Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space |
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Authors: | Shoji Kamimura Fumiaki Kohsaka Wataru Takahashi |
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Affiliation: | (1) Graduate School of International Corporate Strategy, Hitotsubashi University, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8439, Japan;(2) Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan |
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Abstract: | In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim.14 (1976), 877–898] and Kamimura and Takahashi [J. Approx. Theory106 (2000), 226–240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem. |
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Keywords: | convex minimization problem maximal monotone operator proximal point algorithm resolvent uniformly convex Banach space |
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