A new variation on statistically quasi Cauchy sequences

Abstract

A sequence (α k ) of points in R, the set of real numbers, is called ρ-statistically p quasi Cauchy iffor each ε > 0, where ρ = (ρn) is a non-decreasing sequence of positive real numbers tending to ∞ such that lim sup n ρn n < ∞, ∆ρn = O(1), and ∆pα k+p = α k+p − α k for each positive integer k. A real-valued function defined on a subset of R is called ρ-statistically p-ward continuous if it preserves ρ-statistical p-quasi Cauchy sequences. ρ-statistical p-ward compactness is also introduced and investigated. We obtain results related to ρ-statistical p-ward continuity, ρ-statistical p-ward compactness, p-ward continuity, continuity, and uniform continuity.

Variations on the strongly lacunary quasi Cauchy sequences

“…A sequence (α n ) is called quasi Cauchy if lim n→∞ ∆α n = 0, where ∆α n = α n+1 − α n for each n ∈ N ( [1,5,12,20,23]). The set of all bounded quasi-Cauchy sequences is a closed subspace of the space of all bounded sequences with respect to the norm defined for bounded sequences.…”

On λ statistical upward compactness and continuity

“…We acknowledge that the results in this paper were presented at the International Workshop, Mathematical Methods in Engineering, MME-2017, held in Cankaya University, Ankara, Turkey on April 27-29, 2017, and some results were presented in an extended abstract [18].…”

A variation on strongly ideal lacunary ward continuity

“…A function f : R −→ R is continuous if and only if it preserves Cauchy sequences. Using the idea of continuity of a real function in terms of sequences, many kinds of continuities were introduced and investigated, 100 not all but some of them we recall in the following: slowly oscillating continuity ( [12]), quasi-slowly oscillating continuity ( [23]), ∆-quasi-slowly oscillating continuity ( [13]), ward continuity ( [14]), δ-ward continuity ( [15]), δ 2 -ward continuity ( [2]), contra δ − β−continuity ( [1]), statistical ward continuity, ( [8], [9], [7]), lacunary statistical ward continuity, ( [43], [42], [39]), λ-statistically ward continuity ( [16]), ideal ward continuity ( [10]) and Abel continuity ( [17]) which enabled some authors to obtain some characterizations of uniform continuity in terms of sequences in the sense that a function, on a special subset of R, preserves certain types of sequences (see [3], [41], [18], [23]). The concept of lacunary I-convergence of sequences was introduced and investigated in [40].…”