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Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications. (English) Zbl 1247.47073

Summary: We introduce a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of the variational inequality problem for a co-coercive mapping in a real Hilbert space. Then, strong convergence of the scheme to a common element of the three sets is proved. Furthermore, new convergence results are deduced, and finally, we apply our results to solving optimization problems, and present other applications.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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[1] Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994) · Zbl 0888.49007
[2] Browder F.E.: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc. 71, 780–785 (1965) · Zbl 0138.39902
[3] Browder F.E., Petryshyn W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967) · Zbl 0153.45701
[4] Bruck R.E.: On weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977) · Zbl 0423.47023
[5] Bruck, R.E.: Asymptotic behaviour of nonexpansive mappings. In: Sine, R.C. (ed.) Contemporary mathematics, 18, fixed points and nonexpansive mappings. AMS, Providence, RI, (1980) · Zbl 0439.34036
[6] Byrne C.: A unified treatment of some iterative algorithms in signal processing and image construction. Inv. Probl. 20, 103–120 (2004) · Zbl 1051.65067
[7] Cho Y.J., Qin X., Kang J.I.: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 71, 4203–4214 (2009) · Zbl 1219.47105
[8] Combettes P.L., Hirstoaga S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005) · Zbl 1109.90079
[9] Giannessi, F., Maugeri, A., Pardalos, P.M. (eds.): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, vol. 58. Springer, Berlin, (2002). ISBN 978-1-4020-0161-1 · Zbl 0979.00025
[10] Iiduka H., Takahashi W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341–350 (2005) · Zbl 1093.47058
[11] Kinderlehrer D., Stampacchia G.: An introduction to variational inequalities and their applications. Academic Press, New York (1980) · Zbl 0457.35001
[12] Li S.J., Zhao P.: A method of duality for a mixed vector equilibrium problem. Optim. Lett. 4(1), 85–96 (2010) · Zbl 1189.90189
[13] Lim T.C., Xu H.K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. TMA 22, 1345–1355 (1994) · Zbl 0812.47058
[14] Liu Y.: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 71, 4852–4861 (2009) · Zbl 1222.47104
[15] Liu, M., Chang, S., Zuo, P.: On a hybrid method for generalized mixed equilibrium problem and fixed point problem of a family of quasi- $${\(\backslash\)phi}$$ -asymptotically nonexpansive mappings in Banach spaces. J. Fixed Point Theory Appl. Article ID 157278, 18 p (2010) · Zbl 1205.47062
[16] Maugeri A., Raciti F.: On general infinite dimensional complementarity problems. Optim. Lett. 2(1), 71–90 (2008) · Zbl 1145.90095
[17] Moudafi A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37–43 (2008) · Zbl 1167.47049
[18] Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187–201. Springer, Berlin (1999) · Zbl 0944.65080
[19] Nadezhkina N., Takahashi W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006) · Zbl 1130.90055
[20] Noor M.A.: General variational inequalities and nonexpansive mappings. J. Math. Anal. Appl. 331, 810–822 (2007) · Zbl 1112.49013
[21] Pardalos, P.M., Rassias, T.M., Khan, A.A. (eds): Nonlinear Analysis and Variational Problems. Springer, Berlin (2010) · Zbl 1178.49001
[22] Peng J.W., Yao J.C.: Strong convergence theorems of an iterative scheme based on extragradient method for mixed equilibrium problems and fixed points problems. Math. Comput. Model. 49, 1816–1828 (2009) · Zbl 1171.90542
[23] Petrot N., Wattanawitoon K., Kumam P.: A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. Nonlinear Anal. Hybrid Syst. 4, 631–643 (2010) · Zbl 1292.47051
[24] Plubtieng S., Punpaeng R.: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Appl. Math. Comput. 197, 548–558 (2008) · Zbl 1154.47053
[25] Plubtieng S., Kumam P.: Weak convergence theorems for monotone mappings and countable family of nonexpansive mappings. Com. Appl. Math. 224, 614–621 (2009) · Zbl 1161.65042
[26] Plubtieng, S., Sombut, K.: Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space. J. Ineq. Appl. Article ID 246237,12 p. (2010) · Zbl 1189.47067
[27] Podilchuk C.I., Mammone R.J.: Image recovery by convex projections using a least-squares constraint. J. Opt. Soc. Am. A 7, 517–521 (1990)
[28] Qin X., Cho Y.J., Kang S.M.: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 72, 99–112 (2010) · Zbl 1225.47106
[29] 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
[30] Rockafellar R.T.: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053
[31] Shioji S., Takahashi W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641–3645 (1997) · Zbl 0888.47034
[32] Shehu Y.: Fixed point solutions of generalized equilibrium problems for nonexpansive mappings. J. Comput. Appl. Math. 234, 892–898 (2010) · Zbl 1189.47069
[33] Shehu Y.: Fixed point solutions of variational inequality and generalized equilibrium problems with applications. Ann. Univ. Ferrara 56(2), 345–368 (2010) · Zbl 1206.47085
[34] Su Y., Shang M., Qin X.: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69, 2709–2719 (2008) · Zbl 1170.47047
[35] Suzuki T.: Strong convergence of Krasnoselkii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005) · Zbl 1068.47085
[36] Takahashi S., Takahashi W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008) · Zbl 1142.47350
[37] Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–518 (2007) · Zbl 1122.47056
[38] Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003) · Zbl 1055.47052
[39] Wangkeeree R.: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. J. Fixed Point Theory Appl. 2008(134148), 17 (2008) · Zbl 1170.47051
[40] Wangkeeree, R. Wangkeeree, R.: A general iterative method for variational inequality problems, mixed equilibrium problems and fixed point problems of strictly pseudocontractive mappings in Hilbert spaces. J. Fixed Point Theory Appl. Article ID 519065, 32 p. (2009) · Zbl 1186.47079
[41] Xu H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66(2), 1–17 (2002) · Zbl 1013.60050
[42] Yao, Y., Liou, Y.C., Yao, J.C.: A new hybrid iterative algorithm for fixed point problems, variational inequality problems and mixed equilibrium problems. Journal of Fixed Point Theory and Applications 2008(417089), 15 · Zbl 1203.47087
[43] Yao Y., Yao J.C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186(2), 1551–1558 (2007) · Zbl 1121.65064
[44] Youla D.: On deterministic convergence of iterations of related projection operators. J. Vis. Commun. Image Represent 1, 12–20 (1990)
[45] Zhang S.: Generalized mixed equilibrium problems in Banach spaces. Appl. Math. Mech. (English Edition) 30, 1105–1112 (2009) · Zbl 1178.47051
[46] Zhao J., He S.: A new iterative method for equilibrium problems and fixed points problems for infinite nonexpansive mappings and monotone mappings Appl. Math. Comput. 215(2), 670–680 (2009) · Zbl 1179.65064
[47] Zhou H.: Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 69, 456–462 (2008) · Zbl 1220.47139
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