×

zbMATH — the first resource for mathematics

Hybrid algorithms of common solutions of generalized mixed equilibrium problems and the common variational inequality problems with applications. (English) Zbl 1215.49014
Summary: We introduce new iterative algorithms by the hybrid method for finding a common element of the set of solutions of fixed points of infinite family of nonexpansive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequality with inverse-strongly monotone mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative method under some suitable conditions. Finally, we apply our results to complementarity problems and optimization problems. Our results improve and extend the results announced by many others.

MSC:
49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. · Zbl 0997.47002
[2] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007
[3] Chadli, O; Schaible, S; Yao, JC, Regularized equilibrium problems with application to noncoercive hemivariational inequalities, Journal of Optimization Theory and Applications, 121, 571-596, (2004) · Zbl 1107.91067
[4] Chadli, O; Wong, NC; Yao, JC, Equilibrium problems with applications to eigenvalue problems, Journal of Optimization Theory and Applications, 117, 245-266, (2003) · Zbl 1141.49306
[5] Konnov, IV; Schaible, S; Yao, JC, Combined relaxation method for mixed equilibrium problems, Journal of Optimization Theory and Applications, 126, 309-322, (2005) · Zbl 1110.49028
[6] Moudafi, A; Théra, M, Proximal and dynamical approaches to equilibrium problems, No. 477, 187-201, (1999), Berlin, Germany · Zbl 0944.65080
[7] Zeng, L-C; Wu, S-Y; Yao, J-C, Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems, Taiwanese Journal of Mathematics, 10, 1497-1514, (2006) · Zbl 1121.49005
[8] 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, 186-201, (2008) · Zbl 1143.65049
[9] Takahashi, W; Toyoda, M, Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal of Optimization Theory and Applications, 118, 417-428, (2003) · Zbl 1055.47052
[10] Takahashi, S; Takahashi, W, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, Journal of Mathematical Analysis and Applications, 331, 506-515, (2007) · Zbl 1122.47056
[11] Burachik, RS; Lopes, JO; Da Silva, GJP, An inexact interior point proximal method for the variational inequality problem, Computational & Applied Mathematics, 28, 15-36, (2009) · Zbl 1169.65063
[12] Cho, YJ; Petrot, N; Suantai, S, Fixed point theorems for nonexpansive mappings with applications to generalized equilibrium and system of nonlinear variational inequalities problems, Journal of Nonlinear Analysis and Optimization, 1, 45-53, (2010) · Zbl 1413.47076
[13] Flåm, SD; Antipin, AS, Equilibrium programming using proximal-like algorithms, Mathematical Programming, 78, 29-41, (1997) · Zbl 0890.90150
[14] Jitpeera, T; Kumam, P, A composite iterative method for generalized mixed equilibrium problems and variational inequality problems, Journal of Computational Analysis and Applications, 13, 345-361, (2011) · Zbl 1229.47115
[15] Jitpeera, T; Kumam, P, An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings, Journal of Nonlinear Analysis and Optimization, 1, 71-91, (2010) · Zbl 1413.47112
[16] Jaiboon, C; Kumam, P, Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities, No. 2010, 43, (2010) · Zbl 1187.47048
[17] Jaiboon, C; Kumam, P, A general iterative method for addressing mixed equilibrium problems and optimization problems, Nonlinear Analysis. Theory, Methods & Applications, 73, 1180-1202, (2010) · Zbl 1205.49011
[18] Kumam, P, A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear Analysis. Hybrid Systems, 2, 1245-1255, (2008) · Zbl 1163.49003
[19] Kumam, P, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turkish Journal of Mathematics, 33, 85-98, (2009) · Zbl 1223.47083
[20] Kumam, P, A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping, Journal of Applied Mathematics and Computing, 29, 263-280, (2009) · Zbl 1220.47102
[21] Kumam, P; Jaiboon, C, A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems, Nonlinear Analysis. Hybrid Systems, 3, 510-530, (2009) · Zbl 1221.49010
[22] Kumam, P; Jaiboon, C, A system of generalized mixed equilibrium problems and fixed point problems for pseudocontractive mappings in Hilbert spaces, No. 2010, 33, (2010) · Zbl 1205.47061
[23] Kumam, P; Katchang, P, A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Analysis. Hybrid Systems, 3, 475-486, (2009) · Zbl 1221.49011
[24] Kumam, W; Jaiboon, C; Kumam, P; Singta, A, A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings, No. 2010, 25, (2010) · Zbl 1206.47077
[25] Wang, Z; Su, Y, Strong convergence theorems of common elements for equilibrium problems and fixed point problems in Banach paces, Journal of Application Mathematics and Informatics, 28, 783-796, (2010) · Zbl 1295.47098
[26] Wangkeeree, R; Wangkeeree, R, Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings, Nonlinear Analysis. Hybrid Systems, 3, 719-733, (2009) · Zbl 1175.49012
[27] Wangkeeree, R; Petrot, N; Kumam, P; Jaiboon, C, Convergence theorem for mixed equilibrium and variational inequality problems for relaxed cocoercive mappings, Journal of Computational Analysis and Applications, 13, 425-449, (2011) · Zbl 1221.47130
[28] Yao, Y; Noor, MA; Zainab, S; Liou, Y-C, Mixed equilibrium problems and optimization problems, Journal of Mathematical Analysis and Applications, 354, 319-329, (2009) · Zbl 1160.49013
[29] Yao, Y; Cho, YJ; Chen, R, An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems, Nonlinear Analysis. Theory, Methods & Applications, 71, 3363-3373, (2009) · Zbl 1166.49013
[30] Yao, J-C; Chadli, O, Pseudomonotone complementarity problems and variational inequalities, No. 76, 501-558, (2005), New York, NY, USA · Zbl 1106.49020
[31] Zeng, LC; Schaible, S; Yao, JC, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, Journal of Optimization Theory and Applications, 124, 725-738, (2005) · Zbl 1067.49007
[32] Takahashi, W; Takeuchi, Y; Kubota, R, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 341, 276-286, (2008) · Zbl 1134.47052
[33] Kangtunyakarn, A, Iterative methods for finding common solution of generalized equilibrium problems and variational inequality problems and fixed point problems of a finite family of nonexpansive mappings, No. 2010, 29, (2010) · Zbl 1207.65067
[34] Shehu Y: Strong convergence theorems for family of nonexpansive mappings and sys- tem of generalized mixed equilibrium problems and variational inequality problems.International Journal of Mathematics and Mathematical Sciences. In press · Zbl 1107.91067
[35] Chantarangsi, W; Jaiboon, C; Kumam, P, A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces, No. 2010, 39, (2010) · Zbl 1206.47067
[36] Combettes, PL; Hirstoaga, SA, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 117-136, (2005) · Zbl 1109.90079
[37] Peng, J-W; Liou, Y-C; Yao, J-C, An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions, No. 2009, 21, (2009) · Zbl 1163.91463
[38] Xu, H-K, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291, (2004) · Zbl 1061.47060
[39] Iiduka, H; Takahashi, W, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Analysis. Theory, Methods & Applications, 61, 341-350, (2005) · Zbl 1093.47058
[40] Browder, FE; Petryshyn, WV, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228, (1967) · Zbl 0153.45701
[41] Zhou, H, Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis. Theory, Methods & Applications, 69, 456-462, (2008) · Zbl 1220.47139
[42] Takahashi, W, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Annales Universitatis Mariae Curie-Skłodowska, 51, 277-292, (1997) · Zbl 1012.47029
[43] Baillon, J-B; Haddad, G, Quelques propriétés des opérateurs angle-bornés et [inlineequation not available: see fulltext.]-cycliquement monotones, Israel Journal of Mathematics, 26, 137-150, (1977) · Zbl 0352.47023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.