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.

scholarly journals Viscosity approximation method for solving variational inequality problem in real Banach spaces

2021 ◽  
pp. 1-20
Author(s):  
Godwin Ugwunnadi
Keyword(s):  
Variational Inequality ◽  
Banach Spaces ◽  
Approximation Method ◽  
Variational Inequality Problem ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Inequality Problem ◽  
Mild Conditions ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

In this paper, we study the implicit and inertial-type viscosity approximation method for approximating a solution to the hierarchical variational inequality problem. Under some mild conditions on the parameters, we prove that the sequence generated by the proposed methods converges strongly to a solution of the above-mentioned problem in $q$-uniformly smooth Banach spaces. The results obtained in this paper generalize and improve many recent results in this direction.

Convergence Theorems for Accretive Operators in Banach Spaces

2011 ◽  
Vol 50-51 ◽  
pp. 224-228
Author(s):  
Yan Guo ◽  
Ji Wei He ◽  
Yan Mei Yang
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Accretive Operator ◽  
Common Element ◽  
Accretive Operators ◽  
Uniformly Smooth ◽  
Strongly Accretive Operator

In this paper, a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of the variational inequality for an inverse-strongly accretive operator in a 2-uniformly smooth Banach space was introduced . It was verified that the sequence converged to a common element of two sets.

scholarly journals An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems

2021 ◽  
Vol 54 (1) ◽  
pp. 335-358
Author(s):  
Oluwatosin T. Mewomo ◽  
Olawale K. Oyewole
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Variational Inequality Problem ◽  
Optimization Problems ◽  
Monotone Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Iterative Approximation ◽  
Common Solution ◽  
Uniformly Smooth

Abstract In this paper, we study the problem of finding a common solution of split generalized vector mixed equlibrium problem (SGVMEP), fixed point problem (FPP) and variational inequality problem (VIP). We propose an inertial-type iterative algorithm, which uses a projection onto a feasible set and a linesearch, which can be easily calculated. We prove a strong convergence of the sequence generated by the proposed algorithm to a common solution of SGVMEP, fixed point of a quasi- ϕ \phi -nonexpansive mapping and VIP for a general class of monotone mapping in 2-uniformly convex and uniformly smooth Banach space E 1 {E}_{1} and a smooth, strictly convex and reflexive Banach space E 2 {E}_{2} . Some numerical examples are presented to illustrate the performance of our method. Our result improves some existing results in the literature.

scholarly journals Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-18
Author(s):  
Lu-Chuan Ceng ◽  
Abdul Latif ◽  
Abdullah E. Al-Mazrooei
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Approximation Methods ◽  
Uniformly Convex ◽  
Viscosity Approximation ◽  
Uniformly Smooth ◽  
Viscosity Approximation Methods

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

scholarly journals A Cyclic Method for Solutions of a Class of Split Variational Inequality Problem in Banach Space

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Bashir Ali ◽  
Aisha A. Adam ◽  
Yusuf Ibrahim
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Variational Inequality Problem ◽  
Strong Convergence Theorem ◽  
Inequality Problem ◽  
Cyclic Algorithm ◽  
Split Variational Inequality ◽  
Split Variational Inequality Problem

In this paper, a cyclic algorithm for approximating a class of split variational inequality problem is introduced and studied in some Banach spaces. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, unify, and generalize certain recent results in the literature.

scholarly journals An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdulmalik U. Bello ◽  
Monday O. Nnakwe
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Equilibrium Problem ◽  
Real Banach Space ◽  
Nonlinear Problems ◽  
Uniformly Convex ◽  
Convex Minimization Problem ◽  
Common Solution ◽  
Uniformly Smooth ◽  
Contractive Map

AbstractIn this paper, we construct a new Halpern-type subgradient extragradient iterative algorithm. The sequence generated by this algorithm converges strongly to a common solution of a variational inequality, an equilibrium problem, and a J-fixed point of a continuous J-pseudo-contractive map in a uniformly smooth and two-uniformly convex real Banach space. Also, the theorem is applied to approximate a common solution of a variational inequality, an equilibrium problem, and a convex minimization problem. Moreover, a numerical example is given to illustrate the implementability of our algorithm. Finally, the theorem proved complements, improves, and unifies some related recent results in the literature.

scholarly journals A Method for Solving the Variational Inequality Problem and Fixed Point Problems in Banach Spaces

2021 ◽  
Vol 53 ◽  
Author(s):  
Wongvisarut Khuangsatung ◽  
Atid Kangtunyakarn
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Contractive Mapping ◽  
Smooth Banach Space ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Inequality Problem ◽  
Uniformly Smooth Banach Space ◽  
Uniformly Smooth

The purpose of this research is to modify Halpern iteration’s process for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of a strictly pseudo contractive mapping in q-uniformly smooth Banach space. We also introduce a new technique to prove a strong convergence theorem for a finite family of strictly pseudo contractive mappings in q-uniformly smooth Banach space. Moreover, we give a numerical result to illustrate the main theorem.

scholarly journals Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Uamporn Witthayarat ◽  
Yeol Je Cho ◽  
Poom Kumam
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Accretive Operator ◽  
Uniformly Convex ◽  
Common Solution ◽  
Uniformly Smooth ◽  
Convex Feasibility

The aim of this paper is to introduce an iterative algorithm for finding a common solution of the sets(A+M2)−1(0) and(B+M1)−1(0), where M is a maximal accretive operator in a Banach space and, by using the proposed algorithm, to establish some strong convergence theorems for common solutions of the two sets above in a uniformly convex and 2-uniformly smooth Banach space. The results obtained in this paper extend and improve the corresponding results of Qin et al. 2011 from Hilbert spaces to Banach spaces and Petrot et al. 2011. Moreover, we also apply our results to some applications for solving convex feasibility problems.

ADAPTIVE OPERATOR EXTRAPOLATION METHOD FOR VARIATIONAL INEQUALITIES IN BANACH SPACES

2021 ◽  
Vol 5 ◽  
pp. 82-92
Author(s):  
Sergei Denisov ◽  
◽  
Vladimir Semenov ◽  
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Operations Research ◽  
Optimization Problems ◽  
Metric Projection ◽  
Feasible Set ◽  
Uniformly Smooth ◽  
Developing Area ◽  
Lipschitz Operators

Many problems of operations research and mathematical physics can be formulated in the form of variational inequalities. The development and research of algorithms for solving variational inequalities is an actively developing area of applied nonlinear analysis. Note that often nonsmooth optimization problems can be effectively solved if they are reformulated in the form of saddle point problems and algorithms for solving variational inequalities are applied. Recently, there has been progress in the study of algorithms for problems in Banach spaces. This is due to the wide involvement of the results and constructions of the geometry of Banach spaces. A new algorithm for solving variational inequalities in a Banach space is proposed and studied. In addition, the Alber generalized projection is used instead of the metric projection onto the feasible set. An attractive feature of the algorithm is only one computation at the iterative step of the projection onto the feasible set. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.

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