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Strong convergence theorems for families of weak relatively nonexpansive mappings. (English) Zbl 1215.47086

Summary: We construct a new Halpern type iterative scheme by hybrid methods and prove a strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized \(f\)-projection operator. Using this result, we discuss a strong convergence theorem concerning general \(H\)-monotone mappings. Our results extend many known recent results in the literature.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009. · Zbl 1167.47002
[2] W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and Application, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002
[3] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002
[4] 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
[5] Y. Su, H.-K. Xu, and X. Zhang, “Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 12, pp. 3890-3906, 2010. · Zbl 1215.47091 · doi:10.1016/j.na.2010.08.021
[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] Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323-339, 1996. · Zbl 0883.47063 · doi:10.1080/02331939608844225
[9] S.-Y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257-266, 2005. · Zbl 1071.47063 · doi:10.1016/j.jat.2005.02.007
[10] X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 6, pp. 1958-1965, 2007. · Zbl 1124.47046 · doi:10.1016/j.na.2006.08.021
[11] 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
[12] J. Kang, Y. Su, and X. Zhang, “Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp. 755-765, 2010. · Zbl 1292.47049 · doi:10.1016/j.nahs.2010.05.002
[13] H. Zegeye and N. Shahzad, “Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 12, pp. 4496-4503, 2008. · Zbl 1168.47056 · doi:10.1016/j.na.2007.11.005
[14] S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2, pp. 103-115, 2007. · Zbl 1137.47056 · doi:10.1016/j.jat.2007.04.014
[15] X. Li, N.-J. Huang, and D. O’Regan, “Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1322-1331, 2010. · Zbl 1201.65091 · doi:10.1016/j.camwa.2010.06.013
[16] 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, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15-50, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0883.47083
[17] Y. 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
[18] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1990. · Zbl 0712.47043
[19] K.-Q. Wu and N.-J. Huang, “The generalised f-projection operator with an application,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307-317, 2006. · Zbl 1104.47053 · doi:10.1017/S0004972700038892
[20] J. Fan, X. Liu, and J. Li, “Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 11, pp. 3997-4007, 2009. · Zbl 1219.47110 · doi:10.1016/j.na.2008.08.008
[21] Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,” Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 707-717, 2004. · Zbl 1081.47063 · doi:10.1016/S0898-1221(04)90058-2
[22] F.-Q. Xia and N.-J. Huang, “Variational inclusions with a general H-monotone operator in Banach spaces,” Computers & Mathematics with Applications, vol. 54, no. 1, pp. 24-30, 2007. · Zbl 1131.49011 · doi:10.1016/j.camwa.2006.10.028
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