Here we consider the case of infinitely
differentiable functions on [R.sub.+] with the condition that ([[f.sup.m])[sigma])(x) is uniformly bounded with respect to m and x.
Let V: [[t.sub.0], [infinity]) x [R.sup.n] [right arrow] [R.sup.+] be continuously
differentiable function. The solution of the fractional differential equation
where [alpha] is a real parameter which will be determined later and v is a twice continuously
differentiable function. That is, u(x, t) depends on x and t primarily through the term [absolute value of x]/ [absolute value of 4kt].
A
differentiable function [phi] : S [subset or equal to] [R.sup.n] [right arrow] [R.sup.n] is said to be (strictly) pseudoinvex function with respect to [eta] : S x S [right arrow] [R.sup.n] if
Then, for any [sigma] [member of] G, a neighborhood U [subset] [R.sup.l] of [[??].sub.0] and a neighborhood G [subset] [R.sup.r] of [[sigma].sub.0] can make the equation [PSI]([??], [sigma]) = 0 which has a unique solution [??] [member of] U, and the solution can be expressed as [??] = [g.sub.0] ([sigma]), where [g.sub.0](*) is a continuously
differentiable function on [sigma] = [[sigma].sub.0].
Then, for any
differentiable function w : [0,n] [intersection] Z [right arrow] R, and each k = 1,2, ..., n, one has
where f(x) and y(x) are
differentiable functions. Here, we apply fractional integration operational matrix of rational Haar wavelet to solve Abel integral equations as fractional integral equations.
where h(t) = diag{[h.sub.1](t), [h.sub.2](t), ..., [h.sub.n](t)}, [h.sub.i](t) are continuously
differentiable functions, and h(t) is a scaling function matrix.
Let f : I [subset] r [right arrow] R be a twice
differentiable function on [I.sup.[omicron]] ([I.sup.[omicron]] is the interior of I), and let a,b [member of] [I.sup.[omicron]] with a < b.
Suppose that f : [a,b] [right arrow] R be a twice
differentiable function on (a,b) and suppose that [gamma] [less than or equal to] f" (t) [less than or equal to] [gamma] for all t [member of] (a, b).
Let (Eq.) be a
differentiable function on (Eq.) such that (Eq.) where (Eq.).
It is clear that the payoff function [[PHI].sub.i]([p.sub.i], y) is a continuously
differentiable function of [p.sub.i](s) and y(s).