## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2014, Special Issue (2014), Article ID 129379, 25 pages.

### Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

Lu-Chuan Ceng, Cheng-Wen Liao, Chin-Tzong Pang, and Ching-Feng Wen

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#### Abstract

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

#### Article information

**Source**

J. Appl. Math., Volume 2014, Special Issue (2014), Article ID 129379, 25 pages.

**Dates**

First available in Project Euclid: 1 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1412177563

**Digital Object Identifier**

doi:10.1155/2014/129379

**Mathematical Reviews number (MathSciNet)**

MR3216114

#### Citation

Ceng, Lu-Chuan; Liao, Cheng-Wen; Pang, Chin-Tzong; Wen, Ching-Feng. Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems. J. Appl. Math. 2014, Special Issue (2014), Article ID 129379, 25 pages. doi:10.1155/2014/129379. https://projecteuclid.org/euclid.jam/1412177563

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