scholarly journals Convergence Theorems for the Variational Inequality Problems and Split Feasibility Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Panisa Lohawech ◽  
Anchalee Kaewcharoen ◽  
Ali Farajzadeh
Keyword(s):  
Variational Inequality ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Descent Method ◽  
Variational Inequality Problems ◽  
Steepest Descent Method ◽  
Split Feasibility Problems ◽  
The Common ◽  
Split Feasibility

In this paper, we establish an iterative algorithm by combining Yamada’s hybrid steepest descent method and Wang’s algorithm for finding the common solutions of variational inequality problems and split feasibility problems. The strong convergence of the sequence generated by our suggested iterative algorithm to such a common solution is proved in the setting of Hilbert spaces under some suitable assumptions imposed on the parameters. Moreover, we propose iterative algorithms for finding the common solutions of variational inequality problems and multiple-sets split feasibility problems. Finally, we also give numerical examples for illustrating our algorithms.

scholarly journals Iterative methods for generalized split feasibility problems in Banach spaces

2017 ◽  
Vol 33 (1) ◽  
pp. 09-26
Author(s):  
QAMRUL HASAN ANSARI ◽  
◽  
AISHA REHAN ◽  
◽  
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Split Feasibility Problems ◽  
Convergence Results ◽  
Split Feasibility

Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205–221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, SetValued Var. Anal., 23 (2015), 205–221].

scholarly journals A Parallel-Viscosity-Type Subgradient Extragradient-Line Method for Finding the Common Solution of Variational Inequality Problems Applied to Image Restoration Problems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 248 ◽  
Author(s):  
Suthep Suantai ◽  
Pronpat Peeyada ◽  
Damrongsak Yambangwai ◽  
Watcharaporn Cholamjiak
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Hybrid Algorithm ◽  
Variational Inequality Problems ◽  
Strong Convergence Theorem ◽  
Infinite Dimensional ◽  
Common Solution ◽  
Line Method ◽  
The Common ◽  
Monotone Hybrid Algorithm

In this paper, we study a modified viscosity type subgradient extragradient-line method with a parallel monotone hybrid algorithm for approximating a common solution of variational inequality problems. Under suitable conditions in Hilbert spaces, the strong convergence theorem of the proposed algorithm to such a common solution is proved. We then give numerical examples in both finite and infinite dimensional spaces to justify our main theorem. Finally, we can show that our proposed algorithm is flexible and has good quality for use with common types of blur effects in image recovery.

scholarly journals An Inertial Iterative Algorithm to Find Common Solution of a Split Generalized Equilibrium and a Variational Inequality Problem in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Mohammad Farid ◽  
Rehan Ali ◽  
Watcharaporn Cholamjiak
Keyword(s):  
Variational Inequality ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Variational Inequality Problem ◽  
Field Of Study ◽  
Inequality Problem ◽  
Numerical Example ◽  
Common Solution ◽  
Generalized Equilibrium ◽  
The Common

In this paper, we introduce and study an iterative algorithm via inertial and viscosity techniques to find a common solution of a split generalized equilibrium and a variational inequality problem in Hilbert spaces. Further, we prove that the sequence generated by the proposed theorem converges strongly to the common solution of our problem. Furthermore, we list some consequences of our established algorithm. Finally, we construct a numerical example to demonstrate the applicability of the theorem. We emphasize that the result accounted in the manuscript unifies and extends various results in this field of study.

scholarly journals Gap Functions and Algorithms for Variational Inequality Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Congjun Zhang ◽  
Baoqing Liu ◽  
Jun Wei
Keyword(s):  
Variational Inequality ◽  
Error Bounds ◽  
Optimization Problem ◽  
Variational Inequality Problem ◽  
Descent Method ◽  
Variational Inequality Problems ◽  
Steepest Descent Method ◽  
Gap Functions ◽  
Constraint Optimization ◽  
Mixed Variational Inequality

We solve several kinds of variational inequality problems through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis. By generalized gap functions and generalized D-gap functions, we give global bounds for the set-valued mixed variational inequality problems. And through gap function, we equivalently transform the generalized variational inequality problem into a constraint optimization problem, give the steepest descent method, and show the convergence of the method.

scholarly journals Viscosity Approximation Methods with Errors and Strong Convergence Theorems for a Common Point of Pseudocontractive and Monotone Mappings: Solutions of Variational Inequality Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yan Tang
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Variational Inequality Problems ◽  
Monotone Mappings ◽  
Convergence Theorems ◽  
Common Solution ◽  
The Common ◽  
Viscosity Approximation Methods

We introduce two proximal iterative algorithms with errors which converge strongly to the common solution of certain variational inequality problems for a finite family of pseudocontractive mappings and a finite family of monotone mappings. The strong convergence theorems are obtained under some mild conditions. Our theorems extend and unify some of the results that have been proposed for this class of nonlinear mappings.

scholarly journals Inertial Subgradient Extragradient Methods for Solving Variational Inequality Problems and Fixed Point Problems

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 51
Author(s):  
Godwin Amechi Okeke ◽  
Mujahid Abbas ◽  
Manuel de la Sen
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
Variational Inequality Problems ◽  
Subgradient Extragradient Method ◽  
Viscosity Approximation ◽  
Extragradient Methods ◽  
Mann Algorithm

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods.

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