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Implicit Ishikawa approximation methods for nonexpansive semigroups in CAT(0) spaces. (English) Zbl 1364.47035

Summary: This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the \(\Delta\)-convergence results of the implicit Ishikawa iteration sequences for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces. The results presented in this paper extend and generalize some previous results.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H20 Semigroups of nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] Browder, F. E., Nonexpansive nonlinear operators in a Banach space, Proceedings of the National Academy of Sciences of the United States of America, 54, 1041-1044 (1965) · Zbl 0128.35801
[2] Lim, T. C., A fixed point theorem for families on nonexpansive mappings, Pacific Journal of Mathematics, 53, 487-493 (1974) · Zbl 0291.47032
[3] Bridson, M. R.; Haefliger, A., Metric Spaces of Non-Positive Curvature, 319 (1999), New York, NY, USA: Springer, New York, NY, USA · Zbl 0988.53001
[4] Kirk, W. A., A fixed point theorems in CAT(0) spaces and \(R\)-trees, Fixed Point Theory and Applications, 2004, 4, 309-316 (2004) · Zbl 1089.54020
[5] Ćirić, L., Common fixed point theorems for a family of non-self mappings in convex metric spaces, Nonlinear Analysis: Theory, Methods & Applications, 71, 5-6, 1662-1669 (2009) · Zbl 1203.54038
[6] Cho, Y. J.; Ćirić, L.; Wang, S.-H., Convergence theorems for nonexpansive semigroups in CAT(0) spaces, Nonlinear Analysis: Theory, Methods & Applications, 74, 17, 6050-6059 (2011) · Zbl 1237.47070
[7] Dhompongsa, S.; Kaewkhao, A.; Panyanak, B., Lim’s theorems for multivalued mappings in CAT(0) spaces, Journal of Mathematical Analysis and Applications, 312, 2, 478-487 (2005) · Zbl 1086.47019
[8] Dhompongsa, S.; Kirk, W. A.; Sims, B., Fixed points of uniformly Lipschitzian mappings, Nonlinear Analysis: Theory, Methods & Applications, 65, 4, 762-772 (2006) · Zbl 1105.47050
[9] Dhompongsa, S.; Kirk, W. A.; Panyanak, B., Nonexpansive set-valued mappings in metric and Banach spaces, Journal of Nonlinear and Convex Analysis, 8, 1, 35-45 (2007) · Zbl 1120.47043
[10] Dhompongsa, S.; Panyanak, B., On \(\Delta \)-convergence theorems in CAT(0) spaces, Computers & Mathematics with Applications, 56, 10, 2572-2579 (2008) · Zbl 1165.65351
[11] Khan, A. R.; Khamsi, M. A.; Fukhar-ud-din, H., Strong convergence of a general iteration scheme in CAT(0) spaces, Nonlinear Analysis: Theory, Methods & Applications, 74, 3, 783-791 (2011) · Zbl 1202.47076
[12] Kirk, W. A.; Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear Analysis: Theory, Methods & Applications, 68, 12, 3689-3696 (2008) · Zbl 1145.54041
[13] Laowang, W.; Panyanak, B., Strong and \(\Delta\) convergence theorems for multivalued mappings in CAT(0) spaces, Journal of Inequalities and Applications, 2009 (2009) · Zbl 1176.47056
[14] Laowang, W.; Panyanak, B., Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces, Fixed Point Theory and Applications, 2010 (2010) · Zbl 1188.54021
[15] Leustean, L., A quadratic rate of asymptotic regularity for CAT(0)-spaces, Journal of Mathematical Analysis and Applications, 325, 1, 386-399 (2007) · Zbl 1103.03057
[16] Nanjaras, B.; Panyanak, B.; Phuengrattana, W., Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces, Nonlinear Analysis: Hybrid Systems, 4, 1, 25-31 (2010) · Zbl 1225.54021
[17] Saejung, S., Halpern’s iteration in CAT(0) spaces, Fixed Point Theory and Applications, 2010 (2010) · Zbl 1197.54074
[18] Razani, A.; Salahifard, H., Invariant approximation for CAT(0) spaces, Nonlinear Analysis: Theory, Methods & Applications, 72, 5, 2421-2425 (2010) · Zbl 1188.54022
[19] Thong, D. V., An implicit iteration process for nonexpansive semigroups, Nonlinear Analysis: Theory, Methods & Applications, 74, 17, 6116-6120 (2011) · Zbl 1225.47093
[20] Chidume, C. E.; Shahzad, N., Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Analysis: Theory, Methods & Applications, 62, 6, 1149-1156 (2005) · Zbl 1090.47055
[21] Li, X.-S.; Huang, N.-J.; Kim, J. K., General viscosity approximation methods for common fixed points of nonexpansive semigroups in Hilbert spaces, Fixed Point Theory and Applications, 2011 (2011)
[22] Qin, X.; Cho, S. Y., Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions, Applied Mathematics Letters, 23, 10, 1252-1255 (2010) · Zbl 1229.47120
[23] Xu, H.-K.; Ori, R. G., An implicit iteration process for nonexpansive mappings, Numerical Functional Analysis and Optimization, 22, 5-6, 767-773 (2001) · Zbl 0999.47043
[24] Buong, N., Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space, Computers & Mathematics with Applications, 61, 9, 2546-2554 (2011) · Zbl 1221.65131
[25] Tan, K. K.; Xu, H. K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications, 178, 301-308 (1993) · Zbl 0895.47048
[26] Lim, T. C., Remarks on some fixed point theorems, Proceedings of the American Mathematical Society, 60, 179-182 (1976) · Zbl 0346.47046
[27] Suzuki, T., Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory and Applications, 2005 (2005) · Zbl 1123.47308
[28] DeMarr, R., Common fixed points for commuting contraction mappings, Pacific Journal of Mathematics, 13, 1139-1141 (1963) · Zbl 0191.14901
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