Lacunary ward continuity in 2-normed spaces

Abstract

In this paper, we introduce lacunary statistical ward continuity in a 2-normed space. A function f defined on a subset E of a 2-normed space X is lacunary statistically ward continuous if it preserves lacunary statistically quasi-Cauchy sequences of points in E where a sequence (x k ) of points in X is lacunary statistically quasi-Cauchy iffor every positive real number ε and z ∈ X, and (k r ) is an increasing sequence of positive integers such that k 0 = 0 and h r = k r − k r−1 → ∞ as r → ∞, I r = (k r−1 , k r ]. We investigate not only lacunary statistical ward continuity, but also some other kinds of continuities in 2-normed spaces.

On Abel statistical convergence

“…An investigation of Abel sequential continuity and Abel sequential compactness can be done for double sequences (see [35], [41], [10] for basic concepts in the double sequences case). For some further study, we suggest to investigate Abel statistical quasi Cauchy sequences of points in a topological vector space valued cone metric space (see [38], [51], [36], [56], and [57]) or in 2-normed spaces ( [50], [30], [31], [42]).…”

Ward continuity in 2-normed spaces

“…Then using this idea, different types of continuities were defined for real functions in [6,7] as ward continuity, statistically ward continuity, lacunary ward continuity and etc. They were also studied in 2-normed space in [24,9,10].…”

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

“…(f (α k )) is N θ -quasi-Cauchy whenever (α k ) is an N θquasi-Cauchy sequence of points in A. Recently, the concept of the ward continuity in 2-normed spaces was investigated in [28,29,30].…”