Statistically quasi Cauchy sequences in abstract metric spaces

Abstract

In this extended abstract, we introduce a concept of statistically quasi-Cauchyness of a sequence in X in the sense that a sequence (x(k)) is statistically quasi -Cauchy in X if lim(n ->infinity) 1/n vertical bar{k <= n : d(x(k+1), x(k)) - c is an element of P}vertical bar for each c is an element of P where (X, d) is a cone metric space, and p denotes interior of a cone P of X. It turns out that a function f from a totally bounded subset A of X into X is uniformly continuous if f preserves statistically quasi-Cauchy sequences.