([H.sub.1)] f (t, z) is

nondecreasing in z, and there exists a constant 0 < b < 1 such that, for any (t,x) [member of] (0,1) x [0, and r e (0,1),

Moreover, assume that g is

nondecreasing on [R.sub.+], then the solution for problem (1) is unique for any [lambda], [mu] [member of] [R.sub.+].

Suppose that a sequence [{[x.sub.[alpha]]}.sub.[alpha][member of][OMEGA]] [subset or equal to] R is bounded and either nonincreasing or

nondecreasing. Then there exists [beta] [member of] [OMEGA] such that [x.sub.[alpha]] = [x.sub.[beta]] for all [OMEGA] > [alpha] [greater than or equal to] [beta].

(C) The function [mathematical expression not reproducible] is nonincreasing on the left of X = [x.sub.0] and

nondecreasing on the right of x = [x.sub.0].

Let ([OMEGA], [summation], [mu]) be a measure space and let [PSI] : [0, +[infinity]) [right arrow] [0, +[infinity]) be a

nondecreasing function.

Then the map T is said to have mixed monotone property if T(x, y) is monotone

nondecreasing in x and is monotone nonincreasing in y; that is,

where [alpha] [member of] [C.sup.1](I, I) is

nondecreasing with [alpha](t) [less than or equal to] t on I = [0, T) and [u.sub.1] and [u.sub.2] are constants.

An Orlicz function is a function M : [0, [infinity]) [right arrow] [0, [infinity]) which is continuous,

nondecreasing, and convex with M(0) = 0, M(x) > 0 for x > 0 and M(x) [right arrow] [infinity], as x [right arrow] [infinity], (see Krasnoselskii and Rutickii [6]).

Let g(t) be a positive function on (0, [infinity]) such that for some [t.sub.0] > 0, [square root of t]g(t) is

nondecreasing and g(t)/[square root of t] is bounded for all t [greater than or equal to] [t.sub.0].

A multi-valued mapping T : K [right arrow] CB(K) is said to satisfy condition (I) if there exists a

nondecreasing function h : [0, [infinity]) [right arrow] [0, [infinity]) with h(0) = 0 and h(r) > 0 for all r [member of] (0, to) such that dist(x, Tx) [greater than or equal to] h(dist(x, F(T)) for all x [member of] K.

According to the second law of thermodynamics, the sum of the horizon entropy and the entropy of the matter field, that is, the total entropy, is a

nondecreasing function of time.

We shall see in the next section that F(t) is stochastically monotone; that is, [[summation].sub.k[greater than or equal to]j] [f.sub.ik](t) is

nondecreasing in i, for each j.