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A proximal-type algorithm by the hybrid method for maximal monotone operators in a Banach space. (English) Zbl 1152.47056
Takahashi, Wataru (ed.) et al., Nonlinear analysis and convex analysis. Proceedings of the 4th international conference (NACA 2005), Okinawa, Japan, June 30–July 4, 2005. Yokohama: Yokohama Publishers (ISBN 978-4-946552-27-4/hbk). 355-365 (2007).
The authors consider a proximal type algorithm for solving the inclusion $$0\in Tx$$, where $$T:E\to 2^{E^*}$$ is a maximal monotone operator acting in an uniformly convex and smooth real Banach space $$E$$. They first prove that the sequence generated by the algorithm is well-defined and then that a necessary and sufficient condition in order for a zero to exist for the inclusion $$0\in Tx$$ is the boundedness of that sequence. Finally, they apply the result in the study of a convex minimization problem.
For the entire collection see [Zbl 1104.47002].

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H05 Monotone operators and generalizations 49J53 Set-valued and variational analysis