We say that the function f: [OMEGA] [right arrow] Y is H-differentiable (differentiable
in the sense of Hadamard) at the point a [member of] [OMEGA] if there exists u [member of] L(X,Y) such that
(2) the functions [[phi].sub.1[sigma]] and [[phi].sub.2[sigma]] are two differentiable
functions and [[[D.sup.1.sub.2][phi](x)].sup.[sigma]] = [[[phi]'.sub.2[sigma]](x), [[phi]'.sub.1[sigma]](x)], when [phi] is (2)-differentiable.
f is first-order differentiable
, so in convex domain D, first-order difference of f can be written in the following integral form:
Let J be a differentiable
application on [[OMEGA].sub.0].
Recently, Farid  extended Theorems 1 and 2 to functions of two variables that are differentiable
on their coordinates.
For nonnegative and differentiable
functions i, [m.sub.0], and [m.sub.1]: [[a.sub.0], [omega]] [right arrow] R, with [DELTA]m(a) = [m.sub.1](a)-[m.sub.0] (a) [greater than or equal to] 0 for all a [member of] [[a.sub.0], [omega]], the system of ODEs given by equations (1) and (2) with initial conditions S([a.sub.0]) = [S.sub.0] [greater than or equal to] 0, C([a.sub.0]) = [C.sub.0] [greater than or equal to] 0, and S([a.sub.0])+ C([a.sub.0]) > 0 has a unique solution S and C with N(a) = S(a) + C(a) [greater than or equal to] 0 for all a [member of] [[a.sub.0], [omega]].
If f (x) is differentiable
at x = a then it is continuous at x = a.
there exists a twice continuously differentiable
function v :
Let X be a real Banach space, and let [PHI], [PSI] : X [right arrow] R be two continuously Gateaux differentiable
functionals such that [inf.sub.X] [PHI] = [PHI](0) = [PSI](0) = 0.
As an application of the main results, it is shown that if we identify every fuzzy number with the corresponding equivalence class, there would be more differentiable
fuzzy functions than what is found in the literature.
Motivated by the above works, in this paper we give some applications and examples for trapezoidal and midpoint type inequalities when the intended function is differentiable
. Furthermore we consider integral quadrature formula and give an error estimate related to trapezoidal and midpoint formula.
The main disadvantage can be seen immediately by considering (7): the terminal condition [PSI] must be infinitely Malliavin differentiable
. In contradistinction, the viscosity solution given in  necessitates [PSI] to be only bounded and continuous.