Iterative methods for triple hierarchical variational inequalities with mixed equilibrium problems, variational inclusions, and variational inequalities constraints

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• 作者：Lu-Chuan Ceng ; Yen-Cherng Lin ; Ching-Feng Wen
• 关键词：49J30 ; 47H09 ; 47J20 ; 49M05 ; triple hierarchical variational inequalities ; generalized mixed equilibrium problems ; variational inclusions ; general system of variational inequalities ; hybrid steepest ; descent extragradient algorithm ; composite Mann ; type viscosity algorithm ; fixed points
• 刊名：Journal of Inequalities and Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：2015
• 期：1
• 全文大小：1,620 KB
• 参考文献：1. Glowinski, R: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) CrossRef
  2. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
  3. Oden, JT: Quantitative Methods on Nonlinear Mechanics. Prentice Hall, Englewood Cliffs (1986)
  4. Lions, JL: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)
  5. Zeidler, E: Nonlinear Functional Analysis and Its Applications. Springer, New York (1985) CrossRef
  6. Korpelevich, GM: The extragradient method for finding saddle points and other problems. Matecon 12, 747-756 (1976)
  7. Kong, Z-R, Ceng, L-C, Ansari, QH, Pang, C-T: Multistep hybrid extragradient method for triple hierarchical variational inequalities. Abstr. Appl. Anal. 2013, Article ID 718624 (2013)
  8. Zeng, L-C, Yao, J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 10, 1293-1303 (2006)
  9. Peng, J-W, Yao, J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwan. J. Math. 12, 1401-1432 (2008)
  10. Ceng, L-C, Ansari, QH, Schaible, S: Hybrid extragradient-like methods for generalized mixed equilibrium problems, system of generalized equilibrium problems and optimization problems. J. Glob. Optim. 53, 69-96 (2012) CrossRef
  11. Yao, Y, Liou, Y-C, Kang, SM: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Math. Appl. 59, 3472-3480 (2010) CrossRef
  12. Ceng, L-C, Ansari, QH, Wong, M-M, Yao, J-C: Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems. Fixed Point Theory 13(2), 403-422 (2012)
  13. Nadezhkina, N, Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191-201 (2006) CrossRef
  14. Ceng, L-C, Yao, J-C: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Anal. 72, 1922-1937 (2010) CrossRef
  15. Ceng, L-C, Wong, M-M: Relaxed extragradient method for finding a common element of systems of variational inequalities and fixed point problems. Taiwan. J. Math. 17(2), 701-724 (2013)
  16. Ceng, L-C, Wang, C-Y, Yao, J-C: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67(3), 375-390 (2008) CrossRef
  17. Ceng, L-C, Ansari, QH, Yao, J-C: Relaxed extragradient iterative methods for variational inequalities. Appl. Math. Comput. 218, 1112-1123 (2011) CrossRef
  18. Ceng, L-C, Petrusel, A: Relaxed extragradient-like method for general system of generalized mixed equilibria and fixed point problem. Taiwan. J. Math. 16(2), 445-478 (2012)
  19. Ceng, L-C, Guu, S-M, Yao, J-C: Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems. Fixed Point Theory Appl. 2012, 92 (2012) CrossRef
  20. Ceng, L-C, Hu, H-Y, Wong, M-M: Strong and weak convergence theorems for generalized mixed equilibrium problem wi
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X

文摘

In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a?solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.