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On strong convergence by the hybrid method for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings. (English) Zbl 1186.47056
Summary: We introduce two modifications of the Mann iteration, by using hybrid methods, for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then, we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asymptotically nonexpansive mappings. Our results improve and extend the results announced by many others.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] Goebel, K; Kirk, WA, A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society, 35, 171-174, (1972) · Zbl 0256.47045
[2] Combettes, PL; Hirstoaga, SA, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 117-136, (2005) · Zbl 1109.90079
[3] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007
[4] Moudafi, A; Théra, M, Proximal and dynamical approaches to equilibrium problems, No. 477, 187-201, (1999), Berlin, Germany · Zbl 0944.65080
[5] Browder, FE; Petryshyn, WV, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228, (1967) · Zbl 0153.45701
[6] Liu, F; Nashed, MZ, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Analysis, 6, 313-344, (1998) · Zbl 0924.49009
[7] Nakajo, K; Takahashi, W, Strong and weak convergence theorems by an improved splitting method, Communications on Applied Nonlinear Analysis, 9, 99-107, (2002) · Zbl 1050.47049
[8] Podilchuk, CI; Mammone, RJ, Image recovery by convex projections using a least-squares constraint, Journal of the Optical Society of America A, 7, 517-521, (1990)
[9] Byrne, C, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 103-120, (2004) · Zbl 1051.65067
[10] Nakajo, K; Shimoji, K; Takahashi, W, On strong convergence by the hybrid method for families of mappings in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 71, 112-119, (2009) · Zbl 1162.49037
[11] Mann, WR, Mean value methods in iteration, Proceedings of the American Mathematical Society, 4, 506-510, (1953) · Zbl 0050.11603
[12] Ishikawa, S, Fixed points by a new iteration method, Proceedings of the American Mathematical Society, 44, 147-150, (1974) · Zbl 0286.47036
[13] Tan, K-K; Xu, HK, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications, 178, 301-308, (1993) · Zbl 0895.47048
[14] Nakajo, K; Takahashi, W, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, Journal of Mathematical Analysis and Applications, 279, 372-379, (2003) · Zbl 1035.47048
[15] Kim, T-H; Xu, H-K, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Analysis: Theory, Methods & Applications, 64, 1140-1152, (2006) · Zbl 1090.47059
[16] Inchan, I; Plubtieng, S, Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Analysis: Hybrid Systems, 2, 1125-1135, (2008) · Zbl 1203.47051
[17] Takahashi, W; Takeuchi, Y; Kubota, R, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 341, 276-286, (2008) · Zbl 1134.47052
[18] Zegeye, H; Shahzad, N, Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups, Nonlinear Analysis: Theory, Methods & Applications, 69, 4496-4503, (2008) · Zbl 1168.47056
[19] Tada, A; Takahashi, W, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, Journal of Optimization Theory and Applications, 133, 359-370, (2007) · Zbl 1147.47052
[20] Opial, Z, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society, 73, 591-597, (1967) · Zbl 0179.19902
[21] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244. · Zbl 0708.47031
[22] Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. · Zbl 0997.47002
[23] Lin, P-K; Tan, K-K; Xu, HK, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Analysis: Theory, Methods & Applications, 24, 929-946, (1995) · Zbl 0865.47040
[24] Martinez-Yanes, C; Xu, H-K, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Analysis: Theory, Methods & Applications, 64, 2400-2411, (2006) · Zbl 1105.47060
[25] Takahashi, W, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Annales Universitatis Mariae Curie-Sklodowska. Sectio A, 51, 277-292, (1997) · Zbl 1012.47029
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