| Summary: | 碩士 === 國立中央大學 === 數學研究所 === 89 === 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 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.
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