Adv. Math. Econ. 10, 51-64 (2007)

Weak and strong convergence theorems for new resolvents of maximal monotone operators in Banach spaces

Takanori Ibaraki^{1} and Wataru Takahashi^{2}

1 Information Security Promotion Agency, Nagoya University, Furo-cho, Chikusaku, Nagoya, Aichi 464-8601, Japan

2 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan

Received: September 11, 2006 Revised: November 2, 2006

JEL classification: C61, C63, C02

Mathematical Subject Classification (2000): 49M05, 47H05, 47H09

Abstract. 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