Special Sequences on Generalized Metric Spaces

2016

Singh and Srivastava [9] have studied n-normed structure of 񒍽ґ by introducing a new n-norm .,.,…,. 񒑠 on it and have observed that 񒍽ґ, .,.,…,. 񒑠is not complete in general. Here we shall investigate some sufficient condition for a Cauchy sequence to be convergent in the n-normed space 񒍽ґ, .,.,…,. 񒑠and some new results on Cauchy sequences.

For a fixed positive i nteger p, a sequence (x n) in a metric space X is c alled p-quasi-Cauchy if (Δ p x n) is a null sequence where Δ p x n = d(x n+p , x n) for each positive integer n. A subset E of X is called p-ward compact if any sequence (x n) of points in E has a p-quasi-Cauchy subsequence. A subset of X is totally bounded if and only if it is p-ward compact. A function f from a subset E of X into a metric space Y is called p-ward continuous if it preserves p-quasi Cauchy sequences, i.e. (f (x n)) is a p-quasi Cauchy sequence in Y whenever (x n) is a p-quasi Cauchy sequence of points of E. A function f from a totally bounded subset of X into Y preserves p-quasi Cauchy sequences if and only if it is uniformly continuous. If a function is uniformly continuous on a subset E of X into Y, then (f (x n) is p-quasi Cauchy in Y whenever (x n) is a quasi cauchy sequence of points in E.

Fixed Point Theory and Applications, 2010

The concept of weakly quasi-nonexpansive mappings with respect to a sequence is introduced. This concept generalizes the concept of quasi-nonexpansive mappings with respect to a sequence due to Ahmed and Zeyada 2002 . Mainly, some convergence theorems are established and their applications to certain iterations are given.

Universal Journal of Mathematics and Applications

In this paper, our goal is to introduce some new Cauchy sequence spaces. These spaces are defined by Cauchy transforms. We shall use notations C ∞ (s,t), C (s,t) and C 0 (s,t) for these new sequence spaces. We prove that these new sequence spaces C ∞ (s,t), C (s,t) and C 0 (s,t) are the BK−spaces and isomorphic to the spaces l ∞ , c and c 0 , respectively. Besides the bases of these spaces, α−, β − and γ− duals of these spaces will be given. Finally, the matrix classes (C (s,t) : l p) and (C (s,t) : c) have been characterized.

Mathematics and Statistics, 2021

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 (X, p) for orbitally continuous functions f : X → X. To illustrate the obtained results some examples are given.

FOURTH INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2020), 2021

In this extended abstract, we introduce the concept of delta quasi Cauchy sequences in metric spaces. A function f defined on a subset of a metric space X to X is called delta ward continuous if it preserves delta quasi Cauchy sequences, where a sequence (x k) of points in X is called delta quasi Cauchy if lim n→∞ [d(x k+2 , x k+1)−d(x k+1 , x k)] = 0. A new type compactness in terms of δ-quasi Cauchy sequences, namely δ-ward compactness is also introduced, and some theorems related to δ-ward continuity and δ-ward compactness are obtained. Some other types of continuities are also discussed, and interesting results are obtained.

2018

The notion of S-metric space was introduced by Sedghi et al. In this paper we study the ideas of I and I∗-Cauchy sequences in S-metric spaces and investigate their relation following the same approach as done by Das and Ghosal. We then study the ideas of I and I∗-divergent sequences in S-metric spaces and examine their relation under certain general assumption.

Cornell University - arXiv, 2010

Recall that a subset E of a metric space (X, d) is called bounded if δ(A) = sup{d(a, b) : a, b ∈ E} ≤ M where M is a positive real constant number, X is a non empty set, and d : X 2 → R satisfies (M1) d(x, y) = 0 if and only if x = y, (M2) d(x, y) = d(y, x), and (M 3) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X. A metric space (X, d) is said to be an ε-net if X = a∈E B(a, ε). The metric space (X, d) is called totally bounded if it has a finite ε-net for each ε > 0. A subspace (E, d E) of (X, d) is said to be totally bounded if it is totally bounded as a metric space in its own right. A subset E of a metric space (X, d) is said to be totally bounded if it is totally bounded as a metric subspace. The

Applied Mathematics and Computation, 2004

The idea of difference sequence spaces was introduced by Kızmaz [Canad. Math. Bull. 24 (1981) 169] and this concept was generalized by Et and C ß olak [Soochow J. Math. 21 (1995) 337]. In this paper we define the sequence spaces ' 1 ðpÞðD r v Þ, cðpÞðD r v Þ, c 0 ðpÞðD r v Þ and cðpÞðD r v Þ, ðr 2 NÞ, give some topological properties and inclusion relations of these sequence spaces.

We present a survey of fixed point results in generalized metric spaces (g.m.s.) in the sense of Branciari , A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57, 31-37]. Since it may happen that the topology of such space is not Hausdorff, several authors added Hausdorfness (or some other condition) as an additional assumption in order to obtain their results. We show here that such assumptions are usually superfluous. Finally, we state some open questions on the topic.