By S. Kusuoka, A. Yamazaki

Loads of fiscal difficulties can formulated as limited optimizations and equilibration in their strategies. numerous mathematical theories were delivering economists with necessary machineries for those difficulties bobbing up in monetary concept. Conversely, mathematicians were motivated via a variety of mathematical problems raised via monetary theories. The sequence is designed to collect these mathematicians who have been heavily attracted to getting new demanding stimuli from fiscal theories with these economists who're looking for powerful mathematical instruments for his or her researchers.

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**Example text**

2) and hm inf „_»oo ^n > 0. we have lim = 0. If (z*, z) € J5, then it holds from monotonicity of B that ^rii ~ yrii * J \ ^ r\ for all i e N. Since J is weakly sequentially continuous, letting / -^ oo, we get {z, z* — Jv) > 0. Then, the maximality of B imphes Jv e B~^0, That is, i; 6 {BJ)-^0. } and {xnj} be two subsequences of {xn} such that x^ -^ v\ and Xfij -^ V2- As above, we have v\,V2 ^ (BJ)~^0. Put a = lim {V(xn, vi) - V(xn, V2)). Note that V(Xn, Vi) - ViXn, V2) = 2(jC„, JV2 -Jvi) + \\Vi f - \\v2f, W = 1, 2, .

1) in a Hilbert space H is the proximal point algorithm: x\ e H and 52 T. Ibaraki and W. Takahashi Xn+\=Jrn^n. « = 1, 2, . . 2) where {r„} c (0, oo)andyr = (Z+rJ)"^ forallr > 0. This algorithm was first introduced by Martinet [5]. 1). Motivated by Rockafellar's result, Kamimura and Takahashi [6] proved the following two convergence theorems. 1 ([6]). Let H be a Hilbert space and letT c H x H be a maximal monotone operator. Let Jr — {I -\- rT)~^ for all r > 0 and let {xn} be a sequence defined as follows: x\ = x e H and Xn-\-\ = OinXn + (1 - an)^r„^Ai, « = 1, 2, .

Proof By our assumption it is immediate that 4>(M) := {Au, u)i^i^oo^i^\ ^ -h J(u) verifies \^{u)\<{\ + mu\\ii^y F for all u e G. Let (M„) be a minimizing sequence for 4>, that is lim„ 0(M„) = inf^eG ^(w) with Un e G for all n. We may assume that (w„) converges a(Lj^, L^f) (alias weakly) in L]^([0, l],dt) to u e G. By our assumption, the sequence (Aun) is bounded in L^([0, 1], dt) and converges in measure to Au with respect to the norm topology of F\ By virtue of Castaing [10], see also Grothendieck [8] for the one dimensional case, we conclude that l i m ( ^ M „ , Un)ijoo / I \ = {Au, u),joo j \ \.