Infinite Family Of Nonexpansive Mappings Research Articles

Viscosity Approximation Methods for Generalized Modification of the System of Equilibrium Problem and Fixed Point Problems of an Infinite Family of Nonexpansive Mappings

Viscosity Approximation Methods for Generalized Modification of the System of Equilibrium Problem and Fixed Point Problems of an Infinite Family of Nonexpansive Mappings

A modified inertial viscosity algorithm for an infinite family of nonexpansive mappings and its application to image restoration

A modified inertial viscosity algorithm for an infinite family of nonexpansive mappings and its application to image restoration

Iterative Algorithm of Split Monotone Variational Inclusion Problem for New Mappings

In this paper, we developed a new type iterative scheme to approximate a common solution of split monotone variational inclusion, variational inequality and fixed point problems for an infinite family of nonexpansive mappings in the framework of Hilbert spaces. Further, we proved that the sequence generated by the proposed iterative method converges strongly to a common solution of split monotone variational inclusion, variational inequality and fixed point problems. Furthermore, we give some consequences of the main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm. The result presented in this paper unifies and extends some known results in this area.

Strong convergence theorems for split variational inequality problems in Hilbert spaces

<abstract><p>In this paper, we consider the variational inequality problem and the split common fixed point problem. Considering the common fixed points of an infinite family of nonexpansive mappings, instead of just the fixed point of one nonexpansive mapping, we generalize the results of Tian and Jiang. By removing a projection operator, we improve the efficiency of our algorithm. Finally, we propose a very simple modification to the extragradient method, which gives our algorithm strong convergence properties. We also provide some numerical examples to illustrate our main results.</p></abstract>

A Fast Fixed-Point Algorithm for Convex Minimization Problems and Its Application in Image Restoration Problems

In this paper, we introduce a new iterative method using an inertial technique for approximating a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space. The proposed method’s weak convergence theorem was established under some suitable conditions. Furthermore, we applied our main results to solve convex minimization problems and image restoration problems.

An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings

In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction.

An iterative scheme for split monotone variational inclusion, variational inequality and fixed point problems

We propose and analyze a new type iterative algorithm to find a common solution of split monotone variational inclusion, variational inequality, and fixed point problems for an infinite family of nonexpansive mappings in the framework of Hilbert spaces. Further, we show that a sequence generated by the algorithm converges strongly to common solution. Furthermore, we list some consequences of our established theorem. Finally, we provide a numerical example to demonstrate the applicability of the algorithm. We emphasize that the result accounted in manuscript unifies and extends various results in this field of study.

A New Accelerated Viscosity Iterative Method for an Infinite Family of Nonexpansive Mappings with Applications to Image Restoration Problems

The image restoration problem is one of the popular topics in image processing which is extensively studied by many authors because of its applications in various areas of science, engineering and medical image. The main aim of this paper is to introduce a new accelerated fixed algorithm using viscosity approximation technique with inertial effect for finding a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and prove a strong convergence result of the proposed method under some suitable control conditions. As an application, we apply our algorithm to solving image restoration problem and compare the efficiency of our algorithm with FISTA method which is a popular algorithm for image restoration. By numerical experiments, it is shown that our algorithm has more efficiency than that of FISTA.

Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

Hybrid viscosity approximation methods for systems of variational inequalities and hierarchical fixed point problems

In this paper, we propose an implicit iterative method and an explicit iterative method for solving a general system of variational inequalities with a hierarchical fixed point problem constraint for an infinite family of nonexpansive mappings. We show that the proposed algorithms converge strongly to a solution of the general system of variational inequalities, which is a unique solution of the hierarchical fixed point problem.

A hybrid method for solving variational inequalities over the common fixed point sets of infinite families of nonexpansive mappings in Banach spaces

ABSTRACT In this paper, we introduce a hybrid method, a combination of the steepest-descent method and the Krasnosel'skii-Mann one, for solving a variational inequality over the set of common fixed points of an infinite family of nonexpansive mappings in Banach spaces under two different conditions on the Banach space, either a uniformly smooth Banach space or a reflexive and strictly convex one with a uniformly Gâteaux differentiable norm, without imposing the sequential weak continuity of the normalized duality mapping. The method is an improvement and extension of some other published results. We also give a numerical example to illustrate the convergence analysis of the proposed method.

Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.

An inertial forward–backward splitting method for solving combination of equilibrium problems and inclusion problems

In this paper, we prove a weak convergence theorem for finding a common solution of combination of equilibrium problems, infinite family of nonexpansive mappings, and the modified inclusion problems using inertial forward–backward algorithm. Further, we discuss some applications of our obtained results. Furthermore, we provide some numerical results to illustrate the convergence behavior of some of our results, and compare the convergence rate between the existing projection method and the proposed inertial forward–backward algorithm.

Modified Proximal Point Algorithms for Solving Constrained Minimization and Fixed Point Problems in Complete CAT(0) Spaces

In this paper, we propose a new modified proximal point algorithm for a countably infinite family of nonexpansive mappings in complete CAT(0) spaces and prove strong convergence theorems for the proposed process under suitable conditions. We also apply our results to solving linear inverse problems and minimization problems. Several numerical examples are given to show the efficiency of the presented method.

Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems

The purpose of this paper is to introduce a general iterative method for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of split variational inclusion problem in the framework Hilbert spaces. Strong convergence theorem of the sequences generated by the purpose iterative scheme is obtained. In the last section, we present some computational examples to illustrate the assumptions of the proposed algorithms.

A New Simple Parallel Iteration Method for a Class of Variational Inequalities

In this paper, we propose a new simple parallel iterative method to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm. Our parallel iterative method is simpler than the one proposed by Buong et al. (Numer. Algorithms 72, 467–481 2016). An iterative method of Halpern type for common zeros of an infinite family of m-accretive mappings is shown as a special case of our result. Two numerical examples are also given to illustrate the effectiveness and superiority of the proposed algorithm.

General iteration scheme for finding the common fixed points of an infinite family of nonexpansive mappings

General iteration scheme for finding the common fixed points of an infinite family of nonexpansive mappings

Modified general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces

In this paper we introduce an iterative method for finding a common fixed point of an infinite family of nonexpansive mappings in q-uniformly real smooth Banach space which is also uniformly convex. We proved strong convergence of the proposed iterative algorithms to the unique solution of a variational inequality problem.

New ergodic convergence theorems for non-expansive mappings and m-accretive mappings

Two new ergodic convergence theorems for approximating the common element of the set of zero points of an m-accretive mapping and the set of fixed points of an infinite family of non-expansive mappings in a real smooth and uniformly convex Banach space are obtained, which improves some of the previous work. The computational experiments to demonstrate the effectiveness of the proposed iterative algorithms in this paper are conducted.

A new explicit iteration method for a class of variational inequalities

In this paper, we propose a new simple explicit iterative algorithm to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm. Two numerical examples also are given for illustration.