Annals of Functional Analysis

On the weak convergence theorem for nonexpansive semigroups in Banach spaces

Rongjie Yao and Liping Yang

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Abstract

Assume that $K$ is a closed convex subset of a uniformly convex Banach space $E$, and assume that $\{T(s)\}_{s\gt 0}$ is a nonexpansive semigroup on $K$. By using the following implicit iteration sequence $\{x_{n}\}$ defined by \[x_{n}=(1-\alpha_{n})x_{n-1}+\alpha_{n}\cdot\frac{1}{t_{n}}\int _{0}^{t_{n}}T(s)x_{n}\,ds,\quad\forall n\geq1,\] the main purpose of this paper is to establish a weak convergence theorem for the nonexpansive semigroup $\{T(s)\}_{s\gt 0}$ in uniformly convex Banach spaces without the Opial property. Our results are different from some recently announced results.

Article information

Source
Ann. Funct. Anal. Volume 8, Number 3 (2017), 341-349.

Dates
Received: 27 May 2016
Accepted: 26 October 2016
First available in Project Euclid: 22 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.afa/1492826604

Digital Object Identifier
doi:10.1215/20088752-0000018X

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
Secondary: 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
nonexpansive semigroups fixed point implicit iteration scheme uniformly convex Banach spaces

Citation

Yao, Rongjie; Yang, Liping. On the weak convergence theorem for nonexpansive semigroups in Banach spaces. Ann. Funct. Anal. 8 (2017), no. 3, 341--349. doi:10.1215/20088752-0000018X. https://projecteuclid.org/euclid.afa/1492826604.


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References

  • [1] R. P. Agarwal, D. O’Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, New York, 2009.
  • [2] F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces” in Nonlinear Functional Analysis (Chicago, 1968), Amer. Math. Soc., Providence, 1976, 1–308.
  • [3] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image contruction, Inverse Problems 20 (2004), no. 1, 103–120.
  • [4] R. Chen and Y. Song, Convergence to common fixed point of nonexpansive semigroups, J. Comput. Appl. Math. 200 (2007), no. 2, 566–575.
  • [5] C. E. Chidume and B. Ali, Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 330 (2007), no. 1, 377–387.
  • [6] K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math. 28, Cambridge Univ. Press, Cambridge, 1990.
  • [7] W. Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups, J. Math. Anal. Appl. 272 (2002), no. 2, 565–574.
  • [8] A. R. Khan, H. Fukhar-ud-din, and M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012 (2012), art. ID 54.
  • [9] C. I. Podilchuk and R. J. Mammone, Image recovery by convex projections using a least-squares constraint, J. Opt. Soc. Am. A7 (1990), 517–521.
  • [10] J. Schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc. 43 (1991), no. 1, 153–159.
  • [11] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127–1138.
  • [12] H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001), no. 5–6, 767–773.
  • [13] L. P. Yang, Convergence of the new composite implicit iteration process with random errors, Nonlinear Anal. TMA 69 (2008), no. 10, 3591–3600.
  • [14] L. P. Yang and W. M. Kong, Stability and convergence of a new composite implicit iterative sequence in Banach spaces, Fixed Point Theory Appl. 2015 (2015), art. ID 172.
  • [15] E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed Points Theorems, Springer, New York, 1986.