Statistical quasi-Cauchy sequences
Mathematical and Computer Modelling, cilt.54, sa.5-6, ss.1620-1624, 2011 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 54 Sayı: 5-6
- Basım Tarihi: 2011
- Doi Numarası: 10.1016/j.mcm.2011.04.037
- Dergi Adı: Mathematical and Computer Modelling
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Sayfa Sayıları: ss.1620-1624
- Anahtar Kelimeler: Quasi-Cauchy sequences, Slowly oscillating sequences, Summability, Total boundedness, Uniform continuity
- Maltepe Üniversitesi Adresli: Evet
Özet
A subset E of a metric space (X,d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (xn) statistically quasi-Cauchy if st-limn→∞d(xn+1,xn)=0, and lacunary statistically quasi-Cauchy if Sθ-limn→∞d(xn+1,xn)=0. We prove that a subset E of a metric space is totally bounded if and only if any sequence of points in E has a subsequence which is any type of the following: statistically quasi-Cauchy, lacunary statistically quasi-Cauchy, quasi-Cauchy, and slowly oscillating. It turns out that a function defined on a connected subset E of a metric space is uniformly continuous if and only if it preserves either quasi-Cauchy sequences or slowly oscillating sequences of points in E. © 2011 Elsevier Ltd.