Abstract and Applied Analysis

On Asymptotically Quasi- ϕ -Nonexpansive Mappings in the Intermediate Sense

Xiaolong Qin and Lin Wang

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Abstract

A projection iterative process is investigated for the class of asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. Strong convergence theorems of common fixed points of a family of asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense are established in the framework of Banach spaces.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 636217, 13 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475921

Digital Object Identifier
doi:10.1155/2012/636217

Mathematical Reviews number (MathSciNet)
MR2994947

Zentralblatt MATH identifier
1256.47054

Citation

Qin, Xiaolong; Wang, Lin. On Asymptotically Quasi- $\varphi $ -Nonexpansive Mappings in the Intermediate Sense. Abstr. Appl. Anal. 2012 (2012), Article ID 636217, 13 pages. doi:10.1155/2012/636217. https://projecteuclid.org/euclid.aaa/1364475921


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References

  • P. L. Combettes, “The convex feasibility problem in image recovery,” in Advanced in Imaging and Electron Physcis, P. Hawkes, Ed., vol. 95, pp. 155–270, Academic Press, New York, NY, USA, 1996.
  • R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1–6, Springer, New York, NY, USA, 1988–1993.
  • H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, vol. 62 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1999.
  • M. A. Khan and N. C. Yannelis, Equilibrium Theory in Infinite Dimensional Spaces, Springer, New York, NY, USA, 1991.
  • F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.
  • W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004–1006, 1965.
  • K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
  • H. K. Xu, “Existence and convergence for fixed points of mappings of asymptotically nonexpansive type,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1139–1146, 1991.
  • L. E. J. Brouwer, “Über Abbildung von Mannigfaltigkeiten,” Mathematische Annalen, vol. 71, no. 1, pp. 97–115, 1912.
  • J. Schauder, “Der Fixpunktsatz in Funktionalraumen,” Studia Mathematica, vol. 2, pp. 171–180, 1930.
  • A. Tychonoff, “Ein Fixpunktsatz,” Mathematische Annalen, vol. 111, no. 1, pp. 767–776, 1935.
  • E. Casini and E. Maluta, “Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure,” Nonlinear Analysis. Theory, Methods & Applications, vol. 9, no. 1, pp. 103–108, 1985.
  • K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365–374, 1974.
  • A. Vanderluge, Optical Signal Processing, Wiley, New York, NY, USA, 1992.
  • H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.
  • C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
  • W. G. Dotson, Jr., “Fixed points of quasi-nonexpansive mappings,” Australian Mathematical Society A, vol. 13, pp. 167–170, 1972.
  • R. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no. 2, pp. 169–179, 1993.
  • I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
  • Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
  • H. Hudzik, W. Kowalewski, and G. Lewicki, “Approximate compactness and full rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 2, pp. 163–192, 2006.
  • S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applicationsof Nonlinear Operatorsof Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 313–318, Marcel Dekker, New York, NY, USA, 1996.
  • D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001.
  • D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508, 2003.
  • Y. Su and X. Qin, “Strong convergence of modified Ishikawa iterations for nonlinear mappings,” Proceedings of the Indian Academy of Science, vol. 117, no. 1, pp. 97–107, 2007.
  • R. P. Agarwal, Y. J. Cho, and X. Qin, “Generalized projection algorithms for nonlinear operators,” Numerical Functional Analysis and Optimization, vol. 28, no. 11-12, pp. 1197–1215, 2007.
  • X. Qin, Y. Su, C. Wu, and K. Liu, “Strong convergence theorems for nonlinear operators in Banach spaces,” Communications on Applied Nonlinear Analysis, vol. 14, no. 3, pp. 35–50, 2007.
  • H. Zhou, G. Gao, and B. Tan, “Convergence theorems of a modified hybrid algorithm for a family of quasi-$\varphi $-asymptotically nonexpansive mappings,” Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 453–464, 2010.
  • X. Qin, S. Y. Cho, and S. M. Kang, “On hybrid projection methods for asymptotically quasi-$\varphi $-nonexpansive mappings,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 3874–3883, 2010.
  • X. Qin and R. P. Agarwal, “Shrinking projection methods for a pair of asymptotically quasi-$\varphi $-nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 31, no. 7-9, pp. 1072–1089, 2010.
  • X. Qin, S. Huang, and T. Wang, “On the convergence of hybrid projection algorithms for asymptotically quasi-$\varphi $-nonexpansive mappings,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 851–859, 2011.
  • X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.
  • X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, “Convergence of a modified Halpern-type iteration algorithm for quasi-$\varphi $-nonexpansive mappings,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1051–1055, 2009.