On the translations of quasimonotone maps and monotonicity

It is clear that a monotone map is pseudomonotone, while a pseudomonotone map is quasimonotone. The converse is not true. If T is pseudomonotone (quasimonotone) and w ∈ X*\{}, then T + w is not pseudomonotone (quasimonotone) in general. In the case of a single-valued linear map T defined on the whole space Rn, it is known that if T + w is quasimonotone, then T is monotone []. Many authors (see, e.g., [, ]) extended this result for a nonlinearGateaux differentiablemapdefined on a convex subsetK (of aHilbert space) with a nonempty interior. Recently, Hadjisavvas [] extended the above result to the multivalued maps defined on a convex subset of a real topological vector space with no assumption of differentiability