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. Theory 106 (2000), 226–240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.
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Kamimura, S., Kohsaka, F. & Takahashi, W. Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space. Set-Valued Anal 12, 417–429 (2004). https://doi.org/10.1007/s11228-004-8196-4
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DOI: https://doi.org/10.1007/s11228-004-8196-4