İstatistiksel boşluklu delta 2 quasi Cauchy dizileri

Abstract

The notion of a lacunary statistical δ 2 -quasi-Cauchyness of sequence of real numbers is introduce and investigated. In this work, we present interesting theorems related to lacunary statistically δ 2 -ward continuity. A function f, whose domain is included in R, and whose range included in R is called lacunary statistical δ 2 ward continuous if it preserves lacunary statistical δ 2 quasi-Cauchy sequences, i.e. (f(xk)) is a lacunary statistically δ 2 quasi-Cauchy sequence whenever (xk) is a lacunary statistically δ 2 quasi-Cauchy sequence, where a sequence (xk) is called lacunary statistically δ 2 quasi-Cauchy if (∆ 2 xk) is a lacunary statistically quasi-Cauchy sequence. We find out that the set of lacunary statistical δ 2 ward continuous functions is closed as a subset of the set of continuous functions. İstatistiksel boşluklu delta 2 quasi Cauchy dizileri ÖZBu makalede istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi kavramı tanımlanmış ve araştırılmıştır. Bu araştırmada istatistiksel boşluklu δ 2 -süreklilik ile ilgili ilgi çekici teoremler ispatlanmıştır. (∆ 2 xk) istatistiksel boşluklu quasi Cauchy dizisi olduğunda (xk) dizisine istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi dendiğine göre, reel sayılar kümesinin bir alt kümesi üzerinde tanımlı reel değerli bir f fonksiyonuna eğer terimleri A da olan istsatistiksel boşluklu δ 2 -quasi-Cauchy dizilerini koruyor ise, yani (xk) dizisi terimleri A da olan istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi olduğunda (f(xk)) dizisi de istatistiksel boşluklu δ 2 -quasi-Cauchy dizisi oluyor ise istatistiksel boşluklu δ 2 -ward süreklidir denir. İstatistiksel boşluklu δ 2 -ward sürekli fonksiyonların kümesinin sürekli fonksiyonlar uzayının kapalı bir alt kümesi olduğu ortaya çıkarılmıştır.Anahtar Kelimeler: toplanabilme, quasi Cauchy dizisi, istatistiksel boşluklu yakınsaklık, süreklilik 1 Ahi Evran Üniversitesi Fen Edebiyat Fakültesi / Matematik sebnemyildiz@ahievran.edu.tr

Abel statistical quasi Cauchy sequences

“…As a counterexample consider the function defined by f (x) = x 2 and the Abel statistical quasi Cauchy sequence defined by ( √ n). We note that Abel statistical ward continuity implies not only ordinary continuity, but also statistical continuity, which follows from [4,Corollary 4], Lemma 1 and Theorem 8 in [6]; lacunary statistical sequential continuity, which was observed in [14] (see also [36]), λ-statistical continuity ( [18]), ρ-statistical continuity, and ( [11]); G-sequential continuity for any regular subsequential method G, which follows from Theorem 8 in [6](see also [31])…”

A Variation on Statistical Ward Continuity

“…Conversely, suppose that E is not bounded above. Pick an element It follows from the above theorem that if a closed subset E of R is ρ-statistically downward compact, and −A is ρ-statistically downward compact, then any sequence of points in E has a (P n , s)-absolutely almost convergent subsequence (see [36], [52], [62], [3], [64], [63], and [65]…”

“…A sequence (α k ) of points in R, the set of real numbers, is called statistically downward quasi-Cauchy if lim n→∞ 1 n |{k ≤ n : α k+1 − α k ≥ ε}| = 0 for every ε > 0 ( [19]). Recently, many kinds of continuities were introduced and investigated, not all but some of them we recall in the following: slowly oscillating continuity ( [13], [60]), quasi-slowly oscillating continuity ( [37]), ward continuity ( [23], [4]), δ-ward continuity ( [14]), p-ward continuity ( [20]), statistical ward continuity ( [16]), ρ-statistical ward continuity ( [6], lacunary statistical δ 2 -ward continuity ( [65]), arithmetic continuity ( [24], [25]) Abel continuity ( [28]), which enabled some authors to obtain conditions on the domain of a function for some characterizations of uniform continuity in terms of sequences in the sense that a function preserves a certain kind of sequences (see [60,Theorem 6], [4, Theorem 1 and Theorem 2], [37, Theorem 2.3]).…”

Variations on lacunary statistical quasi Cauchy sequences

“…p will always be a fixed element of N. The boldface letters such as α, β, ζ will be used for sequences α = (α n ), β = (β n ), ζ = (ζ n ), ... of points in R. A function f : R −→ R is continuous if and only if it preserves convergent sequences. Using the idea of continuity of a real function in this manner, many kinds of continuities were introduced and investigated, not all but some of them we recall in the following: ward continuity ( [15], [4]), p-ward continuity ( [23]), δ-ward continuity ( [18]), δ 2 -ward continuity ( [3]), statistical ward continuity, ( [19]), λ-statistical ward continuity ( [36]), ρ-statistical ward continuity ( [6], [25]), slowly oscillating continuity ( [12,59,35]), quasi-slowly oscillating continuity ( [42]), ∆quasi-slowly oscillating continuity ( [16]), arithmetic continuity ( [60], [5]), upward and downward statistical continuities ( [24]), lacunary statistical ward continuity ( [7], [66]), lacunary statistical δ ward continuity ( [31]), lacunary statistical δ 2 ward continuity ( [64]), N θ -ward continuity ( [22], [30], [48], [8], [48], [47]), N θ -δ-ward continuity, and ( [8]) , which enabled some authors to obtain interesting results.…”