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Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems. (English) Zbl 1287.90076

Summary: We introduce some new iterative sequences for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of nonspreading mappings and a finite family of nonexpansive multivalued mappings in Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our iterative process. The results obtained extend and improve some recent known results.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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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] Combettes P.L., Hirstoaga S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005) · Zbl 1109.90079
[3] Li X.B., Li S.J.: Existence of solutions for generalized vector quasi-equilibrium problems. Optim. Lett. 4, 17–28 (2010) · Zbl 1183.49006
[4] Giannessi F., Maugeri G., Pardalos P.M.: Equilibrium problems: nonsmooth optimization and variational inequality models. Kluwer Academics Publishers, Dordrecht, Holland (2001) · Zbl 0979.00025
[5] Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems, In: Lecture Note in Economics and Mathematical Systems, vol. 477, pp. 187-201. Springer-Verlag, New York (1999) · Zbl 0944.65080
[6] Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010) · Zbl 1178.49001
[7] Tada A., Takahashi W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007) · Zbl 1147.47052
[8] Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007) · Zbl 1122.47056
[9] Ceng L.C., Al-Homidan S., Ansari Q.H., Yao J.C.: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 223, 967–974 (2009) · Zbl 1167.47307
[10] Wang, S., Marino, G., Liou, Y.C.: Strong convergence theorems for variational inequality, equilibrium and fixed point problems with applications. J. Glob. Optim. doi: 10.1007/s10898-011-9754-6 · Zbl 1267.47105
[11] Qin X., Cho S.Y., Kang S.M.: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 49, 679–693 (2011) · Zbl 1256.47053
[12] Shehu, Y.: A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium problem in Banach spaces. J. Glob. Optim. doi: 10.1007/s10898-011-9775-1 · Zbl 06121387
[13] Qin X., Cho S.Y., Kang S.M.: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 48, 423–445 (2010) · Zbl 1292.47052
[14] Liu Y.: Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings. J. Glob. Optim. 46, 319–329 (2010) · Zbl 1228.47061
[15] Kamraksa, U., Wangkeeree, R.: Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings. J. Glob. Optim. doi: 10.1007/s10898-011-9739-5 · Zbl 1252.47074
[16] Moudafi A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37–43 (2008) · Zbl 1167.47049
[17] 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
[18] Chang S.S., Lee H.W.J., Kim J.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009) · Zbl 1198.47082
[19] 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
[20] Song Y., Wang H.: Convergence of iterative algorithms for multivalued mappings in Banach spaces. Nonlinear Anal. 70, 1547–1556 (2009) · Zbl 1175.47063
[21] Shahzad N., Zegeye H.: On Mann and Ishikawa iteration schemes for multivalued maps in Banach space. Nonlinear Anal. 71, 838–844 (2009) · Zbl 1218.47118
[22] Eslamian M., Abkar A.: One-step iterative process for a finite family of multivalued mappings. Math. Comput. Modell. 54, 105–111 (2011) · Zbl 1225.65059
[23] Kohsaka F., Takahashi W.: Fixed point theorems for a class of nonlinear mappings relate to maximal monotone operators in Banach spaces. Arch. Math. (Basel). 91, 166–177 (2008) · Zbl 1149.47045
[24] Kohsaka F., Takahashi W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008) · Zbl 1168.47047
[25] Iemoto S., Takahashi W.: Approximating commom fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71, 2082–2089 (2009) · Zbl 1239.47054
[26] Kangtunyakarn, A.: Convergence theorem of common fixed points for a family of nonspreading mappings in Hilbert space. Optim. Lett. doi: 10.1007/s11590-011-0326-y · Zbl 1254.90297
[27] Marino G., Xu H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert space. J. Math. Anal. Appl. 329, 336–346 (2007) · Zbl 1116.47053
[28] Osilike M.O., Igbokwe D.I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput. Math. Appl. 40, 559–567 (2000) · Zbl 0958.47030
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