陳　明

(遵義師范學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院，貴州 遵義 563002)

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一個(gè)有限漸近擬非擴(kuò)張映射族的收斂定理*

陳明

(遵義師范學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院，貴州 遵義 563002)

摘要:在一致凸的Banach空間中, 研究有限漸近擬非擴(kuò)張映射族的Mann迭代和多步Ishikawa型迭代序列的收斂性, 并對(duì)一些已有的Mann迭代和多步Ishikawa型迭代序列進(jìn)行進(jìn)一步地推廣和統(tǒng)一. 在實(shí)數(shù)空間中, 構(gòu)造一個(gè)非負(fù)實(shí)序列,使得這個(gè)非負(fù)實(shí)序列是收斂的, 從而利用這個(gè)非負(fù)實(shí)序列的收斂性證明該迭代序列在一定條件下強(qiáng)收斂到有限漸近擬非擴(kuò)張映射族的公共不動(dòng)點(diǎn).

關(guān)鍵詞:Banach空間； 漸近非擴(kuò)張映射； 漸近擬非擴(kuò)張映射； Ishikawa型迭代序列

1引言

設(shè)K為Banach空間E的非空閉凸子集, T是 K上的一個(gè)自映射, F(T)為T的不動(dòng)點(diǎn)集, 且F(T)≠?.

定義 1[1]映射T被稱為

在文獻(xiàn)[1-8]中,都在研究眾所周知的Mann迭代[3]xn+1=(1-αn)xn+αnTxn和修改后的Mann迭代[3]xn+1=(1-αn)xn+αnTnxn, n≥1.

文獻(xiàn)[14]研究了多步Ishikawa型迭代序列的收斂性，如式(1)

(1)

(2)

顯然, 當(dāng)式(2)中 qi≡0和 Ti≡ T時(shí), 即可以得到(1). 因此, 由(2)定義的序列 {xn}是文獻(xiàn)[14]中由式(1)定義的序列 {xn}的推廣.

在適當(dāng)?shù)募僭O(shè)下, 本文證明了由(2)定義的序列{xn}強(qiáng)收斂到有限漸近擬非擴(kuò)張映射族的公共不動(dòng)點(diǎn), 其結(jié)果對(duì)一些學(xué)者所研究的Mann迭代和多步 Ishikawa型迭代序列進(jìn)行了統(tǒng)一和推廣[2,9-10,14].

2主要結(jié)果

引理 2[9]設(shè)p>1,r>0是兩個(gè)固定常數(shù),E是Banach空間，則E是一致凸的當(dāng)且僅當(dāng)存在一個(gè)連續(xù)的強(qiáng)增的凸函數(shù)g∶[0, ∞)→ [0, ∞),g(0)=0，使得對(duì)任意的x,y∈Br(0)={x∈E:‖x‖≤r}, 且λ∈[0,1], 有 ‖λx+(1-λ)y‖p≤λ‖x‖p+(1-λ)‖y‖p-ωp(λ)g(‖x-y‖) 成立, 其中ωp(λ)=λ(1-λ)p+(1-λ)λp.

引理 3設(shè) {an}, {tn}和{ln} 3個(gè)非負(fù)實(shí)序列滿足條件

(3)

由式(3)有an+1≤tnan+lnan-q≤tnbn+lnbn≤(tn+ln) bn≤(1+cn)bn, n=q+1, q+2, …, 因此

現(xiàn)設(shè) a>0, 如果序列{an}不收斂于a.由{bn} 的定義知, 對(duì)任意的n∈ N,有an0,對(duì)任意的j>0, 存在nj>j使得

(4)

(5)

由式(4)可知, 存在n0≥ Nε+2q+1(n0-q>Nε+q+1 ), 使得

(6)

由式(3), 式(5)和式(6)可知

(7)

進(jìn)一步, 由式(7)推出

因此，可以證明

(8)

對(duì)任意的i=1,2,…,m-2, 由式(2)知

(9)

對(duì) i=m-1, 有

(10)

由式(2)，式(9)和式(10)可得

(11)

由式(9)和式(10)知, 對(duì)任意的i=1,2,…,m-1, 有

另一方面, 由式(11)知

(12)

當(dāng)i= 1,2, …, m-2時(shí), 有

(13)

當(dāng)i= m-1時(shí), 有

(14)

由式(12)可得

(15)

(16)

當(dāng)i= 1, 2,…, m-2時(shí), 可推出

綜上所述, 當(dāng) i= 1,2,…, m-1, 推出

故

當(dāng) i=1, 2, …, m, 有

(17)

故當(dāng) i= 1,2,…,m時(shí), 有

這表明

因此, {xn}是Cauchy序列. 又因?yàn)镋是完備的, 從而{xn}在E中是收斂的.

若在定理1中取m=1, qi≡0和Tin≡ T, 將得到迭代序列(1)在漸近擬非擴(kuò)張映射下的收斂定理.

參考文獻(xiàn):

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Convergence Theorem for a Finite Family of Asymptotically Nonexpansive Mappings

CHEN Ming

(School of Mathematics and Computational Science, Zunyi Normal College, Zunyi 563002, China)

Abstract:Finite asymptotically nonexpansive Mann iterative and multi-step Ishikawa type iterative sequence of convergentare are investigated in the uniformly convex Banach space, and some Mann iterations and multi-step Ishikawa type iterative sequences are popularized. In the real space, a convergent non-negative real sequence is constructed, and it’s convergent. Under certain conditions, the iterative sequence is proved that it converges strongly to the common fixed point of a finite family of asymptotically nonexpansive mappings.

Key words:Banach space; asymptotically nonexpansive mapping; asymptotically quasi-nonexpansive mapping; Ishikawa type iterative

文章編號(hào):1673-3193(2016)02-0104-05

*收稿日期:2015-08-02

基金項(xiàng)目：貴州省科技廳自然科學(xué)基金(No.LKZS[2011]2117, No.LKZS[2012]11, No.LKZS[2012]12)

作者簡(jiǎn)介：陳明(1961-)，男，副教授，主要從事函數(shù)論及概率論方面的研究.

中圖分類號(hào):O173

文獻(xiàn)標(biāo)識(shí)碼:A

doi:10.3969/j.issn.1673-3193.2016.02.002