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A general iterative method with strongly positive operators for general variational inequalities. (English) Zbl 1189.49006
Summary: We introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. The explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators. Under two sets of quite mild conditions, we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively.

49J40Variational methods including variational inequalities
47J25Iterative procedures (nonlinear operator equations)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
65K15Numerical methods for variational inequalities and related problems
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Full Text: DOI
[1] Browder, F. E.; Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert space, J. math. Anal. appl. 20, 197-228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[2] Liu, F.; Nashed, M. Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-valued anal. 6, 313-344 (1998) · Zbl 0924.49009 · doi:10.1023/A:1008643727926
[3] Noor, M. A.: General variational inequalities, Appl. math. Lett. 1, 119-122 (1998) · Zbl 0655.49005 · doi:10.1016/0893-9659(88)90054-7
[4] Isac, G.: A special variational inequality and the implicit complementarity problem, J. fac. Sci.univ.tokyo 1A 37, 109-127 (1990) · Zbl 0702.49008
[5] Zeng, L. C.: Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities, J. math. Anal. appl. 187, No. 2, 352-360 (1994) · Zbl 0820.49005 · doi:10.1006/jmaa.1994.1361
[6] Deutsch, F.; Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numer. funct. Anal. optim. 19, 33-56 (1998) · Zbl 0913.47048 · doi:10.1080/01630569808816813
[7] Moudafi, A.: Viscosity approximation methods for fixed point problems, J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615
[8] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings, J. math. Anal. appl. 298, 279-291 (2004) · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[9] Xu, H. K.; Kim, T. H.: Convergence of hybrid steepest-descent methods for variational inequalities, J. optim. Theory appl., 185-201 (2003) · Zbl 1045.49018 · doi:10.1023/B:JOTA.0000005048.79379.b6
[10] Xu, H. K.: An iterative approach to quadratic optimization, J. optim. Theory appl. 116, 659-678 (2003) · Zbl 1043.90063 · doi:10.1023/A:1023073621589
[11] Marino, G.; Xu, H. K.: A general iterative method for nonexpansive mappings in Hilbert spaces, J. math. Anal. appl. 318, 43-52 (2006) · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028
[12] 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
[13] Chen, J. M.; Zhang, L. J.; Fan, T. G.: Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. math. Anal. appl. 334, 1450-1461 (2007) · Zbl 1137.47307 · doi:10.1016/j.jmaa.2006.12.088
[14] Zeng, L. C.; Yao, J. C.: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear anal. 64, 2507-2515 (2006) · Zbl 1105.47061 · doi:10.1016/j.na.2005.08.028
[15] Ceng, L. C.; Yao, J. C.: Relaxed viscosity approximation methods for fixed point problems and variational inequality problems, Nonlinear anal. (2007)
[16] Ceng, L. C.; Yao, J. C.: An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. math. Comput. 190, No. 1, 205-215 (2007) · Zbl 1124.65056 · doi:10.1016/j.amc.2007.01.021
[17] Zeng, L. C.; Wong, N. C.; Yao, J. C.: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. optim. Theory appl. 132, No. 1, 51-69 (2007) · Zbl 1137.47059 · doi:10.1007/s10957-006-9068-x
[18] Zeng, L. C.; Wong, N. C.; Yao, J. C.: Convergence of hybrid steepest-descent methods for generalized variational inequalities, Acta math. Sin. (Engl. Ser.) 22, No. 1, 1-12 (2006) · Zbl 1121.49012 · doi:10.1007/s10114-005-0608-3
[19] Geobel, K.; Kirk, W. A.: Topics in metric fixed point theory, Cambridge stud. Adv. math. 28 (1990) · Zbl 0708.47031