Abstract and Applied Analysis

Strong Convergence of Non-Implicit Iteration Process with Errors in Banach Spaces

Yan Hao, Xiaoshuang Wang, and Aihua Tong

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Abstract

The purpose of this paper is to study the strong convergence of a non-implicit iteration process with errors for asymptotically I-nonexpansive mappings in the intermediate sense in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results recently announced.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 242354, 12 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475945

Digital Object Identifier
doi:10.1155/2012/242354

Mathematical Reviews number (MathSciNet)
MR2999888

Zentralblatt MATH identifier
06134045

Citation

Hao, Yan; Wang, Xiaoshuang; Tong, Aihua. Strong Convergence of Non-Implicit Iteration Process with Errors in Banach Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 242354, 12 pages. doi:10.1155/2012/242354. https://projecteuclid.org/euclid.aaa/1364475945


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References

  • K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mapping,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
  • B. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach space with the uniform Opial property,” Colloquium Mathematicum, vol. 65, pp. 169–179, 1993.
  • W. A. Kirk, “Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type,” Israel Journal of Mathematics, vol. 17, no. 4, pp. 339–346, 1974.
  • X. Qin, S. S. Chang, and Y. J. Cho, “Iterative methods for generalized equilibrium problems and fixed point problems with applications,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2963–2972, 2010.
  • S. Y. Cho and S. M. Kang, “Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process,” Applied Mathematics Letters, vol. 24, no. 2, pp. 224–228, 2011.
  • S. Y. Cho and S. M. Kang, “Approximation of common solutions of variational inequalities via strict pseudocontractions,” Acta Mathematica Scientia, vol. 32, no. 4, pp. 1607–1618, 2012.
  • J. K. Kim, Y. M. Nam, and J. Y. Sim, “Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 12, pp. e2839–e2848, 2009.
  • S. M. Kang, S. Y. Cho, and Z. Liu, “Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings,” Journal of Inequalities and Applications, vol. 2010, Article ID 827082, 16 pages, 2010.
  • J. K. Kim, S. Y. Cho, and X. Qin, “Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings,” Journal of Inequalities and Applications, vol. 2010, Article ID 312602, 18 pages, 2010.
  • S. Y. Cho, S. M. Kang, and X. Qin, “Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions,” Applied Mathematics Letters, vol. 25, no. 5, pp. 854–857, 2012.
  • Y. Su and S. Li, “Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 882–891, 2006.
  • X. Qin, S. Y. Cho, and S. M. Kang, “Iterative algorithms for variational inequality and equilibrium problems with applications,” Journal of Global Optimization, vol. 48, no. 3, pp. 423–445, 2010.
  • S. Yang and W. Li, “Iterative solutions of a system of equilibrium problems in Hilbert spaces,” Advances in Fixed Point Theory, vol. 1, no. 1, pp. 15–26, 2011.
  • X. Qin, S. Y. Cho, and S. M. Kang, “Strong convergence of shrinking projection methods for quasi-$\phi $-nonexpansive mappings and equilibrium problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 750–760, 2010.
  • J. Ye and J. Huang, “Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces,” Journal of Mathematical and Computational Science, vol. 1, no. 1, pp. 1–18, 2011.
  • S. Husain and S. Gupta, “A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities,” Advances in Fixed Point Theory, vol. 2, no. 1, pp. 18–28, 2012.
  • X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.
  • X. Qin, S. Y. Cho, and S. M. Kang, “On hybrid projection methods for asymptotically quasi-$\phi $-nonexpansive mappings,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3874–3883, 2010.
  • F. Gu, “Some convergence theorems of non-implicit iteration process with errors for a finite families of \emphI-asymptotically nonexpansive mappings,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 161–172, 2010.
  • S. Lv and C. Wu, “Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping,” Engineering Mathematics Letters, vol. 1, no. 1, pp. 44–57, 2012.
  • J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.
  • F. Mukhamedov and M. Saburov, “Strong convergence of an explicit iteration process for a totally asymptotically \emphI-nonexpansive mapping in Banach spaces,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1473–1478, 2010.
  • S. Temir, “On the convergence theorems of implicit iteration process for a finite family of \emphI-asymptotically nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 398–405, 2009.
  • F. Mukhamedov and M. Saburov, “Weak and strong convergence of an implicit iteration process for an asymptotically quasi-\emphI-nonexpansive mapping in Banach space,” Fixed Point Theory and Applications, vol. 2010, Article ID 719631, 13 pages, 2010.
  • N. Shahzad, “Generalized \emphI-nonexpansive maps and best approximations in Banach spaces,” Demonstratio Mathematica, vol. 37, pp. 597–600, 2004.
  • K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.