A new method for the obtaining of eigenvalues of variational inequalities of the special type (Preliminary communication) (original) (raw)

Numerical and Analytical Methods for Variational Inequalities and Related Problems 2013

Journal of Applied Mathematics, 2013

The study of variational inequalities and related problems with applications constitutes a rich topic of intensive research efforts within the latest 50 years. Variational inequality theory, which was introduced by Stampacchia in 1964, has emerged as a fascinating branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, ecology, social, regional, pure, and applied sciences. The corresponding iterative methods have witnessed great progress in recent years to handle problems in optimization problems, inverse problems, and differential equations.

Comparison Results for a Class of Variational Inequalities

Revista Matemática Complutense, 1993

In this paper we study a variational inequality related te a linear differential operater of elliptic type. Wc give a pointwise beund for ¡be rearrangement of ¡be solution u, and an estiniate fer the L 2-norm of ¡be gradient of u.

Some aspects of variational inequalities

Journal of Computational and Applied Mathematics, 1993

In this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence, uniqueness, computational properties, various generalizations, sensitivity analysis and their applications. We also propose some open problems with sufficient information and references, so that someone may attempt solution(s) in his/her area of special interest. We also include some new results, which we have recently obtained.

Nonlinear variational inequalities depending on a parameter

1995

This paper develops the results announced in the Note . Using an eigenvalue problem governed by a variational inequality, we try to unify the theory concerning the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions. 1991 Mathematics Subject Classification: primary -47H19, 49J40, secondary -47H11, 47H15, 58C40, 58E07. A one-dimensional manifold of class C 1 such that the domain Ω is located in one side of ∂Ω is enough σ ij (w) = λ · σ • ij +σ ij (w),

GENERAL VARIATIONAL INEQUALITIES

In this paper, we introduce and study a new class of variational inequalities: Projection technique is used to suggest an iterative algorithm for finding the approximate solution of this class. We also discuss the convergence criteria of the iterative algorithm. Several special cases are discussed, which can be obtained from the general result.

On an Approximate Solution of Variational Inequalities

Mathematische Nachrichten, 1985

By means of time discretization, we approximate evolution variational inequalities by the corresponding elliptic variational inequalities. Using ROTHE'S method (method of lines), an approximate solution is constructed by means of direct variational methods. Existence, uniqueness and regularity of solutions as well a8 convergence of the approximate solutions are proved. Introduction. In this paper we shall be concerned with approximate solutions of the following types of variational inequalities 2 (f(t)-W)), 4 0) = uo for all v E I' and for a.e. t E (0, T) (T < oo), where (u, v) is the scalar product in a HILBERT space H and a(u, v), b(u, v) are continuous bilincar forms on the corresponding HILBERT spaces V , V,, respectively, with the continuous imbedding V 4 V , 4 H. In the case (4), A is a monotone operator from a real reflexive BANACH space B into its dual space V*. The functional YJ(v) is convex on V with values in [-oo, 003. The obtained results hold true also in the more general form of fsee Remark 7. Instead of f (t) a LIrscHrrz continuous operator f (t , u) : [0, T] X V-+ H can be considered.

On The Eigenvalues Problem for Hemivariational Inequalities : Existence and Stability

2003

This problem can be considered as a nonconvex generalization of the classical variational inequalities of J. L. Lions and G. Stampacchia. For typical examples in connection with mechanics and engineering we refer to the books of Panagiotopoulos [20, 22] and [18]. The techniques used for resolution of hemivariational inequalities are subsequently based on arranging fixed point theorems, Galerkin methods and the convolution product regularization, see [15]-[17], [21], [22] and the bibliography therein. In the last few years, much attention has been focused to the existence theory of such inequalities by means of the generalized Ky Fan minimax theorem [5, 4]. It is the aim of the present paper to investigate the variational-hemivariational inequality (V HI): find u ∈ D and λ ∈ IR such that ∀v ∈ D λ〈H(u), v − u〉 ≤ α (u, v − u) + 〈C (u) , v − u〉

The truncation method for the solution of a class of variational inequalities

Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique, 1976

The truncation method for the solution of a class of variational inequalities Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique, tome 10, n o 1 (1976), p. 29-42. http://www.numdam.org/item?id=M2AN\_1976\_\_10\_1\_29\_0 © AFCET, 1976, tous droits réservés. L'accès aux archives de la revue « Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ R.A.I.R.O.

Some developments in general variational inequalities

Applied Mathematics and Computation, 2004

General variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection, Wiener-Hopf equations, updating the solution, auxiliary principle, inertial proximal, penalty function, dynamical system and well-posedness. We also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection-type algorithms. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, results presented here continue to hold for these problems. Several open problems have been suggested for further research in these areas.