## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 174318, 20 pages.

### General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

#### Abstract

This paper deals with new methods for approximating a solution to the fixed point problem; find $\stackrel{̃}{x}\in F\left(T\right)$, where $H$ is a Hilbert space, $C$ is a closed convex subset of $H$, $f$ is a $\rho$-contraction from $C$ into $H$, $0<\rho <1$, $A$ is a strongly positive linear-bounded operator with coefficient $\stackrel{̅}{\gamma }>0$, $0<\gamma <\stackrel{̅}{\gamma }/\rho$, $T$ is a nonexpansive mapping on $\mathrm{C,}$ and ${P}_{F\left(T\right)}$ denotes the metric projection on the set of fixed point of $T$. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality $〈\left(A-\gamma f\right)\stackrel{̃}{x}+\tau \left(I-S\right)\stackrel{̃}{x},x-\stackrel{̃}{x}〉\ge 0$ for $x\in F\left(T\right)$, where $\tau \in \left[0,\infty \right)$. Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 174318, 20 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357179969

Digital Object Identifier
doi:10.1155/2012/174318

Mathematical Reviews number (MathSciNet)
MR2948145

Zentralblatt MATH identifier
1251.49012

#### Citation

Wairojjana, Nopparat; Kumam, Poom. General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities. J. Appl. Math. 2012, Special Issue (2012), Article ID 174318, 20 pages. doi:10.1155/2012/174318. https://projecteuclid.org/euclid.jam/1357179969

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