Abstract: The purpose of this paper is to study the variational inequality and complementarity problem for set-valued mapping in Banach spaces.The results presented in this paper generalize and improve some recent results on finite dimensional spaces,Hiblert spaces,single-valued mapping.  As applications,we utilize our results to study saddle point problems and two programming problems in infinite dimensional spaces.

Key words: variational inequality, complementarity problem, saddle point, prugranming problem ,