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Theorem 7

Given a set of N maps and probabilities (w, p), satisfying the properties i)- v) along with Wi : [0, 1] →[Ci, D i) then the fixed points of M : M([0, 1]) → M([0, 1]) and Tp : F([0, 1]) → F([0, 1]), say ~µ and ~ F respectively, relate as follows

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Proof. From the contractivity of M and Tp, there exist

and

fixed points of M and Tp, respectively. Let

the thesis consists of proving

So we have:

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By the uniqueness of the fixed points, we get

The previous theorem allows to reuse the results of Forte and Vrscay (1995) and in particular gives another way of finding the solution of ( P ) in terms of ( Q ) at least on the simplex ΠN by letting δi = 0 in CN. This is true in particular if we choose the maps as in TF . To be more explicit: from now on the functional Tp is intended to have fixed maps w and all δi = 0.

Stefano M. Iacus, Davide La Torre