([H.sub.0]) g(0, y) = 0, for ally [greater than or equal to] 0; ([partial derivative]g/[partial derivative]x)(x, y) [greater than or equal to] 0 (or g(x, y) is a strictly monotone increasing function with respect to x when f [equivalent to] 0) and ([partial derivative]g/[partial derivative]y)(x, y) [less than or equal to] 0, for all x [greater than or equal to] 0 and y [greater than or equal to] 0.

([H.sub.2]) f(x, y, v) is a strictly monotone increasing function with respect to x (or ([partial derivative]f/[partial derivative]x)(x, v, y) [greater than or equal to] 0 when g(x, y) is a strictly monotone increasing function with respect to x), for any fixed y [greater than or equal to] 0 and v [greater than or equal to] 0.

We strictly define t as the

monotone increasing function of both [n.sub.j] and [s.sub.j].

For 3 [less than or equal to] k [less than or equal to] n - 1, it is clear that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a strictly

monotone increasing function of t.

(A4) There exists a

monotone increasing function l(*) and k [member of] (0,1) such that

where [phi](t) is a differentiable function, [alpha](t) [member of] [C.sup.1] [0, T] is a strictly

monotone increasing function and satisfies that -[tau] [less than or equal to] [alpha](t) [less than or equal to] t and [alpha](0) = -[tau], there exists [t.sub.1] [member of] [0, T] such that [alpha]([t.sub.1]) = 0, and f:D = [0,T] x R x R x R is a given continuous mapping and satisfies the Lipschitz condition

(ii) [xi] = [xi](x) is a

monotone increasing function about x;

Suppose also that [psi]: [R.sub.+] [right arrow] [R.sub.+] is a

monotone increasing function such that [psi](0) = 0 and [psi]: [R.sub.+] [right arrow] [R.sub.+] is a sublinear comparison function.

is a strictly

monotone increasing function of the state variable y in case of risk complementarity and a monotone decreasing function in case if risk substitutability (cf.

[17] Let [??] and [??] be real-valued nonnegative continuous functions with domain {t : t [less than or equal to] [t.sub.0]} and let [alpha](t) = [[alpha].sub.0]([absolute value of t - [t.sub.0]]), where [[alpha].sub.0] is a

monotone increasing function. If, for t [greater than or equal to] [t.sub.0],

In Imoru and Olatinwo [12], the following contractive definition was employed: there exist a [member of] [0,1) and a

monotone increasing function [partial derivative]: [R.sub.+] [right arrow] R+ with [partial derivative](0) = 0, such that

That is to say, f(x) is a

monotone increasing function if x [member of] [1, [infinity]).