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Iterative selection methods for common fixed point problems. (English) Zbl 1106.47057
The author presents two iterative methods for solving the following problem. Let $(T_n)_{n\in \mathbb{N}}$ and $(S_n)_{n\in \mathbb{N}}$ be two families of quasi-nonexpansive operators from a Hilbert space $H$ to itself such that $\emptyset \neq \bigcap_{n\in \mathbb{N}}\text{Fix}T_n\subset \bigcap_{n\in \mathbb{N}}\text{Fix}S_n$ and $Q:H\rightarrow H$ be a strict contraction. Find (the unique) $\overline{x} \in H$ such that $\overline{x}=P_C(Q\overline{x})$, where $P_C$ is the projection operator on $C$. Finally, applications to monotone inclusions and equilibrium problems are considered.

47J25Iterative procedures (nonlinear operator equations)
65J15Equations with nonlinear operators (numerical methods)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
[1] Aubin, J. P.; Frankowska, H.: Set-valued analysis. (1990) · Zbl 0713.49021
[2] Bauschke, H. H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. math. Anal. appl. 202, 150-159 (1996) · Zbl 0956.47024
[3] Bauschke, H. H.; Borwein, J. M.: On projection algorithms for solving convex feasibility problems. SIAM rev. 38, 367-426 (1996) · Zbl 0865.47039
[4] Bauschke, H. H.; Combettes, P. L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. oper. Res. 26, 248-264 (2001) · Zbl 1082.65058
[5] Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. stud. 63, 123-145 (1994) · Zbl 0888.49007
[6] Bregman, L. M.: The method of successive projection for finding a common point of convex sets. Soviet math. Dokl. 6, 688-692 (1965) · Zbl 0142.16804
[7] Browder, F. E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. ration. Mech. anal. 24, 82-90 (1967) · Zbl 0148.13601
[8] Combettes, P. L.: Construction d’un point fixe commun à une famille de contractions fermes. C. R. Acad. sci. Paris sér. I math. 320, 1385-1390 (1995) · Zbl 0830.65047
[9] Combettes, P. L.: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE trans. Signal process. 51, 1771-1782 (2003)
[10] Combettes, P. L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475-504 (2004) · Zbl 1153.47305
[11] Combettes, P. L.; Hirstoaga, S. A.: Equilibrium programming in Hilbert spaces. J. nonlinear convex anal. 6, 117-136 (2005) · Zbl 1109.90079
[12] Deutsch, F.; Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. funct. Anal. optim. 19, 33-56 (1998) · Zbl 0913.47048
[13] Dotson, W. G.: On the Mann iterative process. Trans. amer. Math. soc. 149, 65-73 (1970) · Zbl 0203.14801
[14] Dunn, J. C.: Convexity, monotonicity, and gradient processes in Hilbert space. J. math. Anal. appl. 53, 145-158 (1976) · Zbl 0321.49025
[15] Halpern, B.: Fixed points of nonexpanding maps. Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101
[16] Kamimura, S.; Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. approx. Theory 106, 226-240 (2000) · Zbl 0992.47022
[17] Knopp, K.: Infinite sequences and series. (1956) · Zbl 0070.05807
[18] Lions, P. -L.: Approximation de points fixes de contractions. C. R. Acad. sci. Paris sér. A 284, 1357-1359 (1977) · Zbl 0349.47046
[19] Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039
[20] Moudafi, A.; Théra, M.: Proximal and dynamical approaches to equilibrium problems. Lecture notes in economics and mathematical systems 477, 187-201 (1999) · Zbl 0944.65080
[21] O’hara, J. G.; Pillay, P.; Xu, H. -K.: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear anal. 54, 1417-1426 (2003) · Zbl 1052.47049
[22] Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control optim. 14, 877-898 (1976) · Zbl 0358.90053
[23] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. math. 58, 486-491 (1992) · Zbl 0797.47036
[24] Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Inherently parallel algorithms in feasibility and optimization and their applications, 473-504 (2001) · Zbl 1013.49005