General Class Of Variational Inequalities Research Articles

A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings

In this paper, a normal S-iterative algorithm is studied and analyzed for solving a general class of variational inequalities involving a set of fixed points of nonexpansive mappings and two nonlinear operators. It is shown that the proposed algorithm converges strongly under mild conditions. The rate of convergence of the proposed iterative algorithm is also studied. An equivalence of convergence between the normal S-iterative algorithm and Algorithm 2.6 of Noor (J. Math. Anal. Appl. 331, 810–822, 2007) is established and a comparison between the two is also discussed. As an application, a modified algorithm is employed to solve convex minimization problems. Numerical examples are given to validate the theoretical findings. The results obtained herein improve and complement the corresponding results in Noor (J. Math. Anal. Appl. 331, 810–822, 2007).

A duality based semismooth Newton framework for solving variational inequalities of the second kind

In an appropriate function space setting, semismooth Newton methods are proposed for iteratively computing the solution of a rather general class of variational inequalities (VIs) of the second kind. The Newton scheme is based on the Fenchel dual of the original VI problem which is regularized if necessary. In the latter case, consistency of the regularization with respect to the original problem is studied. The application of the general framework to specific model problems including Bingham flows, simplified friction, or total variation regularization in mathematical imaging is described in detail. Finally, numerical experiments are presented in order to verify the theoretical results.

A self-adaptive projection method for a class of variant variational inequalities

In this paper, we consider the general variant variational inequality of the type: Find a vector u ∗ ∈ R n , such that Q(u ∗ ) ∈ Ω, � v −Q(u ∗ ),Tu ∗ � 0, ∀v ∈ Ω, where T,Q are operators. We suggest and analyze a very simple self-adaptive iterative method for solving this class of general variational inequalities. Under certain conditions, the global con- vergence of the proposed method is proved. An example is given to illustrate the efficiency and implementation of the proposed method. Preliminary numerical results show that the proposed method is applicable.

Extended general variational inequalities

In this work, we introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. Noor [M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.

A Recurrent Neural Network for Solving a Class of General Variational Inequalities

This paper presents a recurrent neural-network model for solving a special class of general variational inequalities (GVIs), which includes classical VIs as special cases. It is proved that the proposed neural network (NN) for solving this class of GVIs can be globally convergent, globally asymptotically stable, and globally exponentially stable under different conditions. The proposed NN can be viewed as a modified version of the general projection NN existing in the literature. Several numerical examples are provided to demonstrate the effectiveness and performance of the proposed NN.

The treatment of the locking phenomenon for a general class of variational inequalities

We present the analysis of a class of variational inequalities depending on a small nonnegative parameter in a singular way, for which direct numerical approximation yield a numerical locking phenomenon. It consists in extending some robust approaches to variational inequalities, mainly, conforming and nonconforming methods. We give general sufficient conditions on the discrete problem insuring a uniform convergence relatively to the small parameter, and consequently avoiding numerical locking.

Harmonic solutions for noncoercive dynamic unilateral problems

This paper studies the solvability of a general class of variational inequalities. Existence of periodic solutions for noncoercive variational inequalities will be proved.

Existence and multiplicity results for semicoercive unilateral problems

In this paper, we investigate a general class of variational inequalities. Existence and multiplicity results are obtained by using minimax principles for lower semicontinuous functions due to A. Szulkin.

On the existence and uniqueness of solutions for a class of variational inequalities with applications to the signorini problem in mechanics

In this paper, we introduce a new unified and general class of variational inequalities, and show some existence and uniqueness results of solutions for this kind of variational inequalities. As an application, we utilize the results presented in this paper to study the Signorini problem in mechanics.

Mixed variational inequalities

In this paper, we use the fixed-point technique to prove the existence of a unique solution of a new unified and general class of variational inequalities. Several special cases are also discussed.

On a class of variational inequalities

In this paper, we introduce and study a new unified and general class of variational inequalities. We derive the general error estimates for the finite element solutions of variational inequalities. It has been shown that a class of contact problems with friction terms arising in elastostatics can be studied in the framework of variational inequalities. Several special cases, which can be obtained from the general results, are also discussed.

On the Identification of Parameters in General Variational Inequalities by Asymptotic Regularization

A method for the identification of parameters in a general class of variational inequalities from the knowledge of the solutions is proposed. The method consists of embedding the original problem into a sequence of regularizing equations. An a priori estimate is fundamental to prove that limit points of the regularizing sequence are solutions of the original problem. The method is applied to various applications, in particular, to the dam problem and to linear elasticity with friction.

Strongly nonlinear parabolic variational inequalities

An existence and uniqueness result is established for a general class of variational inequalities for parabolic partial differential equations of the form partial differentialu/ partial differentialt + A(u) + g(u) = f with g nondecreasing but satisfying no growth condition. The proof is based upon a type of compactness result for solutions of variational inequalities that should find a variety of other applications.

Über nichtlineare Eigenwertaufgaben in konvexen Mengen

We consider the bifurcation problem of a general class of variational inequalities at the first igenvalue of the associated linear problem. Essentially we use a generalization of LJUSTERNIK's emma, MIERSEMANN [13]. We apply our results to buckling problems for the thin elastic plate.