##
**Computing the fixed points of strictly pseudocontractive mappings by the implicit and explicit iterations.**
*(English)*
Zbl 1252.47076

Summary: It is known that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we study computing the fixed points of strictly pseudocontractive mappings by iterations. Two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings are constructed in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

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

PDF
BibTeX
XML
Cite

\textit{Y.-C. Liou}, Abstr. Appl. Anal. 2012, Article ID 315835, 13 p. (2012; Zbl 1252.47076)

Full Text:
DOI

### References:

[1] | F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82-90, 1967. · Zbl 0148.13601 |

[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,” vol. 284, no. 21, pp. A1357-A1359, 1977. · Zbl 0349.47046 |

[5] | 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 |

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

[7] | S. Reich and A. J. Zaslavski, “Convergence of Krasnoselskii-Mann iterations of nonexpansive operators,” Mathematical and Computer Modelling, vol. 32, no. 11-13, pp. 1423-1431, 2000. · Zbl 0977.47046 |

[8] | A. T. M. Lau and W. Takahashi, “Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces,” Nonlinear Analysis, vol. 70, no. 11, pp. 3837-3841, 2009. · Zbl 1219.47082 |

[9] | P. L. Combettes and T. Pennanen, “Generalized Mann iterates for constructing fixed points in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 275, no. 2, pp. 521-536, 2002. · Zbl 1032.47034 |

[10] | H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150-159, 1996. · Zbl 0956.47024 |

[11] | 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 |

[12] | 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 |

[13] | S. A. Hirstoaga, “Iterative selection methods for common fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1020-1035, 2006. · Zbl 1106.47057 |

[14] | A. Petru\csel and J.-C. Yao, “Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings,” Nonlinear Analysis, vol. 69, no. 4, pp. 1100-1111, 2008. · Zbl 1142.47329 |

[15] | Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triplehierarchical constrained optimization problem,” Mathematical & Computer Modelling, vol. 55, no. 3-4, pp. 1506-1515, 2012. · Zbl 1275.47130 |

[16] | Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 6, no. 4, pp. 621-628, 2012. · Zbl 1280.90097 |

[17] | Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, article 79, 2011. · Zbl 1270.47066 |

[18] | S. Reich and H. K. Xu, “An iterative approach to a constrained least squares problem,” Abstract and Applied Analysis, no. 8, pp. 503-512, 2003. · Zbl 1053.65041 |

[19] | Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 279058, 7 pages, 2009. · Zbl 1186.47080 |

[20] | N. Shahzad, “Approximating fixed points of non-self nonexpansive mappings in Banach spaces,” Nonlinear Analysis, vol. 61, no. 6, pp. 1031-1039, 2005. · Zbl 1089.47058 |

[21] | H. Zegeye and N. Shahzad, “Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 155-163, 2007. · Zbl 1194.47089 |

[22] | 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 |

[23] | N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641-3645, 1997. · Zbl 0888.47034 |

[24] | W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibility problems,” Mathematical and Computer Modelling, vol. 32, no. 11-13, pp. 1463-1471, 2000. · Zbl 0971.47040 |

[25] | 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 |

[26] | G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43-52, 2006. · Zbl 1095.47038 |

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

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

[29] | 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 |

[30] | Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of some algorithms for \lambda -strict pseudo-contractions in Hilbert spaces,” Bulletin of Australian Mathematical Society, vol. 85, no. 2, pp. 232-240, 2012. · Zbl 1236.47080 |

[31] | Y. Yao and J.-C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551-1558, 2007. · Zbl 1121.65064 |

[32] | F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, vol. 2009, Article ID 208692, 13 pages, 2009. · Zbl 1180.47040 |

[33] | M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199-277, 2004. · Zbl 1134.49304 |

[34] | M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182-185, 2009. · Zbl 1163.49303 |

[35] | M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 303569, 16 pages, 2012. · Zbl 1242.49017 |

[36] | O. Scherzer, “Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems,” Journal of Mathematical Analysis and Applications, vol. 194, no. 3, pp. 911-933, 1995. · Zbl 0842.65036 |

[37] | A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative least-squares and regularization,” IEEE Transactions on Signal Processing, vol. 46, no. 9, pp. 2345-2352, 1998. · Zbl 0978.93085 |

[38] | Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis, vol. 72, no. 7-8, pp. 3447-3456, 2010. · Zbl 1183.49012 |

[39] | Y. L. Cui and X. Liu, “Notes on Browder’s and Halpern’s methods for nonexpansive mappings,” Fixed Point Theory, vol. 10, no. 1, pp. 89-98, 2009. · Zbl 1190.47068 |

[40] | X. Liu and Y. Cui, “The common minimal-norm fixed point of a finite family of nonexpansive mappings,” Nonlinear Analysis, vol. 73, no. 1, pp. 76-83, 2010. · Zbl 1214.47050 |

[41] | Y. Yao and H. K. Xu, “Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications,” Optimization, vol. 60, no. 6, pp. 645-658, 2011. · Zbl 1368.47093 |

[42] | 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 |

[43] | H. Zhou, “Convergence theorems of fixed points for \lambda -strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 69, no. 2, pp. 456-462, 2008. · Zbl 1220.47139 |

[44] | 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 |

[45] | 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 |

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.