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 suggestions. quite a few mathematical theories were offering economists with fundamental machineries for those difficulties coming up in fiscal conception. Conversely, mathematicians were prompted by means of quite a few mathematical problems raised through fiscal theories. The sequence is designed to compile these mathematicians who have been heavily drawn to getting new tough stimuli from monetary theories with these economists who're looking for potent mathematical instruments for his or her researchers. contributors of the editorial board of this sequence involves following widespread economists and mathematicians: coping with 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), okay. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), T. Maruyama (Keio Univ.), H. Matano (Univ. Tokyo), ok. 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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**Example text**

Let C he a nonempty closed subset of a smooth and strictly convex Banach space E. Let Re be a retraction ofE onto C. Then Re is sunny and generalized nonexpansive if and only if {x-Rcx,JRcx-Jy) > 0, for each x e E and y e C, where J is the duality mapping of E. Let £ be a reflexive, strictly convex, and smooth Banach space with its dual £*. 1 in [4]). So, for each r > 0 and x e £, we can consider the set JrX = {z e E : x e z + rBJz}. From [4], JrX consists of one point. We denote such ^ Jr by (I -\- rB J)~^.

Bij = 0, we have A{w, p) = A(w, p). The Rybczynski and Stolper-Samuelson theorems are equivalent since [dy/dx] = [dw/dpY. However, in presence of externalities, the Rybczynski effects depend on the input coefficients from the private perspective while the Stolper-Samuelson effects depend on the quasi-input coefficients from Capital-labor substitution and indeterminacy 35 the social perspective. The duality between these two effects is thus destroyed. This property is again explained by the same intuition as the one given above.

OO ^ ^' Therefore, there exists PQ > 0 such that Umpj^o^ > 0 for any po > Pg- Since b > b, we get PQ < PQ. Similarly, we derive from (21) that limpj^+oo^| ^Q < 0 while Umpi^+oo^l _>+oo ^ ^' Therefore, there exists PQ > 0 such that hmpi^+oo ^ > 0 for any po > PQ. Now notice from (20) and (21) the following property: 1 -^+^ ^po=Po y ^Qi J \8 + gj Therefore, we get PQ < PQ. Consider finally equations (24) and (26). Since PQ > 0 is such that ]imp^^ob\ 2 = 0, we derive from (20): b\^^^^^^2 < 0. On the contrary, since Pn > 0 is such that Um^i^+cx) ^1 ^3 = 0, we derive from (21): Therefore, we get b\p^=pQ > 0 for any po > PQConsidering Lemma 5, we then conclude the following results: (c) (c) for any given po € (PQ, PQ), there exists p^ e (0, po] such that Z?