Delta-Quasi-slowly oscillating continuity
CitationÇakallı, H., Çanak, İ. ve Dik, M. (2010). Delta-quasi-slowly oscillating continuity. Applied Mathematics and Computation. s. 2865-2868.
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  to an element ‘ of R if lim n!1 1 n jfk 6 n : jxk ‘j P egj ¼ 0; for every e > 0, and this is denoted by st limn?1xn = ‘. A sequence x = (xn) of points in R is slowly oscillating , denoted by x 2 SO, if lim k!1þ limn max nþ16k6½kn jxk xnj ¼ 0; where [kn] denotes the integer part of kn. This is equivalent to the following: xm xn?0 whenever 1 6 mn ! 1 as m,n?1. In terms of e and d, this is also equivalent to the case when for any given e > 0, there exist d = d (e) > 0 and a positive integer N = N(e) such that jxm xnj < e if nPN(e) and n 6 m 6 (1 + d)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  to ‘ if x 2 cG(R) and G(x) = ‘. 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) = ‘, then there is a subsequence ðxnk Þ of x with limkxnk ¼ ‘. A function f is called G-continuous  if G(f(x)) = f (G(x)) for any Gconvergent sequence x. Here we note that for special G = st lim, f is called statistically continuous . 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 .
SourceApplied Mathematics and Computation
- Makale Koleksiyonu 
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In this paper, a new concept of Delta-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 ...
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