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**New iterative manner involving sunny nonexpansive retractions for pseudocontractive mappings.**
*(English)*
Zbl 1253.65092

Summary: Iterative methods for pseudocontractions have been studied by many authors in the literature. In the present paper, we propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

49J40 | Variational inequalities |

47H10 | Fixed-point theorems |

### Keywords:

sunny nonexpansive retractions; pseudocontractions in Banach spaces; variational inequality
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\textit{Y. Zheng} and \textit{J. Shi}, Abstr. Appl. Anal. 2012, Article ID 207896, 9 p. (2012; Zbl 1253.65092)

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### References:

[1] | F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197-228, 1967. · Zbl 0153.45701 |

[2] | B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957-961, 1967. · Zbl 0177.19101 |

[3] | Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591-597, 1967. · Zbl 0179.19902 |

[4] | P.-L. Lions, “Approximation de points fixes de contractions,” Comptes Rendus de l’Académie des Sciences, vol. 284, no. 21, pp. A1357-A1359, 1977. · Zbl 0349.47046 |

[5] | R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486-491, 1992. · Zbl 0797.47036 |

[6] | S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274-276, 1979. · Zbl 0423.47026 |

[7] | B. E. Rhoades, “Fixed point iterations using infinite matrices,” Transactions of the American Mathematical Society, vol. 196, pp. 161-176, 1974. · Zbl 0303.47036 |

[8] | B. E. Rhoades, “Comments on two fixed point iteration methods,” Journal of Mathematical Analysis and Applications, vol. 56, no. 3, pp. 741-750, 1976. · Zbl 0353.47029 |

[9] | L. C. Ceng, P. Cubiotti, and J. C. Yao, “Strong convergence theorems for finitely many nonexpansive mappings and applications,” Nonlinear Analysis A, vol. 67, no. 5, pp. 1464-1473, 2007. · Zbl 1123.47044 |

[10] | L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces,” Computers & Mathematics with Applications, vol. 58, no. 3, pp. 605-617, 2009. · Zbl 1192.47054 |

[11] | J.-P. Chancelier, “Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 141-153, 2009. · Zbl 1166.65026 |

[12] | S.-S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1402-1416, 2006. · Zbl 1111.47057 |

[13] | C. E. Chidume and C. O. Chidume, “Iterative approximation of fixed points of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 288-295, 2006. · Zbl 1095.47034 |

[14] | K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0708.47031 |

[15] | K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387-404, 2001. · Zbl 0993.47037 |

[16] | T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 103-123, 2005. · Zbl 1123.47308 |

[17] | W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417-428, 2003. · Zbl 1055.47052 |

[18] | 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 |

[19] | 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 |

[20] | P.-E. Maingé, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469-479, 2007. · Zbl 1111.47058 |

[21] | G. Lopez, V. Martin, and H.-K. Xu, “Perturbation techniques for nonexpansive mappings with applications,” Nonlinear Analysis, vol. 10, no. 4, pp. 2369-2383, 2009. · Zbl 1163.47306 |

[22] | Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis A, vol. 68, no. 6, pp. 1687-1693, 2008. · Zbl 1189.47071 |

[23] | Y. Yao, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm for approximating convex minimization problem,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 648-656, 2007. · Zbl 1121.65071 |

[24] | Y. Yao, Y.-C. Liou, and J.-C. Yao, “An extragradient method for fixed point problems and variational inequality problems,” Journal of Inequalities and Applications, vol. 2007, Article ID 38752, 12 pages, 2007. · Zbl 1137.47057 |

[25] | Y. Yao, M. Aslam Noor, K. Inayat Noor, Y.-C. Liou, and H. Yaqoob, “Modified extragradient methods for a system of variational inequalities in Banach spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1211-1224, 2010. · Zbl 1192.47065 |

[26] | Y. Yao, M. A. Noor, and Y.-C. Liou, “Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. · Zbl 1232.49011 |

[27] | G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis A, vol. 67, no. 7, pp. 2258-2271, 2007. · Zbl 1133.47050 |

[28] | G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336-346, 2007. · Zbl 1116.47053 |

[29] | M. O. Osilike, “Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 73-81, 2004. · Zbl 1045.47056 |

[30] | L.-C. Zeng, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 837-849, 2006. · Zbl 1159.47054 |

[31] | H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis A, vol. 16, no. 12, pp. 1127-1138, 1991. · Zbl 0757.46033 |

[32] | 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 |

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