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Ceng, Lu-Chuan; Al-Mezel, Saleh A.; Latif, Abdul

Hybrid viscosity approaches to general systems of variational inequalities with hierarchical fixed point problem constraints in Banach spaces. (English) Zbl 1474.49019

Abstr. Appl. Anal. 2014, Article ID 945985, 18 p. (2014).
Summary: The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings in \(X\), respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.

Cited in 1 Document

MSC:

49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
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\textit{L.-C. Ceng} et al., Abstr. Appl. Anal. 2014, Article ID 945985, 18 p. (2014; Zbl 1474.49019)
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References:

[1] Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Analysis A, 61, 3, 341-350, (2005) · Zbl 1093.47058
[2] Ceng, L.-C.; Ansari, Q. H.; Yao, J.-C., An extragradient method for solving split feasibility and fixed point problems, Computers & Mathematics with Applications, 64, 4, 633-642, (2012) · Zbl 1252.65102
[3] Ceng, L.-C.; Ansari, Q. H.; Yao, J.-C., Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Analysis A, 75, 4, 2116-2125, (2012) · Zbl 1236.47066
[4] Ceng, L.-C.; Ansari, Q. H.; Yao, J.-C., Relaxed extragradient iterative methods for variational inequalities, Applied Mathematics and Computation, 218, 3, 1112-1123, (2011) · Zbl 1229.65109
[5] Ceng, L.-C.; Ansari, Q. H.; Yao, J.-C., Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numerical Functional Analysis and Optimization, 29, 9-10, 987-1033, (2008) · Zbl 1163.49002
[6] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 118, 2, 417-428, (2003) · Zbl 1055.47052
[7] Iiduka, H.; Takahashi, W.; Toyoda, M., Approximation of solutions of variational inequalities for monotone mappings, Panamerican Mathematical Journal, 14, 2, 49-61, (2004) · Zbl 1060.49006
[8] Takahashi, Y.; Hashimoto, K.; Kato, M., On sharp uniform convexity, smoothness, and strong type, cotype inequalities, Journal of Nonlinear and Convex Analysis, 3, 2, 267-281, (2002) · Zbl 1030.46012
[9] Cai, G.; Bu, S., Approximation of common fixed points of a countable family of continuous pseudocontractions in a uniformly smooth Banach space, Applied Mathematics Letters, 24, 12, 1998-2004, (2011) · Zbl 1231.65098
[10] Cai, G.; Bu, S., Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces, Mathematical and Computer Modelling, 55, 3-4, 538-546, (2012) · Zbl 1255.49015
[11] Xu, H. K., Inequalities in Banach spaces with applications, Nonlinear Analysis A, 16, 12, 1127-1138, (1991) · Zbl 0757.46033
[12] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM Journal on Optimization, 13, 3, 938-945, (2002) · Zbl 1101.90083
[13] Xu, H. K.; Kim, T. H., Convergence of hybrid steepest-descent methods for variational inequalities, Journal of Optimization Theory and Applications, 119, 1, 185-201, (2003) · Zbl 1045.49018
[14] Takahashi, W., Nonlinear Functional Analysis—Fixed Point Theory and Its Applications, (2000), Yokohama, Japan: Yokohama Publishers, Yokohama, Japan · Zbl 0997.47002
[15] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 67, 2, 274-276, (1979) · Zbl 0423.47026
[16] Bruck,, R. E., Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Transactions of the American Mathematical Society, 179, 251-262, (1973) · Zbl 0265.47043
[17] Aoyama, K.; Kimura, Y.; Takahashi, W.; Toyoda, M., Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Analysis A, 67, 8, 2350-2360, (2007) · Zbl 1130.47045
[18] Ceng, L.-C.; Yao, J.-C., An extragradient-like approximation method for variational inequality problems and fixed point problems, Applied Mathematics and Computation, 190, 1, 205-215, (2007) · Zbl 1124.65056
[19] Censor, Y.; Gibali, A.; Reich, S., Two extensions of Korpelevich’s extragradient method for solving the variational inequality problem in Euclidean space, Technical Report, (2010)
[20] Ceng, L.-C.; Wang, C.-y.; Yao, J.-C., Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Mathematical Methods of Operations Research, 67, 3, 375-390, (2008) · Zbl 1147.49007
[21] Yao, Y.; Liou, Y.-C.; Chen, R., Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Analysis A, 67, 12, 3311-3317, (2007) · Zbl 1129.47059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.