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Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. (English) Zbl 1281.47050

Summary: Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in [Y. Kurokawa and W. Takahashi, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 6, 1562–1568 (2010; Zbl 1229.47117)]. We explore an auxiliary mapping in our theorems and proofs and this also yields a strong convergence theorem of Halpern type for our class of mappings and hence resolves in the affirmative an open problem posed by Kurokawa and Takahashi [loc.cit.]in their final remark for the case where the mapping \(T\) is averaged.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H25 Nonlinear ergodic theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1229.47117
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References:

[1] Kohsaka, F.; Takahashi, W., Fixed point theorems for a class of nonlinear mappings relate to maximal monotone operators in Banach spaces, Arch. Math. (Basel), 91, 166-177 (2008) · Zbl 1149.47045
[2] 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
[3] Igarashi, T.; Takahashi, W.; Tanaka, K., Weak convergence theorems for nonspreading mappings and equilibrium problems, (Akashi, S.; Takahashi, W.; Tanaka, T., Nonlinear Analysis and Optimization (2009), Yokohama), 75-85
[4] Iemoto, S.; Takahashi, W., Approximating commom fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Anal., 71, 2080-2089 (2009)
[5] Kurokawa, Y.; Takahashi, W., Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal., 73, 1562-1568 (2010) · Zbl 1229.47117
[6] Baillon, J., Un théorème de type ergodique pour les contractions nonlinéaires dans un espace de Hilbert, C. R. Acad. Sci., Paris Ser. A-B, 280, Aii, A1511-A1514 (1975), (in French)
[7] Halpern, B., Fixed points of nonexpanding mappings, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101
[8] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118, 417-428 (2003) · Zbl 1055.47052
[9] 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
[10] Xu, H. K., Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66, 2, 240-256 (2002) · Zbl 1013.47032
[11] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20, 197-228 (1967) · Zbl 0153.45701
[12] Hicks, T. L.; Kubicek, J. R., On the Mann iterative process in Hilbert spaces, J. Math. Anal. Appl., 59, 498-504 (1977) · Zbl 0361.65057
[13] Naimpally, S. A.; Singh, K. L., Extensions of some fixed point theorems of Rhoades, J. Math. Anal. Appl., 96, 437-446 (1983) · Zbl 0524.47033
[14] Song, Y.; Chai, X., Halpern iteration for firmly type nonexpansive mappings, Nonlinear Anal., 71, 4500-4506 (2009) · Zbl 1169.49010
[15] Saejung, S., Halpern’s iteration in Banach spaces, Nonlinear Anal., 73, 3431-3439 (2010) · Zbl 1234.47054
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