Generalized -complementarity problems in Banach spaces

In this paper, a class of generalized f-complementarity problems and three classes of variational inequalities are introduced in real Banach spaces, and the equivalences among them are established under certain conditions. Several coercivity conditions are introduced for the existence of solutions of the generalized f-complementarity problem. Under some suitable assumptions, it is shown that each of these coercivity conditions is equivalent to the nonemptyness and boundedness of the solution set for the generalized f-complementarity problem in infinite-dimensional Banach spaces, and even the nonemptyness and compactness of the solution set for the generalized f-complementarity problem in finite-dimensional spaces. The existence of least elements for the feasible set of the generalized f-complementarity problem is also presented under suitable conditions.