Abstract and Applied Analysis

A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

Mei Yuan, Xi Li, Xue-song Li, and John J. Liu

Full-text: Open access

Abstract

Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalized f-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 498487, 12 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475822

Digital Object Identifier
doi:10.1155/2012/498487

Mathematical Reviews number (MathSciNet)
MR2970011

Zentralblatt MATH identifier
1254.47037

Citation

Yuan, Mei; Li, Xi; Li, Xue-song; Liu, John J. A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 498487, 12 pages. doi:10.1155/2012/498487. https://projecteuclid.org/euclid.aaa/1364475822


Export citation

References

  • Y. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Proceedings of the Israel Seminar, vol. 1 of Functional Differential Equation, pp. 1–21, Ariel, Israel, 1994.
  • J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55–71, 2005.
  • K.-q. Wu and N.-j. Huang, “The generalised f-projection operator with an application,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006.
  • K.-q. Wu and N.-j. Huang, “Properties of the generalized f-projection operator and its applications in Banach spaces,” Computers & Mathematics with Applications, vol. 54, no. 3, pp. 399–406, 2007.
  • Y. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
  • Y. Alber, “Proximal projection methods for variational inequalities and Cesro averaged approximations,” Computers & Mathematics with Applications, vol. 43, no. 8-9, pp. 1107–1124, 2002.
  • Y. Alber and S. Guerre-Delabriere, “On the projection methods for fixed point problems,” Analysis, vol. 21, no. 1, pp. 17–39, 2001.
  • K.-Q. Wu and N.-J. Huang, “The generalized f-projection operator and set-valued variational inequalities in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2481–2490, 2009.
  • 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.
  • X. Li, N.-j. Huang, and D. O'Regan, “Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1322–1331, 2010.
  • E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
  • P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.
  • 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.
  • W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009.
  • K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
  • S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
  • S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.