Pii: S0893-9659(01)00137-9

Keywords--Variat ional inequalities, Projection method, Fixed point, Convergence. 1. I N T R O D U C T I O N Variational inequalities.have had a great impact and influence in the development of almost all branches of pure and applied sciences. There are a substantial number of numerical methods including projection, the WieneroHopf equations, auxiliary principle techniques for solving variational inequalities, see for example, [1-16]. It is well known that the convergence of the projection method requires the operator T to be strongly monotone and Lipschitz continuous. Gabay [4] and Tseng [16] have shown that the convergence of the projection method can be proved for the co-coercive operators. Note that co-coercivity is weaker than strongly monotonicity, but is stronger than the monotonicity. These strict conditions rule out many applications of the projection method for a wide class of problems. These facts motivated to modify the projection method and its variant forms. The extragradient method [2,16] overcomes this difficulty by using the technique of updating the solution, which modified the projection method by performing additional step and projection at each step according to double projection formula. Solodov and Tseng [14] and He [7] suggested another modified projection type-method involving only one project ion. It is worth mentioning that the convergence of both the extragradient method and the modified projection type method requires that the solution exists and the operator to be monotone and Lipschitz continuous. These facts motivated us to modify the extragradient method by using the ideas of Noor [11,12] and Tseng [16]. Our proposed modification is in the spirit of the extragradient method performing an additional forward step and projection at each step. We would like to point out tha t our modified projection method still involves two projections as in the extragradient method. One advantage of this approach is tha t one can suggest a splitting 0893-9659/02/$ see front matter © 2002 Elsevier Science Ltd. All rights reserved. Typeset by .A.MS-TEX PII: S0893-9659 (01)00137-9