Taiwanese Journal of Mathematics

PROXIMAL POINT ALGORITHMS AND FOUR RESOLVENTS OF NONLINEAR OPERATORS OF MONOTONE TYPE IN BANACH SPACES

Wataru Takahashi

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Abstract

In this article, motivated by Rockafellar’s proximal point algorithm in Hilbert spaces, we discuss various weak and strong convergence theorems for resolvents of accretive operators and maximal monotone operators which are connected with the proximal point algorithm. We first deal with proximal point algorithms in Hilbert spaces. Then, we consider weak and strong convergence theorems for resolvents of accretive operators in Banach spaces which generalize the results in Hilbert spaces. Further, we deal with weak and strong convergence theorems for three types of resolvents of maximal monotone operators in Banach spaces which are related to proximal point algorithms. Finally, in Section 7, we apply some results obtained in Banach spaces to the problem of finding minimizers of convex functions in Banach spaces.

Article information

Source
Taiwanese J. Math., Volume 12, Number 8 (2008), 1883-1910.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405125

Digital Object Identifier
doi:10.11650/twjm/1500405125

Mathematical Reviews number (MathSciNet)
MR2449952

Zentralblatt MATH identifier
1215.47092

Subjects
Primary: 47H05: Monotone operators and generalizations 47J25: Iterative procedures [See also 65J15]

Keywords
Banach space proximal point algorithm resolvent nonexpansive mapping maximal monotone operator retraction projection convex optimization

Citation

Takahashi, Wataru. PROXIMAL POINT ALGORITHMS AND FOUR RESOLVENTS OF NONLINEAR OPERATORS OF MONOTONE TYPE IN BANACH SPACES. Taiwanese J. Math. 12 (2008), no. 8, 1883--1910. doi:10.11650/twjm/1500405125. https://projecteuclid.org/euclid.twjm/1500405125


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