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Strong convergence of the composite Halpern iteration. (English) Zbl 1135.47052

A three-step fixed point iteration procedure of Halpern type is used to approximate fixed points of nonexpansive mappings in uniformly smooth Banach spaces. A strong convergence theorem is proved for this iteration procedure.

MSC:

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
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References:

[1] Bruck, R. E., Nonexpansive projections on subsets of Banach spaces, Pacific J. Math., 47, 341-355 (1973) · Zbl 0274.47030
[2] Browder, F. E., Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Arch. Ration. Mech. Anal., 24, 82-90 (1967) · Zbl 0148.13601
[3] Browder, F. E., Fixed points theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. USA, 53, 1272-1276 (1965) · Zbl 0125.35801
[4] Chang, S. S., On Halpern’s open question, Acta Math. Sinica, 48, 979-984 (2005) · Zbl 1125.47315
[5] Goebel, K.; Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984), Marcel Dekker: Marcel Dekker New York · Zbl 0537.46001
[6] Halpern, B., Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. (N.S.), 73, 957-961 (1967) · Zbl 0177.19101
[7] Kim, T. H.; Xu, H. K., Strong convergence of modified Mann iterations, J. Math. Anal. Appl., 61, 51-60 (2005) · Zbl 1091.47055
[8] Lions, P. L., Approximation de points fixes de contractions, C. R. Acad. Sci. Paris Sér. A-B, 284, 1357-1359 (1977) · Zbl 0349.47046
[9] Reich, S., Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl., 44, 57-70 (1973) · Zbl 0275.47034
[10] Reich, S., Approximating fixed points of nonexpansive mappings, Panamer. Math. J., 4, 486-491 (1994)
[11] Reich, S., Strong convergence theorems for resolvent of accretive operators in Banach spaces, J. Math. Anal. Appl., 75, 287-292 (1980) · Zbl 0437.47047
[12] Wittmann, R., Approximation of fixed points nonexpansive mappings, Arch. Math. (Basel), 59, 486-491 (1992) · Zbl 0797.47036
[13] Xu, H. K., Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66, 240-256 (2002) · Zbl 1013.47032
[14] Xu, H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 659-678 (2003) · Zbl 1043.90063
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