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A new iterative method for the set of solutions of equilibrium problems and of operator equations with inverse-strongly monotone mappings. (English) Zbl 1473.47037

Summary: The purpose of the paper is to present a new iteration method for finding a common element for the set of solutions of equilibrium problems and of operator equations with a finite family of \(\lambda_i\)-inverse-strongly monotone mappings in Hilbert spaces.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
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[1] Antipin, A. S., Equilibrium programming: gradient methods, Automation and Remote Control, 58, 8, 1337-1347 (1997) · Zbl 0945.90063
[2] Antipin, A. S., Equilibrium programming: proximal methods, Zhurnal Vychislite’noi Matematiki i Matematicheskoi Fiziki, 37, 11, 1327-1339 (1997) · Zbl 0944.90083
[3] Antipin, A. S., Solution methods for variational inequalities with coupled constraints, Computational Mathematics and Mathematical Physics, 40, 9, 1239-1254 (2000) · Zbl 0999.65055
[4] Antipin, A. S., Solving variational inequalities with coupling constraints with the use of differential equations, Differential Equations, 36, 11, 1587-1596 (2000) · Zbl 1016.49011 · doi:10.1007/BF02757358
[5] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 1-4, 123-145 (1994) · Zbl 0888.49007
[6] Bounkhel, M.; Al-Senan, B. R., An iterative method for nonconvex equilibrium problems, Journal of Inequalities in Pure and Applied Mathematics, 7, 2, article 75 (2006) · Zbl 1135.49005
[7] Chadli, O.; Schaible, S.; Yao, J. C., Regularized equilibrium problems with application to noncoercive hemivariational inequalities, Journal of Optimization Theory and Applications, 121, 3, 571-596 (2004) · Zbl 1107.91067 · doi:10.1023/B:JOTA.0000037604.96151.26
[8] Chadli, O.; Konnov, I. V.; Yao, J. C., Descent methods for equilibrium problems in a Banach space, Computers & Mathematics with Applications, 48, 3-4, 609-616 (2004) · Zbl 1057.49009 · doi:10.1016/j.camwa.2003.05.011
[9] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 1, 117-136 (2005) · Zbl 1109.90079
[10] Anh, P. N.; Kim, J. K., An interior proximal cutting hyperplane method for equilibrium problems, Journal of Inequalities and Applications, 2012, article 99 (2012) · Zbl 1276.65032 · doi:10.1186/1029-242X-2012-99
[11] Kim, J. K.; Nam, Y. M.; Sim, J. Y., Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonxpansive type mappings, Nonlinear Analysis: Theory, Methods & Applications, 71, 12, 2839-2848 (2009) · Zbl 1239.47056 · doi:10.1016/j.na.2009.06.090
[12] Kim, J. K.; Cho, S. Y.; Qin, X., Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Mathematica Scientia. Series B, 31, 5, 2041-2057 (2011) · Zbl 1247.47061 · doi:10.1016/S0252-9602(11)60380-9
[13] Kim, J. K., Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-\( \varphi \)-nonexpansive mappings, Fixed Point Theory and Applications, 2011, article 10 (2011) · Zbl 1390.47020 · doi:10.1186/1687-1812-2011-10
[14] Kim, J. K.; Lim, W. H., A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces, Journal of Inequalities and Applications, 2013, article 128 (2013) · Zbl 1455.47022 · doi:10.1186/1029-242X-2013-128
[15] Konnov, I. V.; Pinyagina, O. V., D-gap functions and descent methods for a class of monotone equilibrium problems, Lobachevskii Journal of Mathematics, 13, 57-65 (2003) · Zbl 1041.65054
[16] Konnov, I. V.; Pinyagina, O. V., D-gap functions for a class of equilibrium problems in Banach spaces, Computational Methods in Applied Mathematics, 3, 2, 274-286 (2003) · Zbl 1051.47046 · doi:10.2478/cmam-2003-0018
[17] Marino, G.; Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, Journal of Mathematical Analysis and Applications, 329, 1, 336-346 (2007) · Zbl 1116.47053 · doi:10.1016/j.jmaa.2006.06.055
[18] Mastroeni, G., Gap functions for equilibrium problems, Journal of Global Optimization, 27, 4, 411-426 (2003) · Zbl 1061.90112 · doi:10.1023/A:1026050425030
[19] Mastroeni, G., On auxiliary principle for equilibrium problems, 3.244.1258 (2000), Pisa, Italy: Department of Mathematics of Pisa University, Pisa, Italy
[20] Moudafi, A., Second-order differential proximal methods for equilibrium problems, Journal of Inequalities in Pure and Applied Mathematics, 4, 1, article 18 (2003) · Zbl 1175.90413
[21] Moudafi, A.; Théra, M., Proximal and dynamical approaches to equilibrium problems, Ill-Posed Variational Problems and Regularization Techniques. Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Mathematical Systems, 477, 187-201 (1999), Berlin, Germay: Springer, Berlin, Germay · Zbl 0944.65080 · doi:10.1007/978-3-642-45780-7_12
[22] Noor, M. A.; Noor, K. I., On equilibrium problems, Applied Mathematics E-Notes, 4, 125-132 (2004) · Zbl 1064.49009
[23] Oettli, W., A remark on vector-valued equilibria and generalized monotonicity, Acta Mathematica Vietnamica, 22, 1, 213-221 (1997) · Zbl 0914.90235
[24] Stukalov, A. S., A regularized extragradient method for solving equilibrium programming problems in a Hilbert space, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 45, 9, 1538-1554 (2005) · Zbl 1117.90323
[25] Takahashi, W.; Toyoda, M., Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 118, 2, 417-428 (2003) · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[26] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, Journal of Mathematical Analysis and Applications, 331, 1, 506-515 (2007) · Zbl 1122.47056 · doi:10.1016/j.jmaa.2006.08.036
[27] Wang, G.; Peng, J.; Lee, H.-W. J., Implicit iteration process with mean errors for common fixed points of a finite family of strictly pseudocontrative maps, International Journal of Mathematical Analysis, 1, 1-4, 89-99 (2007) · Zbl 1145.47053
[28] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[29] Göpfert, A.; Riahi, H.; Tammer, C.; Zălinescu, C., Variational Methods in Partially Ordered Spaces, xiv+350 (2003), New York, NY, USA: Springer, New York, NY, USA · Zbl 1140.90007
[30] Kim, J. K.; Tuyen, T. M., Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, 2011, article 52 (2011) · Zbl 1315.47067 · doi:10.1186/1187-1812-2011-52
[31] Kim, J. K.; Buong, N., Regularization inertial proximal point algorithm for monotone hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces, Journal of Inequalities and Applications, 2010 (2010) · Zbl 1184.49015 · doi:10.1155/2010/451916
[32] Kim, J. K.; Buong, N., An iteration method for common solution of a system of equilibrium problems in Hilbert spaces, Fixed Point Theory and Applications, 2011 (2011) · Zbl 1221.65153 · doi:10.1155/2011/780764
[33] Kim, J. K.; Anh, P. N.; Nam, Y. M., Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems, Journal of the Korean Mathematical Society, 49, 1, 187-200 (2012) · Zbl 1317.65146 · doi:10.4134/JKMS.2012.49.1.187
[34] Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Analysis: Theory, Methods & Applications, 61, 3, 341-350 (2005) · Zbl 1093.47058 · doi:10.1016/j.na.2003.07.023
[35] Nadezhkina, N.; Takahashi, W., Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM Journal on Optimization, 16, 4, 1230-1241 (2006) · Zbl 1143.47047 · doi:10.1137/050624315
[36] Noor, M. A.; Yao, Y.; Chen, R.; Liou, Y.-C., An iterative method for fixed point problems and variational inequality problems, Mathematical Communications, 12, 1, 121-132 (2007) · Zbl 1149.49013
[37] Zeng, L.-C.; Yao, J.-C., Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics, 10, 5, 1293-1303 (2006) · Zbl 1110.49013
[38] Buong, N., Approximation methods for equilibrium problems and common solution for a finite family of inverse strongly-monotone problems in Hilbert spaces, Applied Mathematical Sciences, 2, 13-16, 735-746 (2008) · Zbl 1186.47071
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