Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces

Volume 12, Issue 10, pp 621--633 http://dx.doi.org/10.22436/jnsa.012.10.01

Publication Date: May 30, 2019 Submission Date: July 18, 2018 Revision Date: April 23, 2019 Accteptance Date: May 10, 2019

Download PDF

Download XML

1615 Downloads
3042 Views

Authors

Issara Inchan - Department of Mathematics, Uttaradit Rajabhat University, Uttaradit, Thailand.

Abstract

The main objective of this work is to modify two hybrid projection algorithm. First, we prove the strongly convergence to common fixed points of a sequence \(\{x_{n}\}\) generated by the hybrid projection algorithm of two asymptotically nonexpansive mappings, second, we prove the strongly convergence of a sequence \(\{x_{n}\}\) generated by the hybrid projection algorithm of two asymptotically nonexpansive semigroups. Our main results extend and improve the results of Dong et al. [Q.-L. Dong, S. N. He, Y. J. Cho, Fixed Point Theory Appl., \(\textbf{2015}\) (2015), 12 pages].

Share and Cite

ISRP Style

Issara Inchan, Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 10, 621--633

AMA Style

Inchan Issara, Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces. J. Nonlinear Sci. Appl. (2019); 12(10):621--633

Chicago/Turabian Style

Inchan, Issara. "Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces." Journal of Nonlinear Sciences and Applications, 12, no. 10 (2019): 621--633

Keywords

Asymptotically nonexpansive mappings
asymptotically nonexpansive semigroup
fixed point

MSC

46C05
47D03
47H09
47H10
47H20

References

[1] Q.-L. Dong, S. N. He, Y. J. Cho, A new hybrid algorithm and numerical realization for two nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015), 12 pages

[2] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171--174

[3] I. Inchan, S. Plubtieng, Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. Hybrid Syst., 2 (2008), 1125--1135

[4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147--150

[5] T.-H. Kim, H.-K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear. Anal., 64 (2006), 1140--1152

[6] P.-K. Lin, K.-K. Tan, H. K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear. Anal., 24 (1995), 929--946

[7] W. A. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506--510

[8] C. Martinez-Yanes, H.-K. Xu, Strong convergence of the CQ method for fixed point processes, Nonlinear Anal., 64 (2006), 2400--2411

[9] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591--597

[10] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Amer. Math. Soc., 43 (1991), 153--159

[11] W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341 (2008), 276--286

[12] H.-K. Xu, Strong asymptotic behavior of almost-orbits of nonlinear semigroups, Nonlinear. Anal., 46 (2001), 135--151