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

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 773 ◽  
Author(s):  
Mihai Postolache ◽  
Ashish Nandal ◽  
Renu Chugh
Keyword(s):  
Fixed Point ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Special Cases ◽  
Implicit Rule

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.

scholarly journals Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 655 ◽  
Author(s):  
Ashish Nandal ◽  
Renu Chugh ◽  
Mihai Postolache
Keyword(s):  
Fixed Point ◽  
Recent Literature ◽  
Maximal Monotone ◽  
Common Element ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Monotone Mappings ◽  
Point Problem ◽  
Maximal Monotone Mappings ◽  
Special Cases

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature.

scholarly journals Approximating Iterations for Nonexpansive and Maximal Monotone Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Zhangsong Yao ◽  
Sun Young Cho ◽  
Shin Min Kang ◽  
Li-Jun Zhu
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Minimum Norm ◽  
Nonexpansive Operator

We present two algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive operator in Hilbert spaces. We show that these two algorithms converge strongly to the minimum norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive operator.

scholarly journals A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Young-Ye Huang ◽  
Chung-Chien Hong
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Zero Point ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Special Cases ◽  
Split Feasibility

We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.

Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

2021 ◽  
pp. 2250106
Author(s):  
A. A. Mebawondu ◽  
L. O. Jolaoso ◽  
H. A. Abass ◽  
O. K. Narain
Keyword(s):  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Inertial Method ◽  
Split Feasibility

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

scholarly journals Relaxed Inertial Tseng’s Type Method for Solving the Inclusion Problem with Application to Image Restoration

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 818 ◽  
Author(s):  
Jamilu Abubakar ◽  
Poom Kumam ◽  
Abdulkarim Hassan Ibrahim ◽  
Anantachai Padcharoen
Keyword(s):  
Lipschitz Constant ◽  
Monotone Mapping ◽  
Splitting Method ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Variable Step Size ◽  
Step Size ◽  
Type Method

The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods.

scholarly journals Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, Variational Inequalities and Mixed Equilibrium Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-31 ◽  
Author(s):  
Kriengsak Wattanawitoon ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Zero Point ◽  
Monotone Operators ◽  
Common Element ◽  
Proximal Point ◽  
Strong And Weak Convergence ◽  
Mixed Equilibrium Problems ◽  
Inverse Strongly Monotone

We prove strong and weak convergence theorems of modified hybrid proximal-point algorithms for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to complementarity problems are given. Our results modify and improve the recently announced ones by Li and Song (2008) and many authors.

scholarly journals On an Iterative Method for Finding a Zero to the Sum of Two Maximal Monotone Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hongwei Jiao ◽  
Fenghui Wang
Keyword(s):  
Weak Convergence ◽  
Iterative Method ◽  
Variational Inequalities ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Gradient Projection ◽  
Projection Algorithms

In this paper we consider a problem that consists of finding a zero to the sum of two monotone operators. One method for solving such a problem is the forward-backward splitting method. We present some new conditions that guarantee the weak convergence of the forward-backward method. Applications of these results, including variational inequalities and gradient projection algorithms, are also considered.

A weakly convergent fully inexact Douglas-Rachford method with relative error tolerance

2019 ◽  
Vol 25 ◽  
pp. 57
Author(s):  
Benar Fux Svaiter
Keyword(s):  
Relative Error ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Error Tolerance ◽  
Inclusion Problem ◽  
Splitting Algorithm ◽  
Sequential Solution

Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Each of its iterations requires the sequential solution of two proximal subproblems. The aim of this work is to present a fully inexact version of Douglas-Rachford method wherein both proximal subproblems are solved approximately within a relative error tolerance. We also present a semi-inexact variant in which the first subproblem is solved exactly and the second one inexactly. We prove that both methods generate sequences weakly convergent to the solution of the underlying inclusion problem, if any.

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