## Abstract and Applied Analysis

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

#### 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

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

#### 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.