By C. Castaing, M. Saadoune (auth.), Shigeo Kusuoka, Akira Yamazaki (eds.)

ISBN-10: 4431727337

ISBN-13: 9784431727330

ISBN-10: 4431727612

ISBN-13: 9784431727613

A lot of monetary difficulties can formulated as restricted optimizations and equilibration in their ideas. quite a few mathematical theories were offering economists with critical machineries for those difficulties bobbing up in financial conception. Conversely, mathematicians were inspired via a variety of mathematical problems raised via financial theories. The sequence is designed to collect these mathematicians who have been heavily attracted to getting new not easy stimuli from financial theories with these economists who're looking for powerful mathematical instruments for his or her researchers. participants of the editorial board of this sequence includes following fashionable economists and mathematicians: handling Editors: S. Kusuoka (Univ. Tokyo), A. Yamazaki (Hitotsubashi Univ.) - Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier II), F. H. Clarke (Univ. Lyon I), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Fukuoka Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Hitotsubashi Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), okay. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), T. Maruyama (Keio Univ.), H. Matano (Univ. Tokyo), okay. Nishimura (Kyoto Univ.), M. ok. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

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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, . .

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. Key words: Banach space, generalized nonexpansive retraction, proximal point algorithm, convex minimization problem 1. Introduction Let / / be a Hilbert space and let T be a maximal monotone operator from H to / / .

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, . . , where {an\ C [0, 1] and {r^} C (0, oo) satisfy limsupa„ < 1 and liminf r^ > 0. If T~^0 ^ 0, then the sequence {xn'S converges weakly to an element v of r~^0, where v — Um„_>oo ^r-io^n ^^^ ^T-^O ^^ ^^^ metric projection of H onto I'^O. 2 ([6]). Let H be a Hilbert space and let T c H x H be a maximal monotone operator Let Jr = (I + rT)~^ for all r > 0 and let {JC„} be a sequence defined as follows: x\ = x e H and Xn-^\ = anX + (1 - an)Jr„Xn, « = 1, 2, .

### Advances in Mathematical Economics by C. Castaing, M. Saadoune (auth.), Shigeo Kusuoka, Akira Yamazaki (eds.)

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