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

In this paper, we study -generalized weak subdierential for vector valued functions defined on a real ordered topological vector space X:We give various characterizations of -generalized weak subdierential for this class of functions. It is well known that if the function f : X ! R is subdierentiable at x0 2 X; then f has a global minimizer at x0 if and only if 0 2 @ f (x0): We show that a similar result can be obtained for -generalized weak subdierential. Finally, we investigate some relations between -directional derivative and -generalized weak subdifferential. In fact, in the classical subdierential theory, it is well known that if the function f : X ! R is subdierentiable at x0 2 X and it has directional derivative at x0 in the direction u 2 X; then the relation f ′(x0; u) ⟨u; x⟩; 8 x 2 @ f (x0) is satisfied. We prove that a similar result can be obtained for - generalized weak subdierential.