Shrinking projection algorithm for fixed points of firmly nonexpansive mappings and its applications. (English) Zbl 1252.47075

Summary: The purpose of this paper is to study the shrinking projection method for finding common fixed points of firmly nonexpansive mappings. Some strong convergence theorems are proved. The main convergence theorem is also applied to equilibrium and optimization problems.
The results of this paper improve and extend the results of K. Aoyama, F. Kohsaka and W. Takahashi [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e-Suppl., e1626–e1632 (2009; Zbl 1238.47043)] in the following respects: (1) the main convergence theorem is proved by using the new method; (2) the condition of a family of firmly nonexpansive mappings \(\{T_n\}^{\infty}_{n=1}\) is relaxed from the condition \((Z)\) to uniformly closed; (3) an application is given to find the solution of equilibrium and optimization problems.


47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems


Zbl 1238.47043
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