scholarly journals A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-29 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Banach Spaces ◽  
Monotone Mapping ◽  
Maximal Monotone ◽  
General System ◽  
Common Element ◽  
Variational Inclusions ◽  
Common Solution ◽  
Uniformly Smooth ◽  
Nonexpansive Semigroups ◽  
General Iterative Method

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given.

scholarly journals Algorithms for General System of Generalized Resolvent Equations with Corresponding System of Variational Inclusions

2012 ◽  
Vol 2012 ◽  
pp. 1-24
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Banach Spaces ◽  
General System ◽  
Approximate Solutions ◽  
Variational Inclusions ◽  
Convergence Criteria ◽  
Generalized Resolvent ◽  
Resolvent Equations ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces ◽  
Generalized Resolvent Equations

Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones.

scholarly journals System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 5
Author(s):  
Lu-Chuan Ceng ◽  
Mihai Postolache ◽  
Xiaolong Qin ◽  
Yonghong Yao
Keyword(s):  
Banach Spaces ◽  
Infinite Family ◽  
General System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Variational Inclusions ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Mild Conditions ◽  
Pseudocontractive Mappings

The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point problem, we construct an implicit multiple-viscosity approximation method for solving the GSVI. Under very mild conditions, we prove the strong convergence of the proposed method to a solution of the GSVI with the HVI constraint, for infinitely many pseudocontractions.

scholarly journals On Variational Inclusion and Common Fixed Point Problems inq-Uniformly Smooth Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Yanlai Song ◽  
Huiying Hu ◽  
Luchuan Ceng
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Algorithm ◽  
Infinite Family ◽  
General System ◽  
Iterative Sequence ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Fixed Point Set ◽  
Uniformly Smooth

We introduce a general iterative algorithm for finding a common element of the common fixed-point set of an infinite family ofλi-strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in aq-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent.

Three-Step Iterative Method for Completely Generalized Set-Valued Strongly Nonlinear Quasi-Variational Inclusions

2013 ◽  
Vol 710 ◽  
pp. 598-602
Author(s):  
Bao Di Fang
Keyword(s):  
Iterative Method ◽  
Monotone Mapping ◽  
Maximal Monotone ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Point Problem ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Strongly Nonlinear ◽  
Resolvent Operator Technique

In this paper, we introduce and study a new class of completely generalized set-valued strongly nonlinear variational inclusions in Hilbert spaces and establish the equivalence between this variational inclusion and the fixed-point problem by using the resolvent operator technique for maximal monotone mapping. We construct a new three-step iterative algorithm and show the existence of solution for this variational inclusion and the convergence of the iterative method generated by the iterative method.

scholarly journals Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hongjie Liu ◽  
Junqing Wang ◽  
Qiansheng Feng
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Monotone Mapping ◽  
Maximal Monotone ◽  
Iterative Algorithms ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Convergence Theorems ◽  
Nonspreading Mappings

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mappingTand the solution sets of zero of a maximal monotone mapping and anα-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.

scholarly journals Modified Inertial Forward–Backward Algorithm in Banach Spaces and Its Application

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1365
Author(s):  
Yanlai Song ◽  
Mihai Postolache
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Unique Solution ◽  
Variational Inequality Problem ◽  
Inclusion Problem ◽  
Numerical Examples ◽  
Common Solution ◽  
Variational Inclusion Problem ◽  
Uniformly Smooth

In this paper, we present a new modified inertial forward–backward algorithm for finding a common solution of the quasi-variational inclusion problem and the variational inequality problem in a q-uniformly smooth Banach space. The proposed algorithm is based on descent, splitting and inertial ideas. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the abovementioned problems. Numerical examples are also given to demonstrate our results.

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