Document Zbl 1515.47083

Chang, Shih-sen; Lee, H. W. Joseph; Chan, Chi Kin

A block hybrid method for solving generalized equilibrium problems and convex feasibility problem. (English) Zbl 1515.47083

Adv. Comput. Math. 38, No. 3, 563-580 (2013).

Summary: A block hybrid projection algorithm for solving the convex feasibility problem and the generalized equilibrium problems for an infinite family of total quasi-\(\phi \)-asymptotically nonexpansive mappings is introduced. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

Cited in 6 Documents

MSC:

47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49J40	Variational inequalities

Keywords:

block iterative algorithm; convex feasibility problem; generalised equilibrium problem; total quasi-\(\phi \)-asymptotically nonexpansive mapping; quasi-\(\phi \)-nonexpansive mapping; relatively nonexpansive mapping; generalised projection; strong convergence

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References:

[1]	Combettes, PL; Hawkes, P. (ed.), The convex feasibility problem in inage recovery, 155-270 (1996), New York · doi:10.1016/S1076-5670(08)70157-5
[2]	Kitahara, S., Takahashi, W.: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 2, 333-342 (1993) · Zbl 0815.47068
[3]	O’Hara, J.G., Pillay, P., Xu, H.K.: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal. 64(9), 2022-2042 (2006) · Zbl 1139.47056 · doi:10.1016/j.na.2005.07.036
[4]	Chang, S.S., Yao, J.C., Kim, J.K., Yang, L.: Iterative approximation to convex feasibility problems in Banach space. Fixed Point Theory and Applications, Article ID 46797, 19 p. (2008) · Zbl 1155.47312
[5]	Matsushita, S., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 134, 257-266 (2005) · Zbl 1071.47063 · doi:10.1016/j.jat.2005.02.007
[6]	Qin, X., Cho, Y.J., Kang, S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problem in Banach spaces. J. Comput. Appl. Math. 225, 20-30 (2009) · Zbl 1165.65027 · doi:10.1016/j.cam.2008.06.011
[7]	Zhou, H.Y., Gao, G., Tan, B.: Convergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings. J. Appl. Math. Comput. (2009). doi:10.1007/s12190-009-0263-4 · Zbl 1203.47091 · doi:10.1007/s12190-009-0263-4
[8]	Wattanawitoon, K., Kumam, P.: Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings. Nonlinear Anal., Hybrid Systems 3, 11-20 (2009) · Zbl 1166.47060 · doi:10.1016/j.nahs.2008.10.002
[9]	Li, H.Y., Su, Y.F.: Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems. Nonlinear Anal. 72(2), 847-855 (2010) · Zbl 1225.47101
[10]	Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45-57 (2009) · Zbl 1170.47049 · doi:10.1016/j.na.2007.11.031
[11]	Kikkawa, M., Takahashi, W.: Approximating fixed points of nonexpansive mappings by the block iterative method in Banach spaces. Int. J. Comput. Numer., Anal. Appl. 5(1), 59-66 (2004) · Zbl 1083.47056
[12]	Aleyner, A., Reich, S.: Block-iterative algorithms for solving convex feasibilitys in Hilbert and in Banach spaces. J. Math. Anal. Appl. 343(1), 427-435 (2008) · Zbl 1167.90010 · doi:10.1016/j.jmaa.2008.01.087
[13]	Plubtieng, S., Ungchittrakool, K.: Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications Article ID 583082, 19 p. (2008) doi:10.1155/2008/583082 · Zbl 1173.47051
[14]	Plubtieng, S., Ungchittrakool, K.: Strong convergence theorems of block iterative methods for a fonite family of relatively nonexpansive mappings in Banach spaces. J. Nonlinear Convex Anal. 8(3), 431-450 (2009) · Zbl 1142.47041
[15]	Sahu, D.R., Xu, H.K., Yao, J.C.: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal. TMA 79(10), 3502-3511 (2009) · Zbl 1221.47122 · doi:10.1016/j.na.2008.07.007
[16]	Chang, S.S., Joseph Lee, H.W., Chan, C.K.: A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications. Nonlinear Anal. TMA, 73, 2260-2270 (2010) · Zbl 1236.47068 · doi:10.1016/j.na.2010.06.006
[17]	Chang, S.S.: On the generalized mixed equilibrium problem in Banach spaces. Appl. Math. Mech. 30(9), 1105-1112 (2009) · Zbl 1178.47051 · doi:10.1007/s10483-009-0904-6
[18]	Chang, S.S., Kim, J.K., Wang, X.R.: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. Article ID 869684, 14 p. (2010) doi:10.1155/2010/869684 · Zbl 1187.47045
[19]	Kang, J., Su, Y., Zhang, X.: Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications. Nonlinear Anal. HS, 4(2), 847-855 (2010)
[20]	Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990) · Zbl 0712.47043 · doi:10.1007/978-94-009-2121-4
[21]	Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1/4), 123-145 (1994) · Zbl 0888.49007
[22]	Alber, YI; Kartosator, AG (ed.), Metric and generalized projection operators in Banach spaces: properties and applications, 15-50 (1996), New York · Zbl 0883.47083
[23]	Su, Y.F., Xu, H.K., Zhang, X.: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal. 73, 3890-3906 (2010) · Zbl 1215.47091 · doi:10.1016/j.na.2010.08.021
[24]	Chang, S.S., Joseph Lee, H.W., Chan, C.K., Yang, L.: Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications. Applied Math. Cmput. 218, 2921-2931 (2011) · Zbl 1297.47072 · doi:10.1016/j.amc.2011.08.036

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