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(Some Results about Weakly Almost-Convergence on Banach Spaces)

 ★ 關於某些正線性近似算子的收斂速度 ★ 一些關於sσ-limit的結果與一個數列在reflexive Banach space 中的表現定理 ★ 格瑪算子函數的應用

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∥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)

∥Tn+iu- Tnv∥存在，均勻在i 0上。 (1.2)
∥Tku- Tkv∥ ak∥u-v∥ 對所有C內的元素u,v及k 0 (1.3)

∥Tku+ Tkv∥ ak∥u+v∥ 對所有C內的元素u,v及k 0 (1.4)

｜fn｜≦g a.e. IL1(μ) 而且σ-lim fn (ω)=f (ω) a.e.[μ]，則f是μ-可積且 = = 。

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 of nonexpansive 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.

SECTION 1 INTRODUCTION………………………………………………? 1
SECTION 2 BASIC PROPERTIES OF GENERALIZED σ- LIMITS ………? 2
SECTION 3 SCALAR-VALUED FUNCTIONS………………………………… 6
SECTION 4 EXAMPLES…………………………………………………… 13
REFERENCE………………………………………………………………… 14

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