A variation on strongly lacunary delta ward continuity in 2-normed spaces

Sibel Ersan

doi:10.5269/bspm.v38i7.45496

Abstract

A sequence $(x_{k})$ of points in a subset E of a 2-normed space $X$ is called strongly lacunary $\delta$-quasi-Cauchy, or $N_\theta$-$\delta$-quasi-Cauchy if $(\Delta x_k)$ is $N_\theta$-convergent to 0, that is $\lim_{r\rightarrow\infty}\frac{1}{h_r}\sum_{k\in I_r}||\Delta^2 x_k, z||=0$ for every fixed $z\in X$. A function defined on a subset $E$ of $X$ is called strongly lacunary $\delta$-ward continuous if it preserves $N_{\theta}$-$\delta$-quasi-Cauchy sequences, i.e. $(f(x_{k}))$ is an $N_{\theta}$-$\delta$-quasi-Cauchy sequence whenever $(x_{k})$ is. In this study we obtain some theorems related to strongly lacunary $\delta$-quasi-Cauchy sequences.

Highlights

A Variation on Strongly Lacunary delta Ward Continuity in 2-normed SpacesSibel Ersan abstract: A sequence (xk) of points in a subset E of a 2-normed space X is called strongly lacunary δ-quasi-Cauchy, or Nθ-δ-quasi-Cauchy if (∆xk) is Nθconvergent to
1 hr k∈Ir ||∆2xk, z|| = 0 for every fixed z ∈ X
Using the idea of Freedman, Sember, and Raphael; Fridy and Orhan introduced the concept of lacunary statistical convergence of a sequence of real numbers in

Summary

A Variation on Strongly Lacunary delta Ward Continuity in 2-normed Spaces

Sibel Ersan abstract: A sequence (xk) of points in a subset E of a 2-normed space X is called strongly lacunary δ-quasi-Cauchy, or Nθ-δ-quasi-Cauchy if (∆xk) is Nθconvergent to. A function defined on a subset E of X is called strongly lacunary δ-ward continuous if it preserves Nθ-δ-quasi-Cauchy sequences, i.e. A sequence (αk) of points in R, the set of real numbers, is called statistically convergent to. This is denoted by st − lim αk = L (see [14]). In [15], the concept of a strongly lacunary convergent sequence of real numbers, or an Nθ convergent sequence, was defined by Freedman, Sember, and Raphael. A function defined on a subset A of R is called strongly lacunary ward continuous or Nθ-ward continuous if it preserves Nθ-quasi-Cauchy sequences of points in A, i.e. The purpose of this paper is to introduce the concept of strongly lacunary delta ward continuity in 2-normed spaces and prove some related theorems

Strongly Lacunary δ-ward continuity

Conclusion