## A new composite general iterative scheme for nonexpansive semigroups in Banach spaces.(English)Zbl 1221.47127

Summary: We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the proposed iterative approximation method is established under certain control conditions. Our results improve and extend those announced by many others.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H20 Semigroups of nonlinear operators
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### References:

 [1] F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272-1276, 1965. · Zbl 0125.35801 [2] F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041-1044, 1965. · Zbl 0128.35801 [3] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279-291, 2004. · Zbl 1061.47060 [4] F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization. An International Journal, vol. 19, no. 1-2, pp. 33-56, 1998. · Zbl 0913.47048 [5] H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659-678, 2003. · Zbl 1043.90063 [6] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46-55, 2000. · Zbl 0957.47039 [7] S. Plubtieng and T. Thammathiwat, “A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities,” Journal of Global Optimization, vol. 46, no. 3, pp. 447-464, 2010. · Zbl 1203.47064 [8] G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43-52, 2006. · Zbl 1095.47038 [9] T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51-60, 2005. · Zbl 1091.47055 [10] Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1687-1693, 2008. · Zbl 1189.47071 [11] R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566-575, 2007. · Zbl 1204.47076 [12] P. Sunthrayuth and P. Kumam, “A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces,” Journal of Nonlinear Analysis and Optimization: Theory and Applications, vol. 1, no. 1, pp. 139-150, 2010. · Zbl 1413.47141 [13] P. Kumam and K. Wattanawitoon, “A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 998-1006, 2011. · Zbl 1218.47105 [14] P. Sunthrayuth, K. Wattanawitoon, and P. Kumam, “Convergence theorems of a general composite iterative method for nonexpansive semigroups in Banach spaces,” Mathematical Analysis, vol. 2011, Article ID 576135, 24 pages, 2011. · Zbl 1215.65113 [15] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002 [16] F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,” Mathematische Zeitschrift, vol. 100, pp. 201-225, 1967. · Zbl 0149.36301 [17] R. Wangkeeree, N. Petrot, and R. Wangkeeree, “The general iterative methods for nonexpansive mappings in Banach spaces,” Journal of Global Optimization. In press. · Zbl 1471.65041 [18] T.-C. Lim and H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 11, pp. 1345-1355, 1994. · Zbl 0812.47058 [19] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032 [20] J.-B. Baillon, “Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert,” vol. 280, no. 22, pp. A1511-A1514, 1975. · Zbl 0307.47006 [21] W. Kaczor, T. Kuczumow, and S. Reich, “A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense,” Journal of Mathematical Analysis and Applications, vol. 246, no. 1, pp. 1-27, 2000. · Zbl 0981.47037 [22] S. Reich, “Almost convergence and nonlinear ergodic theorems,” Journal of Approximation Theory, vol. 24, no. 4, pp. 269-272, 1978. · Zbl 0404.47032 [23] S. Reich, “A note on the mean ergodic theorem for nonlinear semigroups,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 547-551, 1983. · Zbl 0521.47034
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