, we obtained some results related to fixed figures such as fixed circles or fixed discs which are symmetrical under the effect of self mappings on metric spaces, we proposed some new types of

{(ψ, φ)}_{c} -

rational contractions and obtained the corresponding fixed figure theorems on metric spaces. Several examples are provided to indicate the validity of the results presented. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Analysis and Fixed Point Theory)

21 pages, 340 KB

Open AccessArticle

A New Approach to Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces

by Mi Zhou, Naeem Saleem, Xiaolan Liu, Andreea Fulga and Antonio Francisco Roldán López de Hierro

Mathematics 2021, 9(23), 3001; https://doi.org/10.3390/math9233001 - 23 Nov 2021

Cited by 26 | Viewed by 2699

Abstract

Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form

ψ (d (T x, T y)) \leq φ (d (x, y))

, Proinov [...] Read more.

Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form

ψ (d (T x, T y)) \leq φ (d (x, y))

, Proinov proved some fixed-point theorems, which extended and unified many existing results in the literature. Accordingly, inspired by Proinov-type contraction conditions, Roldán López de Hierro et al. introduced a novel family of contractions in fuzzy metric spaces (in the sense of George and Veeramani), whose main advantage is the very weak constraints imposed on the auxiliary functions that appear in the contractivity condition. They also proved the existence and uniqueness of fixed points for the discussed family of fuzzy contractions in the setting of non-Archimedean fuzzy metric spaces. In this paper, we introduce a new family of fuzzy contractions based on Proinov-type contractions for which the involved auxiliary functions are not supposed to satisfy any monotonicity assumptions; further, we establish some new results about the existence and uniqueness of fixed points. Furthermore, we show how the main results in the above-mentioned paper can be deduced from our main statements. In this way, our conclusions provide a positive partial solution to one of the open problems posed by such authors for deleting or weakening the hypothesis of the nondecreasingness character of the auxiliary functions. Full article

(This article belongs to the Special Issue General Metric Spaces: Usage and Applications in Statistics and Fixed Point Theory)

12 pages, 281 KB

Open AccessArticle

A New Approach of Some Contractive Mappings on Metric Spaces

by Ion Marian Olaru and Nicolae Adrian Secelean

Mathematics 2021, 9(12), 1433; https://doi.org/10.3390/math9121433 - 19 Jun 2021

Cited by 9 | Viewed by 2410

Abstract

In this paper, we introduce a new contraction-type mapping and provide a fixed-point theorem which generalizes and improves some existing results in the literature. Thus, we prove that the Boyd and Wong theorem (1969) and, more recently, the fixed-point results due to Wardowski (2012), Turinici (2012), Piri and Kumam (2016), Secelean (2016), Proinov (2020), and others are consequences of our main result. An application in integral equations and some illustrative examples are indicated. Full article

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

13 pages, 281 KB

Open AccessArticle

Fixed Points of Proinov E-Contractions

by Maryam A. Alghamdi, Selma Gulyaz-Ozyurt and Andreea Fulga

Symmetry 2021, 13(6), 962; https://doi.org/10.3390/sym13060962 - 28 May 2021

Cited by 7 | Viewed by 2331

Abstract

In this paper, we consider a new type of Proinov contraction on the setting of a symmetrical abstract structure, more precisely, the metric space. Our goal is to expand on some results from the literature using admissible mappings and the concept of E [...] Read more.

(This article belongs to the Special Issue Symmetry in Continuum Mechanics and Dynamical Systems)

10 pages, 269 KB

Open AccessArticle

On Some New Jungck–Fisher–Wardowski Type Fixed Point Results

by Jelena Vujaković, Eugen Ljajko, Slobodan Radojević and Stojan Radenović

Symmetry 2020, 12(12), 2048; https://doi.org/10.3390/sym12122048 - 10 Dec 2020

Cited by 9 | Viewed by 1949

Abstract

Many authors used the concept of

F -

Many authors used the concept of

F -

contraction introduced by Wardowski in 2012 in order to define and prove new results on fixed points in complete metric spaces. In some later papers (for example Proinov P.D., J. Fixed Point Theory Appl. (2020)22:21, doi:10.1007/s11784-020-0756-1) it is shown that conditions (F2) and (F3) are not necessary to prove Wardowski’s results. In this article we use a new approach in proving that the Picard–Jungck sequence is a Cauchy one. It helps us obtain new Jungck–Fisher–Wardowski type results using Wardowski’s condition (F1) only, but in a way that differs from the previous approaches. Along with that, we came to several new contractive conditions not known in the fixed point theory so far. With the new results presented in the article, we generalize, extend, unify and enrich methods presented in the literature that we cite. Full article

(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)

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