A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exists non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a ra...

A function f defined on a subset E of a 2-normed space X is strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points in E; that is, (f(x k)) is a strongly lacunary quasi-Cauchy sequence whenever (x k) is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also some other kinds of continuities are investigated...

The purpose of this article is to investigate lacunary ideal convergence sequences in neutrosophic normed space (NNS). Also, an original notion, named sequence NNS, defined. $% \mathcal{I}$-limit points and $\mathcal{I}$-cluster NNS have been examined. Furthermore, Cauchy $\mathcal{I}$-Cauchy are introduced some properties these notions studied.

The notion of ideal convergence is a process generalizing statistical which dependent on the idea $I$ subsets set positive integer numbers. In this study we also present concept for triple sequences in fuzzy metric spaces (FMS) manner George and Veeramani terms Cauchy sequence $I^{∗}$-Cauchy FMS their certain properties.

In this paper, we deal with the notion of fuzzy metric space (X,M,∗), or simply X, due to George and Veeramani. It is well known that such spaces, in general, are not completable also there exist p-Cauchy sequences which Cauchy. We prove if every sequence X Cauchy, then principal, observe converse false, general. Hence, introduce study a stronger concept than called strongly principal. Moreover...

The terminology and notation used in this paper are introduced in the following papers: [22], [3], [20], [9], [5], [12], [10], [11], [15], [2], [18], [4], [1], [21], [16], [17], [14], [13], [19], [6], [7], and [8]. For simplicity, we follow the rules: X denotes a complex unitary space, s1, s2, s3 denote sequences of X, R1 denotes a sequence of real numbers, C1, C2, C3 denote complex sequences, ...

and Applied Analysis 3 In order to prove that the sequence {xn} is a Cauchy sequence with respect to norm ‖·‖C, we introduce an equivalent norm and show that {xn} is a Cauchy sequence with respect to the new one. Basing on the condition H2 , we see that there are two positive constantsM and m such that m ≤ y t ≤ M for all t ∈ R. Define the new norm ‖ · ‖1 by ‖u‖1 sup { 1 y t ‖u t ‖E : t ∈ R } ,...

In this study, we introduce the notion of pointwise and uniform statistical convergence of double sequences of real-valued functions. We also give the relations between these convergences and pointwise,uniform convergence. Furthermore, we introduce the concept of statistically Cauchy sequence and study statistical analogue of convergence and Cauchy criterion for double sequences of real-valued ...

It is well-known that on quasi-pseudometric space $(X,q)$, every $q^s$-Cauchy sequence left (or right) $K$-Cauchy but the converse does not hold in general. In this article, we study a class of maps preserve (right) sequences call sequentially-regular maps. Moreover, characterize totally bounded sets terms and right uniformly locally semi-Lipschitz