A new study on the strongly lacunary quasi Cauchyness

TL;DR

This paper introduces the concept of $N_{\theta}^{2}$ quasi-Cauchy sequences, explores their properties, and defines $N_{\theta}^{2}$ ward continuous functions that preserve such sequences, expanding the theory of quasi-Cauchy sequences.

Contribution

It defines and investigates $N_{\theta}^{2}$ quasi-Cauchy sequences and introduces $N_{\theta}^{2}$ ward continuity, a new form of sequence preservation in real analysis.

Findings

Established properties of $N_{\theta}^{2}$ quasi-Cauchy sequences

Proved theorems relating to $N_{\theta}^{2}$ ward continuous functions

Connected $N_{\theta}^{2}$ quasi-Cauchy sequences with existing sequence concepts

Abstract

In this paper, the concept of an $N_{\theta}^{2}$ quasi-Cauchy sequence is introduced. We proved interesting theorems related to $N_{\theta}^{2}$-quasi-Cauchy sequences. A real valued function $f$ defined on a subset $A$ of $\mathbb{R}$, the set of real numbers, is $N_{\theta}^{2}$ ward continuous on $A$ if it preserves $N_{\theta}^{2}$ quasi-Cauchy sequences of points in $A$, i.e. $(f( \alpha_{k}))$ is an $N_{\theta}^{2}$ quasi-Cauchy sequence whenever $(\alpha_{k})$ is an $N_{\theta}^{2}$ quasi-Cauchy sequences of points in $A$, where a sequence $(\alpha_{k})$ is called $N_{\theta}^{2}$ quasi-Cauchy if $(\Delta^{2} \alpha_{k})$ is an $N_{\theta}$ quasi-Cauchy sequence where $\Delta^{2}\alpha_{k}=\alpha_{k+2}-2\alpha_{k+1}+\alpha_{k}$ for each positive integer $k$.