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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2013
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| Online-Zugang: | https://arxiv.org/abs/1301.2469 |
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| _version_ | 1866910308192223232 |
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| author | Song, Yisheng Wang, Hongjun |
| author_facet | Song, Yisheng Wang, Hongjun |
| contents | In this paper, for an $λ$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}=β_{n}u+γ_nx_n+(1-β_{n}-γ_n)[α_{n}Tx_n+(1-α_{n})x_n],$$ where $\{α_{n}\}$, $ \{β_{n}\}$ and $\{γ_n\}$ in $(0,1)$ satisfy: (i) $0 \leq α_{n}\leq \fracλ{K^2}$ with $\liminf\limits_{n\to\infty}α_n(λ-K^2α_n)> 0$; (ii) $\lim\limits_{n\to\infty}β_n= 0$ and $\sum\limits_{n=1}^\inftyβ_n=\infty$; (iii) $\limsup\limits_{n\to\infty}γ_n<1$.Our results unify and improve some existing results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1301_2469 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Strong convergence for the modified Mann's iteration of $λ$-strict pseudocontraction Song, Yisheng Wang, Hongjun Functional Analysis 49J40, 47H05, 47H04, 65J15 In this paper, for an $λ$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}=β_{n}u+γ_nx_n+(1-β_{n}-γ_n)[α_{n}Tx_n+(1-α_{n})x_n],$$ where $\{α_{n}\}$, $ \{β_{n}\}$ and $\{γ_n\}$ in $(0,1)$ satisfy: (i) $0 \leq α_{n}\leq \fracλ{K^2}$ with $\liminf\limits_{n\to\infty}α_n(λ-K^2α_n)> 0$; (ii) $\lim\limits_{n\to\infty}β_n= 0$ and $\sum\limits_{n=1}^\inftyβ_n=\infty$; (iii) $\limsup\limits_{n\to\infty}γ_n<1$.Our results unify and improve some existing results. |
| title | Strong convergence for the modified Mann's iteration of $λ$-strict pseudocontraction |
| topic | Functional Analysis 49J40, 47H05, 47H04, 65J15 |
| url | https://arxiv.org/abs/1301.2469 |