The existence of zeros of set-valued mappings in reflexive Banach spaces

In this paper, we establish the equivalence between the existence of zeros for set‐valued mappings and the solvability of variational inequality, under some conditions involving the generalized projection operator. Basing on this result, we obtain some existence theorems of zeros for quasimonotone set‐valued mappings. As application, we derive several fixed point theorems for generalized inward mappings in Hilbert spaces.