Abstract and Applied Analysis

${N}_{\theta }$-Ward Continuity

Huseyin Cakalli

Abstract

A function $f$ is continuous if and only if $f$ preserves convergent sequences; that is, $(f({\alpha }_{n}))$ is a convergent sequence whenever $({\alpha }_{n})$ is convergent. The concept of ${N}_{\theta }$-ward continuity is defined in the sense that a function $f$ is ${N}_{\theta }$-ward continuous if it preserves ${N}_{\theta }$-quasi-Cauchy sequences; that is, $(f({\alpha }_{n}))$ is an ${N}_{\theta }$-quasi-Cauchy sequence whenever $({\alpha }_{n})$ is ${N}_{\theta }$-quasi-Cauchy. A sequence $({\alpha }_{k})$ of points in $\mathbf{R}$, the set of real numbers, is ${N}_{\theta }$-quasi-Cauchy if ${\mathrm{lim}}_{r\to \infty }(1/{h}_{r}){\sum }_{k\in {I}_{r}}|\Delta {\alpha }_{k}|=0$, where $\Delta {\alpha }_{k}={\alpha }_{k+1}-{\alpha }_{k}$, ${I}_{r}=({k}_{r-1},{k}_{r}],$ and $\theta =({k}_{r})$ is a lacunary sequence, that is, an increasing sequence of positive integers such that ${k}_{0}=0$ and ${h}_{r}:{k}_{r}-{k}_{r-1}\to \infty$. A new type compactness, namely, ${N}_{\theta }$-ward compactness, is also, defined and some new results related to this kind of compactness are obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 680456, 8 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495749

Digital Object Identifier
doi:10.1155/2012/680456

Mathematical Reviews number (MathSciNet)
MR2935154

Zentralblatt MATH identifier
1260.40001

Citation

Cakalli, Huseyin. ${N}_{\theta }$ -Ward Continuity. Abstr. Appl. Anal. 2012 (2012), Article ID 680456, 8 pages. doi:10.1155/2012/680456. https://projecteuclid.org/euclid.aaa/1355495749