### Abstract

Let K be a nonempty closed convex subset of a real Banach space E. Let T {colon equals} {T (t) : t ∈ R^{+}} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {L_{t}} ⊂ [1, ∞). Suppose F (T) ≠ 0{combining long solidus overlay}. Then, for a given u_{0} ∈ K and t_{n} > 0 there exists a sequence {u_{n}} ⊂ K such that u_{n} = (1 - α_{n}) T (t_{n}) u_{n} + α_{n} u_{0}, for n ∈ N such that {α_{n}} ⊂ (0, 1) and L_{tn} - 1 < α_{n}, where t_{n} ∈ R^{+}. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that lim_{n → ∞} t_{n} = ∞, lim_{n → ∞} α_{n} = lim_{n → ∞} frac(L_{tn} - 1, α_{n}) = 0. Then the sequence {u_{n}} converges strongly to a point of F (T). Moreover, it is proved that an explicit sequence {x_{n}} generated from x_{1} ∈ K by x_{n + 1} {colon equals} α_{n} u + (1 - α_{n}) T (t_{n}) x_{n}, n ≥ 1, converges to a fixed point of T.

Original language | English |
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Pages (from-to) | 2308-2315 |

Number of pages | 8 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 71 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - Sep 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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*Nonlinear Analysis, Theory, Methods and Applications*, vol. 71, no. 5-6, pp. 2308-2315. https://doi.org/10.1016/j.na.2009.01.065

**Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings.** / Zegeye, Habtu; Shahzad, Naseer.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings

AU - Zegeye, Habtu

AU - Shahzad, Naseer

PY - 2009/9/1

Y1 - 2009/9/1

N2 - Let K be a nonempty closed convex subset of a real Banach space E. Let T {colon equals} {T (t) : t ∈ R+} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {Lt} ⊂ [1, ∞). Suppose F (T) ≠ 0{combining long solidus overlay}. Then, for a given u0 ∈ K and tn > 0 there exists a sequence {un} ⊂ K such that un = (1 - αn) T (tn) un + αn u0, for n ∈ N such that {αn} ⊂ (0, 1) and Ltn - 1 < αn, where tn ∈ R+. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that limn → ∞ tn = ∞, limn → ∞ αn = limn → ∞ frac(Ltn - 1, αn) = 0. Then the sequence {un} converges strongly to a point of F (T). Moreover, it is proved that an explicit sequence {xn} generated from x1 ∈ K by xn + 1 {colon equals} αn u + (1 - αn) T (tn) xn, n ≥ 1, converges to a fixed point of T.

AB - Let K be a nonempty closed convex subset of a real Banach space E. Let T {colon equals} {T (t) : t ∈ R+} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {Lt} ⊂ [1, ∞). Suppose F (T) ≠ 0{combining long solidus overlay}. Then, for a given u0 ∈ K and tn > 0 there exists a sequence {un} ⊂ K such that un = (1 - αn) T (tn) un + αn u0, for n ∈ N such that {αn} ⊂ (0, 1) and Ltn - 1 < αn, where tn ∈ R+. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that limn → ∞ tn = ∞, limn → ∞ αn = limn → ∞ frac(Ltn - 1, αn) = 0. Then the sequence {un} converges strongly to a point of F (T). Moreover, it is proved that an explicit sequence {xn} generated from x1 ∈ K by xn + 1 {colon equals} αn u + (1 - αn) T (tn) xn, n ≥ 1, converges to a fixed point of T.

UR - http://www.scopus.com/inward/record.url?scp=67349235538&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349235538&partnerID=8YFLogxK

U2 - 10.1016/j.na.2009.01.065

DO - 10.1016/j.na.2009.01.065

M3 - Article

AN - SCOPUS:67349235538

VL - 71

SP - 2308

EP - 2315

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 5-6

ER -