Exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in Banach spaces

In this paper, we generalize the concept of exceptional family of elements for a completely continuous field from Hilbert spaces to Banach spaces and we study the solvability of the variational inequalities with respect to a mapping f that is from a closed convex cone of a Banach space B to the dual space B∗ by applying the generalized projection operator πK and by using the Leray-Schauder type alternative.