### Abstract

Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let T ℑ: ={T(t): t ∈ ℝ^{+}} be a strongly continuous semigroup of asymptotically nonexpansive self-mappings on K with a sequence {L_{t}} ⊂ [1,∞ ). Then, for a given μ_{0} ∈ K and s_{n} ∈(0, 1), 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 each n,∈ Ndbl; atisfying||u_{n}-T (t)u_{n}||→0 as→n ∞, for any t+, where [image omitted]. If, in addition, E is uniformly convex with uniformly G[image omitted]teaux differentiable norm, then it is proved that F(ℑ) ≠ ∅ and the sequence {u_{n}} converges strongly to a point of F(ℑ) under certain mild conditions on {L_{t}}, {t _{n}} and {s_{n}}. Moreover, it is proved that an explicit sequence {x_{n}} generated from x1 ∈ K by x_{n}+1: α _{n}u_{0}+(1-α_{n}n)T(t_{n})x _{n}, n ≥1,converges to a fixed point of T under appropriate assumption imposed upon the sequence {x_{n}}.

Original language | English |
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Pages (from-to) | 833-848 |

Number of pages | 16 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 30 |

Issue number | 7-8 |

DOIs | |

Publication status | Published - Jul 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Signal Processing
- Computer Science Applications
- Control and Optimization

### Cite this

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*Numerical Functional Analysis and Optimization*, vol. 30, no. 7-8, pp. 833-848. https://doi.org/10.1080/01630560903123197

**Strong convergence theorems for continuous semigroups of asymptotically nonexpansive mappings.** / Zegeye, Habtu; Shahzad, Naseer.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong convergence theorems for continuous semigroups of asymptotically nonexpansive mappings

AU - Zegeye, Habtu

AU - Shahzad, Naseer

PY - 2009/7/1

Y1 - 2009/7/1

N2 - Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let T ℑ: ={T(t): t ∈ ℝ+} be a strongly continuous semigroup of asymptotically nonexpansive self-mappings on K with a sequence {Lt} ⊂ [1,∞ ). Then, for a given μ0 ∈ K and sn ∈(0, 1), tn > 0 there exists a sequence {un} ⊂ K such that un=(1-αn)T(t n)un+αnu0, for each n,∈ Ndbl; atisfying||un-T (t)un||→0 as→n ∞, for any t+, where [image omitted]. If, in addition, E is uniformly convex with uniformly G[image omitted]teaux differentiable norm, then it is proved that F(ℑ) ≠ ∅ and the sequence {un} converges strongly to a point of F(ℑ) under certain mild conditions on {Lt}, {t n} and {sn}. Moreover, it is proved that an explicit sequence {xn} generated from x1 ∈ K by xn+1: α nu0+(1-αnn)T(tn)x n, n ≥1,converges to a fixed point of T under appropriate assumption imposed upon the sequence {xn}.

AB - Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let T ℑ: ={T(t): t ∈ ℝ+} be a strongly continuous semigroup of asymptotically nonexpansive self-mappings on K with a sequence {Lt} ⊂ [1,∞ ). Then, for a given μ0 ∈ K and sn ∈(0, 1), tn > 0 there exists a sequence {un} ⊂ K such that un=(1-αn)T(t n)un+αnu0, for each n,∈ Ndbl; atisfying||un-T (t)un||→0 as→n ∞, for any t+, where [image omitted]. If, in addition, E is uniformly convex with uniformly G[image omitted]teaux differentiable norm, then it is proved that F(ℑ) ≠ ∅ and the sequence {un} converges strongly to a point of F(ℑ) under certain mild conditions on {Lt}, {t n} and {sn}. Moreover, it is proved that an explicit sequence {xn} generated from x1 ∈ K by xn+1: α nu0+(1-αnn)T(tn)x n, n ≥1,converges to a fixed point of T under appropriate assumption imposed upon the sequence {xn}.

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U2 - 10.1080/01630560903123197

DO - 10.1080/01630560903123197

M3 - Article

VL - 30

SP - 833

EP - 848

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 7-8

ER -