Variations on lacunary statistical quasi Cauchy sequences

Abstract

In this paper, we introduce a concept of lacunary statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (α k ) is lacunary statistically p-quasi-Cauchy if limr→∞ 1 hr |{k ∈ Ir : |α k+p − α k | ≥ ε}| = 0 for each ε > 0. A function f is called lacunary statistically p-ward continuous on a subset A of the set of real numbers R if it preserves lacunary statistically p-quasi-Cauchy sequences, i.e. the sequence (f (αn)) is lacunary statistically p-quasi-Cauchy whenever α = (αn) is a lacunary statistically pquasi-Cauchy sequence of points in A. It turns out that a real valued function f is uniformly continuous on a bounded subset A of R if there exists a positive integer p such that f preserves lacunary statistically p-quasi-Cauchy sequences of points in A.

Lacunary Statistical Delta 2 2-Quasi Cauchy Double Sequences

“…Savas and Patterson [17] extended the concept of lacunary statistical convergence of single sequence to double sequences. Yildiz [19] introduced the concept of lacunary statistical delta 2 quasi Cauchy sequences. In this paper we have further extended this concept to double sequences and established some essential results using analogy.…”

A variation on strongly ideal lacunary ward continuity

“…A function f : R −→ R is continuous if and only if it preserves Cauchy sequences. Using the idea of continuity of a real function in terms of sequences, many kinds of continuities were introduced and investigated, 100 not all but some of them we recall in the following: slowly oscillating continuity ( [12]), quasi-slowly oscillating continuity ( [23]), ∆-quasi-slowly oscillating continuity ( [13]), ward continuity ( [14]), δ-ward continuity ( [15]), δ 2 -ward continuity ( [2]), contra δ − β−continuity ( [1]), statistical ward continuity, ( [8], [9], [7]), lacunary statistical ward continuity, ( [43], [42], [39]), λ-statistically ward continuity ( [16]), ideal ward continuity ( [10]) and Abel continuity ( [17]) which enabled some authors to obtain some characterizations of uniform continuity in terms of sequences in the sense that a function, on a special subset of R, preserves certain types of sequences (see [3], [41], [18], [23]). The concept of lacunary I-convergence of sequences was introduced and investigated in [40].…”