Some relations between ε-directional derivative and ε-generalized weak subdifferential

In this paper, we study ε-generalized weak subdifferential forvector valued functions defined on a real ordered topologicalvector space x. we give various characterizations of ε-generalizedweak subdifferential for this class of functions. it is well knownthat if the function f : x → r is subdifferentiable at x0 ∈ x,then f has a global minimizer at x0 if and only if 0 ∈ ∂ f(x0).we show that a similar result can be obtained for ε-generalizedweak subdifferential. finally, we investigate some relations between ε-directional derivative and ε-generalized weak subdifferential. in fact, in the classical subdifferential theory, it is wellknown that if the function f : x → r is subdifferentiable atx0 ∈ x and it has directional derivative at x0 in the directionu ∈ x, then the relation f ′(x0, u) ≥ ⟨u, x∗⟩, ∀ x∗ ∈ ∂ f(x0) issatisfied. we prove that a similar result can be obtained for ε-generalized weak subdifferential.