##### Title: A Generalized Iterative Algorithm for Hierarchical Fixed Points Problems and Variational Inequalities

##### Pages: 54-63

##### Cite as:

Vahid Dadashi, Somayeh Amjadi, A Generalized Iterative Algorithm for Hierarchical Fixed Points Problems and Variational Inequalities, Int. J. Anal. Appl., 13 (1) (2017), 54-63.#### Abstract

In this paper we propose a method for approximating of the common fixed point in $\bigcap\limits_{n=1}^\infty F(T_n)$ where $\{T_n\}$ is a countable family of nonexpansive mappings on a closed convex subset $C$ of a real Hilbert space $\mathcal{H}$. Then, we prove strong convergence theorems with less control conditions for $\{T_n\}$ which solves some variational inequality. The main results improve and extend the corresponding results of "F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities, J. Inequal. Appl., 2009 (2009), Article ID 208692" and "Y. Yao, Y.J. Cho, and Y.C. Liou, Iterative algorithms for hierarchical fixed points problems and variational inequalities, Mathematical and Computer Modelling, 52(9) (2010), 1697--1705".

##### Full Text: PDF

#### References

- M. Alimohammady and V. Dadashi, Convergence of a generalized iterations for a countable family of nonexpansive mappings, TJMM. 4(1) (2012), 15–24.
- F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities, J. Inequal. Appl., 2009 (2009), Article ID 208692.
- V. Dadashi, S. Ghafari, Convergence theorems of iterative approximation for finding zeros of accretive operator and fixed points problems, Int. J. Nonlinear Anal. Appl., 4(2) (2013), 53–61.
- Y. Dong and X. Zhang, New step lengths in projection method for variational inequality problems, Appl. Math. Comput. 220(1) (2013), 239–245.
- K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, 1990.
- B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 220(1) (1967), 957–961.
- P.E. Mainge and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed point problems, Pacific J. Optim. 3 (2007), 529–538.
- A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241 (2000), 46–55.
- X.Qin, M. Shang, H. Zhou, Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces, Appl. Math. Comput., 200(1) (2008), 242–253.
- Y. Yao, Y.J. Cho, and Y.C. Liou, Iterative algorithms for hierarchical fixed points problems and variational in- equalities, Mathematical and Computer Modelling, 52(9) (2010), 1697–1705.
- H. K. Xu, Iterative algorithm for nonlinear operators, J. Lond. Math. Soc., 2 (2002), 1–17.