scholarly journals An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Wanna Sriprad ◽  
Somnuk Srisawat
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Finite Family ◽  
Variational Inequality Problems ◽  
Variational Inclusion ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Variational Inclusion Problem ◽  
Convex Minimization Problems

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.

scholarly journals Projection Algorithms for Variational Inclusions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Youli Yu ◽  
Pei-Xia Yang ◽  
Khalida Inayat Noor
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Projection Algorithm ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Variational Inclusions ◽  
Projection Algorithms ◽  
Variational Inclusion Problem

We present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality.

scholarly journals Modified subgradient extragradient method for system of variational inclusion problem and finite family of variational inequalities problem in real Hilbert space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Araya Kheawborisut ◽  
Atid Kangtunyakarn
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Finite Family ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Subgradient Extragradient Method ◽  
Generalized System

AbstractFor the purpose of this article, we introduce a modified form of a generalized system of variational inclusions, called the generalized system of modified variational inclusion problems (GSMVIP). This problem reduces to the classical variational inclusion and variational inequalities problems. Motivated by several recent results related to the subgradient extragradient method, we propose a new subgradient extragradient method for finding a common element of the set of solutions of GSMVIP and the set of a finite family of variational inequalities problems. Under suitable assumptions, strong convergence theorems have been proved in the framework of a Hilbert space. In addition, some numerical results indicate that the proposed method is effective.

scholarly journals An inertial iterative scheme for solving variational inclusion with application to Nash-Cournot equilibrium and image restoration problems

2021 ◽  
Vol 37 (3) ◽  
pp. 361-380
Author(s):  
JAMILU ABUBAKAR ◽  
◽  
POOM KUMAM ◽  
ABOR ISA GARBA ◽  
MUHAMMAD SIRAJO ABDULLAHI ◽  
...  
Keyword(s):  
Variational Inequality ◽  
Image Restoration ◽  
Split Feasibility Problem ◽  
Variational Inclusion ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Relaxation Techniques ◽  
Cournot Equilibrium ◽  
Monotone Equations ◽  
Variational Inclusion Problem

Variational inclusion is an important general problem consisting of many useful problems like variational inequality, minimization problem and nonlinear monotone equations. In this article, a new scheme for solving variational inclusion problem is proposed and the scheme uses inertial and relaxation techniques. Moreover, the scheme is self adaptive, that is, the stepsize does not depend on the factorial constants of the underlying operator, instead it can be computed using a simple updating rule. Weak convergence analysis of the iterates generated by the new scheme is presented under mild conditions. In addition, schemes for solving variational inequality problem and split feasibility problem are derived from the proposed scheme and applied in solving Nash-Cournot equilibrium problem and image restoration. Experiments to illustrate the implementation and potential applicability of the proposed schemes in comparison with some existing schemes in the literature are presented.

scholarly journals Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 255
Author(s):  
Yan Tang ◽  
Yeol Cho
Keyword(s):  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Point Problem ◽  
Step Size ◽  
Convergence Theorems ◽  
Common Solution ◽  
Variational Inclusion Problem

In this paper, the split variational inclusion problem (SVIP) and the system of equilibrium problems (EP) are considered in Hilbert spaces. Inspired by the works of Byrne et al., López et al., Moudafi and Thukur, Sobumt and Plubtieng, Sitthithakerngkiet et al. and Eslamian and Fakhri, a new self-adaptive step size algorithm is proposed to find a common element of the solution set of the problems SVIP and EP. Convergence theorems are established under suitable conditions for the algorithm and application to the common solution of the fixed point problem, and the split convex optimization problem is considered. Finally, the performances and computational experiments are presented and a comparison with the related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.

scholarly journals Multi-Step Inertial Hybrid and Shrinking Tseng’s Algorithm with Meir–Keeler Contractions for Variational Inclusion Problems

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1548
Author(s):  
Yuanheng Wang ◽  
Mingyue Yuan ◽  
Bingnan Jiang
Keyword(s):  
Projection Method ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Inertial Method ◽  
Inclusion Problems ◽  
Variational Inclusion Problem ◽  
Convex Minimization Problems ◽  
Tseng’S Method

In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng’s method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong convergence of the proposed iterative algorithms is proven. Using our results, we can solve convex minimization problems.

scholarly journals Solving Split Variational Inclusion Problem and Fixed Point Problem for Nonexpansive Semigroup without Prior Knowledge of Operator Norms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Haitao Che ◽  
Meixia Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Point Problem ◽  
Split Variational Inclusion Problem ◽  
Variational Inclusion Problem ◽  
Nonexpansive Semigroups ◽  
Operator Norms

We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, which may be viewed as a refinement and improvement of the previously known results announced by many other researchers.

scholarly journals Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 749 ◽  
Author(s):  
Mujahid Abbas ◽  
Yusuf Ibrahim ◽  
Abdul Rahim Khan ◽  
Manuel De la Sen
Keyword(s):  
Fixed Point ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Multivalued Mappings ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Point Problem ◽  
Strictly Pseudocontractive Mapping ◽  
Split Variational Inclusion Problem ◽  
Variational Inclusion Problem

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

scholarly journals The Hybrid Steepest Descent Method for Split Variational Inclusion and Constrained Convex Minimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Jitsupa Deepho ◽  
Poom Kumam
Keyword(s):  
Descent Method ◽  
Convex Minimization ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Constrained Convex Minimization ◽  
Hybrid Steepest Descent Method ◽  
Variational Inclusion Problem

We introduced an implicit and an explicit iteration method based on the hybrid steepest descent method for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem.

scholarly journals An efficient iterative method for solving split variational inclusion problem with applications

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jamilu Abubakar ◽  
Poom Kumam ◽  
Abor Isa Garba ◽  
Muhammad Sirajo Abdullahi ◽  
Abdulkarim Hassan Ibrahim ◽  
...  
Keyword(s):  
Iterative Method ◽  
Strong Convergence ◽  
Iterative Scheme ◽  
Variational Inclusion ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Monotone Mappings ◽  
Bounded Linear ◽  
Split Variational Inclusion Problem ◽  
Variational Inclusion Problem

<p style='text-indent:20px;'>A new strong convergence iterative method for solving a split variational inclusion problem involving a bounded linear operator and two maximally monotone mappings is proposed in this article. The study considers an iterative scheme comprised of inertial extrapolation step together with the Mann-type step. A strong convergence theorem of the iterates generated by the proposed iterative scheme is given under suitable conditions. In addition, methods for solving variational inequality problems and split convex feasibility problems are derived from the proposed method. Applications of solving Nash-equilibrium problems and image restoration problems are solved using the derived methods to demonstrate the implementation of the proposed methods. Numerical comparisons with some existing iterative methods are also presented.</p>

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