N θ-Ward continuity
Abstract and Applied Analysis, cilt.2012, 2012 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 2012
- Basım Tarihi: 2012
- Doi Numarası: 10.1155/2012/680456
- Dergi Adı: Abstract and Applied Analysis
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Maltepe Üniversitesi Adresli: Evet
Özet
A function f is continuous if and only if f preserves convergent sequences; that is, (f (α n)) is a convergent sequence whenever (α n) is convergent. The concept of N θ-ward continuity is defined in the sense that a function f is N theta;-ward continuous if it preserves N θ-quasi-Cauchy sequences; that is, (f (α n)) is an N θ-quasi-Cauchy sequence whenever (α n) is N θ-quasi-Cauchy. A sequence (α k) of points in R, the set of real numbers, is N θ-quasi-Cauchy if lim r → ∞ (1 / h r) ∑k∈I r |Δα k | = 0, where Δα k = α k + 1 - α k, I r = (k r - 1, k r ], and θ = (k r) is a lacunary sequence, that is, an increasing sequence of positive integers such that k 0 = 0 and h r: k r - k r-1 → ∞. A new type compactness, namely, N θ -ward compactness, is also, defined and some new results related to this kind of compactness are obtained. Copyright © 2012 Huseyin Cakalli.