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Generalized Mann iterates for constructing fixed points in Hilbert spaces. (English) Zbl 1032.47034
Authors’ abstract: “The mean iteration scheme originally proposed by Mann is extended to a broad class of relaxed, inexact fixed point algorithms in Hilbert spaces. Weak and strong convergence results are established under general conditions on the underlying averaging process and the type of operators involved. This analysis significantly widens the range of applications of mean iteration methods. Several examples are given”.

MSC:
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
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References:
[1] Bauschke, H. H.; Borwein, J. M.: On projection algorithms for solving convex feasibility problems. SIAM rev. 38, 367-426 (1996) · Zbl 0865.47039
[2] 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
[3] Borwein, D.; Borwein, J.: Fixed point iterations for real functions. J. math. Anal. appl. 157, 112-126 (1991) · Zbl 0742.26006
[4] Borwein, J.; Reich, S.; Shafrir, I.: Krasnoselski--Mann iterations in normed spaces. Canad. math. Bull. 35, 1-28 (1992) · Zbl 0712.47050
[5] Browder, F. E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201-225 (1967) · Zbl 0149.36301
[6] Chidume, C. E.; Mutangadura, S. A.: An example on the Mann iteration method for Lipschitz pseudocontractions. Proc. amer. Math. soc. 129, 2359-2363 (2001) · Zbl 0972.47062
[7] 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
[8] Combettes, P. L.: Hilbertian convex feasibility problem: convergence of projection methods. Appl. math. Optim. 35, 311-330 (1997) · Zbl 0872.90069
[9] Combettes, P. L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE trans. Image process. 6, 493-506 (1997)
[10] Combettes, P. L.: Quasi-Fejérian analysis of some optimization algorithms. Inherently parallel algorithms for feasibility and optimization, 115-152 (2001) · Zbl 0992.65065
[11] Dotson, W. G.: On the Mann iterative process. Trans. amer. Math. soc. 149, 65-73 (1970) · Zbl 0203.14801
[12] J. Eckstein, Splitting methods for monotone operators with applications to parallel optimization, Thesis, Massachusetts Institute of Technology (1989)
[13] Eckstein, J.: The alternating step method for monotropic programming on the connection machine CM-2. ORSA J. Comput. 5, 84-96 (1993) · Zbl 0773.90055
[14] Eckstein, J.; Bertsekas, D. P.: On the Douglas--Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. programming 55, 293-318 (1992) · Zbl 0765.90073
[15] Engl, H. W.; Leitão, A.: A Mann iterative regularization method for elliptic Cauchy problems. Numer. funct. Anal. optim. 22, 861-884 (2001) · Zbl 0998.65114
[16] Goebel, K.; Kirk, W. A.: Topics in metric fixed point theory. (1990) · Zbl 0708.47031
[17] Groetsch, C. W.: A note on segmenting Mann iterates. J. math. Anal. appl. 40, 369-372 (1972) · Zbl 0244.47042
[18] Hicks, T. L.; Kubicek, J. D.: On the Mann iteration process in a Hilbert space. J. math. Anal. appl. 59, 498-504 (1977) · Zbl 0361.65057
[19] Kiwiel, K. C.; łopuch, B.: Surrogate projection methods for finding fixed points of firmly nonexpansive mappings. SIAM J. Optim. 7, 1084-1102 (1997) · Zbl 0905.47044
[20] Knopp, K.: Infinite sequences and series. (1956) · Zbl 0070.05807
[21] Lions, P. L.; Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. anal. 16, 964-979 (1979) · Zbl 0426.65050
[22] Mann, W. R.: Mean value methods in iteration. Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603
[23] Măruşter, Şt.: The solution by iteration of nonlinear equations in Hilbert spaces. Proc. amer. Math. soc. 63, 69-73 (1977)
[24] Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. amer. Math. soc. 73, 591-597 (1967) · Zbl 0179.19902
[25] Ortega, J. M.: Numerical analysis--A second course. (1972) · Zbl 0248.65001
[26] Park, J. Y.; Jeong, J. U.: Weak convergence to a fixed point of the sequence of Mann type iterates. J. math. Anal. appl. 184, 75-81 (1994) · Zbl 0811.47067
[27] T. Pennanen, B.F. Svaiter, Solving monotone inclusions with linear multi-step methods, submitted · Zbl 1033.65051
[28] Petryshyn, W. V.: Construction of fixed points of demicompact mappings in Hilbert space. J. math. Anal. appl. 14, 276-284 (1966) · Zbl 0138.39802
[29] Pierra, G.: Decomposition through formalization in a product space. Math. programming 28, 96-115 (1984) · Zbl 0523.49022
[30] Reinermann, J.: Über toeplitzsche iterationsverfahren und einige ihrer anwendungen in der konstruktiven fixpunkttheorie. Studia math. 32, 209-227 (1969) · Zbl 0176.12302
[31] Rhoades, B. E.: Fixed point iterations using infinite matrices. Trans. amer. Math. soc. 196, 161-176 (1974) · Zbl 0285.47038
[32] Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control optim. 14, 877-898 (1976) · Zbl 0358.90053