## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2013, Special Issue (2013), Article ID 926078, 8 pages.

### Viscosity Approximation Methods and Strong Convergence Theorems for the Fixed Point of Pseudocontractive and Monotone Mappings in Banach Spaces

Yan Tang

#### Abstract

Suppose that $C$ is a nonempty closed convex subset of a real reflexive Banach space $E$ which has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.

#### Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 926078, 8 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807684

Digital Object Identifier
doi:10.1155/2013/926078

Mathematical Reviews number (MathSciNet)
MR3068180

Zentralblatt MATH identifier
1271.47061

#### Citation

Tang, Yan. Viscosity Approximation Methods and Strong Convergence Theorems for the Fixed Point of Pseudocontractive and Monotone Mappings in Banach Spaces. J. Appl. Math. 2013, Special Issue (2013), Article ID 926078, 8 pages. doi:10.1155/2013/926078. https://projecteuclid.org/euclid.jam/1394807684

#### References

• B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967.
• Y. Yao, “A general iterative method for a finite family of non-expansive mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 66, no. 12, pp. 2676–2687, 2007.
• Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis, Theory, Methods and Applications, vol. 68, no. 6, pp. 1687–1693, 2008.
• S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “On Reich's strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 66, no. 11, pp. 2364–2374, 2007.
• H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
• S.-Y. Matsushita and W. Takahashi, “Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,” Nonlinear Analysis, Theory, Methods and Applications, vol. 68, no. 2, pp. 412–419, 2008.
• L. Deng and Q. Liu, “Iterative scheme for nonself generalized asymptotically quasi-nonexpansive mappings,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 317–324, 2008.
• Y. Song, “A new sufficient condition for the strong convergence of Halpern type iterations,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 721–728, 2008.
• N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in danach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997.
• W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively non-expansive mappings in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 1, pp. 45–57, 2009.
• H. Zegeye, “An iterative approximation for a common fixed point of two Pseudo-contractive mappings,” ISRN Mathematical Analysis, vol. 2011, Article ID 621901, 14 pages, 2011.
• W. Takahashi, Non-Linear Functional Analysis-Fixed Point Theory and Its Applications, Yokohama Publishers Inc, Yokohama, Japan, 2000, (in Japanese).
• J. Lou, L.-J. Zhang, and Z. He, “Viscosity approximation meth-ods for asymptotically nonexpansive mappings,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 171–177, 2008.
• L.-C. Ceng, H.-K. Xu, and J.-C. Yao, “The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces,” Nonlinear Analysis, Theory, Methods and Appli-cations, vol. 69, no. 4, pp. 1402–1412, 2008.
• Y. Song and R. Chen, “Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 275–287, 2006.
• H. Zegeye and N. Shahzad, “Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces,” Optimization Letters, vol. 5, no. 4, pp. 691–704, 2011.
• D.-J. Wen, “Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators,” Nonlinear Analysis, Theory, Methods and Applications, vol. 73, no. 7, pp. 2292–2297, 2010.
• A. Moudifi, “Viscosity approximation methods for fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
• W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.
• W. Takahashi, Nonlinear Functional Analysis, Yokohama, Yokohama, Japan, 2000.
• H. Zegeye and N. Shahzad, “Strong convergence of an iterative method for pseudo-contractive and monotone mappings,” Journal of Global Optimization, vol. 54, no. 1, pp. 173–184, 2011.
• H. Zegeye and N. Shahzad, “Strong convergence theorems for a common zero of a countably infinite family of $\alpha$-inverse strongly accretive mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 1-2, pp. 531–538, 2009.
• T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.