Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI(T-ξ,K) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399-406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307-317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321-337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55-71].