By S. Kusuoka, A. Yamazaki

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

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**Extra info for Advances in Mathematical Economics, Vol. 10**

**Example text**

Po)» ^ < 0 for all p\ > 0; for any given po > PQ' ^^^^^ exists p | e (0, po) such that Z? < 0 for all P\ > PI Consider now the quasi capital intensity difference at the social level b. Under Assumption 2 and PioPox/^wPm < 1, we derive from (20) that hm^j^o ^1 >o ^ ^ while hmp^^o ^| _^_i < 0- Therefore, there exists PQ € (-1,0) such that hmp^^o ^ > 0 for any po > PQ. Similarly, we derive from (21) that limpi^+oo ^l^^^o ^ ^ ^^^^^^ ^^^^^' ^^^^^ hmp,-^+oo ^|po^+oo ^ ^• Therefore, under (22), there exists PQ > 0 such that Ump^-^+oo ^ > 0 for any Po > PQ- Notice also that ^1 _ ^ 1 _ ^^Q^Q^ /^00^1 A ^^^0 ^24) is an increasing function of po since ^QQPU/P\QPQ\ < 1.

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?

Therefore, {JC„} converges strongly to R(BJ)-^OM' 5. Applications In this section, we study the problem of finding a minimizer of a proper lower semicontinuous convex function; see [2,16,21]. 2, we obtain the following result. 1. Let E be a smooth and uniformly convex Banach space whose duality mapping J is weakly sequentially continuous. Let f* : E* -> (—cx), oo] be a proper lower semicontinuous convex function such that (3/*)~H0) ^ 0.