Fixed point theorem for generic 2-generalized hybrid mappings in Hilbert spaces
DOI:
https://doi.org/10.12775/TMNA.2021.038Keywords
Fixed point, generic 2-generalized hybrid mapping, Hilbert spaceAbstract
We establish a fixed point theorem for a class of mappings called generic 2-generalized hybrid mappings in the setting of a real Hilbert space. Two examples of that class of mappings are presented herein. The mappings are not quasi-nonexpansive even though they have fixed points. One of these maps is even not continuous. The fixed point theorem proved in this article improves many previous works in the literature.References
S. Alizadeh and F. Moradlou, A weak convergence theorem for 2-generalized hybrid mappings, ROMAI J. 11 (2015), no. 1, 131–138.
S. Alizadeh and F. Moradlou, Weak and strong convergence theorems for m-generalized hybrid mappings in Hilbert spaces, Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 315–328.
F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965), no. 4, 1041.
D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258.
M. Hojo, W. Takahashi and I. Termwuttipong, Strong convergence theorems for 2generalized hybrid mappings in Hilbert spaces, Nonlinear Anal. 75 (2012), 2166–2176.
T. Igarashi, W. Takahashi and K. Tanaka, Weak convergence theorems for nonspreading mappings and equilibrium problems, Nonlinear Analysis and Optimization (S. Akashi, W. Takahashi and T. Tanaka, eds.), Yokohama Publishers, Yokohama, 2008, pp. 75–85.
T. Kawasaki, Fixed point and acute point theorems for new mappings in a Banach space, J. Math. (2019).
T. Kawasaki, Generalized acute point theorems for generalized pseudocontractions in a Banach space, Linear Nonlinear Anal. 6 (2020), 73–90.
T. Kawasaki, Fixed point and acute point theorems for generalized pseudocontractions in a Banach space, J. Nonlinear Convex Anal. 22 (2021), 1057–1075.
T. Kawasaki and T. Kobayashi, Existence and mean approximation of fixed points of generalized hybrid non-self mappings in Hilbert spaces, Sci. Math. Jpn. 77 (2014), 13–26.
T. Kawasaki and W. Takahashi, Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 14 (2013), 71–87.
W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006.
P. Kocourek, W. Takahashi and J.-C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math. 14 (2010), 2497–2511.
F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008), 824–835.
F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. 91 (2008), 166–177.
A. Kondo and W. Takahashi, Attractive point and nonlinear ergodic theorems for generic 2-generalized hybrid mappings in Hilbert spaces, Linear Nonlinear Anal. 5 (2019), 87–103.
A. Kondo and W. Takahashi, Attractive point, weak and strong convergence theorems for generic 2-generalized hybrid mappings in Hilbert spaces, Linear Nonlinear Anal. 6 (2020), 103–133.
T. Maruyama, W. Takahashi and M. Yao, Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal. 12 (2011), 185–197.
B.D. Rouhani, Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space, Demonstr. Math. 51 (2018), no. 1, 27–36.
P. Sadeewong, T. Saleewong, P. Kumam, and Y.J. Cho, The modified viscosity iteration with m-generalized hybrid mappings and (a, b)-monotone mappings for equilibrium problems, Thai J. Math. 16 (2018), no. 1, 243–265.
J. Schauder, Der fixpunktsatz in funktionalräumen, Studia Math. 2 (1930), 171–180.
W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and its Applications, Yokohama Publishes, Yokohama, 2000.
W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010), 79–88.
W. Takahashi, N.-C. Wong and J.-C. Yao, Fixed point theorems for new generalized hybrid mappings in Hilbert spaces and applications, Taiwanese J. Math. 17 (2013), no. 5, 1597–1611.
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Copyright (c) 2022 Atsumasa Kondo
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