Δ-quasi-slowly oscillating continuity
Introduction
Firstly, some definitions and notations will be given in the following. Throughout this paper, N will denote the set of all positive integers. We will use boldface letters x, y, z, … for sequences x = (xn), y = (yn), z = (zn), … of terms in R, the set of all real numbers. Also, s and c will denote the set of all sequences of points in R and the set of all convergent sequences of points in R, respectively.
A sequence x = (xn) of points in R is called statistically convergent [1] to an element ℓ of R iffor every ε > 0, and this is denoted by st − limn→∞xn = ℓ.
A sequence x = (xn) of points in R is slowly oscillating [2], denoted by x ∈ SO, ifwhere [λn] denotes the integer part of λn. This is equivalent to the following: xm − xn → 0 whenever as m, n → ∞. In terms of ε and δ, this is also equivalent to the case when for any given ε > 0, there exist δ = δ (ε) > 0 and a positive integer N = N(ε) such that ∣xm − xn∣ < ε if n ⩾ N (ε) and n ⩽ m ⩽ (1 + δ)n.
By a method of sequential convergence, or briefly a method, we mean a linear function G defined on a sublinear space of s, denoted by cG (R), into R. A sequence x = (xn) is said to be G-convergent [3] to ℓ if x ∈ cG (R) and G (x) = ℓ. In particular, lim denotes the limit function lim x = limnxn on the linear space c. A method G is called regular if every convergent sequence x = (xn) is G-convergent with G (x) = lim x. A method G is called subsequential if whenever x is G-convergent with G (x) = ℓ, then there is a subsequence of x with . A function f is called G-continuous [3] if G (f(x)) = f (G(x)) for any G-convergent sequence x. Here we note that for special G = st − lim, f is called statistically continuous [3]. For real and complex number sequences, we note that the most important transformation class is the class of matrix methods. For more information for classical and modern summability methods see [4].
A function f is slowly oscillating continuous, denoted by f ∈ SOC, if it transforms slowly oscillating sequences to slowly oscillating sequences, i.e. (f(xn)) is slowly oscillating whenever (xn) is slowly oscillating. A sequence x = (xn) is called quasi-slowly oscillating, denoted by x ∈ QSO, if (Δxn) = (xn − xn + 1) is a slowly oscillating sequence. A subset F of R is called slowly oscillating compact [5] if whenever x = (xn) is a sequence of points in F, there is a slowly oscillating subsequence of x. We see that quasi-slowly oscillating compactness can not be obtained by any G-sequential compactness in the manner of [6].
The purpose of this work is to introduce a new concept of Δ-quasi-slowly oscillating continuity and show that this kind of continuity implies ordinary continuity. We also define a new type of compactness and prove some new results related to compactness.
Section snippets
Δ-quasi-slowly oscillating continuity
We now introduce the concept of Δ-quasi-slowly oscillating of a sequence of points in R and the concept of Δ-quasi-slowly oscillating continuity. Definition 2.1 A sequence (xn) is called Δ-quasi-slowly oscillating, denoted by x ∈ ΔQSO, if (Δxn) = (xn − xn + 1) is a quasi-slowly oscillating sequence [7]. Definition 2.2 A function f is Δ-quasi-slowly oscillating continuous, denoted by f ∈ ΔQSOC, if it transforms Δ-quasi-slowly oscillating sequences to Δ-quasi-slowly oscillating sequences, i.e. (f(xn)) is Δ-quasi-slowly oscillating
Acknowledgement
Our thanks go to Prof. Scott whose suggestions made us enable to present this paper in this finalized form.
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