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Iterative algorithms for systems of generalized equilibrium problems with the constraints of variational inclusion and fixed point problems. (English) Zbl 1472.47064

Summary: We introduce and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.

MSC:

47J25 Iterative procedures involving nonlinear operators
47J22 Variational and other types of inclusions
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] Lions, J. L., Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaire (1969), Paris, France: Dunod, Paris, France · Zbl 0189.40603
[2] Peng, J.-W.; Yao, J.-C., A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics, 12, 6, 1401-1432 (2008) · Zbl 1185.47079
[3] Ansari, Q. H.; Yao, J.-C., Systems of generalized variational inequalities and their applications, Applicable Analysis, 76, 3-4, 203-217 (2000) · Zbl 1018.49005 · doi:10.1080/00036810008840877
[4] Ansari, Q. H.; Schaible, S.; Yao, J.-C., The system of generalized vector equilibrium problems with applications, Journal of Global Optimization, 22, 1-4, 3-16 (2002) · Zbl 1041.90069 · doi:10.1023/A:1013857924393
[5] Lin, L.-J.; Ansari, Q. H., Collective fixed points and maximal elements with applications to abstract economies, Journal of Mathematical Analysis and Applications, 296, 2, 455-472 (2004) · Zbl 1051.54028 · doi:10.1016/j.jmaa.2004.03.067
[6] Al-Homidan, S.; Ansari, Q. H.; Schaible, S., Existence of solutions of systems of generalized implicit vector variational inequalities, Journal of Optimization Theory and Applications, 134, 3, 515-531 (2007) · Zbl 1135.49004 · doi:10.1007/s10957-007-9236-7
[7] Al-Homidan, S.; Ansari, Q. H.; Yao, J.-C., Collectively fixed point and maximal element theorems in topological semilattice spaces, Applicable Analysis, 90, 6, 865-888 (2011) · Zbl 1225.49012 · doi:10.1080/00036811.2010.492503
[8] Al-Homidan, S.; Ansari, Q. H., Fixed point theorems on product topological semilattice spaces, generalized abstract economies and systems of generalized vector quasi-equilibrium problems, Taiwanese Journal of Mathematics, 15, 1, 307-330 (2011) · Zbl 1237.47060
[9] Cai, G.; Bu, S. Q., Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces, Journal of Computational and Applied Mathematics, 247, 34-52 (2013) · Zbl 1266.65110 · doi:10.1016/j.cam.2013.01.004
[10] Ceng, L. C.; Hu, H.-Y.; Wong, M. M., Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings, Taiwanese Journal of Mathematics, 15, 3, 1341-1367 (2011) · Zbl 1239.49005
[11] Ceng, L.-C.; Ansari, Q. H.; Schaible, S., Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems, Journal of Global Optimization, 53, 1, 69-96 (2012) · Zbl 1275.90102 · doi:10.1007/s10898-011-9703-4
[12] Ceng, L.-C.; Guu, S.-M.; Yao, J.-C., Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems, Fixed Point Theory and Applications, 2012, article 92 (2012) · Zbl 1275.47120 · doi:10.1186/1687-1812-2012-92
[13] Ceng, L.-C.; Petruşel, A., Relaxed extragradient-like method for general system of generalized mixed equilibria and fixed point problem, Taiwanese Journal of Mathematics, 16, 2, 445-478 (2012) · Zbl 1243.49037
[14] Yao, Y.; Liou, Y.-C.; Yao, J.-C., New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems, Optimization, 60, 3, 395-412 (2011) · Zbl 1296.47104 · doi:10.1080/02331930903196941
[15] Yao, Y.; Cho, Y. J.; Liou, Y.-C., Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, European Journal of Operational Research, 212, 2, 242-250 (2011) · Zbl 1266.90186 · doi:10.1016/j.ejor.2011.01.042
[16] Ceng, L.-C.; Ansari, Q. H.; Schaible, S.; Yao, J.-C., Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces, Fixed Point Theory, 12, 2, 293-308 (2011) · Zbl 1242.49013
[17] Ceng, L.-C.; Yao, J.-C., A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Analysis. Theory, Methods & Applications, 72, 3-4, 1922-1937 (2010) · Zbl 1179.49003 · doi:10.1016/j.na.2009.09.033
[18] Ceng, L.-C.; Wang, C.-Y.; Yao, J.-C., Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Mathematical Methods of Operations Research, 67, 3, 375-390 (2008) · Zbl 1147.49007 · doi:10.1007/s00186-007-0207-4
[19] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14, 5, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[20] Ceng, L.-C.; Yao, J.-C., A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, Journal of Computational and Applied Mathematics, 214, 1, 186-201 (2008) · Zbl 1143.65049 · doi:10.1016/j.cam.2007.02.022
[21] O’Hara, J. G.; Pillay, P.; Xu, H.-K., Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications, 64, 9, 2022-2042 (2006) · Zbl 1139.47056 · doi:10.1016/j.na.2005.07.036
[22] Yao, Y.; Liou, Y.-C.; Yao, J.-C., Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory and Applications, 2007 (2007) · Zbl 1153.54024
[23] Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory, 28 (1990), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[24] Suzuki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, Journal of Mathematical Analysis and Applications, 305, 1, 227-239 (2005) · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[25] Xu, H.-K., Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66, 1, 240-256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[26] Huang, N.-J., A new completely general class of variational inclusions with noncompact valued mappings, Computers & Mathematics with Applications, 35, 10, 9-14 (1998) · Zbl 0999.47056 · doi:10.1016/S0898-1221(98)00067-4
[27] Zeng, L.-C.; Guu, S.-M.; Yao, J.-C., Characterization of \(H\)-monotone operators with applications to variational inclusions, Computers & Mathematics with Applications, 50, 3-4, 329-337 (2005) · Zbl 1080.49012 · doi:10.1016/j.camwa.2005.06.001
[28] Ceng, L.-C.; Ansari, Q. H.; Wong, M. M.; Yao, J.-C., Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems, Fixed Point Theory, 13, 2, 403-422 (2012) · Zbl 1280.49015
[29] Yao, Y.; Noor, M. A.; Zainab, S.; Liou, Y. C., Mixed equilibrium problems and optimization problems, Journal of Mathematical Analysis and Applications, 354, 319-329 (2009) · Zbl 1160.49013
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