By S. Kusuoka, A. Yamazaki

Loads of financial difficulties can formulated as restricted optimizations and equilibration in their recommendations. a variety of mathematical theories were providing economists with integral machineries for those difficulties coming up in monetary idea. Conversely, mathematicians were encouraged via a number of mathematical problems raised by way of monetary theories. The sequence is designed to compile these mathematicians who have been heavily attracted to getting new difficult stimuli from financial theories with these economists who're looking for powerful mathematical instruments for his or her researchers.

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56 T. Ibaraki and W. 1. Let E be a reflexive, strictly convex, and smooth Banach space, let B C E* X E be a maximal monotone operator with B~^0 ^ 0, and J, = (I-\- rBJ)-^ for all r > 0. Then V(JC, JrX) + VUrX, U) < V(x, w), for all r > 0, u e (Bjy^O, andx e E, Proof Let r > 0,u e (BJ)~^0, and x e Ehc given. 3 (5), we have V(X, U) = V{X, JrX) + V{JrX, u) + 2{x — JrX, J JrX — J u) = V(X, JrX) + VUrX, U) + 2r T > V(X, JrX)-\- ~ '^ - 0, J JrX - JU V{JrX,u). 1). 2. Let E be a smooth and uniformly convex Banach space whose duality mapping J is weakly sequentially continuous.

4. 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.

Intertemporal equilibrium and steady state A representative agent optimizes a hnear additively separable utility function with discount rate 8 > 0. 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)).

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