ON $\mathcal{I}$-CAUCHY SEQUENCES

IK-Cauchy functions

“…If we take S = N and K = Fin in Theorem 5.1, we get [25,Theorem 4]. Also in view of the fact that a uniform space, which has a countable base of the uniformity, is metrizable, we get [7, Theorem 5] as a special case.…”

“…In the case that K = Fin we obtain the notion of I * -Cauchy sequences, which was studied in [25]. The I * -Cauchy nets introduced in [7] are precisely the I I D -Cauchy functions.…”

Intuitionistic fuzzy Zweier I-convergent double sequence spaces

“…The notion of I-convergence, which is a generalization of statistical convergence [4], was introduced by Kostyrko, Salat and Wilczynski [11] by using the idea of I of subsets of the set of natural numbers N and further studied in [16]. Recently, the notion of statistical convergence of double sequences x = (x i j ) has been defined and studied by Mursaleen and Edely [14]; and for fuzzy numbers by Savaş and Mursaleen [20].…”

Extending asymmetric convergence and Cauchy condition using ideals

“…Since 1951 when Steinhaus [28] and Fast [11] defined statistical convergence for sequences of real numbers, several generalizations and applications of this notion have been investigated (see [1], [2], [4]- [10], [12]- [14], [16], [20]- [22], [24]- [27], [29] where many more references can be found). In particular two interesting generalizations of statistical convergence were introduced by Kostyrko et al [13], using the notion of ideals of the set N of positive integers who named them as I and I * -convergence.…”

“…Corresponding I-Cauchy condition was first introduced and studied by Dems [10]. I * -Cauchy sequences has been very recently introduced by Nabiev et al [22] where they showed that I * -Cauchy sequences are I-Cauchy and they are equivalent if the ideal I satisfies the condition (AP)…”