Symmetry

Research

14 pages, 322 KiB

Open AccessArticle

On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems

by Liang-Ju Chu and Wei–Shih Du

Symmetry 2019, 11(6), 789; https://doi.org/10.3390/sym11060789 - 13 Jun 2019

Viewed by 2003

Two existence theorems of maximal elements in H-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of

L

-majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we [...] Read more.

Two existence theorems of maximal elements in H-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of

L

-majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we obtain an equilibrium existence theorem for general abstract economies, a new fixed point theorem, and give a sufficient condition for the existence of solutions of the eigenvector problem (EIVP). Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

12 pages, 282 KiB

Open AccessArticle

Best Proximity Points for Alternative Maps

by Yi Chou Chen

Symmetry 2019, 11(6), 750; https://doi.org/10.3390/sym11060750 - 3 Jun 2019

Cited by 2 | Viewed by 2143

Abstract

Let

(X, d)

be a metric space and

Ω_{i}

,

i = 1, 2, \dots, m

, be a nonempty subset of

(X, d)

. An operator [...] Read more.

Let

(X, d)

be a metric space and

Ω_{i}

,

i = 1, 2, \dots, m

, be a nonempty subset of

(X, d)

. An operator

T : \cup_{1 \leq i \leq m} Ω_{i} \to \cup_{1 \leq i \leq m} Ω_{i}

is called an alternative map if

T (Ω_{j}) \subseteq \cup_{i \neq j} Ω_{i}

,

j = 1, 2, \dots, m

. In addition, if for any x,

y \in \cup_{1 \leq i \leq m} Ω_{i}

, there exists a constant

α \in [0, 1)

such that

d (T x, T y) \leq α d (x, y) + (1 - α) d (Ω_{j}, Ω_{k})

for some

Ω_{j}

and

Ω_{k} \in {Ω_{i}}_{i = 1}^{m}

with

x \in Ω_{j}

and

y \in Ω_{k}

, then we call T an alternative contraction. Moreover, if

(X, d)

has an alternative UC property and T is an alternative contraction, then the best proximity point of T exists. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

14 pages, 295 KiB

Open AccessArticle

Terminal Value Problem for Differential Equations with Hilfer–Katugampola Fractional Derivative

by Mouffak Benchohra, Soufyane Bouriah and Juan J. Nieto

Symmetry 2019, 11(5), 672; https://doi.org/10.3390/sym11050672 - 15 May 2019

Cited by 27 | Viewed by 3310

Abstract

We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical fixed point [...] Read more.

We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Banach contraction principle and Krasnoselskii’s fixed point theorem. We illustrate our main findings, with a particular case example included to show the applicability of our outcomes. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

19 pages, 315 KiB

Open AccessArticle

Generalized Contractive Mappings and Related Results in b-Metric Like Spaces with an Application

by Hasanen A. Hammad and Manuel De la Sen

Symmetry 2019, 11(5), 667; https://doi.org/10.3390/sym11050667 - 14 May 2019

Cited by 26 | Viewed by 2789

Abstract

In this article, a general contractive mapping is presented and some fixed point results in complete b-metric-like spaces are studied. The results obtained here extend and improve some related results in the literature. Also, new common fixed point results for a graphic [...] Read more.

In this article, a general contractive mapping is presented and some fixed point results in complete b-metric-like spaces are studied. The results obtained here extend and improve some related results in the literature. Also, new common fixed point results for a graphic contraction mappings are proved. Some comparative examples are given to support the obtained results. Moreover, an analytical solution of an integral equation has been presented as an application. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

23 pages, 319 KiB

Open AccessArticle

Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators

by Ashish Nandal, Renu Chugh and Mihai Postolache

Symmetry 2019, 11(5), 655; https://doi.org/10.3390/sym11050655 - 10 May 2019

Cited by 16 | Viewed by 3364

Abstract

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in [...] Read more.

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

21 pages, 354 KiB

Open AccessArticle

Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses

by Ravi Agarwal, Snezhana Hristova and Donal O’Regan

Symmetry 2019, 11(5), 614; https://doi.org/10.3390/sym11050614 - 2 May 2019

Cited by 11 | Viewed by 4844

Abstract

In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and [...] Read more.

In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and their mild solutions are given via integral representations. In the linear case the equivalence of the solution and mild solutions is established. Conditions for existence and uniqueness of initial value problems are presented. Several examples are provided to illustrate the influence of impulsive functions and the interpretation of impulses in the RL fractional case. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

14 pages, 289 KiB

Open AccessArticle

Non-Unique Fixed Point Theorems in Modular Metric Spaces

by Sharafat Hussain

Symmetry 2019, 11(4), 549; https://doi.org/10.3390/sym11040549 - 16 Apr 2019

Cited by 3 | Viewed by 2801

Abstract

This paper is devoted to the study of Ćirić-type non-unique fixed point results in modular metric spaces. We obtain various theorems about a fixed point and periodic points for a self-map on modular spaces which are not necessarily continuous and satisfy certain contractive [...] Read more.

This paper is devoted to the study of Ćirić-type non-unique fixed point results in modular metric spaces. We obtain various theorems about a fixed point and periodic points for a self-map on modular spaces which are not necessarily continuous and satisfy certain contractive conditions. Our results extend the results of Ćirić, Pachpatte, and Achari in modular metric spaces. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

18 pages, 282 KiB

Open AccessArticle

Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

by Manatchanok Khonchaliew, Ali Farajzadeh and Narin Petrot

Symmetry 2019, 11(4), 480; https://doi.org/10.3390/sym11040480 - 3 Apr 2019

Cited by 3 | Viewed by 2586

Abstract

This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, [...] Read more.

This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, these two new algorithms show strong convergence. Some numerical experiments are presented to demonstrate the new algorithms. Finally, the two introduced algorithms are compared with a standard, well-known algorithm. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

15 pages, 296 KiB

Open AccessArticle

Anti-Periodic Boundary Value Problems for Nonlinear Langevin Fractional Differential Equations

by Fang Li, Hongjuan Zeng and Huiwen Wang

Symmetry 2019, 11(4), 443; https://doi.org/10.3390/sym11040443 - 27 Mar 2019

Cited by 1 | Viewed by 2334

Abstract

In this paper, we focus on the existence of solutions of the nonlinear Langevin fractional differential equations involving anti-periodic boundary value conditions. By using some techniques, formulas of solutions for the above problem and some properties of the Mittag-Leffler functions [...] Read more.

In this paper, we focus on the existence of solutions of the nonlinear Langevin fractional differential equations involving anti-periodic boundary value conditions. By using some techniques, formulas of solutions for the above problem and some properties of the Mittag-Leffler functions

E_{α, β} (z)

,

α, β \in (1, 2), z \in R

are presented. Moreover, we utilize the fixed point theorem under the weak assumptions for nonlinear terms to obtain the existence result of solutions and give an example to illustrate the result. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

11 pages, 287 KiB

Open AccessArticle

On Suzuki Mappings in Modular Spaces

by Andreea Bejenaru and Mihai Postolache

Symmetry 2019, 11(3), 319; https://doi.org/10.3390/sym11030319 - 3 Mar 2019

Cited by 13 | Viewed by 2883

Abstract

Inspired by Suzuki’s generalization for nonexpansive mappings, we define the

(C)

-property on modular spaces, and provide conditions concerning the fixed points of newly introduced class of mappings in this new framework. In addition, Kirk’s Lemma is extended to modular spaces. [...] Read more.

Inspired by Suzuki’s generalization for nonexpansive mappings, we define the

(C)

-property on modular spaces, and provide conditions concerning the fixed points of newly introduced class of mappings in this new framework. In addition, Kirk’s Lemma is extended to modular spaces. The main outcomes extend the classical results on Banach spaces. The major contribution consists of providing inspired arguments to compensate the absence of subadditivity in the case of modulars. The results herein are supported by illustrative examples. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

9 pages, 252 KiB

Open AccessArticle

Fixed Point Results Using F_t-Contractions in Ordered Metric Spaces Having t-Property

by Hassen Aydi, Tawseef Rashid, Qamrul Haq Khan, Zead Mustafa and Mohammed M. M. Jaradat

Symmetry 2019, 11(3), 313; https://doi.org/10.3390/sym11030313 - 2 Mar 2019

Cited by 6 | Viewed by 2237

Abstract

In this paper, we prove the existence of fixed points of

F_{t}

-contraction mappings in partially ordered metric spaces not necessarily complete. We require that the ordered metric space has the t-property, which is a new concept introduced recently by Rashid [...] Read more.

In this paper, we prove the existence of fixed points of

F_{t}

-contraction mappings in partially ordered metric spaces not necessarily complete. We require that the ordered metric space has the t-property, which is a new concept introduced recently by Rashid et al. We also give some examples to illustrate the new concepts and obtained results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

15 pages, 246 KiB

Open AccessArticle

Some Results for Split Equality Equilibrium Problems in Banach Spaces

by Zhaoli Ma, Lin Wang and Yeol Je Cho

Symmetry 2019, 11(2), 194; https://doi.org/10.3390/sym11020194 - 10 Feb 2019

Cited by 22 | Viewed by 2396

Abstract

In this paper, a new algorithm for finding a common element of a split equality fixed point problem for nonexpansive mappings and split equality equilibrium problem in three Banach spaces is introduced. Also, some strong and weak convergence theorems for the proposed algorithm [...] Read more.

In this paper, a new algorithm for finding a common element of a split equality fixed point problem for nonexpansive mappings and split equality equilibrium problem in three Banach spaces is introduced. Also, some strong and weak convergence theorems for the proposed algorithm are proved. Finally, the main results obtained in this paper are applied to solve the split equality convex minimization problem. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

13 pages, 759 KiB

Open AccessArticle

Generalized Random α-ψ-Contractive Mappings with Applications to Stochastic Differential Equation

by Chayut Kongban, Poom Kumam, Juan Martínez-Moreno and Kanokwan Sitthithakerngkiet

Symmetry 2019, 11(2), 143; https://doi.org/10.3390/sym11020143 - 28 Jan 2019

Viewed by 2617

Abstract

The main purpose in this paper is to define the modification form of random

α

-admissible and random

α

-

ψ

-contractive maps. We establish new random fixed point theorems in complete separable metric spaces. The interpretation of our results provide the main [...] Read more.

The main purpose in this paper is to define the modification form of random

α

-admissible and random

α

-

ψ

-contractive maps. We establish new random fixed point theorems in complete separable metric spaces. The interpretation of our results provide the main theorems of Tchier and Vetro (2017) as directed corollaries. In addition, some applications to second order random differential equations are presenred here to interpret the usability of the obtained results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

20 pages, 338 KiB

Open AccessArticle

Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations

by Zhenhua Ma, Azhar Hussain, Muhammad Adeel, Nawab Hussain and Ekrem Savas

Symmetry 2019, 11(1), 93; https://doi.org/10.3390/sym11010093 - 15 Jan 2019

Cited by 5 | Viewed by 3798

Abstract

In this paper, we introduce the notion of

\overset{´}{C}

iri

\overset{´}{c}

type

α

-

ψ

-

Θ

-contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially [...] Read more.

In this paper, we introduce the notion of

\overset{´}{C}

iri

\overset{´}{c}

type

α

-

ψ

-

Θ

-contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation

X = Q + \sum_{i = 1}^{m} A_{i}^{*} γ (X) A_{i}

and give a numerical example. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

► Show Figures

Figure 1

24 pages, 368 KiB

Open AccessArticle

Hybrid Multivalued Type Contraction Mappings in α_K-Complete Partial b-Metric Spaces and Applications

by Eskandar Ameer, Hassen Aydi, Muhammad Arshad, Habes Alsamir and Mohd Selmi Noorani

Symmetry 2019, 11(1), 86; https://doi.org/10.3390/sym11010086 - 14 Jan 2019

Cited by 74 | Viewed by 4101

Abstract

In this paper, we initiate the notion of generalized multivalued

(α_{K}^{*}, Υ, Λ)

-contractions and provide some new common fixed point results in the class of

α_{K}

-complete partial b-metric spaces. The obtained results are [...] Read more.

In this paper, we initiate the notion of generalized multivalued

(α_{K}^{*}, Υ, Λ)

-contractions and provide some new common fixed point results in the class of

α_{K}

-complete partial b-metric spaces. The obtained results are an improvement of several comparable results in the existing literature. We set up an example to elucidate our main result. Moreover, we present applications dealing with the existence of a solution for systems either of functional equations or of nonlinear matrix equations. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

17 pages, 315 KiB

Open AccessArticle

Common Fixed Point Results for Fuzzy Mappings on Complex-Valued Metric Spaces with Homotopy Results

by Humaira, Muhammad Sarwar and Poom Kumam

Symmetry 2019, 11(1), 61; https://doi.org/10.3390/sym11010061 - 8 Jan 2019

Cited by 21 | Viewed by 3986

Abstract

Owing to the notion of a complex-valued metric space, we prove fixed point results, which generalize some common fixed point results under contractive condition for rational expression in the context of complex-valued metric spaces. In application, we present a homotopy result to apply [...] Read more.

Owing to the notion of a complex-valued metric space, we prove fixed point results, which generalize some common fixed point results under contractive condition for rational expression in the context of complex-valued metric spaces. In application, we present a homotopy result to apply the results obtained herein. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

7 pages, 235 KiB

Open AccessArticle

On Interpolative Hardy-Rogers Type Contractions

by Erdal Karapınar, Obaid Alqahtani and Hassen Aydi

Symmetry 2019, 11(1), 8; https://doi.org/10.3390/sym11010008 - 22 Dec 2018

Cited by 126 | Viewed by 4419

Abstract

By using an interpolative approach, we recognize the Hardy-Rogers fixed point theorem in the class of metric spaces. The obtained result is supported by some examples. We also give the partial metric case, according to our result. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

14 pages, 267 KiB

Open AccessArticle

Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability

by Mostafa Bachar, Osvaldo Mendez and Messaoud Bounkhel

Symmetry 2018, 10(12), 708; https://doi.org/10.3390/sym10120708 - 3 Dec 2018

Cited by 10 | Viewed by 2778

Abstract

We analyze the modular geometry of the Lebesgue space with variable exponent,

L^{p (\cdot)}

. Our central result is that

L^{p (\cdot)}

possesses a modular uniform convexity property. Part of the novelty is that the property holds [...] Read more.

We analyze the modular geometry of the Lebesgue space with variable exponent,

L^{p (\cdot)}

. Our central result is that

L^{p (\cdot)}

possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case

sup_{x \in Ω} p (x) = \infty

. We present specific applications to fixed point theory. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

15 pages, 285 KiB

Open AccessArticle

Radial Symmetry for Weak Positive Solutions of Fractional Laplacian with a Singular Nonlinearity

by Xing Wang and Li Zhang

Symmetry 2018, 10(12), 695; https://doi.org/10.3390/sym10120695 - 3 Dec 2018

Viewed by 2521

Abstract

This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian. The main results in the paper demonstrate the existence and multiplicity of radial symmetry weak positive solutions by Schwarz spherical rearrangement, constrained minimization, and Ekeland’s [...] Read more.

This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian. The main results in the paper demonstrate the existence and multiplicity of radial symmetry weak positive solutions by Schwarz spherical rearrangement, constrained minimization, and Ekeland’s variational principle. It is worth pointing out that our results extend the previous works of T. Mukherjee and K. Sreenadh to a setting in which the testing functions need not have a compact support. Moreover, we weakened one of the conditions used in their papers. Our results improve on existing studies on radial symmetry solutions of nonlocal boundary value problems. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

24 pages, 315 KiB

Open AccessArticle

Some Fixed-Point Theorems in b-Dislocated Metric Space and Applications

by Panda Sumati Kumari, Obaid Alqahtani and Erdal Karapınar

Symmetry 2018, 10(12), 691; https://doi.org/10.3390/sym10120691 - 2 Dec 2018

Cited by 21 | Viewed by 3310

Abstract

In this article, we prove some fixed-point theorems in b-dislocated metric space. Thereafter, we propose a simple and efficient solution for a non-linear integral equation and non-linear fractional differential equations of Caputo type by using the technique of fixed point. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

8 pages, 268 KiB

Open AccessArticle

Some Inequalities of Čebyšev Type for Conformable k-Fractional Integral Operators

by Feng Qi, Gauhar Rahman, Sardar Muhammad Hussain, Wei-Shih Du and Kottakkaran Sooppy Nisar

Symmetry 2018, 10(11), 614; https://doi.org/10.3390/sym10110614 - 8 Nov 2018

Cited by 39 | Viewed by 4409

Abstract

In the article, the authors present several inequalities of the Čebyšev type for conformable k-fractional integral operators. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

17 pages, 295 KiB

Open AccessArticle

Bochner-Like Transform and Stepanov Almost Periodicity on Time Scales with Applications

by Chao-Hong Tang and Hong-Xu Li

Symmetry 2018, 10(11), 566; https://doi.org/10.3390/sym10110566 - 1 Nov 2018

Cited by 9 | Viewed by 2349

Abstract

By using Bochner transform, Stepanov almost periodic functions inherit some basic properties directly from almost periodic functions. Recently, this old work was extended to time scales. However, we show that Bochner transform is not valid on time scales. Then we present a revised [...] Read more.

By using Bochner transform, Stepanov almost periodic functions inherit some basic properties directly from almost periodic functions. Recently, this old work was extended to time scales. However, we show that Bochner transform is not valid on time scales. Then we present a revised version, called Bochner-like transform, for time scales, and prove that a function is Stepanov almost periodic if and only if its Bochner-like transform is almost periodic on time scales. Some basic properties including the composition theorem of Stepanov almost periodic functions are obtained by applying Bochner-like transform. Our results correct the recent results where Bochner transform is used on time scales. As an application, we give some results on dynamic equations with Stepanov almost periodic terms. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

15 pages, 269 KiB

Open AccessFeature PaperArticle

Convergence Analysis of an Inexact Three-Operator Splitting Algorithm

by Chunxiang Zong, Yuchao Tang and Yeol Je Cho

Symmetry 2018, 10(11), 563; https://doi.org/10.3390/sym10110563 - 1 Nov 2018

Cited by 24 | Viewed by 3377

Abstract

The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, [...] Read more.

The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

11 pages, 264 KiB

Open AccessArticle

Some Globally Stable Fixed Points in b-Metric Spaces

by Umar Yusuf Batsari, Poom Kumam and Kanokwan Sitthithakerngkiet

Symmetry 2018, 10(11), 555; https://doi.org/10.3390/sym10110555 - 30 Oct 2018

Cited by 4 | Viewed by 3108

Abstract

In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction [...] Read more.

In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction mapping that generalizes that of Banach, Kanan, and Chatterjea. Using our new introduced contraction mapping, we establish some results on the existence and uniqueness of fixed points. In obtaining some of our results, we assume that the space is associated with a partial order, and the b-metric function has the regularity property. Our results improve, and generalize some current results in the literature. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

13 pages, 265 KiB

Open AccessArticle

A New Approach to the Solution of the Fredholm Integral Equation via a Fixed Point on Extended b-Metric Spaces

by Erdal Karapınar, Panda Sumati Kumari and Durdana Lateef

Symmetry 2018, 10(10), 512; https://doi.org/10.3390/sym10100512 - 16 Oct 2018

Cited by 37 | Viewed by 3899

Abstract

It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of

f (x) = x

. (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the [...] Read more.

It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of

f (x) = x

. (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the correctness of programs. (2) The Brouwer fixed point theorem is used to prove the existence of Nash equilibria in games. (3) Dlala et al. proposed a solution for magnetic field problems via the fixed point approach. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

10 pages, 272 KiB

Open AccessArticle

Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces

by Buthinah A. Bin Dehaish and Mohamed A Khamsi

Symmetry 2018, 10(10), 481; https://doi.org/10.3390/sym10100481 - 11 Oct 2018

Cited by 2 | Viewed by 3047

Abstract

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by [...] Read more.

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by

x_{n + 1} = t_{n} T^{ϕ (n)} (x_{n}) + (1 - t_{n}) x_{n},

for

n \in N

, when T is a monotone asymptotically nonexpansive self-mapping. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

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