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).