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AN APPROXIMATION OF SOLUTIONS OF VARIATIONAL INEQUALITIES | AN APPROXIMATION OF SOLUTIONS OF VARIATIONAL INEQUALITIES JINLU LI AND B. E. RHOADES Received 27 October 2004 and in revised form 15 December 2004 We use a Mann-type iteration scheme and the metric projection operator the nearest-point projection operator to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces. 1. Introduction Let B II II be a Banach space with the topological dual space B and let p x denote the duality pairing of B and B where p e B and x e B. Let f B B be a mapping and let K be a nonempty closed and convex subset of B. The general variational inequality defined by the mapping f and the set K is VI f K find x e K such that f x x - x 0 for every x e K. 1.1 The nonlinear complementarily problem defined by f and K is by definition as follows NCP f K find x e K such that f x x 0 1.2 for every x e K and f x x 0. It is known see 5 6 that when K is a closed convex cone problems NCP f K and VI f K are equivalent. To study the existence of solutions of the NCP f K and VI f K problems many authors have used the techniques of KKM mappings and the Fan-KKM theorem from fixed point theory see 1 5 6 7 8 9 10 . In case B is a Hilbert space Isac and other authors have used the notion of exceptional family of elements EFE and the Leray-Schauder alternative theorem see 5 6 . In 1 2 Alber generalized the metric projection operator PK to a generalized projection operator nK B K from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces and Alber used this operator to study VI f K problems and to Copyright 2005 Hindawi Publishing Corporation Fixed Point Theoryand Applications 2005 3 2005 377-388 DOI 10.1155 FPTA.2005.377 378 Solutions of variational inequalities approximate the solutions by an iteration sequence. In 7 the author used the generalized projection operator and a Mann-type iteration sequence to approximate the solutions of the VI f K problems. In case B is a uniformly convex and .