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Search Results (180)

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Keywords = nonexpansive mapping

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19 pages, 1691 KiB

Open AccessArticle

The Application of Generalized Viscosity Implicit Midpoint Rule for Nonexpansive Mappings

by Huancheng Zhang

Symmetry 2024, 16(11), 1528; https://doi.org/10.3390/sym16111528 - 14 Nov 2024

Viewed by 469

Abstract

This paper proposes new iterative algorithms by using the generalized viscosity implicit midpoint rule in Banach space, which is also a symmetric space. Then, this paper obtains strong convergence conclusions. Moreover, the results generalize the related conclusions of some researchers. Finally, this paper [...] Read more.

(This article belongs to the Section Mathematics)

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Figure 1

Figure 1
Numerical results. Full article ">Figure 2
Numerical results. Full article ">Figure 3
Numerical results. Full article ">Figure 4
Numerical results. Full article ">Figure 5
Numerical results. Full article ">

16 pages, 287 KiB

Open AccessArticle

An Averaged Halpern-Type Algorithm for Solving Fixed-Point Problems and Variational Inequality Problems

by Vasile Berinde and Khairul Saleh

Axioms 2024, 13(11), 756; https://doi.org/10.3390/axioms13110756 - 31 Oct 2024

Viewed by 439

Abstract

In this paper, we propose and study an averaged Halpern-type algorithm for approximating the solution of a common fixed-point problem for a couple of nonexpansive and demicontractive mappings with a variational inequality constraint in the setting of a Hilbert space. The strong convergence of the sequence generated by the algorithm is established under feasible assumptions on the parameters involved. In particular, we also obtain the common solution of the fixed point problem for nonexpansive or demicontractive mappings and of a variational inequality problem. Our results extend and generalize various important related results in the literature that were established for two pairs of mappings: (nonexpansive, nonspreading) and (nonexpansive, strongly quasi-nonexpansive). Numerical tests to illustrate the superiority of our algorithm over the ones existing in the literature are also reported. Full article

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

17 pages, 412 KiB

Open AccessArticle

An Iterative Approach to Common Fixed Points of G-Nonexpansive Mappings with Applications in Solving the Heat Equation

by Raweerote Suparatulatorn, Payakorn Saksuriya, Teeranush Suebcharoen and Khuanchanok Chaichana

Axioms 2024, 13(11), 729; https://doi.org/10.3390/axioms13110729 - 22 Oct 2024

Viewed by 742

Abstract

This study presents an iterative method for approximating common fixed points of a finite set of G-nonexpansive mappings within a real Hilbert space with a directed graph. We establish definitions for left and right coordinate convexity and demonstrate both weak and strong convergence results based on reasonable assumptions. Furthermore, our algorithm’s effectiveness in solving the heat equation is highlighted, contributing to energy optimization and sustainable development. Full article

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

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Figure 1

Figure 1
The exact and approximate solutions for the heat equation at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>x</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Full article ">Figure 2
The comparison of the average number of iterations between the literature and our proposed algorithm. Full article ">

22 pages, 842 KiB

Open AccessArticle

Fixed Point Results for Fuzzy Enriched Contraction in Fuzzy Banach Spaces with Applications to Fractals and Dynamic Market Equillibrium

by Muhammad Shaheryar, Fahim Ud Din, Aftab Hussain and Hamed Alsulami

Fractal Fract. 2024, 8(10), 609; https://doi.org/10.3390/fractalfract8100609 - 18 Oct 2024

Cited by 1 | Viewed by 1061

Abstract

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems. Full article

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

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17 pages, 301 KiB

Open AccessFeature PaperArticle

A Projection-Type Implicit Algorithm for Finding a Common Solution for Fixed Point Problems and Variational Inequality Problems

by Vasile Berinde

Mathematics 2024, 12(20), 3187; https://doi.org/10.3390/math12203187 - 11 Oct 2024

Viewed by 766

Abstract

This paper deals with the problem of finding a common solution for a fixed point problem for strictly pseudocontractive mappings and for a certain variational inequality problem. We propose a projection-type implicit averaged algorithm and establish the strong convergence of the sequences generated by this method to the common solution for the fixed point problem and the variational inequality problem. In order to illustrate the feasibility of the hypotheses and the superiority of our theoretical results over the existing literature, an example is also presented. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications: 2nd Edition)

19 pages, 312 KiB

Open AccessArticle

Modified Double Inertial Extragradient-like Approaches for Convex Bilevel Optimization Problems with VIP and CFPP Constraints

by Yue Zeng, Lu-Chuan Ceng, Liu-Fang Zheng and Xie Wang

Symmetry 2024, 16(10), 1324; https://doi.org/10.3390/sym16101324 - 8 Oct 2024

Viewed by 898

Abstract

Convex bilevel optimization problems (CBOPs) exhibit a vital impact on the decision-making process under the hierarchical setting when image restoration plays a key role in signal processing and computer vision. In this paper, a modified double inertial extragradient-like approach with a line search procedure is introduced to tackle the CBOP with constraints of the CFPP and VIP, where the CFPP and VIP represent a common fixed point problem and a variational inequality problem, respectively. The strong convergence analysis of the proposed algorithm is discussed under certain mild assumptions, where it constitutes both sections that possess a mutual symmetry structure to a certain extent. As an application, our proposed algorithm is exploited for treating the image restoration problem, i.e., the LASSO problem with the constraints of fractional programming and fixed-point problems. The illustrative instance highlights the specific advantages and potential infect of the our proposed algorithm over the existing algorithms in the literature, particularly in the domain of image restoration. Full article

(This article belongs to the Special Issue Model-Driven, Data-Driven and Symmetry Methods in Hyperspectral Image Processing)

27 pages, 398 KiB

Open AccessArticle

Mann-Type Inertial Accelerated Subgradient Extragradient Algorithm for Minimum-Norm Solution of Split Equilibrium Problems Induced by Fixed Point Problems in Hilbert Spaces

by Manatchanok Khonchaliew, Kunlanan Khamdam and Narin Petrot

Symmetry 2024, 16(9), 1099; https://doi.org/10.3390/sym16091099 - 23 Aug 2024

Viewed by 1315

Abstract

This paper presents the Mann-type inertial accelerated subgradient extragradient algorithm with non-monotonic step sizes for solving the split equilibrium and fixed point problems relating to pseudomonotone and Lipschitz-type continuous bifunctions and nonexpansive mappings in the framework of real Hilbert spaces. By sufficient conditions on the control sequences of the parameters of concern, the strong convergence theorem to support the proposed algorithm, which involves neither prior knowledge of the Lipschitz constants of bifunctions nor the operator norm of the bounded linear operator, is demonstrated. Some numerical experiments are performed to show the efficacy of the proposed algorithm. Full article

(This article belongs to the Section Mathematics)

18 pages, 392 KiB

Open AccessArticle

Method for Approximating Solutions to Equilibrium Problems and Fixed-Point Problems without Some Condition Using Extragradient Algorithm

by Anchalee Sripattanet and Atid Kangtunyakarn

Axioms 2024, 13(8), 525; https://doi.org/10.3390/axioms13080525 - 2 Aug 2024

Viewed by 585

Abstract

The objective of this research is to present a novel approach to enhance the extragradient algorithm’s efficiency for finding an element within a set of fixed points of nonexpansive mapping and the set of solutions for equilibrium problems. Specifically, we focus on applications involving a pseudomonotone, Lipschitz-type continuous bifunction. Our main contribution lies in establishing a strong convergence theorem for this method, without relying on the assumption of

lim_{n \to \infty} ∥ x_{n + 1} - x_{n} ∥ = 0

. Moreover, the main theorem can be applied to effectively solve the combination of variational inequality problem (CVIP). In support of our main result, numerical examples are also presented. Full article

(This article belongs to the Section Mathematical Analysis)

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Figure 1

Figure 1
The convergence of <math display="inline"><semantics> <mrow> <mrow> <mo>{</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <msub> <mi>y</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </semantics></math> with Theorem 1. Full article ">Figure 2
The convergence of <math display="inline"><semantics> <mrow> <mrow> <mo>{</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <msub> <mi>y</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> </semantics></math> with algorithm 1 in [<a href="#B12-axioms-13-00525" class="html-bibr">12</a>]. Full article ">

10 pages, 268 KiB

Open AccessArticle

Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁_p(·)

by Buthinah A. Bin Dehaish and Mohamed A. Khamsi

Symmetry 2024, 16(7), 799; https://doi.org/10.3390/sym16070799 - 25 Jun 2024

Cited by 2 | Viewed by 984

Abstract

Let C denote a convex subset within the vector space

𝓁_{p (\cdot)}

, and let T represent a mapping from C onto itself. Assume

α = (α_{1}, \dots, α_{n})

is a multi-index in [...] Read more.

Let C denote a convex subset within the vector space

𝓁_{p (\cdot)}

, and let T represent a mapping from C onto itself. Assume

α = (α_{1}, \dots, α_{n})

is a multi-index in

{[0, 1]}^{n}

such that

\sum_{i = 1}^{n} α_{i} = 1

, where

α_{1} > 0

and

α_{n} > 0

. We define

T_{α} : C \to C

T_{α} = \sum_{i = 1}^{n} α_{i} T^{i}

, known as the mean average of the mapping T. While every fixed point of T remains fixed for

T_{α}

, the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for

T_{α}

and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in

R

. Our focus then shifts to variable exponent modular vector spaces

𝓁_{p (\cdot)}

, where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework. Full article

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

14 pages, 255 KiB

Open AccessArticle

Three Existence Results in the Fixed Point Theory

by Alexander J. Zaslavski

Axioms 2024, 13(7), 425; https://doi.org/10.3390/axioms13070425 - 25 Jun 2024

Viewed by 861

Abstract

In the present paper, we obtain three results on the existence of a fixed point for nonexpansive mappings. Two of them are generalizations of the result for F-contraction, while third one is a generalization of a recent result for set-valued contractions. Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

19 pages, 787 KiB

Open AccessEditor’s ChoiceArticle

Nonexpansiveness and Fractal Maps in Hilbert Spaces

by María A. Navascués

Symmetry 2024, 16(6), 738; https://doi.org/10.3390/sym16060738 - 13 Jun 2024

Cited by 1 | Viewed by 737

Abstract

Picard iteration is on the basis of a great number of numerical methods and applications of mathematics. However, it has been known since the 1950s that this method of fixed-point approximation may not converge in the case of nonexpansive mappings. In this paper, an extension of the concept of nonexpansiveness is presented in the first place. Unlike the classical case, the new maps may be discontinuous, adding an element of generality to the model. Some properties of the set of fixed points of the new maps are studied. Afterwards, two iterative methods of fixed-point approximation are analyzed, in the frameworks of b-metric and Hilbert spaces. In the latter case, it is proved that the symmetrically averaged iterative procedures perform well in the sense of convergence with the least number of operations at each step. As an application, the second part of the article is devoted to the study of fractal mappings on Hilbert spaces defined by means of nonexpansive operators. The paper considers fractal mappings coming from

φ

-contractions as well. In particular, the new operators are useful for the definition of an extension of the concept of

α

-fractal function, enlarging its scope to more abstract spaces and procedures. The fractal maps studied here have quasi-symmetry, in the sense that their graphs are composed of transformed copies of itself. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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Figure 1

Figure 1
Graph of the function <math display="inline"><semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>π</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in the interval <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math>. Full article ">Figure 2
Graph of the fractal version of the function <math display="inline"><semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>t</mi> </mrow> </msup> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>π</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math>. Full article ">

10 pages, 213 KiB

Open AccessArticle

Existence of a Fixed Point and Convergence of Iterates for Self-Mappings of Metric Spaces with Graphs

by Alexander J. Zaslavski

Symmetry 2024, 16(6), 705; https://doi.org/10.3390/sym16060705 - 6 Jun 2024

Cited by 1 | Viewed by 600

Abstract

In the present paper, under certain assumptions, we establish the convergence of iterates for self-mappings of complete metric spaces with graphs which are of a contractive type. The class of mappings considered in the paper contains the so-called cyclical mappings introduced by W. A. Kirk, P. S. Srinivasan and P. Veeramani in 2003 and the class of monotone nonexpansive operators. Our results hold in the case of a symmetric graph. Full article

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

10 pages, 281 KiB

Open AccessArticle

Common Fixed-Point Theorem and Projection Method on a Hadamard Space

by Yasunori Kimura

Symmetry 2024, 16(4), 483; https://doi.org/10.3390/sym16040483 - 16 Apr 2024

Cited by 1 | Viewed by 1333

Abstract

In this paper, we obtain an equivalent condition to the existence of a common fixed point of a given family of nonexpansive mappings defined on a Hadamard space. Moreover, if the space is bounded, we show that the generating process of the approximate sequence by a specific projection method will stop in finite steps if there is no common fixed point. It is a significant advantage to reveal the nonexistence of a common fixed point in a finite time. Full article

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

►▼ Show Figures

Figure 1

Figure 1
The graph of f. Full article ">Figure 2
Generating <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math> from <math display="inline"><semantics> <msub> <mi>x</mi> <mi>n</mi> </msub> </semantics></math>. Full article ">

20 pages, 363 KiB

Open AccessArticle

Modified Tseng Method for Solving Pseudomonotone Variational Inequality Problem in Banach Spaces

by Rose Maluleka, Godwin Chidi Ugwunnadi, Maggie Aphane, Hammed A. Abass and Abdul Rahim Khan

Symmetry 2024, 16(3), 363; https://doi.org/10.3390/sym16030363 - 18 Mar 2024

Cited by 1 | Viewed by 1098

Abstract

This article examines the process for solving the fixed-point problem of Bregman strongly nonexpansive mapping as well as the variational inequality problem of the pseudomonotone operator. Within the context of p-uniformly convex real Banach spaces that are also uniformly smooth, we introduce a modified Halpern iterative technique combined with an inertial approach and Tseng methods for finding a common solution of the fixed-point problem of Bregman strongly nonexpansive mapping and the pseudomonotone variational inequality problem. Using our iterative approach, we develop a strong convergence result for approximating the solution of the aforementioned problems. We also discuss some consequences of our major finding. The results presented in this paper complement and build upon many relevant discoveries in the literature. Full article

(This article belongs to the Section Mathematics)

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Figure 1

Figure 1
Example 1, (Top left): Case 1; (Top right): Case 2; (Bottom left): Case 3; (Bottom right): Case 4. Full article ">

19 pages, 342 KiB

Open AccessArticle

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

by James Abah Ugboh, Joseph Oboyi, Mfon Okon Udo, Hossam A. Nabwey, Austine Efut Ofem and Ojen Kumar Narain

Fractal Fract. 2024, 8(3), 166; https://doi.org/10.3390/fractalfract8030166 - 14 Mar 2024

Cited by 2 | Viewed by 1216

Abstract

In this paper, we consider a faster iterative method for approximating the fixed points of generalized

α

-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our [...] Read more.

In this paper, we consider a faster iterative method for approximating the fixed points of generalized

α

-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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Figure 1

Figure 1
Graph corresponding to <a href="#fractalfract-08-00166-t001" class="html-table">Table 1</a>. Full article ">Figure 2
Graph corresponding to <a href="#fractalfract-08-00166-t002" class="html-table">Table 2</a>. Full article ">

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