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

A lot of monetary difficulties can formulated as restricted optimizations and equilibration in their strategies. quite a few mathematical theories were providing economists with fundamental machineries for those difficulties bobbing up in financial concept. Conversely, mathematicians were motivated by way of a number of mathematical problems raised by way of financial theories. The sequence is designed to compile these mathematicians who have been heavily drawn to getting new not easy stimuli from fiscal theories with these economists who're looking for potent mathematical instruments for his or her researchers. participants of the editorial board of this sequence comprises following admired 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.), okay. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), T. Maruyama (Keio Univ.), H. Matano (Univ. Tokyo), ok. Nishimura (Kyoto Univ.), M. okay. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

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**Sample text**

If the following condition holds we get with hmpi_^+oo ^p^^o < ^ and hmpi_^+oo ^|po_^+oo ^ ^- Therefore, under (22), there exists PQ > 0 such that hmpj^+oo ^ > 0 for any po > PQConsidering Lemma 5 under (22), we then conclude the following results: - for any given po e (p^, PQ), there exists p^ > 0 such that ^ > 0 for all Pi € [0, p^). b > ^ for all p\ > po- Notice that under Assumption 2, (22) imphes (19). Therefore, under (22), we conclude that local indeterminacy occurs in the following cases: (i) (ii) for any given po e (max{p(J, p^}, p^) and p\ e {p_^,p\) with p^ = p\ > max{0, Po} and px = p^ > p_^\ for any given po > PQ and p\ > p ^ with p ^ = p^ > po.

G ^^ W a mapping transforming weakly convergent sequences in G into sequences in W converging in measure with respect to the norm topology of F\ Let j be a convex normal integrand defined on[0, I] x F satisfying 0

4. Let F be a separable Banach space. Let G be a nonempty weakly compact subset of L\^{[0, 1], dt) and let W be the closed unit ball of L^,([0, 1], dt). Let A '. G ^^ W a mapping transforming weakly convergent sequences in G into sequences in W converging in measure with respect to the norm topology of F\ Let j be a convex normal integrand defined on[0, I] x F satisfying 0