Φ- α(⋅)-K-monotone multifunctions with values in ordered Banach space with increasing norm

Let (X, d) be a metric space. Let Y be an ordered Banach space with increasing norm. Let Φ be a separable linear family (a class) of Lipschitz functions defined on X and with values in Y . Let α(⋅) be a nondecreasing function mapping the interwal [0,+∞) into itself such that limt↓0 α(t) / t = 0. We say that a multifunction mapping X into Φ is Φ -α(⋅)-K-monotone if for all k in the interior of K, k ∈ Int K, there is a constant Ck > 0 such that for all φx ∈Γ (x),φy ∈Γ (y) we have φx(x) + φy(y) − φx(y) − φy(x) ≥K −Ckα(d(x, y))k.It is shown in the paper that under certain conditions on each Φ - Φα(⋅)-K-monotone multifunction is single-valued and continuous on a dense G δ-set..