Variations on strong lacunary quasi-Cauchy sequences
We introduce a new function space, namely the space of N??(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??(p)-ward continuous if it preserves N??(p)-quasi-Cauchy sequences, that is, (f(xn)) is an N??(p)-quasi-Cauchy sequence whenever (xn) is N??(p)-quasi-Cauchy sequence of points in A, where a sequence (xk) of points in R is called N??(p)-quasi-Cauchy if (Formula Presented.) where ?xk= xk+1 – xkfor each positive integer k, p is a constant positive integer, ? is a constant in ]0, 1], Ir = (kr–1, kr], and ? = (kr) is a lacunary sequence, that is, an increasing sequence of positive integers such that k0 ? 0, and hr: kr– kr–1 › ?. Some other function spaces are also investigated. © 2016 All rights reserved.