# Advances in Mathematical Economics (Volume 3) by Charles Castaing

By Charles Castaing

Loads of monetary difficulties can formulated as limited optimizations and equilibration in their strategies. a number of mathematical theories were offering economists with critical machineries for those difficulties bobbing up in fiscal conception. Conversely, mathematicians were inspired via numerous mathematical problems raised via fiscal theories. The sequence is designed to assemble these mathematicians who have been heavily attracted to getting new difficult 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 contains following well-liked economists and mathematicians: coping with Editors: S. Kusuoka (Univ. Tokyo), T. Maruyama (Keio Univ.) Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier), F. H. Clarke (Univ. Lyon I), G. Debreu (U.C. Berkeleyer), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Okayama Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Ohio nation Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), ok. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ. Tokyo), okay. 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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X n E C+(K); cI, ... , cn > O}. The collection of images ~ = UK X dJ-L)xEC+(K) of this set, lying in D, is {~ E im(¢) : 1~(Xi) - ~o(xi)1 < cd for the same finite set {Xl, ... ,X n } ~ C+ (K) and the same CI, ... ,Cn > 0, which is exactly a basic neighbourhood of~o = UK XdJ-LO)XEC+(K) in the subspace topology on im( ¢) induced from the product topology on D. Therefore open sets map bijectively onto open sets of the subspace topology which proves that when the codomain is restricted to im(¢) both functions ¢ and ¢-l are continuous.

A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. : A general denseness result for relaxed control theory, Bull. Austral. Math. Soc. 30463-475 (1984) [BaI88] : Generalized equilibrium results for games with incomplete information, Math. Oper. Res. 13 no. 11 (1989) [BaI90] : New sequential compactness results for spaces of scalarly integrable functions, J. Math. Anal. Appl. 151 1-16 (1990) [BaIOO] : New fundamentals of Young measure convergence, pp.

L 1 E A such that Fiber product of Young measures, application to a control problem 35 where UX,pl " I (/1,1) is the solution trajectory defined on [T, T + a] associated with b 1(p,1), p,1) b 1 E tl, p,1 E A). (p,2)t(z)] dt. JIZ Js T+U Let us define'Y E tl by setting, foraB p, E A: 'Y(p,)t = ,/1 (p,)t fort E [T, a] and 'Y(p,)t = ,/2 (p,)t for t E [T + a, 1]. Coming back to the definition of H J (T, x) HJ(T, x) ::; sup ILEA = 11 [JIZf [Jsf 1 [1[1 T + J(t, UX,P,'Y(p) , s, z) dp,t(s)] d-::y(p,)t(z)] dt T u + J(t, UX,ILI"I(IL I ), s, z) dp,;(s)] d,/l(p,lMz)] dt 11 [f [ f J(t, Vx,p2,,2(1L)' s, z) dP,Z(s)] d,/2(p,2)t(Z)] dt Jz Js T+U ::; KJ(T,X) + 2E.