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**New iterative schemes for asymptotically quasi-nonexpansive mappings.**
*(English)*
Zbl 1187.47056

Summary: We consider an iterative scheme for approximating the common fixed points of two asymptotically quasi-nonexpansive mappings in the intermediate sense in Banach spaces. The present results improve and extend some recent corresponding results of H.-Y. Lan [Comput. Math. Appl. 52, No. 10–11, 1403–1412 (2006; Zbl 1137.47054)] and many others.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

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

### Citations:

Zbl 1137.47054
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\textit{Y. Yao} and \textit{Y.-C. Liou}, J. Inequal. Appl. 2010, Article ID 934692, 9 p. (2010; Zbl 1187.47056)

### References:

[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] | Liu, Q, Iterative sequences for asymptotically quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications, 259, 1-7, (2001) · Zbl 1033.47047 |

[3] | Lan, H-Y, Common fixed-point iterative processes with errors for generalized asymptotically quasi-nonexpansive mappings, Computers & Mathematics with Applications, 52, 1403-1412, (2006) · Zbl 1137.47054 |

[4] | Chang, SS; Kim, JK; Kang, SM, Approximating fixed points of asymptotically quasi-nonexpansive type mappings by the Ishikawa iterative sequences with mixed errors, Dynamic Systems and Applications, 13, 179-186, (2004) · Zbl 1099.47518 |

[5] | Cho, YJ; Zhou, H; Guo, G, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Computers & Mathematics with Applications, 47, 707-717, (2004) · Zbl 1081.47063 |

[6] | Tan, K-K; Xu, HK, Fixed point iteration processes for asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society, 122, 733-739, (1994) · Zbl 0820.47071 |

[7] | Xu, B; Noor, MA, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 267, 444-453, (2002) · Zbl 1011.47039 |

[8] | Huang, Z, Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings, Computers & Mathematics with Applications, 37, 1-7, (1999) · Zbl 0942.47046 |

[9] | Ceng, L-C; Xu, H-K; Yao, J-C, The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 69, 1402-1412, (2008) · Zbl 1142.47326 |

[10] | Ceng, L-C; Petruşel, A; Yao, J-C, Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings, Applied Mathematics and Computation, 209, 162-176, (2009) · Zbl 1168.65350 |

[11] | Yao, Y; Liou, Y-C; Yao, J-C, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory and Applications, 2007, 12, (2007) · Zbl 1153.54024 |

[12] | Honda T, Takahashi W, Yao JC: Nonexpansive retractions onto closed convex cones in Banach spaces. to appear in Taiwanese Journal of Mathematics · Zbl 1193.47068 |

[13] | Liou, Y-C; Yao, Y; Chen, R, Iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Fixed Point Theory and Applications, 2007, 10, (2007) · Zbl 1159.47042 |

[14] | Ceng LC, Petrusel A, Yao JC: Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense. to appear in Taiwanese Journal of Mathematics · Zbl 1437.47046 |

[15] | Wu, X; Yao, J-C; Zeng, L-C, Uniform normal structure and strong convergence theorems for asymptotically pseudocontractive mappings, Journal of Nonlinear and Convex Analysis, 6, 453-463, (2005) · Zbl 1100.46008 |

[16] | Ceng, LC; Wong, NC; Yao, JC, Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings, Journal of Inequalities and Applications, 2006, 11, (2006) · Zbl 1132.65048 |

[17] | Yao, J-C; Zeng, L-C, Strong convergence of averaged approximants for asymptotically pseudocontractive mappings in Banach spaces, Journal of Nonlinear and Convex Analysis, 8, 451-462, (2007) · Zbl 1142.47043 |

[18] | Yao, Y; Liou, YC; Marino, G, Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Applications, 2009, 7, (2009) · Zbl 1186.47080 |

[19] | Ceng, L-C; Al-Homidan, S; Ansari, QH; Yao, J-C, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, Journal of Computational and Applied Mathematics, 223, 967-974, (2009) · Zbl 1167.47307 |

[20] | Sahu, DR; Xu, H-K; Yao, J-C, Asymptotically strict pseudocontractive mappings in the intermediate sense, Nonlinear Analysis: Theory, Methods & Applications, 70, 3502-3511, (2009) · Zbl 1221.47122 |

[21] | Peng, J-W; Yao, J-C, Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings, Taiwanese Journal of Mathematics, 13, 1537-1582, (2009) · Zbl 1193.47068 |

[22] | Liou, Y-C; Yao, Y, Iterative algorithms for nonexpansive mappings, Fixed Point Theory and Applications, 2008, 10, (2008) · Zbl 1203.47058 |

[23] | Ceng, LC; Shyu, DS; Yao, JC, Relaxed composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive mappings, Fixed Point Theory and Applications, 2009, 16, (2009) · Zbl 1186.47057 |

[24] | Ceng LC, Petrusel A, Yao JC: A hybrid method for Lipschitz continuous monotone mappings and asymptotically strict pseudocontractive mappings in the intermediate sense. Journal of Nonlinear and Convex Analysis 2010., 11(1): · Zbl 0820.47071 |

[25] | Yao, Y; Chen, R; Zhou, H, Strong convergence to common fixed points of nonexpansive mappings without commutativity assumption, Fixed Point Theory and Applications, 2006, 8, (2006) · Zbl 1143.47310 |

[26] | Ceng, LC; Sahu, DR; Yao, JC, Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz continuous monotone mappings, Journal of Computational and Applied Mathematics, 233, 2902-2915, (2010) · Zbl 1188.65076 |

[27] | Liou, Y-C; Yao, Y; Kimura, K, Strong convergence to common fixed points of a finite family of nonexpansive mappings, Journal of Inequalities and Applications, 2007, 10, (2007) · Zbl 1129.47053 |

[28] | Ceng LC, Ansari QH, Yao JC: Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense. to appear in Journal of Nonlinear and Convex Analysis · Zbl 1153.54024 |

[29] | Takahashi W, Yao JC: Weak and strong convergence theorems for positively homogenuous nonexpansive mappings in Banach spaces. to appear in Taiwanese Journal of Mathematics · Zbl 1099.47518 |

[30] | Zhou, H; Cho, YJ; Grabiec, M, Iterative processes for generalized asymptotically nonexpansive mappings in Banach spaces, PanAmerican Mathematical Journal, 13, 99-107, (2003) · Zbl 1067.47085 |

[31] | Lin, Y-C; Wong, N-C; Yao, J-C, Strong convergence theorems of Ishikawa iteration process with errors for fixed points of Lipschitz continuous mappings in Banach spaces, Taiwanese Journal of Mathematics, 10, 543-552, (2006) · Zbl 1118.47053 |

[32] | Zeng, L-C; Wong, N-C; Yao, J-C, Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type, Taiwanese Journal of Mathematics, 10, 837-849, (2006) · Zbl 1159.47054 |

[33] | Lin, Y-C, Three-step iterative convergence theorems with errors in Banach spaces, Taiwanese Journal of Mathematics, 10, 75-86, (2006) · Zbl 1106.47058 |

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