### Abstract

Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E^{*}. Let A : E^{*} → E be a Lipschitz continuous monotone mapping with A^{-1} (0) ≠ ∅. For given u, x_{1} ∈ E, let {x_{n}} be generated by the algorithm x_{n + 1} : = β_{n} u + (1 - β_{n}) (x_{n} - α_{n} A J x_{n}), n ≥ 1, where J is the normalized duality mapping from E into E^{*} and {λ_{n}} and {θ_{n}} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {x_{n}} converges strongly to x^{*} ∈ E where J x^{*} ∈ A^{-1} (0). Finally, we apply our convergence theorems to the convex minimization problems.

Original language | English |
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Pages (from-to) | 663-671 |

Number of pages | 9 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 343 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 15 2008 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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**Strong convergence theorems for maximal monotone mappings in Banach spaces.** / Zegeye, Habtu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong convergence theorems for maximal monotone mappings in Banach spaces

AU - Zegeye, Habtu

PY - 2008/7/15

Y1 - 2008/7/15

N2 - Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E*. Let A : E* → E be a Lipschitz continuous monotone mapping with A-1 (0) ≠ ∅. For given u, x1 ∈ E, let {xn} be generated by the algorithm xn + 1 : = βn u + (1 - βn) (xn - αn A J xn), n ≥ 1, where J is the normalized duality mapping from E into E* and {λn} and {θn} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {xn} converges strongly to x* ∈ E where J x* ∈ A-1 (0). Finally, we apply our convergence theorems to the convex minimization problems.

AB - Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E*. Let A : E* → E be a Lipschitz continuous monotone mapping with A-1 (0) ≠ ∅. For given u, x1 ∈ E, let {xn} be generated by the algorithm xn + 1 : = βn u + (1 - βn) (xn - αn A J xn), n ≥ 1, where J is the normalized duality mapping from E into E* and {λn} and {θn} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {xn} converges strongly to x* ∈ E where J x* ∈ A-1 (0). Finally, we apply our convergence theorems to the convex minimization problems.

UR - http://www.scopus.com/inward/record.url?scp=41949096570&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41949096570&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2008.01.076

DO - 10.1016/j.jmaa.2008.01.076

M3 - Article

VL - 343

SP - 663

EP - 671

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -