Variations on strong lacunary quasi-Cauchy sequences

Volume 9, Issue 6, pp 4371--4380 http://dx.doi.org/10.22436/jnsa.009.06.77

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Authors

Huseyin Kaplan - Department of Mathematics, Faculty of science and letters, Nigde University, Nigde, Turkey. Huseyin Cakalli - Graduate School of Science and Engineering, Maltepe University, Marmara Egitim Koyu, Maltepe, Istanbul, Turkey.

Abstract

We introduce a new function space, namely the space of \(N^\alpha_\theta(p)\)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions. A real valued function f defined on a subset \(A\) of \(\mathbb{R}\), the set of real numbers, is \(N^\alpha_\theta(p)\)-ward continuous if it preserves \(N^\alpha_\theta(p)\)-quasi-Cauchy sequences, that is, \((f(x_n))\) is an \(N^\alpha_\theta(p)\)-quasi-Cauchy sequence whenever \((x_n)\) is \(N^\alpha_\theta(p)\)-quasi-Cauchy sequence of points in \(A\), where a sequence \((x_k)\) of points in \(\mathbb{R}\) is called \(N^\alpha_\theta(p)\)-quasi-Cauchy if \[\lim_{r\rightarrow\infty}\frac{1}{h^\alpha_r}\Sigma_{k\in I_r}|\Delta x_k|^p=0,\] where \(\Delta x_k = x_{k+1} - x_k\) for each positive integer \(k, p\) is a constant positive integer, \(\alpha\) is a constant in \(]0; 1], I_r = (k_{r-1}; k_r]\), and \(\theta = (k_r)\) is a lacunary sequence, that is, an increasing sequence of positive integers such that \(k_0 \neq 0\), and \(h_r : k_r - k_{r-1} \rightarrow\infty\). Some other function spaces are also investigated.

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Keywords

• Summability
• strongly lacunary convergence
• quasi-Cauchy sequences
• boundedness
• uniform continuity.

MSC

•  40A05
•  40G15

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