×

zbMATH — the first resource for mathematics

A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems. (English) Zbl 1221.49010
Summary: We introduce and study a new hybrid iterative method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a \(\xi \)-Lipschitz continuous and relaxed \((m,v)\)-cocoercive mappings in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm which solves some optimization problems under some suitable conditions. Our results extend and improve the recent results of Y. Yao, M. A. Noor, S. Zainab and Y. C. Liou [J. Math. Anal. Appl. 354, No. 1, 319–329 (2009; Zbl 1160.49013), doi:10.1016/j.jmaa.2008.12.005] and X. Gao and Y. Guo [J. Inequal. Appl. 2008, Article ID 454181, 23 p. (2008; Zbl 1153.49011), doi:10.1155/2008/454181] and many others.

MSC:
49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Combettes, P.L., Hilbertian convex feasibility problem: convergence of projection methods, Appl. math. optim., 35, 311-330, (1997) · Zbl 0872.90069
[2] Deutsch, F.; Yamada, I., Minimizing certain convex functions over the intersection of the fixed point set of nonexpansive mappings, Numer. funct. anal. optim., 19, 33-56, (1998) · Zbl 0913.47048
[3] Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 659-678, (2003) · Zbl 1043.90063
[4] Yamada, I.; Ogura, N.; Yamashita, Y.; Sakaniwa, K., Quadratic optimization of fixed point of nonexpansive mapping in Hilbert space, Numer. funct. anal. optim., 19, 1,2, 165-190, (1998) · Zbl 0911.47051
[5] Aslam Noor, M.; Ottli, W., On general nonlinear complementarity problems and quasi equilibria, Le mathematics (Catania), 49, 313-331, (1994) · Zbl 0839.90124
[6] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. student, 63, 123-145, (1994) · Zbl 0888.49007
[7] Browder, F.E.; Petryshyn, W.V., Construction of fixed points of nonlinear mappings in Hilbert space, J. math. anal. appl., 20, 197-228, (1967) · Zbl 0153.45701
[8] Changa, S.S.; Lee, H.W.J.; Chan, C.K., A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear anal., (2008)
[9] Gabay, D., Applications of the method of multipliers to variational inequalities, (), 299-331
[10] Konnov, I.V.; Schaible, S.; Yao, J.C., Combined relaxation method for mixed equilibrium problems, J. optim. theory appl., 126, 309-322, (2005) · Zbl 1110.49028
[11] Liu, F.; Nashed, M.Z., Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-valued anal., 6, 313-344, (1998) · Zbl 0924.49009
[12] Stampacchia, G., Formes bilineaires coercivites sur LES ensembles convexes, Comptes rendus acad. sci. Paris, 258, 4413-4416, (1964) · Zbl 0124.06401
[13] Verma, RU., General convergence analysis for two – step projection methods and application to variational problems, J. optim. theory appl., 18, 11, 1286-1292, (2005) · Zbl 1099.47054
[14] Cai, G.; Hu, C.S., A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping, Nonlinear anal.: hybrid syst., (2009)
[15] Combettes, P.L.; Hirstoaga, S.A., Equilibrium programming using proximal-like algorithms, Math. program., 78, 29-41, (1997) · Zbl 0890.90150
[16] Ceng, L.C.; Yao, J.C., A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. comput. appl. math., 214, 186-201, (2008) · Zbl 1143.65049
[17] Gao, X.; Guo, Y., Strong convergence theorem of a modified iterative algorithm for mixed equilibrium problems in Hilbert spaces, J. inequal. appl., (2008)
[18] Jaiboon, C.; Kumam, P.; Humphries, U.W., Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems, Bull. malays. math. sci. soc. (2), 32, 2, 173-185, (2009) · Zbl 1223.47080
[19] Jaiboon, C.; Kumam, P.; Humphries, U.W., Convergence theorem by viscosity approximation method for equilibrium problems and variational inequality problems, J. comput. math. optim., 5, 1, 29-56, (2009), SAS International Publications · Zbl 1194.47084
[20] Kumam, P., A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear anal.: hybrid syst., 2, 1245-1255, (2008) · Zbl 1163.49003
[21] Kangtunyakarn, A.; Suantai, S., Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear anal.: hybrid syst., (2009) · Zbl 1226.47076
[22] Qin, X.; Shang, M.; Su, Y., A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear anal., 69, 8, 3897-3909, (2008) · Zbl 1170.47044
[23] Qin, X.; Shang, M.; Su, Y., Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. comput. model, 48, 1033-1046, (2008) · Zbl 1187.65058
[24] 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 appl., (2008)
[25] Yao, Y.H.; Liou, Y.C.; Yao, J.C., A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems, Fixed point theory and applications, (2008), Article ID 417089, 15 pages · Zbl 1203.47087
[26] Yao, Y.; Noor, M.A.; Liou, Y.C., On iterative methods for equilibrium problems, Nonlinear anal., 70, 1, 479-509, (2009)
[27] Yao, Y.; Noor, M.A.; Zainab, S.; Liouc, Y.C., Mixed equilibrium problems and optimization problems, J. math. anal. appl., (2009)
[28] Ceng, L.C.; Yao, J.C., Iterative algorithm for generalized set-valued strong nonlinear mixed variational-like inequalities, J. optim. theory appl., 124, 725-738, (2005) · Zbl 1067.49007
[29] Yao, J.C.; Chadli, O., Pseudomonotone complementarity problems and variational in- equalities, (), 501-558 · Zbl 1106.49020
[30] Verma, RU., Generalized system for relaxed cocoercive variational inequality and its projection method, J. optim. theory appl., 21, 1, 203-210, (2004) · Zbl 1056.49017
[31] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers Yokohama
[32] Rockafellar, R.T., On the maximality of sums of nonlinear monotone operators, Trans. amer. math. soc., 149, 75-88, (1970) · Zbl 0222.47017
[33] Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear anal., 61, 341-350, (2005) · Zbl 1093.47058
[34] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 595-597, (1967) · Zbl 0179.19902
[35] Marino, G.; Xu, H.K., A general iterative method for nonexpansive mapping in Hilbert spaces, J. math. anal. appl., 318, 43-52, (2006) · Zbl 1095.47038
[36] Yao, Y., A general iterative method for a finite family of nonexpansive mappings, Nonlinear anal., 66, 12, 2676-2687, (2007) · Zbl 1129.47058
[37] Shimoji, K.; Takahashi, W., Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. math, 5, 387-404, (2001) · Zbl 0993.47037
[38] Suzuki, T., Strong convergence of Krasnoselskii and manns type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085
[39] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl, 298, 279-291, (2004) · Zbl 1061.47060
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.