Common fixed points for asymptotic pointwise nonexpansive mappings in metric and Banach spaces.

*(English)*Zbl 1295.47060Summary: Let \(C\) be a nonempty bounded closed convex subset of a complete CAT(0) space \(X\). We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on \(C\) is nonempty closed and convex. We also show that, under some suitable conditions, the sequence \(\{x_k\}^\infty_{k=1}\) defined by

\[ \begin{aligned} x_{k+1} &= (1 - t_{mk})x_k \oplus t_{mk} T^{n_k}_m y_{(m-1)k},\\ y_{(m-1)k} &= (1 - t_{(m-1)k}) x_k \oplus t_{(m-1)k} T^{n_k}_{m-1} y_{(m-2)k},\\ y_{(m-2)k} &= (1 - t_{(m-2)k}) x_k \oplus t_{(m-2)k} T^{n_k}_{m-2} y_{(m-3)k},\\ &\vdots\\ y_{2k} &= (1 - t_{2k}) x_k \oplus t_{2k} T^{n_k}_2 y_{1k},\\ y_{1k} &= (1 - t_{1k}) x_k \oplus t_{1k} T^{n_k}_1 y_{0k},\\ y_{0k} &= x_k,\end{aligned} \]

\(k \in \mathbb N\), converges to a common fixed point of \(T_1, T_2, \dots, T_m\) which are asymptotic pointwise nonexpansive mappings on \(C\), while \(\{t_{ik}\}^\infty_{k=1}\) are sequences in [0,1] for all \(i = 1, 2, \dots, m\), and \(\{n_k\}\) is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.

\[ \begin{aligned} x_{k+1} &= (1 - t_{mk})x_k \oplus t_{mk} T^{n_k}_m y_{(m-1)k},\\ y_{(m-1)k} &= (1 - t_{(m-1)k}) x_k \oplus t_{(m-1)k} T^{n_k}_{m-1} y_{(m-2)k},\\ y_{(m-2)k} &= (1 - t_{(m-2)k}) x_k \oplus t_{(m-2)k} T^{n_k}_{m-2} y_{(m-3)k},\\ &\vdots\\ y_{2k} &= (1 - t_{2k}) x_k \oplus t_{2k} T^{n_k}_2 y_{1k},\\ y_{1k} &= (1 - t_{1k}) x_k \oplus t_{1k} T^{n_k}_1 y_{0k},\\ y_{0k} &= x_k,\end{aligned} \]

\(k \in \mathbb N\), converges to a common fixed point of \(T_1, T_2, \dots, T_m\) which are asymptotic pointwise nonexpansive mappings on \(C\), while \(\{t_{ik}\}^\infty_{k=1}\) are sequences in [0,1] for all \(i = 1, 2, \dots, m\), and \(\{n_k\}\) is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.

##### MSC:

47H10 | Fixed-point theorems |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

54E40 | Special maps on metric spaces |

PDF
BibTeX
Cite

\textit{P. Pasom} and \textit{B. Panyanak}, J. Appl. Math. 2012, Article ID 327434, 17 p. (2012; Zbl 1295.47060)

Full Text:
DOI

##### References:

[1] | W. A. Kirk and H.-K. Xu, “Asymptotic pointwise contractions,” Nonlinear Analysis, vol. 69, no. 12, pp. 4706-4712, 2008. · Zbl 1172.47038 |

[2] | N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in metric spaces,” Nonlinear Analysis, vol. 71, no. 10, pp. 4423-4429, 2009. · Zbl 1176.54031 |

[3] | W. M. Kozlowski, “Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 377, no. 1, pp. 43-52, 2011. · Zbl 1210.47096 |

[4] | A. R. Khan, A.-A. Domlo, and H. Fukhar-ud-din, “Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 1-11, 2008. · Zbl 1137.47053 |

[5] | M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, vol. 319 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999. · Zbl 0988.53001 |

[6] | W. A. Kirk, “Fixed point theorems in CAT(0) spaces and \Bbb R-trees,” Fixed Point Theory and Applications, no. 4, pp. 309-316, 2004. · Zbl 1089.54020 |

[7] | K. S. Brown, Buildings, Springer, New York, NY, USA, 1989. · Zbl 0715.20017 |

[8] | K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984. · Zbl 0537.46001 |

[9] | W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of Mathematical Analysis, vol. 64 of Colecc. Abierta, pp. 195-225, Universidad de Sevilla Secr. Publ., Seville, Spain, 2003. · Zbl 1058.53061 |

[10] | W. A. Kirk, “Geodesic geometry and fixed point theory. II,” in International Conference on Fixed Point Theory and Applications, pp. 113-142, Yokohama Publishers, Yokohama, Japan, 2004. · Zbl 1083.53061 |

[11] | S. Dhompongsa, A. Kaewkhao, and B. Panyanak, “Lim’s theorems for multivalued mappings in CAT(0) spaces,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 478-487, 2005. · Zbl 1086.47019 |

[12] | P. Chaoha and A. Phon-on, “A note on fixed point sets in CAT(0) spaces,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 983-987, 2006. · Zbl 1101.54040 |

[13] | L. Leustean, “A quadratic rate of asymptotic regularity for CAT(0)-spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 386-399, 2007. · Zbl 1103.03057 |

[14] | S. Dhompongsa and B. Panyanak, “On \Delta -convergence theorems in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2572-2579, 2008. · Zbl 1165.65351 |

[15] | N. Shahzad and J. Markin, “Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1457-1464, 2008. · Zbl 1137.47043 |

[16] | N. Shahzad, “Fixed point results for multimaps in CAT(0) spaces,” Topology and Its Applications, vol. 156, no. 5, pp. 997-1001, 2009. · Zbl 1175.47049 |

[17] | R. Espínola and A. Fernández-León, “CAT(k)-spaces, weak convergence and fixed points,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 410-427, 2009. · Zbl 1182.47043 |

[18] | A. Razani and H. Salahifard, “Invariant approximation for CAT(0) spaces,” Nonlinear Analysis, vol. 72, no. 5, pp. 2421-2425, 2010. · Zbl 1188.54022 |

[19] | S. Saejung, “Halpern’s iteration in CAT(0) spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 471781, 13 pages, 2010. · Zbl 1197.54074 |

[20] | A. R. Khan, M. A. Khamsi, and H. Fukhar-ud-din, “Strong convergence of a general iteration scheme in CAT(0) spaces,” Nonlinear Analysis, vol. 74, no. 3, pp. 783-791, 2011. · Zbl 1202.47076 |

[21] | S. H. Khan and M. Abbas, “Strong and \Delta -convergence of some iterative schemes in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 61, no. 1, pp. 109-116, 2011. · Zbl 1207.65069 |

[22] | A. Abkar and M. Eslamian, “Common fixed point results in CAT(0) spaces,” Nonlinear Analysis, vol. 74, no. 5, pp. 1835-1840, 2011. · Zbl 1269.54018 |

[23] | M. Bestvina, “\Bbb R-trees in topology, geometry, and group theory,” in Handbook of Geometric Topology, pp. 55-91, North-Holland, Amsterdam, The Netherlands, 2002. · Zbl 0998.57003 |

[24] | C. Semple and M. Steel, Phylogenetics, vol. 24 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, UK, 2003. · Zbl 1043.92026 |

[25] | R. Espínola and W. A. Kirk, “Fixed point theorems in \Bbb R-trees with applications to graph theory,” Topology and Its Applications, vol. 153, no. 7, pp. 1046-1055, 2006. · Zbl 1095.54012 |

[26] | W. A. Kirk, “Some recent results in metric fixed point theory,” Journal of Fixed Point Theory and Applications, vol. 2, no. 2, pp. 195-207, 2007. · Zbl 1139.05315 |

[27] | S. Dhompongsa, W. A. Kirk, and B. Sims, “Fixed points of uniformly Lipschitzian mappings,” Nonlinear Analysis, vol. 65, no. 4, pp. 762-772, 2006. · Zbl 1105.47050 |

[28] | W. A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Analysis, vol. 68, no. 12, pp. 3689-3696, 2008. · Zbl 1145.54041 |

[29] | T. C. Lim, “Remarks on some fixed point theorems,” Proceedings of the American Mathematical Society, vol. 60, pp. 179-182, 1976. · Zbl 0346.47046 |

[30] | S. Dhompongsa, W. A. Kirk, and B. Panyanak, “Nonexpansive set-valued mappings in metric and Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 35-45, 2007. · Zbl 1120.47043 |

[31] | W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004-1006, 1965. · Zbl 0141.32402 |

[32] | K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171-174, 1972. · Zbl 0256.47045 |

[33] | R. Espínola and N. Hussain, “Common fixed points for multimaps in metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 204981, 14 pages, 2010. · Zbl 1188.54016 |

[34] | P. Pasom and B. Panyanak, “Common fixed points for asymptotic pointwise nonexpansive mappings,” Fixed Point Theory. to appear. · Zbl 1306.47067 |

[35] | R. E. Bruck, Jr., “A common fixed point theorem for a commuting family of nonexpansive mappings,” Pacific Journal of Mathematics, vol. 53, pp. 59-71, 1974. · Zbl 0312.47045 |

[36] | Z.-H. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 351-358, 2003. · Zbl 1095.47046 |

[37] | M. A. Khamsi and A. R. Khan, “Inequalities in metric spaces with applications,” Nonlinear Analysis, vol. 74, no. 12, pp. 4036-4045, 2011. · Zbl 1246.46012 |

[38] | B. Nanjaras and B. Panyanak, “Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 268780, 14 pages, 2010. · Zbl 1197.54069 |

[39] | R. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no. 2, pp. 169-179, 1993. · Zbl 0849.47030 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.