On λ statistical upward compactness and continuity

Huseyin Cakalli
2018 Filomat  
A sequence (α k ) of real numbers is called λ-statistically upward quasi-Cauchy if for every ε > 0 lim n→∞ 1 λn |{k ∈ I n : α k − α k+1 ≥ ε}| = 0, where (λ n ) is a non-decreasing sequence of positive numbers tending to ∞ such that λ n+1 ≤ λ n + 1, λ 1 = 1, and I n = [n − λ n + 1, n] for any positive integer n. A real valued function f defined on a subset of R, the set of real numbers is λ-statistically upward continuous if it preserves λ-statistical upward quasi-Cauchy sequences.
more » ... nces. λ-statistically upward compactness of a subset in real numbers is also introduced and some properties of functions preserving such quasi Cauchy sequences are investigated. It turns out that a function is uniformly continuous if it is λ-statistical upward continuous on a λ-statistical upward compact subset of R. t n (α) := 1 λ n k∈I n α k where I n = [n − λ n + 1, n]. A sequence α = (α k ) is said to be (V, λ)-summable to a number L if t n (α) −→ L as n −→ ∞, which is denoted by V λ − limα k = L. A sequence α = (α k ) is said to be [V, λ]-summable to a number L or strongly (V, λ)-summable to a number L (see [27]) if lim n→∞ 1 λ n k∈I n |α k − L| = 0, 2010 Mathematics Subject Classification. Primary: 40A05; Secondaries: 40G15, 26A05, 26A15
doi:10.2298/fil1812435c fatcat:ezxusjtv7jh35lpymj7ozepruu