Wiener-Hopf Equations Technique ForGeneral Variational Inequalities

Abstract

In this paper, we show that the general variational inequality problem is equivalent to solving the Wiener-Hopf equations. We use this equivalence to suggest and analyze a number of iterative algorithms for solving general variational inequalities. INTRODUCTION Variational inequality theory, which was introduced by Stampacchia l] and Fichera [2] in the early sixties, describes a broad spectrum of very interesting and important developments involving a link among various fields of pure and applied sciences. It has been shown by many research workers that a wide class of contact, obstacle, unilateral, free, moving and general equilibrium problems arising in elasticity, fluid flow through porous media, economics, transportation, regional and engineering sciences can be studied in the general and unified framework of variational inequalities, see [3,4,5,6,7,8] for more details. Recently considerable efforts are being made to develop an efficient and implementable algorithm for computing the approximate solution of variational inequalities. There is a substantial number of iterative algorithms including the projection and auxiliary principle techniques, see [3,4,5,6,7,8,9,10] and the references cited therein. Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 362 Contact Mechanics In this paper, using essentially a variant form of the projection techniques, we show that the general variational inequality problem is equivalent to solving the equations, which are called the Wiener-Hopf equations. This technique is mainly due to Shi [11] and Noor [12,6 . This alternate formulation is quite general and flexible. We use this equivalence to suggest and analyze a number of new iterative algorithms for variational inequalities and related complementarity problems. We also consider the convergence criteria of these algorithms. FORMULATION AND BASIC RESULTS Let H be a real Hilbert space, whose inner product and norm are denoted by and . respectively. Let K be a nonempty closed convex set in H. Given T, g : H —> H nonlinear operators, we consider the problem of finding ueH such that g(u}eK and >0, forallvcJf. (1) The inequality (l) is known as the general nonlinear variational inequality, which was introduced by Noor [9]. REMARK 1. I. We note that if g = /, then problem (l) is equivalent to finding ueK such that >0, for all veK, (2) a problem originally introduced and studied by Stampacchia 1]. II. If K* — {veH : > 0, for all ueK} is a polar cone of the convex cone K in H and K C g(K), then the problem (l) is equivalent to finding ueH such that 2(w)f#, TucJC and = 0, (3) which is known as the general nonlinear complementarity problem. For suitable and appropriate choices of the operators T, g and the convex set K, we can obtain many known classes of linear and nonlinear complementarity problems as special cases from the problem (3). III. If K = H, then the problem (l) is equivalent to finding ueH such that g(u)eH and Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 Contact Mechanics 363 >= 0, for all gĤ ?, (4 which is known as the weak formulation of the odd order and nonsymmetric boundary value problems. From the above observations, it is clear that the problem (l) is more general and includes many known variational inequality and optimization problems as special cases. LEMMA 1 [7]. Given zeH, ueK satisfies the inequality > 0 for all veK, if and only if, u = PKZ, where PK is the projection of H into K. Furthermore, PK is a nonexpansive operator. Related to the general variational inequality (l), we now introduce a new class of the Wiener-Hopf equations. Let QK = / PK, where I is the identity operator. If g~* exists, then we consider the problem of finding zeH such that The equation of the type (7) are called the general Wiener-Hopf equations, see Speck [13] for the general treatment and physical formulations. DEFINITION 1. An operator T : H -> H is said to be: (a) Strongly monotone, if there exists a constant a > 0 such that > a u v || for all u.veH. (b) Lipschitz continuous, if there exists a constant /3 > 0 such that T% Tt, ||< /? % % ||, for all %,