By C. Castaing, M. Saadoune (auth.), Shigeo Kusuoka, Akira Yamazaki (eds.)

A lot of financial difficulties can formulated as restricted optimizations and equilibration in their strategies. a variety of mathematical theories were offering economists with necessary machineries for those difficulties coming up in monetary thought. Conversely, mathematicians were motivated by means of numerous mathematical problems raised via financial theories. The sequence is designed to collect these mathematicians who have been heavily drawn to getting new demanding stimuli from fiscal theories with these economists who're looking for powerful mathematical instruments for his or her researchers. contributors of the editorial board of this sequence involves following well-liked economists and mathematicians: handling Editors: S. Kusuoka (Univ. Tokyo), A. Yamazaki (Hitotsubashi Univ.) - Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier II), F. H. Clarke (Univ. Lyon I), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Fukuoka Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Hitotsubashi Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), ok. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), T. Maruyama (Keio Univ.), H. Matano (Univ. Tokyo), okay. Nishimura (Kyoto Univ.), M. ok. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

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If B~^0 7^ 0, then the sequence {xn} converges weakly to an element of {BJ)-^0. Proof Note that ^"^0 y^ 0 implies (BJ)-^O i^ 0. In fact, if w* e ^"^0, we obtain 0 G 5M* and hence 0 e BJJ-^u\ So, we have / " ^ M * 6 {BJ)-^0. Put yn = Jr^Xn for all w € N and let z e (BJ)~^0 be given. We first prove that {xn} is bounded. z) + {l-an)V{yr,,z) «n V(;c„, z) + (1 - an){V(xr,, z) - V(jcn, yn)] < anV(Xn, Z) + (l- = V{Xr,,z), an)V(Xn, z) Weak and strong convergence theorems for new resolvents 57 for all n eN.

T. Ibaraki and W. Takahashi If E is reflexive, then 7 is a mapping of E onto £"*. If £" is smooth, then the duality mapping J is single valued. If E has a Frechet differentiable norm, then J is norm to norm continuous. If £^ has a uniformly Gateaux differentiable norm, then J is norm to weak* uniformly continuous on each bounded subset of E. E is strictly convex if and only if 7 is a strictly monotone operator. E is uniformly convex if and only if £"* is uniformly smooth. 2) for each x,y e E. 3) e £'(see [9]).

6. Proof of Proposition 2 (i) From (4) we derive /an aiiaoo - aio^oi = ^oofloi I aio\ \«oi am/ xu (, I = ^^oo^oi — I 1 ^00-^11 / and the result follows after substitution of (13) and (14) into the previous expression, From (6) we derive (ii) <^ii«oo - «io«oi = ^oo^^oi ^o\ P\\ aox ^n Po\ -^11 /< Po\ p\\ -^01 \ PooPioaooJ i^ioii^oi i^ooi^ii -^10^01 \ PooPu Piopoi ^00^11 y and the result follows after substitution of (13) and (14) into the previous expression, The result is derived as in (i) but considering instead the input coefficients at the social level (5).