By S. Kusuoka, A. Yamazaki

Published every year lower than the auspices of the study heart of Mathematical Economics in Tokyo, this sequence brings jointly mathematicians attracted to monetary theories and economists looking potent mathematical instruments to help their study. Articles set forth unique effects and specified overviews of the issues less than dialogue, providing readers a transparent knowing of either fiscal and mathematical theories.

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**Example text**

5. D{Jr) = E for each r > 0. (BJ)~'^0 = F{Jr) for each r > 0, where F{Jr) is the set of fixed points of Jr. (BJ)-^O is closed Jr is generalized nonexpansive for each r > 0. Forr > Oandx e E, j{x — Jrx) e BJJrX. 4 ([4]). Let E be a uniformly convex Banach space with a Frechet differentiable norm and let B C E* x E be a maximal monotone operator with B~^0 7^ 0. Then the following hold: L 2. For each x e E, lim^-^oo JrX exists and belongs to (BJ)~^0. If Rx := lim^^oo JrX for each x e E, then R is a sunny generalized nonexpansive retraction of E onto (BJ)~^0.

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]). The following lemma is well known. 1 ([9]). Let E be a smooth and uniformly convex Banach space and let [Xn] and {yn} he sequences in E such that either [Xn] or {yn} is bounded.

Jp) Received: September 11, 2006 Revised: November 2, 2006 JEL classification: C61, C63, C02 Mathematics Subject Classification (2000): 49M05,47H05,47H09 Abstract. In this paper, we prove weak and strong convergence theorems for new resolvents of a maximal monotone operator in a Banach space which are connected with the proximal point algorithm of Rockafellar (SIAM J. Control. Optim. 14:877-898, 1976). Using these results, we consider the problem of finding minimizers of convex functions defined on Banach spaces.