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Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces. (English) Zbl 1229.47117
Summary: We first obtain a weak mean convergence theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem for nonspreading mappings in a Hilbert space.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H25 Nonlinear ergodic theorems
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[1] Baillon, J.-B., Un théorème de type ergodique pour LES contractions non linéaires dans un espace de Hilbert, C. R. acad. sci. Paris Sér. A-B, 280, (1975), Aii, A1511-A1514 (in French) · Zbl 0307.47006
[2] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[3] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026
[4] Takahashi, W., Convex analysis and approximation of fixed points, (2000), Yokohama Publishers Yokohama, (in Japanese)
[5] Halpern, B., Fixed points of nonexpanding maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[6] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036
[7] Shioji, N.; Takahashi, W., Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. amer. math. soc., 125, 3641-3645, (1997) · Zbl 0888.47034
[8] Shioji, N.; Takahashi, W., A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Arch. math. (basel), 72, 354-359, (1999) · Zbl 0940.47045
[9] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-378, (2003) · Zbl 1035.47048
[10] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2002) · Zbl 1101.90083
[11] Browder, F.E., Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z., 100, 201-225, (1967) · Zbl 0149.36301
[12] Goebel, K.; Kirk, W.A., Topics in metric fixed point theory, () · Zbl 0708.47031
[13] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. student, 63, 123-145, (1994) · Zbl 0888.49007
[14] Combettes, P.L.; Hirstoaga, S.A., Equilibrium programming in Hilbert spaces, J. nonlinear convex anal., 6, 117-136, (2005) · Zbl 1109.90079
[15] Kohsaka, F.; Takahashi, W., Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. math. (basel), 91, 166-177, (2008) · Zbl 1149.47045
[16] Kohsaka, F.; Takahashi, W., Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. optim., 19, 824-835, (2008) · Zbl 1168.47047
[17] Takahashi, W., Fixed point theorems for new nonlinear mappings in a Hilbert space, J. nonlinear convex anal., 11, 79-88, (2010) · Zbl 1200.47078
[18] Igarashi, T.; Takahashi, W.; Tanaka, K., Weak convergence theorems for nonspreading mappings and equilibrium problems, (), 75-85
[19] Matsushita, S.; Takahashi, W., Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed point theory appl., 37-47, (2004) · Zbl 1088.47054
[20] Matsushita, S.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. approx. theory, 134, 257-266, (2005) · Zbl 1071.47063
[21] Takahashi, W., Introduction to nonlinear and convex analysis, (2009), Yokohama Publishers Yokohama
[22] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers Yokohama
[23] Iemoto, S.; Takahashi, W., Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear anal., 71, 2082-2089, (2009)
[24] Itoh, S.; Takahashi, W., The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific J. math., 79, 493-508, (1978) · Zbl 0371.47042
[25] Aoyama, K.; Kimura, Y.; Takahashi, W.; Toyoda, M., Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear anal., 67, 2350-2360, (2007) · Zbl 1130.47045
[26] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. optim. theory appl., 118, 417-428, (2003) · Zbl 1055.47052
[27] Akatsuka, M.; Aoyama, K.; Takahashi, W., Mean ergodic theorems for a sequence of nonexpansive mappings in Hilbert spaces, Sci. math. jpn., 68, 233-239, (2008) · Zbl 1188.47047
[28] Takahashi, W., A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. amer. math. soc., 81, 253-256, (1981) · Zbl 0456.47054
[29] Shimizu, T.; Takahashi, W., Strong convergence theorem for asymptotically nonexpansive mappings, Nonlinear anal., 26, 265-272, (1996) · Zbl 0861.47030
[30] Shimizu, T.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. math. anal. appl., 211, 71-83, (1997) · Zbl 0883.47075
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