Abstract
In this paper, we first introduce some semigroups of mappings called quasi-nonexpansive, nonspreading, hybrid, TJ-1, TJ-2, and generalized hybrid semigroup. Then, using the theory of invariant means, we prove fixed point theorems, weak convergence theorem of Mann’s type and generalized nonlinear ergodic theorem for such semigroups in a Hilbert space. Furthermore, we prove a strong convergence theorem of Halpern’s type for the proposed semigroups in a Hilbert space. The results presented in this paper mainly extend and improved some well-known results in the literature.
Similar content being viewed by others
References
Lau, A.T., Takahashi, W.: Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings. Pac. J. Math. 126, 277–294 (1987)
Baillon, J.B.: Un theorem de type ergodique pour les contractions non lineaires dans un espace de Hilbert. C. R. Acad. Sci. A-B 280, 1511–1514 (1975)
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, 2330–2360 (2007)
Hojo, M., Takahashi, W.: Weak and strong convergence theorems for generalized hybrid mappings in Hilbert space. Sci. Math. Jpn. 73, 31–40 (2011)
Hirano, N., Kido, K., Takahashi, W.: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces. Nonlinear Anal. 12, 1269–1281 (1988)
Hussain, N., Takahashi, W.: Weak and strong convergence theorems for semigroups of mappings without continuity in Hilbert spaces. J. Nonlinear Convex Anal. 14, 769–783 (2013)
Kada, O., Lau, A.T., Takahashi, W.: Asymptotically invariant net and fixed point set for semigroup of nonexpansive mappings. Nonlinear Anal. 29, 539–550 (1997)
Kocourek, P., Takahashi, W., Yao, J.C.: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 14, 2497–2511 (2010)
Atsushiba, S., Takahashi, W.: Approximating common fixed points of nonexpansive semigroups by the Mann iteration process. Ann. Univ. Mariae Curie-Skodowska Sect. 51, 1–16 (1997)
Jitpeera, T., Kumam, P.: An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed point of finitely many nonexpansive mappings. J. Nonlinear Anal. Optim. 1, 71–91 (2010)
Shimizu, T., Takahashi, W.: Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl. 211, 71–83 (1997)
Takahashi, W.: A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space. Proc. Am. Math. Soc. 97, 55–58 (1986)
Takahashi, W., Yao, J.C.: fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert space. Taiwan. J. Math. 15, 457–472 (2011)
Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)
Takahashi, W.: Nonlinear Functional Analysis. Yokohoma, Yokohoma (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Piri, H., Daraby, B., Rahimi, A. et al. Fixed point, convergence and nonlinear ergodic theorems for some semigroups in Hilbert spaces. J. Fixed Point Theory Appl. 22, 18 (2020). https://doi.org/10.1007/s11784-019-0752-5
Published:
DOI: https://doi.org/10.1007/s11784-019-0752-5