### Abstract

Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.

Original language | English |
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Pages (from-to) | 544-555 |

Number of pages | 12 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 74 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 15 2011 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Nonlinear Analysis, Theory, Methods and Applications*,

*74*(2), 544-555. https://doi.org/10.1016/j.na.2010.09.008

}

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 74, no. 2, pp. 544-555. https://doi.org/10.1016/j.na.2010.09.008

**Four parameter proximal point algorithms.** / Boikanyo, O. A.; Moroşanu, G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Four parameter proximal point algorithms

AU - Boikanyo, O. A.

AU - Moroşanu, G.

PY - 2011/1/15

Y1 - 2011/1/15

N2 - Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.

AB - Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.

UR - http://www.scopus.com/inward/record.url?scp=77958001016&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77958001016&partnerID=8YFLogxK

U2 - 10.1016/j.na.2010.09.008

DO - 10.1016/j.na.2010.09.008

M3 - Article

AN - SCOPUS:77958001016

VL - 74

SP - 544

EP - 555

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 2

ER -