If E is a weakly sequentially complete space and x : T → E is a function such that for every x ∗ ∈ E ∗, the real function t ↦ 〈 x ∗, x ( t ) 〉 is differentiable T, then x is weakly differentiable on T. Proposition 2.2 [[14], Theorem 1.2].
If f is weakly differentiable at (x_{0}), then it is gH-differentiable at (x_{0}), the converse is not true.
(iii) If f is weakly differentiable at (x_{0}), then it is gH-differentiable at (x_{0}), the converse is not true. .
However, in the terminology of [19], the L p norm is weakly differentiable at every point except the zero vector, that is ⋅ A α p is not weakly differentiable at (W u,φ 1)'.
} end{aligned} (2.31 which is weakly differentiable everywhere, so (widetilde{phi } in H^1 [0,1];mathbb {C}^n)) since (phi _1) is.
weakly differentiable on T, then x w ′ is Pettis integrable on T and x ( t ) = x ( 0 ) + ∫ 0 t x w ′ ( s ) d s, t ∈ T. In 1994 Kadets [17] proved that there exists a strongly measurable and Pettis integrable function x : T → E such that the indefinite Pettis integral y ( t ) = ∫ 0 t x ( s ) d s, t ∈ T, (4). is not weakly differentiable on a set of positive Lebesgue measures (see also [18, 19]).
weakly differentiable on T, then (x cdot)) is pseudo-differentiable on T and (x_{p}^{prime}(cdot )=x_{w}^{prime}(cdot)) a.e. on T. The concept of a Bochner integral and a Pettis integral are well known [12 14].
weakly differentiable on T, then x w ′ is Pettis integrable on T and x ( t ) = x ( 0 ) + ∫ 0 t x w ′ ( s ) d s, t ∈ T. .
If it exists, (x_{w}^{prime }(cdot)) is uniquely determined and it is called the weak derivative of (x cdot)) on T. Obviously, if (x cdot):Trightarrow E) is a weakly differentiable function on T, then the real function (tmapsto langle x^{ast},x t) rangle) is differentiable on T.
A pseudo-derivative (x_{p}^{prime}(cdot)) of a pseudo-differentiable function (x cdot):Trightarrow E) is weakly measurable on T (see [11]).
