2,105 research outputs found
A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces
We present an iterative method for fixed point
problems, generalized mixed equilibrium problems, and
variational inequality problems. Our method is based
on the so-called viscosity hybrid steepest descent
method. Using this method, we can find the common
element of the set of fixed points of a nonexpansive
mapping, the set of solutions of generalized mixed
equilibrium problems, and the set of solutions of
variational inequality problems for a relaxed
cocoercive mapping in a real Hilbert space. Then, we
prove the strong convergence of the proposed iterative
scheme to the unique solution of variational
inequality. The results presented in this paper
generalize and extend some well-known strong
convergence theorems in the literature
A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems
AbstractIn this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of finite mixed equilibrium problems, the set of solutions of variational inequalities for two cocoercive mappings, the set of common fixed points of an infinite family of nonexpansive mappings and the set of common fixed points of a nonexpansive semigroup in Hilbert space. Then we prove a strong convergence theorem under some suitable conditions. The results obtained in this paper extend and improve many recent ones announced by many others
Iterative schemes for approximating common solutions of certain optimization and fixed point problems in Hilbert spaces.
Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a shrinking projection method of an inertial type with
self-adaptive step size for finding a common element of the set of solutions of Split Gen-
eralized Equilibrium Problem (SGEP) and the set of common fixed points of a countable
family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step
size incorporated helps to overcome the difficulty of having to compute the operator norm
while the inertial term accelerates the rate of convergence of the propose algorithm. Under
standard and mild conditions, we prove a strong convergence theorem for the sequence
generated by the proposed algorithm and obtain some consequent results. We apply our
result to solve Split Mixed Variational Inequality Problem (SMVIP) and Split Minimiza-
tion Problem (SMP), and present numerical examples to illustrate the performance of
our algorithm in comparison with other existing algorithms. Moreover, we investigate the
problem of finding common solutions of Equilibrium Problem (EP), Variational Inclusion
Problem (VIP)and Fixed Point Problem (FPP) for an infinite family of strict pseudo-
contractive mappings. We propose an iterative scheme which combines inertial technique
with viscosity method for approximating common solutions of these problems in Hilbert
spaces. Under mild conditions, we prove a strong theorem for the proposed algorithm and
apply our results to approximate the solutions of other optimization problems. Finally,
we present a numerical example to demonstrate the efficiency of our algorithm in comparison with other existing methods in the literature. Our results improve and complement
contemporary results in the literature in this direction
Existence and Iterative Approximation Methods for Generalized Mixed Vector Equilibrium Problems with Relaxed Monotone Mappings
We first consider an auxiliary problem for the generalized mixed vector equilibrium problem with a relaxed monotone mapping and prove the existence and uniqueness of the solution for the auxiliary problem. We then introduce a new iterative scheme for approximating a common element of the set of solutions of a generalized mixed vector equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings. The results presented in this paper can be considered as a generalization of some known results due to Wang et al. (2010)
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