Inequalities In Hilbert Spaces Research Articles

Ergodic approximations via matrix regularization approach

In this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence.

Approximate Controllability for Nonlinear Evolution Hemivariational Inequalities in Hilbert Spaces

Here we study the approximate controllability for control problems driven by a class of nonlinear evolution hemivariational inequalities in Hilbert spaces. Actually, our results cover a broader class of inclusion problems involving time-dependent operators. First, by using a fixed point approach and nonsmooth analysis, we show the existence of mild solutions for the inequality problem. Next, we provide a sufficient condition guaranteeing the approximate controllability of our problem in terms of an associated linear control system. An example demonstrates the applicability of our results.

On some new converses of convex inequalities in Hilbert space

In this paper we study some new converses of the operator version for the Jensen inequality and the scalar Lah-Ribariˇc inequality for continuous convex functions and we will give applications of those results to quasiarithmetic means and power means for selfadjoint operators. Furthermore, we will also give improvements of the obtained results.

A unified extragradient method for systems of hierarchical variational inequalities in a Hilbert space

Abstract In this paper, we introduce and analyze a multistep Mann-type extragradient iterative algorithm by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann’s iteration method, and the projection method. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings and a strict pseudocontraction, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the solution set of a variational inequality problem (VIP), which is just a unique solution of a system of hierarchical variational inequalities (SHVI) in a real Hilbert space. The results obtained in this paper improve and extend the corresponding results announced by many others. MSC:49J30, 47H09, 47J20, 49M05.

Regularity for nonlinear evolution variational inequalities with delay terms

In this paper, we deal with the regularity and a variation of constant formula for solutions of the nonlinear differential equation with delay nonlinear terms governed by the variational inequality in Hilbert spaces. Without the conditions of the uniform boundedness of the nonlinear terms and the compactness of the principal operators, we obtain the wellposedness and the norm estimate of the given equation by converting the problem into the contraction mapping principle on state space.MSC:35K85, 35F25.

Strong convergence of extragradient method for generalized variational inequalities in Hilbert space

In this paper, we present a new type of extra-gradient method for generalized variational inequalities with multi-valued mapping in an infinite-dimensional Hilbert space. For this method, the generated sequence possesses an expansion property with respect to the initial point, and the existence of the solution to the problem can be verified through the behavior of the generated sequence. Furthermore, under mild conditions, we show that the generated sequence of the method strongly converges to the solution of the problem which is closest to the initial point.MSC:90C30, 15A06.

Error Bound for Conic Inequality in Hilbert Spaces

We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds.

The generalized of Selberg’s inequalities in C*-module

We obtain a Generalized of Selberg?s type inequalities in Hilbert spaces and their extensions in operators algebras, in C*-modules and in algebras of adjointable A-linear maps. Some applications for improving the Bessel type inequality result are given.

A Direct Splitting Method for Nonsmooth Variational Inequalities

We propose a direct splitting method for solving a nonsmooth variational inequality in Hilbert spaces. The weak convergence is established when the operator is the sum of two point-to-set and monotone operators. The proposed method is a natural extension of the incremental subgradient method for nondifferentiable optimization, which strongly explores the structure of the operator using projected subgradient-like techniques. The advantage of our method is that any nontrivial subproblem must be solved, like the evaluation of the resolvent operator. The necessity to compute proximal iterations is the main difficulty of other schemes for solving this kind of problem.

A Simpler Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces

In the present paper, we propose a simpler explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces. A strong convergence theorem is proved under fewer restrictions imposed on the mappings and parameters. An extension and numerical result are also given to illustrate the effectiveness and superiority of the proposed algorithm.

On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.

Converses of convex inequalities in Hilbert space

In this paper we give converses of the operator version for the Jensen inequality and the scalar Lah–Ribaric inequality. We also apply those results to quasi-arithmetic means and power means for selfadjoint operators.

The convergence analysis of the projection methods for a system of generalized relaxed cocoercive variational inequalities in Hilbert spaces

In this paper, the approximate solvability for a system of generalized relaxed cocoercive nonlinear variational inequalities in Hilbert spaces is studied, based on the convergence of the projection methods. The results presented in this paper extend and improve the main results of Refs. (Verma in Comput. Math. Appl. 41:1025-1031, 2001; Verma in Int. J. Differ. Equ. Appl. 6:359-367, 2002; Verma in J. Optim. Theory Appl. 121(1):203-210, 2004; Verma in Appl. Math. Lett. 18(11):1286-1292, 2005; Chang, Lee and Chan in Appl. Math. Lett. 20:329-334, 2007).MSC:90C33, 65K15, 58E36.

Exceptional family of elements for generalized variational inequalities

This paper introduces the notion of exceptional family of elements for generalized variational inequalities in Hilbert spaces. The set-valued mapping is assumed to be upper semi-continuous compact with nonempty closed convex values. Based on topological degree for set-valued mappings, instead of the technique of continuous selection, an alternative theorem is obtained which says that the generalized variational inequalities have either a solution or an exceptional family of elements. In addition, an existing result of a solution for generalized variational inequalities is obtained.MSC:90C25, 90C33.

GAP FUNCTIONS AND GLOBAL ERROR BOUNDS FOR SET-VALUED MIXED VARIATIONAL INEQUALITIES

In this paper, we introduce some gap functions for set-valued mixed variational inequalities under suitable conditions. We further use these gap functions to study global error bounds for the solutions of set-valued mixed variational inequalities in Hilbert spaces. The results presented in this paper generalize and improve some corresponding known results in literatures.

A Subgradient-Like Algorithm for Solving Vector Convex Inequalities

In this paper, we propose a strongly convergent variant of Robinson's subgradient algorithm for solving a system of vector convex inequalities in Hilbert spaces. The advantage of the proposed method is that it converges strongly, when the problem has solutions, under mild assumptions. The proposed algorithm also has the following desirable property: the sequence converges to the solution of the problem, which lies closest to the starting point and remains entirely in the intersection of three balls with radius less than the initial distance to the solution set.

Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.

The subgradient extragradient method extended to equilibrium problems

A globally convergent algorithm for equilibrium problems with pseudomonotone bifunctions is proposed. The algorithm is based on the idea of the subgradient extragradient method for solving variational inequalities proposed by Censor et al. [Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148 (2011), 318–335.] and Armijo linesearch techniques. In addition, we give a modified version of our algorithm for finding a common point of the solution set of equilibrium problems and the fixed point set of a nonexpansive mapping. We also analyse the weak convergence of both algorithms in a real Hilbert space.

Iterative algorithms for a system of generalized variational inequalities in Hilbert spaces

In this paper, a new system of generalized nonlinear variational inequalities involving three operators is introduced. A three-step iterative algorithm is considered for the system of generalized nonlinear variational inequalities. Strong convergence theorems of the three-step iterative algorithm are established.MSC:47H05, 47H09, 47J25.

An explicit algorithm for monotone variational inequalities

We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions.