Finite Family Of Nonexpansive Mappings Research Articles

Superiorization with a Projected Subgradient Algorithm on the Solution Sets of Common Fixed Point Problems

In this work, we investigate a minimization problem with a convex objective function on a domain, which is the solution set of a common fixed point problem with a finite family of nonexpansive mappings. Our algorithm is a combination of a projected subgradient algorithm and string-averaging projection method with variable strings and variable weights. This algorithm generates a sequence of iterates which are approximate solutions of the corresponding fixed point problem. Additionally, either this sequence also has a minimizing subsequence for our optimization problem or the sequence is strictly Fejer monotone regarding the approximate solution set of the common fixed point problem.

A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

In this paper, based on the viscosity approximation method and the hybrid steepest-descent iterative method, a new implicit iterative algorithm is presented for finding the common fixed points set of a finite family of nonexpansive mappings in a reflexive Hilbert space, which is called a symmetric space. We prove that the sequence generated by this new implicit rule strongly converges to the unique solution of a class of variational inequalities under certain appropriate conditions of the parameters. Moreover, we also study the applications to a broader family of strictly pseudo-contractive mappings and generalized equilibrium problems that involve several variational inequality problems, optimization problems, and fixed-point problems. Finally, numerical results are provided to clarify the stability and effectiveness of the algorithm and to compare with some existing iterative algorithms.

Generalized viscosity approximation method for solving split generalized mixed equilibrium problem with application to compressed sensing

<abstract><p>In this study, we establish a new inertial generalized viscosity approximation method and prove that the resulting sequence strongly converges to a common solution of a split generalized mixed equilibrium problem, fixed point problem for a finite family of nonexpansive mappings and hierarchical fixed point problem in real Hilbert spaces. As an application, we demonstrate the use of our main finding in compressed sensing in signal processing. Additionally, we include numerical examples to evaluate the efficiency of the suggested method and then conduct a comparative analysis of its efficiency with different methods. Our findings can be used in a variety of contexts to improve results.</p></abstract>

Convergence theorem for an intermixed iteration in $p$-uniformly convex metric space

In this paper, we first introduce the intermixed algorithm in $p$-uniformly convex metric spaces, and then we prove $\Delta$-convergence of the proposed iterative method for finding a common element of the sets of fixed points of finite families of nonexpansive mappings in the framework of complete $p$-uniformly convex metric spaces. Furthermore, we apply our main theorem to prove $\Delta$-convergence to solve the minimization problems in the framework of complete $p$-uniformly convex metric spaces. Finally, we give two examples in $L^p$ spaces and numerical examples to support our main results.

A Modified Inertial Parallel Viscosity-Type Algorithm for a Finite Family of Nonexpansive Mappings and Its Applications

In this work, we aim to prove the strong convergence of the sequence generated by the modified inertial parallel viscosity-type algorithm for finding a common fixed point of a finite family of nonexpansive mappings under mild conditions in real Hilbert spaces. Moreover, we present the numerical experiments to solve linear systems and differential problems using Gauss–Seidel, weight Jacobi, and successive over relaxation methods. Furthermore, we provide our algorithm to show the efficiency and implementation of the LASSO problems in signal recovery. The novelty of our algorithm is that we show that the algorithm is efficient compared with the existing algorithms.

A simple proof for Kazmi et al.'s iterative scheme

In this paper, a simple proof for the existence iterative scheme using two Hilbert spaces due to Kazmi et al.[K.R. Kazmi, R. Ali, M. Furkan, Hybrid iterative method for split monotone variational inclusion problemand hierarchical fixed point problem for a finite family of nonexpansive mappings, Numer. Algor., 2017] isprovided.

Modified Mann Subgradient-like Extragradient Rules for Variational Inequalities and Common Fixed Points Involving Asymptotically Nonexpansive Mappings

In a real Hilbert space, we aim to investigate two modified Mann subgradient-like methods to find a solution to pseudo-monotone variational inequalities, which is also a common fixed point of a finite family of nonexpansive mappings and an asymptotically nonexpansive mapping. We obtain strong convergence results for the sequences constructed by these proposed rules. We give some examples to illustrate our analysis.

On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint.

Convergence results for modified SP-iteration in uniformly convex metric spaces

In this paper, we prove a strong convergence theorem of a Modified SP-iteration for finding a common fixed point of the combination of a finite family of nonexpansive mappings in a convex metric space. Moreover, we give some numerical example for supporting our main theorem and compare convergence rate between the modified SP-iteration and the Ishikawa iteration.

Mann-Type Inertial Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points of Nonexpansive and Quasi-Nonexpansive Mappings

Suppose that in a real Hilbert space H, the variational inequality problem with Lipschitzian and pseudomonotone mapping A and the common fixed-point problem of a finite family of nonexpansive mappings and a quasi-nonexpansive mapping with a demiclosedness property are represented by the notations VIP and CFPP, respectively. In this article, we suggest two Mann-type inertial subgradient extragradient iterations for finding a common solution of the VIP and CFPP. Our iterative schemes require only calculating one projection onto the feasible set for every iteration, and the strong convergence theorems are established without the assumption of sequentially weak continuity for A. Finally, in order to support the applicability and implementability of our algorithms, we make use of our main results to solve the VIP and CFPP in two illustrating examples.

An inertial iterative method for split generalized vector mixed equilibrium and fixed point problems.

In this paper, we introduce an inertial-type algorithm for approximating a common solution of split generalized mixed vector equilibrium and fixed point problems. In the framework of real Hilbert spaces, we state and prove a strong convergence theorem for obtaining a common solution of split generalized mixed vector equilibrium problem and fixed point of a finite family of nonexpansive mappings. Furthermore, we give some consequences of our main result and also report some numerical illustrations to display the performance of our method. The result obtained in this paper unifies and generalizes other corresponding results in the literature.

An Iterative Method for a Common Solution of a Combination of the Split Equilibrium Problem, a Finite Family of Nonexpansive Mapping and a Combination of Variational Inequality Problem

The present paper aims to deal with an iterative algorithm for finding common solution of the combination of the split equilibrium problem and a finite family of non-expansive mappings and the combination of variational inequality problem in the setting of real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution to these problems. A numerical example is presented to illustrate the proposed method and convergence result. The results and method presented in this paper generalize, extend and unify some known results in the literatures.

TWO KINDS OF CONVERGENCES IN HYPERBOLIC SPACES IN THREE-STEP ITERATIVE SCHEMES

In this paper, we introduce a new three-step iterative scheme for three finite families of nonexpansive mappings in hyperbolic spaces. And, we establish a strong convergence and a Δ-convergence of a given iterative scheme to a common fixed point for three finite families of nonexpansive mappings in hyperbolic spaces. Our results generalize and unify the several main results of [1, 4, 5, 9].

General Iterative Scheme for Split Mixed Equilibrium Problems, Variational Inequality Problems and Fixed Point Problems in Hilbert Spaces

The purpose in this paper is to study the strong convergence of general iterative scheme to find a common element of the set of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping and the set of solutions of split mixed equilibrium problem. Our results extend recent results announced by many others.

Multistep hybrid viscosity method for split monotone variational inclusion and fixed point problems in Hilbert spaces

In this paper, we present a multi-step hybrid iterative method. It is proven that under appropriate assumptions, the proposed iterative method converges strongly to a common element of fixed point of a finite family of nonexpansive mappings, the solution set of split monotone variational inclusion problem and the solution set of triple hierarchical variational inequality problem (THVI) in real Hilbert spaces. In addition, we give a numerical example of a triple hierarchical system derived from our generalization.

Iterative Methods for Finding Solutions of a Class of Split Feasibility Problems over Fixed Point Sets in Hilbert Spaces

We consider the split feasibility problem in Hilbert spaces when the hard constraint is common solutions of zeros of the sum of monotone operators and fixed point sets of a finite family of nonexpansive mappings, while the soft constraint is the inverse image of a fixed point set of a nonexpansive mapping. We introduce iterative algorithms for the weak and strong convergence theorems of the constructed sequences. Some numerical experiments of the introduced algorithm are also discussed.

Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward–backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

A new mapping for finding a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem

In this paper, we introduce and study a general iterative algorithm to approximate a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. Further, we prove a strong convergence theorem for the sequences generated by the proposed iterative scheme. Finally, we derive some consequences from our main result. The results presented in this paper extended and unify many of the previously known results in this area.

Iterative methods for fixed points and zero points of nonlinear mappings with applications

ABSTRACT The purpose of this article is to investigate a descent-type iterative algorithm for finding a common element of fixed-point sets of a finite family of nonexpansive mappings and zero-point sets of pseudomonotone mappings in Hilbert spaces. With suitable assumptions, we derive necessary and sufficient conditions for the strong convergence of the algorithm. Numerical examples and applications to signal processing problems are provided to support the main results.

Fixed point results in convex metric spaces

The purpose of this paper is to establish and prove a strong convergence theorem of Ishikawa iterative for finding a common fixed point of the combination of a finite family of nonexpansive mappings in a convex metric space. Moreover, our result generalizes and modifies the other result, Phuengrattana and Suantai’s result Phuengrattana and Suantai (Ind J Pure Appl Math 45(1):121–136, 2014).