Family Of Variational Inequality Problems Research Articles

Inertial self-adaptive Bregman projection method for finite family of variational inequality problems in reflexive Banach spaces

This paper considers an iterative approximation of a common solution of a finite family of variational inequailties in a real reflexive Banach space. By employing the Bregman distance and projection methods, we propose an iterative algorithm which uses a newly constructed adaptive step size to avoid a dependence on the Lipschitz constants of the families of the cost operators. The algorithm is carefully constructed so that the need to find a farthest element in any of its iterate is avoided. A strong convergence theorem was proved using the proposed method. We report some numerical experiments to illustrate the performance of the algorithm and also compare with existing methods in the literature.

A hybrid scheme for fixed points of a countable family of generalized nonexpansive-type maps and finite families of variational inequality and equilibrium problems, with applications

"Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^*$. We present a novel hybrid method for finding a common solution of a family of equilibrium problems, a common solution of a family of variational inequality problems and a common element of fixed points of a family of a general class of nonlinear nonexpansive maps. The sequence of this new method is proved to converge strongly to a common element of the families. Our theorem and its applications complement, generalize, and extend various results in literature."

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

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.

On a Viscosity Iterative Method for Solving Variational Inequality Problems in Hadamard Spaces

In this paper, we propose and study an iterative algorithm that comprises of a finite family of inverse strongly monotone mappings and a finite family of Lipschitz demicontractive mappings in an Hadamard space. We establish that the proposed algorithm converges strongly to a common solution of a finite family of variational inequality problems, which is also a common fixed point of the demicontractive mappings. Furthermore, we provide a numerical experiment to demonstrate the applicability of our results. Our results generalize some recent results in literature.

On fixed point results for hemicontractive-type multi-valued mapping, finite families of split equilibrium and variational inequality problems

In this article, we introduced an iterative scheme for finding a common element of the set of fixed points of a multi-valued hemicontractive-type mapping, the set of common solutions of a finite family of split equilibrium problems and the set of common solutions of a finite family of variational inequality problems in real Hilbert spaces. Moreover, the sequence generated by the proposed algorithm is proved to be strongly convergent to a common solution of these three problems under mild conditions on parameters. Our results improve and generalize many well-known recent results existing in the literature in this field of research.

An Iterative Method for Finding Common Solution of the Fixed Point Problem of a Finite Family of Nonexpansive Mappings and a Finite Family of Variational Inequality Problems in Hilbert Space

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.

A Modified Iterative Method for a Finite Collection of Non-self Mappings and a Family of Variational Inequality Problems

This paper deals with an iterative algorithm with the help of averaged mappings for the common solution of a fixed point problem of a finite collection of k-strictly pseudocontractive non-self mappings and a system of variational inequality problems in the setting of a real Hilbert space. Under some suitable conditions, the sequence generated by the proposed algorithm converges strongly to the common solution of the above said problems. Also, a numerical example is given to establish the superiority of the proposed algorithm over some existing methods.

A hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problems

In this article, we prove some properties of a demicontractive mapping defined on a nonempty closed convex subset of a Hilbert space. By using these properties, we obtain strong convergence theorems of a hybrid shrinking projection method for finding a common element of the set of common fixed points of a finite family of demicontractive mappings and the set of common solutions of a finite family of variational inequality problems in a Hilbert space. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.

Construction of a common solution of a finite family of variational inequality problems for monotone mappings

Construction of a common solution of a finite family of variational inequality problems for monotone mappings

A hybrid iterative method for common solutions of variational inequality problems and fixed point problems for single-valued and multi-valued mappings with applications

AbstractIn this article, we propose a new iterative method for approximating a common element of the set of common fixed points of a finite family of k-strictly pseudononspreading single-valued mappings, the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings, and the set of common solutions of a finite family of variational inequality problems in Hilbert spaces. Furthermore, we prove that the proposed iterative method converges strongly to a common element of the above three sets, and we also apply our results to complementarity problems. Finally, we give two numerical examples to support our main result.

Convergence theorems for Bregman strongly nonexpansive mappings in reflexive Banach spaces

In this paper, we study a strong convergence theorem for a common fixed point of a finite family of Bregman strongly nonexpansive mappings in the framework of reflexive real Banach spaces. As a consequence, we prove convergence theorem for a common fixed point of a finite family of Bergman relatively nonexpansive mappings. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common zero of a finite family of Bregman inverse strongly monotone mappings and a solution of a finite family of variational inequality problems.

Convergence results for a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function

In this paper, we introduce an iterative process which converges strongly to a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function. Our convergence theorem is applied to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.

Iterative scheme for a nonexpansive mapping, an η-strictly pseudo-contractive mapping and variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space

In this paper, we introduce an iterative scheme by the modification of Mann's iteration process for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of an η-strictly pseudo-contractive mapping and a nonexpansive mapping. Moreover, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of ηi-strictly pseudo-contractive mappings for every i =1 , 2, ... , N in uniformly convex and 2-uniformly smooth Banach spaces.

Strong convergence of the hybrid method for a finite family of nonspreading mappings and variational inequality problems

In this paper, we prove a strong convergence theorem by the hybrid method for finding a common element of the set of fixed points of a finite family of nonspreading mappings and the set of solutions of a finite family of variational inequality problems.

Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of κ i -strictly pseudocontractive mappings

In this article, by using the S-mapping and hybrid method we prove a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of generalized equilibrium defined by Ceng et al., which is a solution of two sets of variational inequality problems. Moreover, by using our main result we have a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of solution of a finite family of generalized equilibrium defined by Ceng et al., which is a solution of a finite family of variational inequality problems.

Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems

We introduce an iterative process for finding an element of common solutions of finite family of equilibrium problems and finite family of variational inequality problems for continuous monotone mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of non-linear operators.

Strong convergence of an iterative method for pseudo-contractive and monotone mappings

In this paper, we introduce an iterative process which converges strongly to a common element of fixed points of pseudo-contractive mapping and solutions of variational inequality problem for monotone mapping. As a consequence, we provide an iteration scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

Strong convergence theorems for variational inequality problems and quasi- $${\phi}$$ -asymptotically nonexpansive mappings

In this paper, we introduce an iterative process which converges strongly to a common solution of finite family of variational inequality problems for ?-inverse strongly monotone mappings and fixed point of two continuous quasi- $${\phi}$$ -asymptotically nonexpansive mappings in Banach spaces. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.

New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces

We introduced a new iterative scheme for finding a common element in the set of common fixed points of a finite family of quasi-ϕ-nonexpansive mappings, the set of common solutions of a finite family of equilibrium problems, and the set of common solutions of a finite family of variational inequality problems in Banach spaces. The proof method for the main result is simplified under some new assumptions on the bifunctions.

A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems

We introduce an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems and common solutions of finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators.