Variations on strong lacunary quasi-Cauchy sequences

Huseyin Kaplan, Huseyin Cakalli
2016 Journal of Nonlinear Science and its Applications  
We introduce a new function space, namely the space of N α θ (p)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions. A real valued function f defined on a subset A of R, the set of real numbers, is N α θ (p)-ward continuous if it preserves N α θ (p)-quasi-Cauchy sequences, that is, (f (x n )) is an N α θ (p)-quasi-Cauchy sequence whenever (x n ) is N α θ (p)-quasi-Cauchy sequence of points in A, where a sequence (x k ) of points in R is
more » ... oints in R is called N α θ (p)-quasi-Cauchy if where ∆x k = x k+1 − x k for each positive integer k, p is a constant positive integer, α is a constant in ]0, 1], I r = (k r−1 , k r ], and θ = (k r ) is a lacunary sequence, that is, an increasing sequence of positive integers such that k 0 = 0, and h r : k r − k r−1 → ∞. Some other function spaces are also investigated.
doi:10.22436/jnsa.009.06.77 fatcat:mjo57j2vg5cdvaxgx3zszgadri