×

Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in Banach spaces. (English) Zbl 1206.47069

Summary: We introduce a new hybrid iterative scheme for finding a common element in the solution set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove strong convergence by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117-136, 2005. · Zbl 1109.90079
[2] S.-y. Matsushita and W. Takahashi, “Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 37-47, 2004. · Zbl 1088.47054 · doi:10.1155/S1687182004310089
[3] W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008. · Zbl 1187.47054 · doi:10.1155/2008/528476
[4] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002
[5] S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 313-318, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0943.47040
[6] D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151-174, 2001. · Zbl 1010.47032 · doi:10.1515/JAA.2001.151
[7] D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489-508, 2003. · Zbl 1071.47052 · doi:10.1081/NFA-120023869
[8] S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938-945, 2002. · Zbl 1101.90083 · doi:10.1137/S105262340139611X
[9] P. Cholamjiak, “A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 719360, 18 pages, 2009. · Zbl 1167.65379 · doi:10.1155/2009/719360
[10] X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, “Convergence of a modified Halpern-type iteration algorithm for quasi-\varphi -nonexpansive mappings,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1051-1055, 2009. · Zbl 1179.65061 · doi:10.1016/j.aml.2009.01.015
[11] Y. Su, D. Wang, and M. Shang, “Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 284613, 8 pages, 2008. · Zbl 1203.47078 · doi:10.1155/2008/284613
[12] K. Wattanawitoon and P. Kumam, “Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings,” Nonlinear Analysis: Hybrid System, vol. 3, no. 1, pp. 11-20, 2009. · Zbl 1166.47060 · doi:10.1016/j.nahs.2008.10.002
[13] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 15-50, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0883.47083
[14] Ya. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39-54, 1994. · Zbl 0851.47043
[15] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. · Zbl 0712.47043
[16] X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20-30, 2009. · Zbl 1165.65027 · doi:10.1016/j.cam.2008.06.011
[17] E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1-4, pp. 123-145, 1994. · Zbl 0888.49007
[18] W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45-57, 2009. · Zbl 1170.47049 · doi:10.1016/j.na.2007.11.031
[19] F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 824-835, 2008. · Zbl 1168.47047 · doi:10.1137/070688717
[20] K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372-379, 2003. · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[21] A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359-370, 2007. · Zbl 1147.47052 · doi:10.1007/s10957-007-9187-z
[22] W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276-286, 2008. · Zbl 1134.47052 · doi:10.1016/j.jmaa.2007.09.062
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.