Strongly Cesaro type quasi-Cauchy sequences

TL;DR

This paper introduces a new class of functions called $N_{\theta}$-ward continuous functions that preserve a specific type of quasi-Cauchy sequences, and establishes related inclusion, compactness, and continuity theorems.

Contribution

It defines $N_{\theta}$-quasi-Cauchy sequences and $N_{\theta}$-ward continuous functions, providing foundational theorems on their properties and relationships.

Findings

Established inclusion and compactness theorems for $N_{\theta}$-ward continuous functions.

Proved continuity theorems related to $N_{\theta}$-quasi-Cauchy sequences.

Characterized the behavior of functions preserving $N_{\theta}$-quasi-Cauchy sequences.

Abstract

In this paper we call a real-valued function $N_{\theta}$-ward continuous if it preserves $N_{\theta}$-quasi-Cauchy sequences where a sequence $\boldsymbol{\alpha}=(\alpha_{k})$ is defined to be $N_{\theta}$-quasi-Cauchy when the sequence $\Delta \boldsymbol{\alpha}$ is in $N^{0}_{\theta}$. We prove not only inclusion and compactness type theorems, but also continuity type theorems.