##
**Hybrid method with perturbation for Lipschitzian pseudocontractions.**
*(English)*
Zbl 1251.65082

Summary: Assume that \(F\) is a nonlinear operator which is Lipschitzian and strongly monotone on a nonempty closed convex subset \(C\) of a real Hilbert space \(H\). Assume also that \(\Omega\) is the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings on \(C\). By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterative algorithm with perturbation \(F\), which generates two sequences from an arbitrary initial point \(x_0 \in H\). These two sequences are shown to converge in norm to the same point \(P_\Omega x_0\) under very mild assumptions.

### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

47H10 | Fixed-point theorems |

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

PDF
BibTeX
XML
Cite

\textit{L.-C. Ceng} and \textit{C.-F. Wen}, J. Appl. Math. 2012, Article ID 250538, 20 p. (2012; Zbl 1251.65082)

Full Text:
DOI

### References:

[1] | 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 |

[2] | D. Wu, S.-s. Chang, and G. X. Yuan, “Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. 987-999, 2005. · Zbl 1153.65340 |

[3] | T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51-60, 2005. · Zbl 1091.47055 |

[4] | 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 |

[5] | H.-K. Xu, “Strong convergence of an iterative method for nonexpansive and accretive operators,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 631-643, 2006. · Zbl 1086.47060 |

[6] | K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372-379, 2003. · Zbl 1035.47048 |

[7] | J. G. O’Hara, P. Pillay, and H.-K. Xu, “Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 8, pp. 1417-1426, 2003. · Zbl 1052.47049 |

[8] | 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 |

[9] | 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 |

[10] | S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257-266, 2005. · Zbl 1071.47063 |

[11] | L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2507-2515, 2006. · Zbl 1105.47061 |

[12] | L.-C. Zeng, “A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 245-250, 1998. · Zbl 0916.47047 |

[13] | 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 |

[14] | H. Zegeye and N. Shahzad, “Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 7, pp. 2005-2012, 2008. · Zbl 1153.47059 |

[15] | 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 |

[16] | J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509-520, 2005. · Zbl 1062.47069 |

[17] | 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 |

[18] | 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 |

[19] | 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 |

[20] | K. Q. Lan and J. H. Wu, “Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 49, no. 6, pp. 737-746, 2002. · Zbl 1019.47040 |

[21] | S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147-150, 1974. · Zbl 0286.47036 |

[22] | C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitz pseudocontractions,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2359-2363, 2001. · Zbl 0972.47062 |

[23] | C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400-2411, 2006. · Zbl 1105.47060 |

[24] | T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 1140-1152, 2006. · Zbl 1090.47059 |

[25] | H. Zhou, “Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 546-556, 2008. · Zbl 1140.47058 |

[26] | K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365-374, 1974. · Zbl 0288.47047 |

[27] | H. Zhou, “Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 10, pp. 2977-2983, 2008. · Zbl 1145.47055 |

[28] | M. O. Osilike and A. Udomene, “Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 431-445, 2001. · Zbl 1009.47067 |

[29] | C. E. Chidume and H. Zegeye, “Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831-840, 2004. · Zbl 1051.47041 |

[30] | Y. Yao, Y.-C. Liou, and R. Chen, “Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3311-3317, 2007. · Zbl 1129.47059 |

[31] | 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 |

[32] | A. Udomene, “Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2403-2414, 2007. · Zbl 1132.47056 |

[33] | W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276-286, 2008. · Zbl 1134.47052 |

[34] | Y. Yao, Y.-C. Liou, and G. Marino, “A hybrid algorithm for pseudo-contractive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4997-5002, 2009. · Zbl 1222.47128 |

[35] | K. Geobel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University, Cambridge, UK, 1990. |

[36] | H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185-201, 2003. · Zbl 1045.49018 |

[37] | L. C. Ceng, D. S. Shyu, and J. C. Yao, “Relaxed composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive mappings,” Fixed Point Theory and Applications, vol. 2009, Article ID 402602, 16 pages, 2009. · Zbl 1186.47057 |

[38] | L. C. Ceng, A. Petru\csel, and J. C. Yao, “A hybrid method for lipschitz continuous monotone mappings and asymptotically strict pseudocontractive mappings in the intermediate sense,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 1, 2010. |

[39] | L.-C. Ceng, A. Petru\csel, and J.-C. Yao, “Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense,” Taiwanese Journal of Mathematics, vol. 15, no. 2, pp. 587-606, 2011. · Zbl 1437.47046 |

[40] | L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 2, pp. 283-308, 2010. · Zbl 1195.49011 |

[41] | L.-C. Ceng, A. Petru\csel, S. Szentesi, and J.-C. Yao, “Approximation of common fixed points and variational solutions for one-parameter family of Lipschitz pseudocontractions,” Fixed Point Theory, vol. 11, no. 2, pp. 203-224, 2010. · Zbl 1203.49013 |

[42] | D. R. Sahu, N. C. Wong, and J. C. Yao, “A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings,” SIAM Journal on Control and Optimization. In press. · Zbl 1262.47091 |

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.