Huseyin Kaplan

A new study on the strongly lacunary quasi Cauchyness

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