Variations on lacunary statistical quasi Cauchy sequences

In this paper, we introduce a concept of lacunary statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (αk) is lacunary statistically p-quasi-Cauchy if limr→∞1hr|{k ∈ Ir:|αk+p−αk|≥ε}| = 0 for each ε > 0. A function f is called lacunary statistically p-ward continuous o...

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Bibliographic Details
Main Author: Yildiz, Sebnem
Format: Conference Proceeding
Language:English
Subjects:
Continuity (mathematics)
Numbers
Real numbers
Online Access:Get full text
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Summary:In this paper, we introduce a concept of lacunary statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (αk) is lacunary statistically p-quasi-Cauchy if limr→∞1hr|{k ∈ Ir:|αk+p−αk|≥ε}| = 0 for each ε > 0. A function f is called lacunary statistically p-ward continuous on a subset A of the set of real numbers ℝ if it preserves lacunary statistically p-quasi-Cauchy sequences, i.e. the sequence f (x) = (f (αn)) is lacunary statistically p-quasi-Cauchy whenever α = (αn) is a lacunary statistically p-quasi-Cauchy sequence of points in A. It turns out that a real valued function f is uniformly continuous on a bounded subset A of ℝ if there exists a positive integer p such that f preserves lacunary statistically p-quasi-Cauchy sequences of points in A.
ISSN:0094-243X
1551-7616
DOI:10.1063/1.5095130