Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems. (English) Zbl 1235.65071

Summary: We introduce a new general hybrid iterative algorithm for finding a common element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of generalized mixed equilibrium problem, and the set of solution of the variational inclusion for a \(\beta\)-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of G. Marino and H. K. Xu [J. Math. Anal. Appl. 318, No. 1, 43–52 (2006; Zbl 1095.47038)], Y.-H. Yao and Y.-C. Liou [Abstr. Appl. Anal. 2010, Article ID 763506 (2010; Zbl 1203.49048)], J. F. Tan and S. S. Chang [Fixed Point Theory Appl. 2011, Article ID 915629, 17 p. (2011; Zbl 1214.47076)] and other authors.


65K15 Numerical methods for variational inequalities and related problems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
Full Text: DOI


[1] E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1-4, pp. 123-145, 1994. · Zbl 0888.49007
[2] P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117-136, 2005. · Zbl 1109.90079
[3] S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29-41, 1997. · Zbl 0890.90150 · doi:10.1007/BF02614504
[4] S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506-515, 2007. · Zbl 1122.47056 · doi:10.1016/j.jmaa.2006.08.036
[5] P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” Acta Mathematica, vol. 115, pp. 271-310, 1966. · Zbl 0142.38102 · doi:10.1007/BF02392210
[6] J.-C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,” in Handbook of Generalized Convexity and Generalized Monotonicity, J. P. Crouzeix, N. Haddjissas, and S. Schaible, Eds., vol. 76 of Nonconvex Optim. Appl., pp. 501-558, Springer, New York, NY, USA, 2005. · Zbl 1106.49020 · doi:10.1007/0-387-23393-8_12
[7] 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
[8] W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004-1006, 1965. · Zbl 0141.32402 · doi:10.2307/2313345
[9] Y. Hao, “Some results of variational inclusion problems and fixed point problems with applications,” Applied Mathematics and Mechanics. English Edition, vol. 30, no. 12, pp. 1589-1596, 2009. · Zbl 1182.47052 · doi:10.1007/s10483-009-1210-x
[10] M. Liu, S. S. Chang, and P. Zuo, “An algorithm for finding a common solution for a system of mixed equilibrium problem, quasivariational inclusion problem, and fixed point problem of nonexpansive semigroup,” Journal of Inequalities and Applications, vol. 2010, Article ID 895907, 23 pages, 2010. · Zbl 1206.47078 · doi:10.1155/2010/895907
[11] J. F. Tan and S. S. Chang, “Iterative algorithms for finding common solutions to variational inclusion equilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 2011, Article ID 915629, 17 pages, 2011. · Zbl 1214.47076 · doi:10.1155/2011/915629
[12] S.-S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics. English Edition, vol. 29, no. 5, pp. 571-581, 2008. · Zbl 1196.47047 · doi:10.1007/s10483-008-0502-y
[13] 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
[14] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002
[15] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46-55, 2000. · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615
[16] Y. Yao and Y.-C. Liou, “Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems,” Abstract and Applied Analysis, vol. 2010, Article ID 763506, 19 pages, 2010. · Zbl 1203.49048 · doi:10.1155/2010/763506
[17] H. Brézis, “Opérateur maximaux monotones,” in Mathematics Studies, vol. 5, North-Holland, Amsterdam, The Netherlands, 1973.
[18] 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
[19] 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
[20] F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” in Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), pp. 1-308, Amer. Math. Soc., Providence, RI, USA, 1976. · Zbl 0327.47022
[21] J.-W. Peng, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions,” Fixed Point Theory and Applications, vol. 2009, Article ID 794178, 21 pages, 2009. · Zbl 1163.91463 · doi:10.1155/2009/794178
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