Δ-quasi-slowly oscillating continuity

Abstract

In this paper, a new concept of Δ-quasi-slowly oscillating continuity is introduced. Furthermore, it is shown that this kind of continuity implies ordinary continuity. A new type of compactness is also defined and some new results related to compactness are proved.

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 = (x_n), y = (y_n), z =  (z_n), … 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 = (x_n) of points in R is called statistically convergent [1] to an element ℓ of R if

$$\underset{n\to \infty }{\lim }\frac{1}{n}|\{k⩽n:|x_k-\ell |⩾\epsilon \}|=0\text{,}$$

for every ε > 0, and this is denoted by st − lim_n→∞x_n = ℓ.

A sequence x = (x_n) of points in R is slowly oscillating [2], denoted by x ∈ SO, if

$$\underset{\lambda \to 1^{+}}{\lim }{\overline{\lim }}_n\underset{n+1⩽k⩽[\lambda n]}{\max }|x_k-x_n|=0\text{,}$$

where [λn] denotes the integer part of λn. This is equivalent to the following: x_m − x_n → 0 whenever $1⩽\frac{m}{n}\to 1$ 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 ∣x_m − x_n∣ < 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 c_G (R), into R. A sequence x = (x_n) is said to be G-convergent [3] to ℓ if x ∈ c_G (R) and G (x) = ℓ. In particular, lim denotes the limit function lim x = lim_nx_n on the linear space c. A method G is called regular if every convergent sequence x = (x_n) 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 $(x_{n_k})$ of x with ${\mathit{lim}}_k{x}_{n_k}=\ell$. 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(x_n)) is slowly oscillating whenever (x_n) is slowly oscillating. A sequence x = (x_n) is called quasi-slowly oscillating, denoted by x ∈ QSO, if (Δx_n) = (x_n − x_n + 1) is a slowly oscillating sequence. A subset F of R is called slowly oscillating compact [5] if whenever x = (x_n) is a sequence of points in F, there is a slowly oscillating subsequence $\mathbf{y}=(y_k)=(x_{n_k})$ 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.