They have created a setting to introduce the operator inverse to [D.sup.(k).sub.t] through the concept of Completely Positive kernels [5]: a kernel M [member of] [L.sub.1, loc] ([R.sub.+]) is called completely positive if there are [k.sub.0] [greater than or equal to] 0 and nonnegative and

nonincreasing [k.sub.1] [member of] [L.sub.1, loc]([R.sub.+]) such that M * ([k.sub.0][delta] + [k.sub.1]) = 1 holds.

Thus, the sequence {d([y.sub.n], [y.sub.n+1])} is

nonincreasing and converges to the greatest lower bound of its range, which we denote by l.

([f.sub.0]) f [member of] C((0,[infinity]),[R.sub.+]) satisfies that there exists [sigma] > 0, such that f is

nonincreasing on (0, [sigma]], [[integral].sup.[sigma].sub.0] f(s)ds < [infinity], and there exists [alpha], [gamma] [member of] (0,1) such that

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

nonincreasing or nondecreasing.

Incentive compatibility constraints (7) and (8) can be reduced to the

nonincreasing functions [mathematical expression not reproducible] and [mathematical expression not reproducible].

(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 w : [0, +[infinity]) [right arrow] [0, +[infinity]) be an increasing function with [omega](0) = 0; we say that [omega] is a majorant if [omega](t)/t is

nonincreasing for t > 0 (cf.

Moreover, we assume

nonincreasing returns on self-protection.

It is easy to check that [d.sub.f] and f* are nonnegative and

nonincreasing functions.

nonincreasing) sequence in E such that [x.sub.n] [right arrow] [x.sup.*] as n [right arrow] [infinity], then [x.sub.n] [??] [x.sup.*] (resp.

where [lambda] is a positive regularization vector in

nonincreasing order.

It can be obtained that [mathematical expression not reproducible] (x, [DELTA]) is

nonincreasing for the variable [DELTA].