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    Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/7786


    Title: 關於在Banach空間上的弱幾乎收斂的一些結果;Some Results about Weakly Almost-Convergence on Banach Spaces
    Authors: 李姍妮;Sanny Li
    Contributors: 數學研究所
    Keywords: 弱幾乎收斂
    Date: 2001-06-28
    Issue Date: 2009-09-22 11:05:22 (UTC+8)
    Publisher: 國立中央大學圖書館
    Abstract: 在這篇論文,我們主要的目的是要研究關於弱幾乎收斂序列以σ-極限的觀點來呈現的一些基本性質。在1996年,李源泉與蕭勝彥已證明σ-極限與弱幾乎收斂是等價的觀點。 弱幾乎收斂已經被很多數學家應用到nonexpansive函數的定點定理上,如Baillon,Bruck,Reich,Hirano,Brezis及Browder等等。 在1975年,Baillon證明了下述的非線性ergodic定理:令C是一個Hilbert空間X的封閉凸子集。若T是從C對應到C的nonexpansive函數,即對所有在C內的元素u,v,T滿足 ‖Tu-Tv‖ ‖u-v‖,有一個定點C內的元素y,則對所有C內的元素x,{ Tnx }會弱幾乎收斂到y。一般來說,即使X是一個Hilber空間,{ Tnx }也不會弱幾乎收斂到一個定點。Bruc和Reich把這個結果推展到X是均勻凸Banach空間有Frechet可微norm及不同的充分條件。另一方面,Baillon也證明了下列的強ergodic定理:若X是一個Hilbert空間,-C = C,及T是奇函數,則對每個C內的元素x,{ Tnx }會強幾乎收斂到T的定點y,即 ∥ Tk+mx-y∥= 0 均勻在m 0上。 Brezis及Browder證明即使T函數的條件消弱如下,則Baillon的結果仍然是對的: 0是C內的元素,對C內的元素u,v ‖Tu+Tv‖ ‖u+v‖2+c[‖u‖2-‖Tu‖2+‖u‖2-‖Tv‖2] (1.1) 其中c 是一個非負的常數。我們知道如果T是在Hilbert空間X的一個閉凸子集C上的一個nonexpansive函數,而且滿足(1.1)式,則T會滿足: 對所有C內元素u,v, ∥Tn+iu- Tnv∥存在,均勻在i 0上。 (1.2) Bruck證明如果C是一個Hilbert空間X上的一個閉凸子集,T是一個滿足(1.2)式的nonexpansive函數且有一個定點,則對每個C內的元素x,{ Tnx }會強幾乎收斂到T的定點。不久,Kobayasi及Miyadera證明即使X是一個均勻凸Banach空間,Bruck的結果仍然是對的。Hirano,Takahashi及Oka證明上述T的條件可以消弱如下: ∥Tku- Tkv∥ ak‖u-v‖ 對所有C內的元素u,v及k 0 (1.3) 其中ak 是一個滿足 ak =1的非負常數。在這種情形,T被稱為asymptoticallyxpansive。若T是從C對應到C的asymptoticallyxpansive奇函數,則它會滿足 ∥Tku+ Tkv∥ ak‖u+v‖ 對所有C內的元素u,v及k 0 (1.4) 其中ak 是一個滿足 ak =1的非負常數。 在這篇論文的第二節,我們證明了如果N是一個實Banach空間X 的proper closed cone,f是從N對應到N的函數,在0點弱連續且滿足f(0)=0,則對每個N內的序列{ x n},滿足{ x n}的σ-極限為0且{f ( x n)}是有界的,則{f ( x n)} 的σ-極限為0。 已知若(Ω,Σ,μ)是一個測度空間,且對所有的n=1,2,3…,fn是從Ω對應到複數值的Lebesgue可測函數,使得 fn =f a.e. 則f是可測函數。由弱幾乎收斂的定義可知,如果σ-lim fn=f a.e. 則f也是可測函數。在第三節,我們提供了一個和dominated收斂定理等價的另一個定理,敘述如下:假設(Ω,Σ,μ)是一個測度空間,且g,f,f1,f2,…都是從Ω對應到複數值的的可測函數。對所有的n=1,2,3…, |fn|≦g a.e. ÎL1(μ) 而且σ-lim fn (ω)=f (ω) a.e.[μ],則f是μ-可積且 = = 。 我們很容易看出弱幾乎收斂的條件比弱收斂還弱,所以若f在點x上弱連續,則{ x n}的σ-極限為x不一定會導致{ f (x n)}的σ-極限為f (x)。在定理3.5我們證明了若x是一個複數,f是複數值函數,則f在x連續若且惟若對所有複數序列{ x n},{|x n -x|}的σ-極限為0可推至{f ( x n)}的σ-極限為f (x)。從定理3.6到序理3.9,我們研究在怎樣的充分條件下,純數x及有界數列{ x n }滿足{ x n}的σ-極限為x可以推至{ f (x n)}的σ-極限為f (x)。例如,若a為實數且{ a n }是實數有界數列,滿足{ a n }的σ-極限為a,且對所有的n 1,a n a都成立,則對所有的p=1,2…, an p = a p都成立。最後,在第四節我們給了兩個例子。 In this paper, our primary objective is to study basic poroperties about weakly almost-convergent squence in terms of the conception of σ-limits. In 1996, Li and Shaw [11] showed that the conception ofσ-limit is equivalent to the weak almost-convergence (see Definition 2.3). The weak almost-convergence had been applied to the fixed point theory ofxpansive mappings by many mathematicians, for example, Baillon[1], Bruck[3,4], Reich, Hirano[7], Brezis and Browder[2], etc. In section 2, we show that if N is a proper closed cone of a real Banach space X and if f:N->N is weak-weak continuous at 0 with f(0)=0, then for every sequence {xn} in N such thatσ-lim xn=0 and {f(xn)} is bounded implyσ-lim f(xn)=0.(see Proposition 2.9) It is well known that if (Ω,Σ,μ) is a measure space and fn:Ω->C, for n=1,2,… , are Lebesgue measurable functions such that limn fn=f a.e. then f is measurable. By the definition of weakly almost-convergence, f is also measurable if σ-lim fn = f a.e. [μ]. In section 3, we give another version of the dominated convergence theorem stated as following: Suppose (Ω,Σ,μ) is a measure space and g,f,f1, f2,… : Ω->C are measurable. Suppose fn≦ g (a.e.) in L1 (μ) for all n=1,2,… and σ-lim fn (ω) = f(ω) a.e. [μ]. Then f is integrable . It is easy to see that the weakly almost convergence is weaker than the weak convergence . From Proposition 3.7 to Corollary 3.10, we study under which sufficient conditions at a scalar x and a bounded sequece { xn } with σ-lim xn = x we have σ-lim f(xn) = f(x). Finally, we give two examples in section 4.
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