By S. Kusuoka, A. Yamazaki

Published annually lower than the auspices of the study heart of Mathematical Economics in Tokyo, this sequence brings jointly mathematicians drawn to financial theories and economists looking powerful mathematical instruments to assist their examine. Articles set forth unique effects and designated overviews of the issues below dialogue, supplying readers a transparent figuring out of either financial and mathematical theories.

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4. 5. D{Jr) = E for each r > 0. (BJ)~'^0 = F{Jr) for each r > 0, where F{Jr) is the set of fixed points of Jr. (BJ)-^O is closed Jr is generalized nonexpansive for each r > 0. Forr > Oandx e E, j{x — Jrx) e BJJrX. 4 ([4]). Let E be a uniformly convex Banach space with a Frechet differentiable norm and let B C E* x E be a maximal monotone operator with B~^0 7^ 0. Then the following hold: L 2. For each x e E, lim^-^oo JrX exists and belongs to (BJ)~^0. If Rx := lim^^oo JrX for each x e E, then R is a sunny generalized nonexpansive retraction of E onto (BJ)~^0.

Wethen conclude from all this that for any given po > Po, there exists p\ e [p\, 0) such that g(p\) is a monotone increasing function for all p\ > p\. u It clearly appears from Proposition 2 that under the conditions of Lemma 5, h and b are monotone decreasing functions of p\. We may now prove Proposition 4: in the first part of the proof we consider positive values of p\ only. Notice that UHopital's rule gives -_p^ ]im g{pi) (l+P0)^01 (16) V^ + ^y Consider the capital intensity difference at the private level b.

Let E be a separable reflexive Banach space, and Z be a compact metric space. ) is continuous on Z, (Hi) there is a constant c > 0 such that 0 < j{t,x,z) < c(l + ||jc||)/(9r all (t,x,z) G [0, 1] x £ X Z. (Z) being the compact metrizable space of all probability Radon measures defined on Z, endowed with the vague topology and let Hbe a convex weakly compact subset 6>/L^([0, 1], dt). Let us consider the integral fiinctional J{u, k) := \ Jo Uz j(t, u(t), z)Xt(dz) dt; J V(M, k) eHxIZ. Then the following hold: (a) there is a pair (u,X) eH x7Z such that maxmin/(M, X) < J(u, X) < XeTlueH vmnm3xJ(u,X).