Cauchy Sequences Research Articles

Statistically Convergent Sequences in Neutrosophic Metric Spaces

In this study, the introduction of statistical convergence and statistical Cauchy sequences with respect to neutrosophic metric spaces is motivated by the notion of statistical convergence in fuzzy metric spaces. We offer useful characterizations for statistically convergent and statistically Cauchy sequences.

Cauchy-subregular functions vis-à-vis different types of continuity

A function between two metric spaces is said to be Cauchy-regular if it takes Cauchy sequences to Cauchy sequences. This well-studied class of functions lies strictly in between the class of continuous functions and that of uniformly continuous functions. In this paper, we exhibit another class of functions between metric spaces which is a weaker form of Cauchy-regular functions, called Cauchy-subregular functions. We call a function f:(X,d)→(Y,ρ) to be Cauchy-subregular if for every Cauchy sequence (xn) in (X,d), (f(xn)) has a Cauchy subsequence. We investigate how much it differs from Cauchy-regularity, while using this weaker notion profitably to characterize complete metric spaces. We also compare Cauchy-subregular functions to various other types of continuity such as locally Lipschitz functions and subcontinuous functions. Furthermore, we study relation between Cauchy-subregular functions and the functions that preserve cofinally Cauchy sequences called cofinally Cauchy regular functions.

Approximation types describing extensions of valuations to rational function fields

AbstractWe introduce the notion of approximation type for the partial, and in certain cases the total description of extensions of a given valuation from a field K to the rational function field K(x). To every extension, a unique approximation type of x over K is associated, while x may be the limit of many pseudo Cauchy sequences. Approximation types also provide information in cases where the extensions are not immediate, and we prove that they correspond bijectively to the extensions when K is algebraically closed or, more generally, lies dense in its algebraic closure with respect to the topology induced by the valuation.

Some further studies on strong ℐλ-statistical convergence in probabilistic metric spaces

Abstract We prove some basic properties of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence of sequences in probabilistic metric spaces and introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical cluster point. We also introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences in probabilistic metric spaces. Further, we establish a connection between strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence and strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences.

A unification of Geraghty type and Ciric type fixed point theorems

In the framework of metric spaces, we introduce the concept of Geraghty-Ciric type contractions and show the existence and uniqueness of the fixed point of such mappings. This result improves and unifies those obtained by Geraghty (Proc. Amer. Math. Soc. 40, 604-608 (1973)) and Ciric (Proc. Amer. Math. Soc. 45, 267-273, (1974)). Several technical lemmas are employed to ensure that a Picard sequence is a Cauchy sequence. In addition, two illustrative examples are provided to indicate the validity of the obtained results.

Near fixed point, near fixed interval circle and their equivalence classes in a $b-$interval metric space

We introduce a novel distance structure called a b−interval metric space to generalize and extend metric interval space. Also, we demonstrate that the collection of open balls, which forms a basis of a b−interval metric space, generates a T0−topology on it. Further, we define topological notions like an open ball, closed ball, b−convergence, b−Cauchy sequence and completeness of the space on a b−interval metric space to create an environment for the survival of a near fixed point and a unique equivalence class of near fixed point. Towards the end, we introduce notions of interval circle, fixed interval circle, its equivalence class and established the existence of a near fixed interval circle and its equivalence interval C−class of near fixed interval circle to study the geometric properties of non-unique equivalence C−classes of nearly fixed interval circles.

Global well-posedness for fractional Sobolev-Galpern type equations

<p style='text-indent:20px;'>This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.</p>

Fuzzifying completeness and compactness in fuzzifying bornological linear spaces

<abstract><p>The notions of completeness and compactness play important role in classical functional analysis. The main purpose of this paper is to generalize these notions to the setting of fuzzifying bornological linear spaces. At first, the concepts of fuzzifying Cauchy sequences and fuzzifying completeness are introduced and some interesting properties of them are studied. The relationships among fuzzifying completeness, separation axiom and fuzzifying bornological closed set are discussed. Then the notions of fuzzifying compactness and precompactness are presented, several properties of them are discussed. Particularly, it is demonstrated that a subset is fuzzifying bornological compact if and only if it is fuzzifying bornological precompact and bornological complete.</p></abstract>

An Introduction to Neutrosophic Real Banach and Hillbert Spaces

Banach and Hillbert spaces are the main important concepts in the study of classical functional analysis. This paper generalizes these two kinds of functional spaces into neutrosophic systems, where the concept of neutrosophic Banach space and neutrosophic Hillbert space will be defined and discussed for the first time over partial ordered neutrosophic spaces. Also, many related concepts such as neutrosophic Cauchy sequence, neutrosophic Bessel's inequality, and neutrosophic Parseval's identity will be established and proved.

Rough Cauchy sequences in a cone metric space

In this paper, we have introduced the idea of rough Cauchyness of sequences in a cone metric space. We have also discussed several basic properties of rough Cauchy sequences in a cone metric space using the idea of Phu [8].

Approximate controllability of fractional neutral evolution systems of hyperbolic type

<p style='text-indent:20px;'>In this paper, we deal with fractional neutral evolution systems of hyperbolic type in Banach spaces. We establish the existence and uniqueness of the mild solution and prove the approximate controllability of the systems under different conditions. These results are mainly based on fixed point theorems as well as constructing a Cauchy sequence and a control function. In the end, we give an example to illustrate the validity of the main results.</p>

On (λ,μ,ζ)-Zweier ideal convergence in intuitionistic fuzzy normed spaces

In this paper, we introduce and study a new type of convergence which is namely (?,?,?)-Zweier convergence and (?,?,?)-Zweier ideal convergence of triple sequences x = (xijk) in intuitionistic fuzzy normed spaces (IFNS), where ?= (?n), ? = (?m) and ? = (? p) are three non-decreasing sequences of positive real numbers such that each tend to infinity. Besides, we define and study (?,?,?)-Zweier Cauchy and (?,?,?)- Zweier ideal Cauchy sequences on the said space and establish some relations among Them. problem.

Some remarks on ∆m (Iλ )-summability on neutrosophic normed spaces

In the present paper, we use the difference operator ∆m (Iλ )-summability to define some new summability concepts on neutrosophic normed spaces. We also introduce concepts of generalized limit point, and cluster point and obtain some relationships among these notions. Finally, we define generalized Cauchy sequences on these spaces and present a characterization of a new summability method that preserves linear operators on neutrosophic normed spaces.

Maps that preserve left (right) $K$-Cauchy sequences

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

On Fundamental Properties in Fuzzy Metric Space

Fuzzy metric spaces theory is important in mathematics, statistics, computer science, and other fields. In this paper fundamental characteristics of the fuzzy metric space are examined. The notions of fuzzy convergent sequence, fuzzy Cauchy sequence, fuzzy open ball, are reviewed with theorems associated with these concepts.

A characterization of p-complete fuzzy metric spaces

George and Veeramani characterized complete fuzzy metric spaces (X,M,⁎) by means of nested sequences {An} of closed sets of X which have fuzzy diameter zero. According to the concept of p-convergence due to D. Mihet, an appropriate concept of p-Cauchy sequence was given. In this paper we introduce for {An} a concept of p-fuzzy diameter zero, which is according to the concept of p-convergence. Then, we characterize by means of certain nested sequences {An}, which have p-fuzzy diameter zero, those fuzzy metric spaces in which p-Cauchy sequences are convergent (p-convergent), called p-complete spaces (w-p-complete spaces). As a consequence of our results we obtain the well-known characterization of a complete metric space (X,d) by means of nested sequences of closed sets of (X,d).

Rough Cauchy sequences in a cone metric space

In this paper, we have introduced the idea of rough Cauchyness of sequences in a cone metric space. We have also discussed several basic properties of rough Cauchy sequences in a cone metric space using the idea of Phu [8].

Some New Results on Equivalent Cauchy Sequences and Their Applications to Meir-Keeler Contraction in Partial Rectangular Metric Space

The study of fixed points in the metric spaces plays a crucial role in the development of Functional Analysis. It is evolved by generalizing the metric space or improving the contractive conditions. Recently, the partial rectangular metric space and its topology have been the center of study for many researchers. They have defined open and closed balls the equivalent Cauchy sequences and Cauchy sequences, convergent sequences which are used as tools in many achieved results. In this paper, two facts for equivalent Cauchy sequences in a partial rectangular metric space are provided by using an ultra - altering distance function. Furthermore, some results of Cauchy sequences in a partial rectangular metric space are highlighted. There is proved that under some conditions the equivalent Cauchy sequences are Cauchy sequences in a partial rectangular metric space. Some fixed point results have been taken as applications of our new conditions of Cauchy sequences and equivalent Cauchy sequences in a partial rectangular metric space <img src=image/13424354_01.gif> for orbitally continuous functions <img src=image/13424354_02.gif>. To illustrate the obtained results some examples are given.

ON I- CONVERGENCE OF SEQUENCES IN GRADUAL NORMED LINEAR SPACES

In this paper, we introduce the concepts of $\mathcal{I}$ and $\mathcal{I^{*}}-$convergence of sequences in gradual normed linear spaces. We study some basic properties and implication relations of the newly defined convergence concepts. Also, we introduce the notions of $\mathcal{I}$ and $\mathcal{I^{*}}-$Cauchy sequences in the gradual normed linear space and investigate the relations between them.

On a fuzzy metric space and fuzzy convergence

In this paper we introduce a new type of fuzzy metric space and obtain several properties of it. Also some topological properties and boundedness are investigated. The notion of convergence of fuzzy sequences together with the notion of fuzzy cauchy sequences in a fuzzy metric space are discussed and some basic results related to these notions are investigated.