Generalized projection method for a variational inequality system with different mapping in Banach spaces

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

On the existence of common fixed points for a pair of lipschitzian mappings in banach spaces

The purpose of this article is to study the fixed point and weak convergence problem for the new defined class of point-dependent λ-hybrid mappings relative to a Bregman distance D f in a Banach space. We at first extend the Aoyama-Iemoto-Kohsaka-Takahashi fixed point theorem for λ-hybrid mappings in Hilbert spaces in 2010 to this much wider class of nonlinear mappings in Banach spaces. Secondly, we derive an Opial-like inequality for the Bregman distance and apply it to establish a weak convergence theorem for this new class of nonlinear mappings. Some concrete examples in a Hilbert space showing that our extension is proper are also given.2010 MSC: 47H09; 47H10.

In the paper, we consider bilevel problems: variational inequality problems over the set of solutions of the equilibrium problem. Finding normal Nash equilibrium is an example of such a problem. To solve these problems, an iterative algorithm is proposed that combines the ideas of the two-stage proximal method, adaptability, and iterative regularization. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of information on bifunction’s Lipschitz constants and operator’s Lipschitz and strong monotonicity constants. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz operators, the theorem on strong convergence of sequences generated by the algorithm is proved. The proposed algorithm is shown to be applicable to monotone bilevel variational inequalities in Hilbert spaces.

We study the existence theory for parabolic variational inequalities in weighted L 2 spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L 2 setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.

This work is devoted to the study of new algorithm for solving variational inequalities in Hilbert spaces. The proposed algorithm is a variant of the operator extrapolation method regularized using the Halpern scheme. The algorithm has an advantage over the Korpelevich extragradient method and the method of extrapolation from the past in terms of the amount of calculations required for the iterative step. For variational inequalities with monotone, Lipschitz continuous operators acting in Hilbert space, a theorem on strong convergence of the method is proved.

The purpose of this paper is to introduce a new parallel algorithm for finding solutions to pairs of variational inequalities over the solution sets of systems of split equality null-point problems in Hilbert spaces. More precisely, we propose a new self-adaptive algorithm based on the hybrid steepest descent method and establish a strong convergence theorem for it. Moreover, we also give some applications of our proposed algorithm to solving some related problems, such as systems of split equality minimum point problems and systems of split equality feasibility problems.

In this article, a projection-type method for mixed variational inequalities is proposed in Hilbert spaces. The proposed method has the following nice features: (i) The algorithm is well defined whether the solution set of the problem is nonempty or not, under some mild assumptions; (ii) If the solution set is nonempty, then the sequence generated by the method is strongly convergent to the solution, which is closest to the initial point; (iii) The existence of the solutions to variational inequalities can be verified through the behavior of the generated sequence. The results presented in this article generalize and improve some known results.

In this paper, we consider pseudomonotone equilibrium problems and generalized variational inequalities in Hilbert spaces. We suggest an iterative procedure for solving pseudomonotone equilibrium problems and generalized variational inequalities. Strong convergence result is proved under some mild assumptions.

Chapter 4 Variational Inequalities in Hilbert Spaces

In this manuscript, we study a system of extended general variational inequalities (SEGVI) with several nonlinear operators, more precisely, six relaxed ( α , r ) -cocoercive mappings. Using the projection method, we show that a system of extended general variational inequalities is equivalent to the nonlinear projection equations. This alternative equivalent problem is used to consider the existence and convergence (or approximate solvability) of a solution of a system of extended general variational inequalities under suitable conditions.

ABSTRACTIn this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others.