MDPI - Publisher of Open Access Journals

16 pages, 296 KB

Open AccessArticle

Averaged Iterative Algorithms for Convex Optimization Problems over a Common Fixed-Points Set of Demicontractive Mappings

by Vasile Berinde and Khairul Saleh

Axioms 2026, 15(1), 8; https://doi.org/10.3390/axioms15010008 - 25 Dec 2025

Viewed by 353

Abstract

In this article, we introduce a novel averaged-type iterative scheme designed for solving convex minimization problems over the set of common fixed points of a pair of demicontractive mappings. Under suitable assumptions, we prove that the proposed algorithm converges strongly to the solution [...] Read more.

In this article, we introduce a novel averaged-type iterative scheme designed for solving convex minimization problems over the set of common fixed points of a pair of demicontractive mappings. Under suitable assumptions, we prove that the proposed algorithm converges strongly to the solution of the considered problem in a Hilbert space setting. We further demonstrate the applicability of our method to quadratic optimization problems with a bounded linear operator. In addition, we also report the numerical experiments that were performed in order to demonstrate the convergence behavior of the algorithm and to highlight its superiority over related existing methods. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics, 2nd Edition)

20 pages, 391 KB

Open AccessReview

Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

by Vasile Berinde

Symmetry 2023, 15(10), 1866; https://doi.org/10.3390/sym15101866 - 4 Oct 2023

Cited by 2 | Viewed by 1938

Abstract

Demicontractive operators form an important class of nonexpansive type mappings whose study led researchers to the creation of some beautiful results in the framework of metric fixed-point theory. This article aims to provide an overview of the most relevant results on the approximation [...] Read more.

Demicontractive operators form an important class of nonexpansive type mappings whose study led researchers to the creation of some beautiful results in the framework of metric fixed-point theory. This article aims to provide an overview of the most relevant results on the approximation of fixed points of single-valued demicontractive mappings in Hilbert spaces. Subsequently, we exhibit the role of additional properties of demicontractive operators, as well as the main features of the employed iterative algorithms to ensure weak convergence or strong convergence. We also include commentaries on the use of demicontractive mappings to solve some important nonlinear problems with the aim of providing a comprehensive starting point to readers who are attempting to apply demicontractive mappings to concrete applications. We conclude with some brief statements on our view on relevant and promising directions of research on demicontractive mappings in nonlinear settings (metric spaces) and some application challenges. Full article

20 pages, 458 KB

Open AccessArticle

Novel Multistep Implicit Iterative Methods for Solving Common Solution Problems with Asymptotically Demicontractive Operators and Applications

by Hai-Yang Xu and Heng-You Lan

Mathematics 2023, 11(18), 3871; https://doi.org/10.3390/math11183871 - 11 Sep 2023

Cited by 1 | Viewed by 1259

Abstract

It is very meaningful and challenging to efficiently seek common solutions to operator systems (CSOSs), which are widespread in pure and applied mathematics, as well as some closely related optimization problems. The purpose of this paper is to introduce a novel class of [...] Read more.

It is very meaningful and challenging to efficiently seek common solutions to operator systems (CSOSs), which are widespread in pure and applied mathematics, as well as some closely related optimization problems. The purpose of this paper is to introduce a novel class of multistep implicit iterative algorithms (MSIIAs) for solving general CSOSs. By using Xu’s lemma and Maingé’s fundamental and important results, we first obtain strong convergence theorems for both one-step and multistep implicit iterative schemes for CSOSs, involving asymptotically demicontractive operators. Finally, for the applications and profits of the main results presented in this paper, we give two numerical examples and present an iterative approximation to solve the general common solution to the variational inequalities and operator equations. Full article

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

► Show Figures

Figure 1

27 pages, 401 KB

Open AccessArticle

Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

by Cong-Shan Wang, Lu-Chuan Ceng, Bing Li, Sheng-Long Cao, Hui-Ying Hu and Yun-Shui Liang

Axioms 2023, 12(9), 832; https://doi.org/10.3390/axioms12090832 - 28 Aug 2023

Cited by 3 | Viewed by 1517

Abstract

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in [...] Read more.

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint. Full article

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

16 pages, 307 KB

Open AccessArticle

Approximation of the Solution of Split Equality Fixed Point Problem for Family of Multivalued Demicontractive Operators with Application

by Ismat Beg, Mujahid Abbas and Muhammad Waseem Asghar

Mathematics 2023, 11(4), 959; https://doi.org/10.3390/math11040959 - 13 Feb 2023

Cited by 2 | Viewed by 1753

Abstract

In this paper, a new viscosity type iterative algorithm is used for obtaining a strong convergence result of split equality fixed point solutions for infinite families of multivalued demicontractive mappings in real Hilbert spaces. Our iterative scheme is based on choosing the step-sizes [...] Read more.

In this paper, a new viscosity type iterative algorithm is used for obtaining a strong convergence result of split equality fixed point solutions for infinite families of multivalued demicontractive mappings in real Hilbert spaces. Our iterative scheme is based on choosing the step-sizes without calculating or estimating the operator norms and the condition of hemicompactness was relaxed to prove the strong convergence result. As an application, the solution of split convex minimization problem was approximated. The result presented herein unifies and extends several comparable results in the literature. Full article

12 pages, 274 KB

Open AccessArticle

Algorithms for Approximating Solutions of Split Variational Inclusion and Fixed-Point Problems

by Li-Jun Zhu and Yonghong Yao

Mathematics 2023, 11(3), 641; https://doi.org/10.3390/math11030641 - 27 Jan 2023

Cited by 38 | Viewed by 2054

Abstract

In this paper, the split fixed point and variational inclusion problem is considered. With the help of fixed point technique, Tseng-type splitting method and self-adaptive rule, an iterative algorithm is proposed for solving this split problem in which the involved operators S and [...] Read more.

In this paper, the split fixed point and variational inclusion problem is considered. With the help of fixed point technique, Tseng-type splitting method and self-adaptive rule, an iterative algorithm is proposed for solving this split problem in which the involved operators S and T are demicontractive operators and g is plain monotone. Strong convergence theorem is proved under some mild conditions. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

29 pages, 1245 KB

Open AccessArticle

A New Construction and Convergence Analysis of Non-Monotonic Iterative Methods for Solving ρ-Demicontractive Fixed Point Problems and Variational Inequalities Involving Pseudomonotone Mapping

by Chainarong Khunpanuk, Bancha Panyanak and Nuttapol Pakkaranang

Mathematics 2022, 10(4), 623; https://doi.org/10.3390/math10040623 - 17 Feb 2022

Cited by 4 | Viewed by 2197

Abstract

Two new inertial-type extragradient methods are proposed to find a numerical common solution to the variational inequality problem involving a pseudomonotone and Lipschitz continuous operator, as well as the fixed point problem in real Hilbert spaces with a

ρ

-demicontractive mapping. These inertial-type [...] Read more.

Two new inertial-type extragradient methods are proposed to find a numerical common solution to the variational inequality problem involving a pseudomonotone and Lipschitz continuous operator, as well as the fixed point problem in real Hilbert spaces with a

ρ

-demicontractive mapping. These inertial-type iterative methods use self-adaptive step size rules that do not require previous knowledge of the Lipschitz constant. We also show that the proposed methods strongly converge to a solution of the variational inequality and fixed point problems under appropriate standard test conditions. Finally, we present several numerical examples to show the effectiveness and validation of the proposed methods. Full article

(This article belongs to the Special Issue New Advances in Functional Analysis)

► Show Figures

Figure 1

12 pages, 638 KB

Open AccessArticle

Enhancing Ant-Based Algorithms for Medical Image Edge Detection by Admissible Perturbations of Demicontractive Mappings

by Vasile Berinde and Cristina Ţicală

Symmetry 2021, 13(5), 885; https://doi.org/10.3390/sym13050885 - 17 May 2021

Cited by 10 | Viewed by 3297

Abstract

The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second [...] Read more.

The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second author and her collaborators, where they used admissible perturbations of demicontractive mappings as test functions. To illustrate this fact, we first consider some typical properties of demicontractive mappings and of their admissible perturbations and then present some appropriate numerical tests to illustrate the improvement brought by the admissible perturbations of demicontractive mappings when they are taken as test functions in ant-based algorithms for medical image edge detection. The edge detection process reported in our study considers both symmetric (Head CT and Brain CT) and asymmetric (Hand X-ray) medical images. The performance of the algorithm was tested visually with various images and empirically with evaluation of parameters. Full article

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

► Show Figures

Figure 1

19 pages, 827 KB

Open AccessArticle

A Cyclic Iterative Algorithm for Multiple-Sets Split Common Fixed Point Problem of Demicontractive Mappings without Prior Knowledge of Operator Norm

by Nishu Gupta, Mihai Postolache, Ashish Nandal and Renu Chugh

Mathematics 2021, 9(4), 372; https://doi.org/10.3390/math9040372 - 13 Feb 2021

Cited by 27 | Viewed by 2651

Abstract

The aim of this paper is to formulate and analyze a cyclic iterative algorithm in real Hilbert spaces which converges strongly to a common solution of fixed point problem and multiple-sets split common fixed point problem involving demicontractive operators without prior knowledge of [...] Read more.

The aim of this paper is to formulate and analyze a cyclic iterative algorithm in real Hilbert spaces which converges strongly to a common solution of fixed point problem and multiple-sets split common fixed point problem involving demicontractive operators without prior knowledge of operator norm. Significance and range of applicability of our algorithm has been shown by solving the problem of multiple-sets split common null point, multiple-sets split feasibility, multiple-sets split variational inequality, multiple-sets split equilibrium and multiple-sets split monotone variational inclusion. Full article

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

22 pages, 419 KB

Open AccessArticle

An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

by Lateef Olakunle Jolaoso and Maggie Aphane

Symmetry 2020, 12(11), 1915; https://doi.org/10.3390/sym12111915 - 20 Nov 2020

Cited by 5 | Viewed by 2407

Abstract

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which [...] Read more.

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)

► Show Figures

Figure 1

13 pages, 608 KB

Open AccessArticle

Admissible Perturbation of Demicontractive Operators within Ant Algorithms for Medical Images Edge Detection

by Cristina Ticala, Ioana Zelina and Camelia-M. Pintea

Mathematics 2020, 8(6), 1040; https://doi.org/10.3390/math8061040 - 26 Jun 2020

Cited by 13 | Viewed by 3274

Abstract

Nowadays, demicontractive operators in terms of admissible perturbation are used to solve difficult tasks. The current research uses several demicontractive operators in order to enhance the quality of the edge detection results when using ant-based algorithms. Two new operators are introduced,

χ

-operator [...] Read more.

Nowadays, demicontractive operators in terms of admissible perturbation are used to solve difficult tasks. The current research uses several demicontractive operators in order to enhance the quality of the edge detection results when using ant-based algorithms. Two new operators are introduced,

χ

-operator and

K H

-operator, the latter one is a Krasnoselskij admissible perturbation of a demicontractive operator. In order to test the efficiency of the new operators, a comparison is made with a trigonometric operator. Ant Colony Optimization (ACO) is the solver chosen for the images edge detection problem. Demicontractive operators in terms of admissible perturbation are used during the construction phase of the matrix of ants artificial pheromone, namely the edge information of an image. The conclusions of statistical analysis on the results shows a positive influence of proposed operators for image edge detection of medical images. Full article

(This article belongs to the Special Issue Computational Intelligence)

► Show Figures

Figure 1

14 pages, 322 KB

Open AccessArticle

The Split Equality Fixed Point Problem of Demicontractive Operators with Numerical Example and Application

by Yaqin Wang, Jinzuo Chen and Ariana Pitea

Symmetry 2020, 12(6), 902; https://doi.org/10.3390/sym12060902 - 1 Jun 2020

Cited by 11 | Viewed by 2932

Abstract

This paper aims to propose a new reckoning method for solving the split equality fixed point problem of demicontractive operators in Hilbert spaces, and to establish a theorem with regard to the strong convergence of this new scheme. As an application, we also [...] Read more.

This paper aims to propose a new reckoning method for solving the split equality fixed point problem of demicontractive operators in Hilbert spaces, and to establish a theorem with regard to the strong convergence of this new scheme. As an application, we also consider quasi-pseudo-contractive operators and obtain a result on the solution to the split equality fixed point problem in the framework of Hilbert spaces. A numerical example is also provided. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

► Show Figures

Figure 1

10 pages, 750 KB

Open AccessArticle

Viscosity Methods and Split Common Fixed Point Problems for Demicontractive Mappings

by Yaqin Wang, Xiaoli Fang and Tae-Hwa Kim

Mathematics 2019, 7(9), 844; https://doi.org/10.3390/math7090844 - 12 Sep 2019

Cited by 6 | Viewed by 2856

Abstract

We, first, propose a new method for solving split common fixed point problems for demicontractive mappings in Hilbert spaces, and then establish the strong convergence of such an algorithm, which extends the Halpern type algorithm studied by Wang and Xu to a viscosity [...] Read more.

We, first, propose a new method for solving split common fixed point problems for demicontractive mappings in Hilbert spaces, and then establish the strong convergence of such an algorithm, which extends the Halpern type algorithm studied by Wang and Xu to a viscosity iteration. Above all, the step sizes in this algorithm are chosen without a priori knowledge of the operator norms. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

20 pages, 590 KB

Open AccessArticle

A General Algorithm for the Split Common Fixed Point Problem with Its Applications to Signal Processing

by Wachirapong Jirakitpuwapat, Poom Kumam, Yeol Je Cho and Kanokwan Sitthithakerngkiet

Mathematics 2019, 7(3), 226; https://doi.org/10.3390/math7030226 - 28 Feb 2019

Cited by 22 | Viewed by 4379

Abstract

In 2014, Cui and Wang constructed an algorithm for demicontractive operators and proved some weak convergence theorems of their proposed algorithm to show the existence of solutions for the split common fixed point problem without using the operator norm. By Cui and Wang’s [...] Read more.

In 2014, Cui and Wang constructed an algorithm for demicontractive operators and proved some weak convergence theorems of their proposed algorithm to show the existence of solutions for the split common fixed point problem without using the operator norm. By Cui and Wang’s motivation, in 2015, Boikanyo constructed also a new algorithm for demicontractive operators and obtained some strong convergence theorems for this problem without using the operator norm. In this paper, we consider a viscosity iterative algorithm in Boikanyo’s algorithm to approximate to a solution of this problem and prove some strong convergence theorems of our proposed algorithm to a solution of this problem. Finally, we apply our main results to some applications, signal processing and others and compare our algorithm with five algorithms such as Cui and Wang’s algorithm, Boikanyo’s algorithm, forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm (FISTA). Full article

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

► Show Figures

Figure 1

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI