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Modified relaxed extragradient method for a general system of variational inequalities and nonexpansive mappings in Banach spaces. (English) Zbl 1246.65094

Summary: The purpose of this paper is to introduce a new modified relaxed extragradient method and study for finding some common solutions for a general system of variational inequalities with inversestrongly monotone mappings and nonexpansive mappings in the framework of real Banach spaces. By using the demiclosedness principle, it is proved that the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution for the system of variational inequalities and nonexpansive mappings under quite mild conditions.

MSC:

65K05 Numerical mathematical programming methods
49J40 Variational inequalities
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l’Académie des Sciences, vol. 258, pp. 4413-4416, 1964. · Zbl 0124.06401
[2] M. A. Noor, “Some algorithms for general monotone mixed variational inequalities,” Mathematical and Computer Modelling, vol. 29, no. 7, pp. 1-9, 1999. · Zbl 0991.49004 · doi:10.1016/S0895-7177(99)00058-8
[3] Y. Yao and M. A. Noor, “On viscosity iterative methods for variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 776-787, 2007. · Zbl 1115.49024 · doi:10.1016/j.jmaa.2006.01.091
[4] M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217-229, 2000. · Zbl 0964.49007 · doi:10.1006/jmaa.2000.7042
[5] M. Aslam Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199-277, 2004. · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[6] R. U. Verma, “Generalized system for relaxed cocoercive variational inequalities and projection methods,” Journal of Optimization Theory and Applications, vol. 121, no. 1, pp. 203-210, 2004. · Zbl 1056.49017 · doi:10.1023/B:JOTA.0000026271.19947.05
[7] S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces,” Applied Mathematics Letters, vol. 20, no. 3, pp. 329-334, 2007. · Zbl 1114.49008 · doi:10.1016/j.aml.2006.04.017
[8] L.-C. Ceng and J.-C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205-215, 2007. · Zbl 1124.65056 · doi:10.1016/j.amc.2007.01.021
[9] X. Qin, S. M. Kang, and M. Shang, “Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces,” Applicable Analysis, vol. 87, no. 4, pp. 421-430, 2008. · Zbl 1149.47051 · doi:10.1080/00036810801952953
[10] H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 3, pp. 341-350, 2005. · Zbl 1093.47058 · doi:10.1016/j.na.2003.07.023
[11] X. Qin, S. Y. Cho, and S. M. Kang, “Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 231-240, 2009. · Zbl 1204.65081 · doi:10.1016/j.cam.2009.07.018
[12] L.-C. Ceng, C.-y. Wang, and J.-C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375-390, 2008. · Zbl 1147.49007 · doi:10.1007/s00186-007-0207-4
[13] D. L. Zhu and P. Marcotte, “Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,” SIAM Journal on Optimization, vol. 6, no. 3, pp. 714-726, 1996. · Zbl 0855.47043 · doi:10.1137/S1052623494250415
[14] Y. Yao, M. Aslam Noor, K. Inayat Noor, Y.-C. Liou, and H. Yaqoob, “Modified extragradient methods for a system of variational inequalities in Banach spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1211-1224, 2010. · Zbl 1192.47065 · doi:10.1007/s10440-009-9502-9
[15] J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55-71, 2005. · Zbl 1129.47043 · doi:10.1016/j.jmaa.2004.11.007
[16] H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127-1138, 1991. · Zbl 0757.46033 · doi:10.1016/0362-546X(91)90200-K
[17] R. E. Bruck, Jr., “Nonexpansive retracts of Banach spaces,” Bulletin of the American Mathematical Society, vol. 76, pp. 384-386, 1970. · Zbl 0224.47034 · doi:10.1090/S0002-9904-1970-12486-7
[18] T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227-239, 2005. · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[19] R. E. Bruck, Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol. 179, pp. 251-262, 1973. · Zbl 0265.47043 · doi:10.2307/1996502
[20] S. Reich, “Asymptotic behavior of contractions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 44, pp. 57-70, 1973. · Zbl 0275.47034 · doi:10.1016/0022-247X(73)90024-3
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