φ-statistically quasi Cauchy sequences | ||
| Journal of the Egyptian Mathematical Society | ||
| Volume 24, Issue 2, 2016, Pages 239-243 PDF (342.05 K) | ||
| DOI: 10.1016/j.joems.2015.07.003 | ||
| Author | ||
| Bipan Hazarika | ||
| Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, India | ||
| Abstract | ||
| Let P denote the space whose elements are finite sets of distinct positive integers. Given any element σ of P , we denote by p ( σ) the sequence { p n ( σ)} such that p n (σ ) = 1 for n ∈ σ and p n (σ ) = 0 otherwise. Further P s = { σ ∈ P : ∞ n =1 p n (σ ) ≤ s } , i.e. P s is the set of those σ whose support has cardinality at most s . Let ( φn ) be a non-decreasing sequence of positive integers such that nφn +1 ≤ (n + 1) φn for all n ∈ N and the class of all sequences ( φn ) is denoted by . Let E ⊆ N . The number δφ (E) = lim s →∞ 1 φs |{ k ∈ σ, σ ∈ P s : k ∈ E}| is said to be the φ-density of E . A sequence ( x n ) of points in R is φ-statistically convergent (or S φ-convergent) to a real number for every ε > 0 if the set { n ∈ N : | x n − | ≥ ε} has φ-density zero. We introduce φ-statistically ward continuity of a real function. A real function is φ-statistically ward continuous if it preserves φ-statistically quasi Cauchy sequences where a sequence ( x n ) is called to be φ-statistically quasi Cauchy (or S φ-quasi Cauchy) when (x n ) = (x n +1 − x n ) is φ-statistically convergent to 0. i.e. a sequence ( x n ) of points in R is called φ-statistically quasi Cauchy (or S φ-quasi Cauchy) for every ε > 0 if { n ∈ N : | x n +1 − x n | ≥ ε} has φ-density zero. Also we introduce the concept of φ-statistically ward compactness and obtain results related to φ-statistically ward continuity, φ-statistically ward compactness, statistically ward continuity, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, δ-ward continuity, and slowly oscillating continuity. | ||
| Keywords | ||
| Statistical convergence; Statistical continuity; Quasi-Cauchy sequence; Summability | ||
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