Pii: S0898-1221(97)00049-7

-In this paper, we establish the equivalence between the generalized nonlinear variational inequalities and the generalized Wiener-Hopf equations. This equivalence is used to suggest and analyze a number of iterative algorithms for solving generalized variational inequalities. We also discuss the convergence analysis of the proposed algorithms. As a special case, we obtain various known results from our results. Keywords--variational inequalities, Wiener-Hopf equations, Iterative algorithms, Fixed points, Convergence. 1. I N T R O D U C T I O N In recent years, variational inequality theory has emerged as an interesting branch of applicable mathematics. Variational inequality techniques are being used to study a wide class of linear and nonlinear problems arising in pure and applied sciences in a general and unified framework. For applications, motivation, physical formulation, and numerical methods, see [1-24] and the references therein. This theory has been generalized and extended in many directions using novel and innovative techniques. In recent years, considerable interest has been shown in developing various classes of variational inequalities both for its own sake and for its applications. There are significant recent developments in this theory related to multivalued operators, nonconvex optimization, iterative methods, Wiener-Hopf equations, and structural analysis. Motivated and inspired by the recent research going on in this field, Verma [24] introduced a class of variational inequalities, which is called the generalized nonlinear variational inequality. This class is the most general and includes many classes of variational inequalities and complementarity problems as special cases. Panagiotopoulos and Stavroulakis [17] have shown that if the nonsmooth and nonconvex super potential of the structure is quasidifferentiable, then these problems can be characterized by certain multivalued variational inequalities, which can be considered as special cases of generalized nonlinear variational inequalities. Equally important is the study of the equations known as the Wiener-Hopf equations or normal maps, which were introduced and studied by Shi [21] in 1991, and Robinson [19] in 1992, independently in different areas. Using different techniques, Shi [21] and Robinson [19] established the equivalence between the variational inequalities and the Wiener-Hopf equations. This equivalence The authors would like to thank the referees for their valuable suggestions and comments. Typeset by ¢4jV~q-TEX 2 M. ASLAM NOOR AND E. A. AL-SAID plays an important part in developing efficient numerical methods including iterative methods and homotopy methods for solving variational inequalities and complementarity problems. This technique has been refined and developed by Noor [6-14] to suggest some iterative methods for solving different classes of variational inequalities and complementarity problems. The WienerHopf equations techniques are simpler and easy to implement for developing efficient numerical techniques. In this paper, we introduce and study a new class of the Wiener-Hopf equations, which is called the generalized Wiener-Hopf equations. This class is the most general and unifying one. Using the projection techniques, we establish the equivalence between the generalized nonlinear variational inequalities and the generalized Wiener-Hopf equations. This equivalence allows us to suggest a number of iterative algorithms for solving variational inequalities. We also discuss the convergence analysis of the proposed algorithms. The results obtained in this paper are very general and include many known results as special cases. In Section 2, we formulate the generalized nonlinear variational inequalities and the generalized Wiener-Hopf equations and review some basic facts. The equivalence between these problems is established in Section 3. A number of iterative algorithms are suggested by using this equivalence. In Section 4, convergence criteria is considered. 2. P R E L I M I N A R I E S Let H be a real Hilbert space whose inner product and norm are denoted by (., .) and [[.tl, respectively. Let K be a closed convex set in H. Let C(H) be the family of nonempty compact subset of H. For given multivalued operators T, A : H ~ C(H) and a single-valued operator g : H --* H, we consider the problem of finding u 6 H, w E T(u) , y 6 A(u) such that g(u) 6 K and (g(u) (g(y) w) ,v g(u)) > 0, for all v 6 K. (2.1) The problem (2.1) is called the generalized nonlinear variational inequality problem, which is a variant form of the problem proposed by Verma [24]. EXAMPLE 2. I. To illustrate the applications and importance of the multivalued variational inequality (2.1), we consider a elastoplasticity problem, which is mainly due to Panagiotopoulos and Stavroulakis [17]. For simplicity, it is assumed that a general hyperelastic material law holds for the elastic behaviour of the elastoplastic material under consideration. Moreover, a nonconvex yield function a --* F(a) is introduced for the plasticity. For the basic definitions and concepts, see [17]. Let us assume the decomposition E = E e + E p, (2.2) where E" denotes the elastic, and E p the plastic deformation of the three-dimensional elastoplastic body. We write the complementary virtual work expression for the body in the form (E e, r a) + (E p, r a) = (f, r a ) , for all r 6 Z. (2.3) Here, we have assumed that the body on a part Fu of its boundary has given displacements, that is, #i = U~ on Fu, and that on the rest of its boundary FF = F F u , the boundary tractions are given, that is, Si = Fi on FF, where f (E, a) =/~ e~a~j dn, (2.4) (f, a) = [ U~S~ dF, (2.5) Jr u Z= {r:%,~ +f~=O onn, i,j=l,2,S,T~=F, onr;,i=l,2,3}, (2.6) is the set of statically admissible stresses and f/is the structure of the body.