Strong Convergence Theorems Research Articles

Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems

In this paper, we introduce a new hybrid inertial accelerated algorithm with a line search technique for solving fixed point problems for demimetric mapping and split equilibrium problems in Hilbert spaces. The algorithm is inspired by Tseng’s extragradient method and the viscosity method. Then, we establish and prove the strong convergence theorem under proper conditions. Furthermore, we also give a numerical example to support the main results. The main results are new and the proofs are relatively simple and different from those in early and recent literature.

A Strong Convergence Theorem for a Finite Family of Bregman Demimetric Mappings in a Banach Space under a New Shrinking Projection Method

In this paper, using a new shrinking projection method and new generalizedk-demimetric mappings, we consider the strong convergence for finding a common point of the sets of zero points of maximal monotone mappings, common fixed points of a finite family of Bregmank-demimetric mappings, and common zero points of a finite family of Bregman inverse strongly monotone mappings in a reflexive Banach space. To the best of our knowledge, such a theorem for Bregmank-demimetric mapping is the first of its kind in a Banach space. This manuscript is online on arXiv by the link http://arxiv.org/abs/2107.13254.

A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space

Abstract In this paper, we introduce a Totally Relaxed Self-adaptive Subgradient Extragradient Method (TRSSEM) with Halpern iterative scheme for finding a common solution of a Variational Inequality Problem (VIP) and the fixed point of quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space. The TRSSEM does not require the computation of projection onto the feasible set of the VIP; instead, it uses a projection onto a finite intersection of sub-level sets of convex functions. The advantage of this is that any general convex feasible set can be involved in the VIP. We also introduce a modified TRSSEM which involves the projection onto the set of a convex combination of some convex functions. Under some mild conditions, we prove a strong convergence theorem for our algorithm and also present an application of our theorem to the approximation of a solution of nonlinear integral equations of Hammerstein’s type. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature. Our algorithm is simple and easy to implement for computation.

Weak and Strong Convergence Theorems for Common Attractive Points of Widely More Generalized Hybrid Mappings in Hilbert Spaces

In this work, we study iterative methods for the approximation of common attractive points of two widely more generalized hybrid mappings in Hilbert spaces and obtain weak and strong convergence theorems without assuming the closedness for the domain. A numerical example supporting our main result is also presented. As a consequence, our main results can be applied to solving a common fixed point problem.

Inertial Krasnoselski–Mann Iterative Method for Solving Hierarchical Fixed Point and Split Monotone Variational Inclusion Problems with Its Applications

In this article, we discuss the hierarchical fixed point and split monotone variational inclusion problems and propose a new iterative method with the inertial terms involving a step size to avoid the difficulty of calculating the operator norm in real Hilbert spaces. A strong convergence theorem of the proposed method is established under some suitable control conditions. Furthermore, the proposed method is modified and used to derive a scheme for solving the split problems. Finally, we compare and demonstrate the efficiency and applicability of our schemes for numerical experiments as well as an example in the field of image restoration.

Inertial splitting algorithms for nonlinear operators of pseudocontractive and accretive types

ABSTRACT In this paper, inertial splitting algorithms for nonlinear operators of pseudocontractive and accretive types are proposed. Weak and strong convergence theorems are established in uniformly convex and q-uniformly smooth Banach spaces. Numerical examples are given to illustrate the effectiveness of our proposed algorithms.

On iterative methods for bilevel equilibrium problems

We use the notion of Halpern-type sequence recently introduced by the present authors to conclude two strong convergence theorems for solving the bilevel equilibrium problems proposed by Yuying et al. and some authors. Our result excludes some assumptions as were the cases in their results.

Inertial projection and contraction methods for pseudomonotone variational inequalities with non-Lipschitz operators and applications

In this paper, some new accelerated iterative schemes are proposed to solve the variational inequality problem with a pseudomonotone and uniformly continuous operator in real Hilbert spaces. Strong convergence theorems of the suggested algorithms are obtained without the prior knowledge of the Lipschitz constant of the operator. Some numerical experiments and applications are performed to illustrate the advantages of the proposed methods with respect to several related ones.

An iterative algorithm with inertial technique for solving the split common null point problem in Banach spaces

In this work, we introduce a modified inertial algorithm for solving the split common null point problem without the prior knowledge of the operator norms in Banach spaces. The strong convergence theorem of our method is proved under suitable assumptions. We apply our result to the split feasibility problem, split equilibrium problem and split minimization problem. Finally, we provide some numerical experiments including compressed sensing to illustrate the performances of the proposed method. The result presented in this paper improves and generalizes many recent important results in the literature.

English

The aim of this paper is to propose two modified forward-backward splitting algorithms for zeros of the sum of a maximal monotone operator and a Bregman inverse strongly monotone operator in reflexive Banach spaces. We prove weak and strong convergence theorems of the generated sequences by the proposed methods under some suitable conditions. We apply our results to study the variational inequality problem and the equilibrium problem. Finally, a numerical example is given to illustrate the proposed methods. The results presented in this paper improve and generalize many known results in recent literature.

The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

The generalized viscosity implicit rules of nonexpansive asymptotically mappings in Hilbert spaces are considered. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. An application of it in the convex minimization problem is considered. The results presented in this paper improve and extend some recent corresponding results in the literature.

The subgradient extragradient method for approximation of fixed-point problem and modification of equilibrium problem

ABSTRACT In this paper, we consider the modification of equilibrium problem (MEP) and new subgradient extragradient algorithm by using the concept of the set of solutions of the modified variational inequality problem introduced by [Kangtunyakarn A. A new iterative scheme for fixed-point problems of infinite family of pseudo contractive mappings, equilibrium problem, variational inequality problems. J Optim Theory Appl. 2013;56:1543–1562.]. Then, we establish and prove weak and strong convergence theorem of the new subgradient extragradient algorithm for finding a common element of the set of solutions of the MEP and two sets of the variational inequality problems under some suitable conditions on and with . Moreover, we apply our main theorem to prove weak and strong convergence theorems to solve the generalized equilibrium problem, the system of equilibrium problem, the variational inequality problem and the general system of variational inequality problems. Finally, we give two numerical examples to support our main result.

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Multi-Step Inertial Regularized Methods for Hierarchical Variational Inequality Problems Involving Generalized Lipschitzian Mappings

In this paper, we construct two multi-step inertial regularized methods for hierarchical inequality problems involving generalized Lipschitzian and hemicontinuous mappings in Hilbert spaces. Then we present two strong convergence theorems and some numerical experiments to show the effectiveness and feasibility of our new iterative methods.

A strong convergence theorem for quasi-equilibrium problems in Banach spaces

A strong convergence theorem for quasi-equilibrium problems in Banach spaces

Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

Abstract The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

Inertial Extragradient Methods for Solving Split Equilibrium Problems

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.

Three novel inertial explicit Tseng's extragradient methods for solving pseudomonotone variational inequalities

In this paper, we construct three new extragradient-type iterative methods for solving variational inequalities in real Hilbert spaces. The proposed iterative methods are functionally equivalent to the extragradient method, which is used to solve variational inequalities in an infinite-dimensional real Hilbert space. The main advantage of these iterative methods is that they use a simple step size rule based on operator information instead of the Lipschitz constant or any line search method. Three strong convergence theorems are well proved, corresponding to the proposed methods by allowing certain control parameter conditions. Finally, we present some numerical experiments to verify the efficacy and superiority of the proposed iterative methods.

Self-adaptive inertial single projection methods for variational inequalities involving non-Lipschitz and Lipschitz operators with their applications to optimal control problems

In this paper, four accelerated subgradient extragradient methods are proposed to solve the variational inequality problem with a pseudo-monotone operator in real Hilbert spaces. These iterative schemes employ two new adaptive stepsize strategies that are significant when the Lipschitz constant of the mapping involved is unknown. Strong convergence theorems for the proposed algorithms are established under the condition that the operators are Lipschitz continuous and non-Lipschitz continuous. Numerical experiments on finite- and infinite-dimensional spaces and applications in optimal control problems are reported to demonstrate the advantages and efficiency of the proposed algorithms over some existing results.

Generalized Hybrid Viscosity-Type Forward-Backward Splitting Method with Application to Convex Minimization and Image Restoration Problems

Let E be a uniformly convex and q-uniformly smooth real Banach space. Let be an α- inverse strongly accretive mapping of order q, be a set-valued m- accretive mapping and be a nonexpansive mapping. In this article, a viscosity-type forward-backward splitting method for approximating a zero of (A + B) which is also a fixed point of S is introduced studied. Strong convergence theorem of the method is proved under suitable conditions. Furthermore, the convergence result obtained is applied to convex minimization and image restoration problems. Finally, numerical illustrations are presented to compare the convergence of the sequence of our algorithm and that of some recent important algorithms.