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.


47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47H25 Nonlinear ergodic theorems
47J25 Iterative procedures involving nonlinear operators
Full Text: DOI


[1] Browder, FE, Fifixed point theorems for noncompact mappings in Hilbert spaces, Proc Nat Acad Sci USA, 53, 1272-1276, (1965) · Zbl 0125.35801
[2] Pazy, A, Asymptotic behavior of contractions in Hilbert space, Israel J Math, 9, 235-240, (1971) · Zbl 0225.54032
[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, 280, 1511-1514, (1975) · Zbl 0307.47006
[4] Ray, WO, The fixed point property and unbounded sets in Hilbert space, Trans Amer Math Soc, 258, 531-537, (1980) · Zbl 0433.47026
[5] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990. · Zbl 0708.47031
[6] Kohsaka, F; Takahashi, W, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch Math, 91, 166-177, (2008) · Zbl 1149.47045
[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, 71, e2082-e2089, (2009) · Zbl 1239.47054
[9] Takahashi, W; Yao, JC, Fixed point theorems and ergodic theorems for non-linear mappings in Hilbert spaces, Taiwan J Math, 15, 457-472, (2011) · Zbl 1437.47027
[10] Mann, WR, Mean value methods in iteration, Proc Amer Math Soc, 4, 506-510, (1953) · Zbl 0050.11603
[11] Moudafi, A, Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl, 23, 1635-1640, (2007) · Zbl 1128.47060
[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, 79, 493-508, (1978)
[15] Takahashi, W; Toyoda, M, Weak convergence theorems for nonexpansive mappings and monotone mappings, J Optim Theory Appl, 118, 417-428, (2003) · Zbl 1055.47052
[16] Kurokawa, Y; Takahashi, W, Weak and strong convergence theorems for non-spreading mappings in Hilbert spaces, Nonlinear Anal, 73, 1562-568, (2010) · Zbl 1229.47117
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.