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A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces. (English) Zbl 1206.47067

Summary: We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.

MSC:

47J25 Iterative procedures involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[2] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002
[3] J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401-1432, 2008. · Zbl 1185.47079
[4] J.-W. Peng and J.-C. Yao, “Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1287-1301, 2009. · Zbl 1189.90192 · doi:10.1016/j.camwa.2009.07.040
[5] W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417-428, 2003. · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[6] H. Iiduka, W. Takahashi, and M. Toyoda, “Approximation of solutions of variational inequalities for monotone mappings,” Panamerican Mathematical Journal, vol. 14, no. 2, pp. 49-61, 2004. · Zbl 1060.49006
[7] I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in InhInherently Parallel Algorithm for Feasibility and Optimization, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473-504, North-Holland, Amsterdam, The Netherlands, 2001. · Zbl 1013.49005
[8] M. Shang, Y. Su, and X. Qin, “Strong convergence theorem for nonexpansive mappings and relaxed cocoercive mappings,” International Journal of Applied Mathematics and Mechanics, vol. 3, pp. 24-34, 2007.
[9] S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025-1033, 2008. · Zbl 1142.47350 · doi:10.1016/j.na.2008.02.042
[10] F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33-56, 1998. · Zbl 0913.47048 · doi:10.1080/01630569808816813
[11] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[12] H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659-678, 2003. · Zbl 1043.90063 · doi:10.1023/A:1023073621589
[13] G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43-52, 2006. · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028
[14] X. Qin, M. Shang, and Y. Su, “A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3897-3909, 2008. · Zbl 1170.47044 · doi:10.1016/j.na.2007.10.025
[15] R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75-88, 1970. · Zbl 0222.47017 · doi:10.2307/1995660
[16] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591-597, 1967. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[17] 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
[18] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279-291, 2004. · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[19] L.C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186-201, 2008. · Zbl 1143.65049 · doi:10.1016/j.cam.2007.02.022
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