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Fixed point theorems for some new nonlinear mappings in Hilbert spaces. (English) Zbl 1315.47044

Summary: In this paper, we introduce two new classes of nonlinear mappings in Hilbert spaces. These two classes of nonlinear mappings contain some important classes of nonlinear mappings, like nonexpansive mappings and nonspreading mappings. We prove fixed point theorems, ergodic theorems, demiclosed principles, and Ray’s type theorem for these nonlinear mappings. { }Next, we prove weak convergence theorems for Moudafi’s iteration process for these nonlinear mappings. Finally, we give some important examples for these new nonlinear mappings.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47H25 Nonlinear ergodic theorems
47J25 Iterative procedures involving nonlinear operators
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[1] Browder FE: Fifixed point theorems for noncompact mappings in Hilbert spaces.Proc Nat Acad Sci USA 1965, 53:1272-1276. · Zbl 0125.35801 · doi:10.1073/pnas.53.6.1272
[2] Pazy A: Asymptotic behavior of contractions in Hilbert space.Israel J Math 1971, 9:235-240. · Zbl 0225.54032 · doi:10.1007/BF02771588
[3] Baillon JB: Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert. C. R.Acad Sci Paris Ser A-B 1975, 280:1511-1514. · Zbl 0307.47006
[4] Ray WO: The fixed point property and unbounded sets in Hilbert space.Trans Amer Math Soc 1980, 258:531-537. · Zbl 0433.47026 · doi:10.1090/S0002-9947-1980-0558189-1
[5] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[6] Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces.Arch Math 2008, 91:166-177. · Zbl 1149.47045 · doi:10.1007/s00013-008-2545-8
[7] Takahashi, W., Nonlinear mappings in equilibrium problems and an open problem in fixed point theory, 177-197 (2010) · Zbl 1221.47097
[8] Iemoto S, Takahashi W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space.Nonlinear Anal 2009, 71:e2082-e2089. · Zbl 1239.47054 · doi:10.1016/j.na.2009.03.064
[9] Takahashi W, Yao JC: Fixed point theorems and ergodic theorems for non-linear mappings in Hilbert spaces.Taiwan J Math 2011, 15:457-472. · Zbl 1437.47027
[10] Mann WR: Mean value methods in iteration.Proc Amer Math Soc 1953, 4:506-510. · Zbl 0050.11603 · doi:10.1090/S0002-9939-1953-0054846-3
[11] Moudafi A: Krasnoselski-Mann iteration for hierarchical fixed-point problems.Inverse Probl 2007, 23:1635-1640. · Zbl 1128.47060 · doi:10.1088/0266-5611/23/4/015
[12] Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohoma Publishers, Yokohoma; 2009. · Zbl 1183.46001
[13] Takahashi W: Nonlinear Functional Analysis-Fixed Point Theory and its Applications. Yokohama Publishers, Yokohama; 2000. · Zbl 0997.47002
[14] Itoh S, Takahashi W: The common fixed point theory of single-valued mappings and multi-valued mappings.Pac J Math 1978, 79:493-508.
[15] Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings.J Optim Theory Appl 2003, 118:417-428. · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[16] Kurokawa Y, Takahashi W: Weak and strong convergence theorems for non-spreading mappings in Hilbert spaces.Nonlinear Anal 2010, 73:1562-568. · Zbl 1229.47117 · doi:10.1016/j.na.2010.04.060
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