## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2014, Special Issue (2014), Article ID 728363, 7 pages.

### Hybrid Iterations for the Fixed Point Problem and Variational Inequalities

#### Abstract

A hybrid iterative algorithm with Meir-Keeler contraction is presented for solving the fixed point problem of the pseudocontractive mappings and the variational inequalities. Strong convergence analysis is given as ${\text{l}\text{i}\text{m}}_{n\to \mathrm{\infty }}d(ST{x}_{n},TS{x}_{n})$.

#### Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2014), Article ID 728363, 7 pages.

Dates
First available in Project Euclid: 1 October 2014

https://projecteuclid.org/euclid.jam/1412177961

Digital Object Identifier
doi:10.1155/2014/728363

Mathematical Reviews number (MathSciNet)
MR3256323

#### Citation

Zhu, Li-Jun; Kang, Shin Min; Yao, Zhangsong; Kwun, Young Chel. Hybrid Iterations for the Fixed Point Problem and Variational Inequalities. J. Appl. Math. 2014, Special Issue (2014), Article ID 728363, 7 pages. doi:10.1155/2014/728363. https://projecteuclid.org/euclid.jam/1412177961

#### References

• L. Ceng, A. Petruşel, and J. Yao, “Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 162–176, 2009.
• C. E. Chidume, M. Abbas, and B. Ali, “Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings,” Applied Mathematics and Computation, vol. 194, no. 1, pp. 1–6, 2007.
• L. Ćirić, A. Rafiq, N. Cakić, and J. S. Ume, “Implicit Mann fixed point iterations for pseudo-contractive mappings,” Applied Mathematics Letters, vol. 22, no. 4, pp. 581–584, 2009.
• S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.
• J. Quan, S.-S. Chang, and M. Liu, “Strong and weak convergence of an implicit iterative process for pseudocontractive semigroups in Banach space,” Fixed Point Theory and Applications, vol. 2012, article 16, 2012.
• Y. C. Yao, Y. Liou, and G. Marino, “A hybrid algorithm for pseudo-contractive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4997–5002, 2009.
• H. Zegeye, N. Shahzad, and T. Mekonen, “Viscosity approximation methods for pseudocontractive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 538–546, 2007.
• H. Zegeye and N. Shahzad, “An algorithm for a common fixed point of a family of pseudocontractive mappings,” Fixed Point Theory and Applications, vol. 2013, article 234, 2013.
• H. Zhou, “Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 546–556, 2008.
• A. Bnouhachem, “A hybrid iterative method for a combination of equilibrium problem, a combination of variational inequality problem and a hierarchical fixed point problem,” Fixed Point Theory and Applications, vol. 2014, article 163, 29 pages, 2014.
• A. Bnouhachem, “Strong convergence algorithm for approximating the common solutions of a variational inequality, a mixed equilibrium problem and a hierarchical fixed-point problem,” Journal of Inequalities and Applications, vol. 2014, article 154, 2014.
• M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004.
• G. M. Korpelevich, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.
• J. L. Lions and G. Stampacchia, “Variational inequalities,” Communications on Pure and Applied Mathematics, vol. 20, pp. 493–519, 1967.
• R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
• Y. Yao, R. Chen, and H. K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3447–3456, 2010.
• Y. Yao and M. A. Noor, “On viscosity iterative methods for variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 776–787, 2007.
• Y. Yao, M. A. Noor, and Y. C. Liou, “A new hybrid iterative algorithm for variational inequalities,” Applied Mathematics and Computation, vol. 216, no. 3, pp. 822–829, 2010.
• U. Mosco, “Convergence of convex sets and of solutions of variational inequalities,” Advances in Mathematics, vol. 3, pp. 510–585, 1969.
• M. Tsukada, “Convergence of best approximations in a smooth Banach space,” Journal of Approximation Theory, vol. 40, no. 4, pp. 301–309, 1984.
• A. Meir and E. Keeler, “A theorem on contraction mappings,” Journal of Mathematical Analysis and Applications, vol. 28, pp. 326–329, 1969.
• T. Suzuki, “Moudafi's viscosity approximations with Meir-Keeler contractions,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 342–352, 2007. \endinput