Economic Theory

Advances in Mathematical Economics. Vol, 10 by S. Kusuoka, A. Yamazaki

By S. Kusuoka, A. Yamazaki

Loads of monetary difficulties can formulated as restricted optimizations and equilibration in their recommendations. quite a few mathematical theories were offering economists with quintessential machineries for those difficulties bobbing up in financial idea. Conversely, mathematicians were influenced through numerous mathematical problems raised via monetary theories. The sequence is designed to assemble these mathematicians who have been heavily attracted to getting new not easy stimuli from fiscal theories with these economists who're looking for potent mathematical instruments for his or her researchers.

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T. yi(t) = {Poixoiit)-'' + PuxniO-f^' ^ei(Xoi{t),Xu(t)))~^ x\{t) = y\(t) - gxiit) 1 =^oo(0 + -^oi(0 xi(t) = xioit) + xn(t) xiiO) and {ej(Xoj(t), Xiy(r))},>o, j = 0, 1, given where g > 0 is the depreciation rate of the capital stock. We can write the modified Hamiltonian in current value as n = [Pooxooity + Pioxioity -^eoiXooit), XioW))'^ +woit) (1 - xoo(t) - xoi(0) + mit) (xi(0 - xioit) - xu(t)) +P\{t)({Poixoi(tr^' +Puxnity ^eiiXoiiO^Xuit)))"^ -gxiit)). The staticfirst-orderconditions are given by equation (3).

These results were extended to more general Banach spaces; see Kamimura and Takahashi [7-9], Ohsawa and Takahashi [13], Kohsaka and Takahashi [11] and Kamimura et al [10]. Recently, Ibaraki and Takahashi [4] found new resolvents in a Banach space which are connected with a maximal monotone operator. Our purpose in this paper is to extend Kamimura-Takahashi's theorems to Banach spaces by using new resolvents of a maximal monotone operator. Such problems were posed in [22,23]. 1. 2. Using these theorems, we consider the problem of finding minimizers of convex functions defined on Banach spaces.

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. 3. Weak convergence theorem In this section, we first start with the following lemma. Compare this lemma with the results in Kamimura and Takahashi [9], and Kohsaka and Takahashi [11].

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