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Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. (English) Zbl 1204.65081
A system of two generalized variational inequalities, based on the variational inequalities introduced by {\it G. Stampacchia} [C. R. Acad. Sci., Paris 258, 4413--4416 (1964; Zbl 0124.06401)] in the early sixties, is considered. An iterative algorithm to approximate the solution of the system is developed using the idea of a fixed point for a corresponding nonexpansive operator. Strong convergence theorems are established in the framework of real Banach spaces.

65K15Numerical methods for variational inequalities and related problems
Full Text: DOI
[1] Stampacchia, G.: Formes bilineaires coercivites sur LES ensembles convexes, CR acad. Sci. Paris 258, 4413-4416 (1964) · Zbl 0124.06401
[2] Aoyama, K.; Iiduka, H.; Takahashi, W.: Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed point theory appl. 2006, 35390 (2006) · Zbl 1128.47056 · doi:10.1155/FPTA/2006/35390
[3] Cho, Y. J.; Yao, Y.; Zhou, H.: Strong convergence of an iterative algorithm for accretive operators in Banach spaces, J. comput. Anal. appl. 10, 113-125 (2008) · Zbl 1176.47052
[4] Chang, S. S.; Lee, H. W. J.; Chan, C. K.: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. math. Lett. 20, 329-334 (2007) · Zbl 1114.49008 · doi:10.1016/j.aml.2006.04.017
[5] Ceng, L. C.; Yao, J. C.: An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. math. Comput. 190, 205-215 (2007) · Zbl 1124.65056 · doi:10.1016/j.amc.2007.01.021
[6] 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. meth. Oper. res. 67, 375-390 (2008) · Zbl 1147.49007 · doi:10.1007/s00186-007-0207-4
[7] Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems, (1990) · Zbl 0712.47043
[8] Huang, Z.; Noor, M. A.: An explicit projection method for a system of nonlinear variational inequalities with different ({$\gamma$},r)-cocoercive mappings, Appl. math. Comput. 190, 356-361 (2007) · Zbl 1120.65080 · doi:10.1016/j.amc.2007.01.032
[9] Iiduka, H.; Takahashi, W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear anal. 61, 341-350 (2005) · Zbl 1093.47058 · doi:10.1016/j.na.2003.07.023
[10] Noor, M. Aslam: Some developments in general variational inequalities, Appl. math. Comput. 152, 199-277 (2004) · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[11] Qin, X.; Kang, S. M.; Shang, M.: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. anal. 87, 421-430 (2008) · Zbl 1149.47051 · doi:10.1080/00036810801952953
[12] Qin, X.; Shang, M.; Zhou, H.: Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces, Appl. math. Comput. 200, 242-253 (2008) · Zbl 1147.65048 · doi:10.1016/j.amc.2007.11.004
[13] Qin, X.; Noor, M. A.: General Wiener--Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces, Appl. math. Comput. 201, 716-722 (2008) · Zbl 1157.65041 · doi:10.1016/j.amc.2008.01.007
[14] Verma, R. U.: Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. optim. Theory appl. 121, 203-210 (2004) · Zbl 1056.49017 · doi:10.1023/B:JOTA.0000026271.19947.05
[15] Verma, R. U.: Generalized class of partial relaxed monotonicity and its connections, Adv. nonlinear var. Inequal. 7, 155-164 (2004) · Zbl 1079.49011
[16] Verma, R. U.: General convergence analysis for two-step projection methods and applications to variational problems, Appl. math. Lett. 18, 1286-1292 (2005) · Zbl 1099.47054 · doi:10.1016/j.aml.2005.02.026
[17] Verma, R. U.: On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. sci. Res. hot-line 3, 65-68 (1999) · Zbl 0970.49011
[18] Yao, Y.; Yao, J. C.: On modified iterative method for nonexpansive mappings and monotone mappings, Appl. math. Comput. 186, 1551-1558 (2007) · Zbl 1121.65064 · doi:10.1016/j.amc.2006.08.062
[19] Xu, H. K.: Inequalities in Banach spaces with applications, Nonlinear anal. 16, 1127-1138 (1991) · Zbl 0757.46033 · doi:10.1016/0362-546X(91)90200-K
[20] Reich, S.: Asymptotic behavior of contractions in Banach spaces, J. math. Anal. appl. 44, 57-70 (1973) · Zbl 0275.47034 · doi:10.1016/0022-247X(73)90024-3
[21] Kitahara, S.; Takahashi, W.: Image recovery by convex combinations of sunny nonexpansive retractions, Topol. meth. Nonlinear anal. 2, 333-342 (1993) · Zbl 0815.47068
[22] Browder, F. E.: Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. natl. Acad. sci. USA 53, 1272-1276 (1965) · Zbl 0125.35801 · doi:10.1073/pnas.53.6.1272
[23] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6
[24] Browder, F. E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. symp. Pure. math. 18, 78-81 (1976)
[25] Bruck, R. E.: Properties of fixed point sets of nonexpansive mappings in Banach spaces, Tras. amer. Math. soc. 179, 251-262 (1973) · Zbl 0265.47043 · doi:10.2307/1996502
[26] Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without bochne integrals, J. math. Anal. appl. 305, 227-239 (2005) · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[27] Xu, H. K.: Iterative algorithms for nonlinear operators, J. London math. Soc. 66, 240-256 (2002) · Zbl 1013.47032