Strong convergence theorems of Cesàro-type means for nonexpansive mapping in CAT(0) space. (English) Zbl 1347.65103
Summary: The iterative algorithms with Cesàro-type means for a nonexpansive mapping are proposed in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the sequence generated by the algorithms to a fixed point of the nonexpansive mapping are proved. We also proved that this fixed point is also a unique solution to some kind of variational inequality. The results presented in this paper extend and improve the corresponding results of some others.
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
References:
| [1] | Baillon, JB: Un théorème de type ergodique pour les contractions non linéairs dans un espaces de Hilbert. C. R. Acad. Sci. Paris Sér. A-B 280(22), 1511-1514 (1975) · Zbl 0307.47006 |
| [2] | Bruck, RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math. 32(2-3), 107-116 (1979) · Zbl 0423.47024 · doi:10.1007/BF02764907 |
| [3] | Song, Y, Chen, R: Viscosity approximate methods to Cesàro means for non-expansive mappings. Appl. Math. Comput. 186(2), 1120-1128 (2007) · Zbl 1121.65063 · doi:10.1016/j.amc.2006.08.054 |
| [4] | Yao, Y, Liou, YC, Zhou, H: Strong convergence of an iterative method for nonexpansive mappings with new control conditions. Nonlinear Anal., Theory Methods Appl. 70(6), 2332-2336 (2009) · Zbl 1223.47107 · doi:10.1016/j.na.2008.03.014 |
| [5] | Zhu, Z, Chen, R: Strong convergence on iterative methods of Cesàro means for nonexpansive mapping in Banach space. Abstr. Appl. Anal. 2014, Article ID 205875 (2014) · Zbl 1472.47090 |
| [6] | Bridson, MR, Haefliger, A: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999) · Zbl 0988.53001 |
| [7] | Kirk, WA: Fixed point theory in CAT(0) spaces and R-trees. Fixed Point Theory Appl. 2004(4), 309-316 (2004) · Zbl 1089.54020 · doi:10.1155/S1687182004406081 |
| [8] | Brown, KS (ed.): Buildings. Springer, New York (1989) · Zbl 0715.20017 |
| [9] | Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Monograph and Textbooks in Pure and Applied Mathematics, vol. 83. Dekker, New York (1984) · Zbl 0537.46001 |
| [10] | Kirk, WA, Geodesic geometry and fixed point theory, Malaga/Seville, 2002/2003, Seville · Zbl 1058.53061 |
| [11] | Kirk, WA, Geodesic geometry and fixed point theory. II, 113-142 (2004), Yokohama · Zbl 1083.53061 |
| [12] | Wangkeeree, R, Preechasilp, P: Viscosity approximation methods for nonexpansive semigroups in CAT(0) spaces. Fixed Point Theory Appl. 2013, Article ID 160 (2013). doi:10.1186/1687-1812-2013-160 · Zbl 1318.47099 · doi:10.1186/1687-1812-2013-160 |
| [13] | Dhompongsa, S, Panyanak, B: On Δ-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56(10), 2572-2579 (2008) · Zbl 1165.65351 · doi:10.1016/j.camwa.2008.05.036 |
| [14] | Tang, JF: Viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaces. Abstr. Appl. Anal. 2014, Article ID 389804 (2014) · Zbl 1472.47084 |
| [15] | Berg, ID, Nikolaev, IG: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 133, 195-218 (2008) · Zbl 1144.53045 · doi:10.1007/s10711-008-9243-3 |
| [16] | Dehghan, H.; Rooin, J., A characterization of metric projection in CAT(0) spaces, 10-12th May 2012, Tabriz |
| [17] | Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659-678 (2003) · Zbl 1043.90063 · doi:10.1023/A:1023073621589 |
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