## Abstract and Applied Analysis

### Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space

#### Abstract

Let ${\{T\}}_{i=1}^{N}$ be N quasi-nonexpansive mappings defined on a closed convex subset C of a real Hilbert space H. Consider the problem of finding a common fixed point of these mappings and introduce the parallel and cyclic algorithms for solving this problem. We will prove the strong convergence of these algorithms.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 218341, 27 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475841

Digital Object Identifier
doi:10.1155/2012/218341

Mathematical Reviews number (MathSciNet)
MR2975308

Zentralblatt MATH identifier
1253.65080

#### Citation

Deng, Bin-Chao; Chen, Tong; Xin, Baogui. Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space. Abstr. Appl. Anal. 2012 (2012), Article ID 218341, 27 pages. doi:10.1155/2012/218341. https://projecteuclid.org/euclid.aaa/1364475841

#### References

• S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed pointproblems of nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 548–558, 2008.
• L. C. Zeng, S. Schaible, and J. C. Yao, “Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities,” Journal of Optimization Theory and Applications, vol. 124, no. 3,pp. 725–738, 2005.
• F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
• H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Appli-cations, vol. 116, no. 3, pp. 659–678, 2003.
• G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
• Y. Liu, “A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis. Theory, Methods and Applications A, vol. 71, no. 10, pp. 4852–4861, 2009.
• X. Qin, M. Shang, and S. M. Kang, “Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis. Theory, Methods and Applications A, vol. 70, no. 3, pp. 1257–1264, 2009.
• G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,” Non-linear Analysis. Theory, Methods and Applications A, vol. 67, no. 7, pp. 2258–2271, 2007.
• M. O. Osilike and Y. Shehu, “Cyclic algorithm for common fixed points of finite family of strictly pseudocontractive mappings of Browder-Petryshyn type,” Nonlinear Analysis. Theory, Methods and Applications A, vol. 70, no. 10, pp. 3575–3583, 2009.
• P.-E. Maingé, “The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces,” Computers and Mathematics with Applications, vol. 59, no. 1, pp. 74–79, 2010.
• I. Yamada and N. Ogura, “Hybrid steepest descent method for variational inequality problem over thefixed point set of certain quasi-nonexpansive mappings,” Numerical Functional Analysis and Optimiza-tion, vol. 25, no. 7-8, pp. 619–655, 2004.
• K. Geobel and W. A. Kirk, “Topics in metric fixed point theory,” in Cambridge Studies in Advanced Mathematics, vol. 28, pp. 473–504, Cambridge University Press, Cambridge, UK, 1990.
• P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.
• W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
• H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002.