Viscosity Approximation Method Research Articles

Common Attractive Points of Generalized Hybrid Multi-Valued Mappings and Applications

In this paper, we first propose the concepts of (ζ,η,λ,π)-generalized hybrid multi-valued mappings, the set of all the common attractive points (CAf,g) and the set of all the common strongly attractive points (CsAf,g), respectively for the multi-valued mappings f and g in a CAT(0) space. Moreover, we give some elementary properties in regard to the sets Af, Ff and CAf,g for the multi-valued mappings f and g in a complete CAT(0) space. Furthermore, we present a weak convergence theorem of common attractive points for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in the above space by virtue of Banach limits technique and Ishikawa iteration respectively. Finally, we prove strong convergence of a new viscosity approximation method for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in CAT(0) spaces, which also solves a kind of variational inequality problem.

Krasnoselskii–Mann Viscosity Approximation Method for Nonexpansive Mappings

We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., uniform smoothness) of the underlying space plays a role in relaxing the conditions on the choice of regularization parameters and step sizes in iterative methods.

A self-adaptive inertial subgradient extragradient method for pseudomonotone equilibrium and common fixed point problems

In this paper, we introduce a self-adaptive inertial subgradient extragradient method for solving pseudomonotone equilibrium problem and common fixed point problem in real Hilbert spaces. The algorithm consists of an inertial extrapolation process for speeding the rate of its convergence, a monotone nonincreasing stepsize rule, and a viscosity approximation method which guaranteed its strong convergence. More so, a strong convergence theorem is proved for the sequence generated by the algorithm under some mild conditions and without prior knowledge of the Lipschitz-like constants of the equilibrium bifunction. We further provide some numerical examples to illustrate the performance and accuracy of our method.

On the Monotone Mappings in CAT(0) Spaces

In this paper, we first introduce a monotone mapping and its resolvent in general metric spaces.Then, we give two new iterative methods by combining the resolvent method with Halpern's iterative method and viscosity approximation method for finding a fixed point of monotone mappings and a solution of variational inequalities. We prove convergence theorems of the proposed iterations in CAT(0) metric spaces.

Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces

We propose a parallel iterative scheme with viscosity approximation method which converges strongly to a solution of the multiple-set split equality common fixed point problem for quasi-pseudocontractive mappings in real Hilbert spaces. We also give an application of our result to approximation of minimization problem from intensity-modulated radiation therapy. Finally, we present numerical examples to demonstrate the behaviour of our algorithm. This result improves and generalizes many existing results in literature in this direction.

The Method for Solving Fixed Point Problem of G-Nonexpansive Mapping in Hilbert Spaces Endowed with Graphs and Numerical Example

The main aim of this paper is to study a strong convergence theorem of viscosity approximation method for G-nonexpansive mapping defined on a Hilbert space endowed with a directed graph. By using our main result, we give a numerical expample to approximate the value of π.

Viscosity approximation methods for generalized asymptotically nonexpansive mappings and monotone inclusions in Hadamard spaces

Viscosity approximation methods for generalized asymptotically nonexpansive mappings and monotone inclusions in Hadamard spaces

Viscosity approximation methods for split common fixed point problems without prior knowledge of the operator norm

In this work, we study the split common fixed point problem which was first introduced by Censor and Segal [14]. We introduce an algorithm based on the viscosity approximation method without prior knowledge of the operator norm by selecting the stepsizes in the same adaptive way as L?opez et al. [22] for solving the problem for two attracting quasi-nonexpansive operators in real Hilbert spaces. A strong convergence result of the proposed algorithm is established under some suitable conditions. We also modify our algorithm to extend to the class of demicontractive operators and the class of hemicontractive operators, and obtain strong convergence results. Moreover, we apply our main result to other split problems, that is, the split feasibility problem and the split variational inequality problem. Finally, a numerical result is also given to illustrate the convergence behavior of our algorithm.

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.

Strong convergence of viscosity forward-backward algorithm to the sum of two accretive operators in Banach space

ABSTRACT In recent years, the Forward-Backward algorithm (FBA) received much attention due to its various applications in image recovery, signal processing, and machine learning. In this paper, we consider the FBA in the setting of Banach spaces that are uniformly convex and q-uniformly smooth. We introduce two viscosity FBA, and one of them with weakly contractive mapping, which generalizes many previous results on the viscosity approximation method with fixed contraction. Moreover, we establish their strong convergences under more general conditions.

Convergence Theorems of Variational Inequality for Asymptotically Nonexpansive Nonself Mapping in CAT(0) Spaces

The aim of this manuscript is to get the strong convergence theorems of the Moudafi’s viscosity approximation methods for an asymptotically nonexpansive nonself mapping in C A T ( 0 ) spaces.

Tseng Type Methods for Inclusion and Fixed Point Problems with Applications

An algorithm is introduced to find an answer to both inclusion problems and fixed point problems. This algorithm is a modification of Tseng type methods inspired by Mann’s type iteration and viscosity approximation methods. On certain conditions, the iteration obtained from the algorithm converges strongly. Moreover, applications to the convex feasibility problem and the signal recovery in compressed sensing are considered. Especially, some numerical experiments of the algorithm are demonstrated. These results are compared to those of the previous algorithm.

Bounded perturbation resilience and superiorization techniques for a modified proximal gradient method

ABSTRACTIn this article, we combine the viscosity approximation method and proximal operator to propose modified proximal gradient methods for solving the unconstrained convex optimization problems. The bounded perturbation resilience of these methods is investigated in the general Hilbert space H. Under reasonable parameter conditions, we prove that our algorithms strongly converge to the unique solution of a variational inequality problem. Our results improve and extend the corresponding results reported by many authors recently.

Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Viscosity method for generalized equilibrium problems with perturbation and nonexpansive mappings in Hilbert spaces

In this paper, we introduce a new iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem with perturbation and the fixed points for a new kind of combination of two nonexpansive mapping in Hilber space, we prove that the sequence converges strongly to a common element of two sets. Our results generalize and improve some results in the litretures.

Generalized viscosity implicit scheme with Meir-Keeler contraction for asymptotically nonexpansive mapping in Banach spaces

Our focus in this paper is on introducing an iterative scheme based on the generalized implicit method and viscosity approximation method with Meir-Keeler contraction for solving a fixed point problem of an asymptotically nonexpansive mapping in the framework of Banach spaces. The strong convergence results of the proposed iterative scheme are established. An application based on convex minimization problem is derived from our main result. The numerical applicability and efficiency of our results are demonstrated using some examples. Our results extend, generalize and unify the previously known results given in the literature.

Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings

ABSTRACT In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Approximate solution of zero point problem involving H-accretive maps in Banach spaces and applications

In this manuscript, we introduce two iterative methods for finding the common zeros of two H-accretive mappings in uniformly smooth and uniformly convex Banach spaces. The proposed iterative methods are based on Mann and Halpern iterative methods and viscosity approximation method. Strong convergence results are established for iterative algorithms. Applications based on convex minimization problem, variational inequality problem and equilibrium problem are derived from the main result. Numerical implementation of the main results and application are demonstrated by some examples. Our results extend, generalize, and unify the previously known results given in literature.

Accelerated Subgradient Extragradient Methods for Variational Inequality Problems

In this paper, we introduce two new iterative algorithms for solving monotone variational inequality problems in real Hilbert spaces, which are based on the inertial subgradient extragradient algorithm, the viscosity approximation method and the Mann type method, and prove some strong convergence theorems for the proposed algorithms under suitable conditions. The main results in this paper improve and extend some recent works given by some authors. Finally, the performances and comparisons with some existing methods are presented through several preliminary numerical experiments.

Convergence of a hybrid viscosity approximation method for finding zeros of m-accretive operators

For finding a zero of an m-accretive operator in a real Banach space, we consider an implicit hybrid viscosity approximation method and show that its explicit variant converges strongly to a zero under weaker assumptions than the ones used recently by Ceng et al. (Numer. Func. Anal. Opt. 35(2), 142–165, 2012). First, we examine the case of spaces which are uniformly smooth or reflexive and strictly convex with a uniformly Gâteaux differentiable norm. Afterwards, we improve our convergence results when the space is uniformly convex. Furthermore, we show that the strong convergence of some modified Halpern and Krasnosel’skii–Mann type methods can also be deduced from our results. Finally, a numerical example is also given to illustrate the convergence analysis of the considered method.