and Applied Analysis 3 Proposition 2.5. Let X,G be a G-metric space. Then the following statements are equivalent. 1 The sequence {xn} is G-Cauchy. 2 For every ε > 0, there is k ∈ N such that G xn, xm, xm < ε, for all n,m ≥ k. Definition 2.6. A G-metric space X,G is called G-complete if every G-Cauchy sequence in X,G is G-convergent in X,G . Proposition 2.7. Let X,G be a G-metric space. Then, f...

A subsequence principle is obtained, characterizing Banach spacescontaining co' in the spirit of the author's 1974 characterization of Banachspaces containing II . Defiaition. A sequence(bj )in a Banach space is called strongly summing (s.s.)if (bj ) is a weak-Cauchy basic sequence so that whenever scalars (cj ) satisfysUPn II E~=I cijll < 00, thenECj converges.A...

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...

Theorem A. Let g : [0, T ]× IR 7→ IR be a bounded function. (i) If the map t 7→ g(t, x) is measurable for each x and the map x 7→ g(t, x) is continuous for each t, then the Cauchy problem (1.1) has at least one solution. (ii) If the map t 7→ g(t, x) is measurable for each x and the map x 7→ g(t, x) is Lipschitz continuous for each t, with a uniform Lipschitz constant, then the Cauchy problem (1...

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 } ,...