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The goal of this research article is to study some fixed point results in dislocated quasi metric spaces. These results are proved for certain generalized contraction conditions involving linear and rational expressions. Our results extend and generalize some of the well-known fixed point results of the literature available in fixed point theory. Suitable examples of the established results are also given. [ABSTRACT FROM AUTHOR]

Ideal convergence of sequences in neutrosophic normed spaces is defined by Omer Ki¸si [12]. This ¨ paper defines new sequence spaces using the Zweier matrix and neutrosophic norm. We explore (λ, μ)-Zweier convergence, (λ, μ) ideal convergence of double sequences in neutrosophic norm. We show that a double sequence that is (λ, μ)-Zweier convergent or ideal convergent has a unique limit with respect to neutrosophic norm. Additionally, we prove that (λ, μ)-Zweier ideal convergence is equivalent to (λ, μ)-Zweier ideal Cauchy for a double sequence in neutrosophic normed space. [ABSTRACT FROM AUTHOR]

In this study, motivated by the notion of statistical convergence in intuitionistic fuzzy metric space, we present the ideas of statistical convergence and cauchy sequences with regard to the neutrosophic metric spaces. This study also discusses the statistical completeness on neutrosophic metric space with an example and results. [ABSTRACT FROM AUTHOR]

This research introduces novel concepts in sequence theory, including Bessel convergence, Bessel boundedness, Bessel statistical convergence, and Bessel statistical Cauchy sequences. These concepts establish new inclusion relations and related results within mathematical analysis. Additionally, we extend the first and second Korovkin-type approximation theorems by incorporating Bessel statistical convergence, providing a more robust and comprehensive framework than existing results. The practical implications of these theorems are demonstrated through examples involving the classical Bernstein operator and Fejér convolution operators. This work contributes to the foundational understanding of sequence behavior, with potential applications across various scientific disciplines. [ABSTRACT FROM AUTHOR]

We develop a theory of Cauchy sequences and intertwinings for morphisms of Cuntz semigroups, which generalizes all past approaches to study metric-like properties of the invariant. Further, the techniques presented here can be applied to all known refinements of the Cuntz semigroup, including those that may be used in new classification results. As a particular application, we introduce a Fraïssé theory for abstract Cuntz semigroups akin to the theory of Fraïssé categories developed by Kubiś. We also show that any (Cuntz) Fraïssé category has a unique Fraïssé limit which is both universal and homogeneous. Several examples of such categories and their Fraïssé limits are given throughout the paper. [ABSTRACT FROM AUTHOR]

This research paper aims to introduce the concept of lacunary ideal Cauchy sequences of fuzzy variables in a credibility space. We establish the interrelationships between this notion with lacunary ideal convergent sequences in the same structure from several aspects of credibility. Furthermore, we explore the concepts of strongly lacunary Cauchy, strongly ℐ -lacunary Cauchy, and strongly ℐ ∗ -lacunary Cauchy sequences of fuzzy variables within the context of credibility. We also examine the interconnections between these concepts and analyze their relationships. [ABSTRACT FROM AUTHOR]

The real number system is a fundamental tool for rigorous demonstrations of the differential and integral calculus results. Even after a century of formalization on solid foundations, discussions about the construction of this field are generally omitted in advanced courses such as Real Analysis. In the present work, we present a comprehensive review on the construction and characterization of the real numbers field. The presentation focuses on the construction through Cauchy sequences of rational numbers. The notion of completeness is delimited differently from completeness when Dedekind’s cut construction is used. The results indicate Q and R Archimedean as a necessary condition for these two notions of completeness to be equivalent. To illustrate this, inspired by the work of Leon W. Cohen and Gertrude Ehrlich, we present an example of a Cauchy-complete non-Archimedean ordered field in which the supremum axiom is not equivalent to the nested intervals principle.

Introduction/purpose: This study establishes sufficient conditions for a sequence to be Cauchy within the framework of controlled revised fuzzy metric spaces. It also generalizes the concept of Banach’s contraction principle by introducing several new contraction conditions. The aim is to derive various fixed-point results that enhance the understanding of these mathematical structures. Methods: The researchers employ rigorous mathematical techniques to develop their findings. By defining a set of novel contraction mappings and utilizing properties of controlled revised fuzzy metric spaces, they analyze the implications for sequence convergence. The methodology includes constructing specific examples to illustrate the theoretical results. Results: The study presents several fixed-point theorems derived from the generalized contraction conditions. Additionally, it provides a number of non-trivial examples that substantiate the claims and demonstrate the applicability of the results in practical scenarios. A significant application is explored regarding the conversion of solar energy into electric power, utilizing differential equations to highlight this connection. Conclusion: The findings deepen the understanding of Cauchy sequences in fuzzy metric spaces and offer a broader perspective on the application of the fixed-point theory in real-world scenarios. The results pave the way for further research in both theoretical mathematics and its practical applications, particularly in the field of renewable energy. [ABSTRACT FROM AUTHOR]

Following the definition of interval-valued fuzzy n-normed linear space given by S. Vijayabalaji et al., in this paper, notion of interval-valued generalized fuzzy n-normed linear space is introduced. The notion of convergent sequence, Cauchy sequence and their relation are studied. Some basic results are established on finite dimensional interval-valued generalized fuzzy n-normed linear space. [ABSTRACT FROM AUTHOR]

This paper is about the generalization of the concept of fuzzy metric. Here we introduce g-fuzzy metric space of order n. In particular case, it reduces to George- and Veeramani-type fuzzy metric and Q-fuzzy metric. Some non-trivial examples of g-fuzzy metric space are given and its topological structure, the notion of convergent sequence, Cauchy sequence, etc. are also studied. [ABSTRACT FROM AUTHOR]

The primary objective of this study is to introduce the notion of ideal convergence in quaternion-valued generalized metric spaces. We define I-convergence and I*-convergence in these spaces and establish their equivalence through the definition of property (AP). Furthermore, we introduce Z-Cauchy and I*-Cauchy sequences, adapting classically theorems to suit quaternion-valued generalized metric spaces. We also present I-statistical convergence, I-lacunary statistical convergence, strong I-Cesàro summability, strong I-lacunary summability, [V, λ] (Z)-summability, and I-λ-statistically convergencein quaternion valued generalized metric spaces, along with their associated properties. [ABSTRACT FROM AUTHOR]

Neutrosophication is a useful tool for handling real-world problems with partially dependent, partially independent, and even independent components. By examining some properties related to λ-statistical convergence on neutrosophic normed spaces, we provide some functional tools that are helpful in situations of inconsistency and indeterminacy. Additionally, we establish some related results on λ-Δm-statistical Cauchy sequences on neutrosophic normed spaces. [ABSTRACT FROM AUTHOR]

In this study, we investigate the concept of I * -statistical convergence for sequences of fuzzy numbers. We establish several theorems that provide a comprehensive understanding of this notion, including the uniqueness of limits, the relationship between I * -statistical convergence and classical convergence, and the algebraic properties of I * -statistically convergent sequences. We also introduce the concept of I * -statistical pre-Cauchy and I * -statistical Cauchy sequences and explore its connection to I * -statistical convergence. Our results show that every I * -statistically convergent sequence is I * -statistically pre-Cauchy, but the converse is not necessarily true. Furthermore, we provide a sufficient condition for an I * -statistically pre-Cauchy sequence to be I * -statistically convergent, which involves the concept of I * − l i m i n f . [ABSTRACT FROM AUTHOR]

This work presents the idea of a symmetric Ncone metric space and explore the convergence properties of sequences within this framework. Furthermore, we show fixedpoint theorems for maps that satisfy certain contractive conditions and present examples to illustrate our point. Our findings complement, expand, and consolidate numerous established conclusions in the literature. [ABSTRACT FROM AUTHOR]

Several novel uses of theorems for fixed points in commuting mapping in a fully metric domain are presented. Several conclusions from full metric fixed point theory are improved and extended by our work. Our proofs are inspired by the study of commuting mappings [B. Fisher and S. Sessa, on a fixed point theorem of Gregus, 1986] and [P. Sumati Kumari, Fixed and periodic point theory in certain spaces, 2013]. [ABSTRACT FROM AUTHOR]

The concept of summability is crucial in deriving formal solutions to partial differential equations. This paper explores the connection between the methods of statistical convergence of sequences and statistical Cesàro summability in intuitionistic fuzzy n-normed linear space (IFnNLS). While the existing literature covers Cesàro summability and its statistical variant in fuzzy, intuitionistic fuzzy, and classical normed spaces, this study stands out not only for its methodology but also for its comprehensive approach, encompassing a broader range of spaces and detailing the pathway from the statistical Cesàro summability method to statistical convergence. These results will lead us to Tauberian theorems in IFnNLS. [ABSTRACT FROM AUTHOR]

Recently, the fixed-point theorem for fuzzy contractive mappings has been investigated within the framework of intuitionistic fuzzy metric-like spaces. This interesting topic was explored through the utilization of G-Cauchy sequences as defined by Grabiec. The aim of this study is to enhance the aforementioned results in a few aspects. Initially, the proof of the fixed-point theorem is simplified and condensed, allowing for potential generalization to papers focusing on similar fixed-point analyses. Furthermore, instead of G-Cauchy sequences, the classical Cauchy sequences proposed by George and Veeramani are examined, incorporating an additional condition on the fuzzy metric. Within this context, a solution to an old unresolved question posed by Gregory and Sapena is provided. The findings are reinforced by relevant examples. Finally, the introduced fuzzy metrics are applied to the field of image processing. [ABSTRACT FROM AUTHOR]

In the framework of Gb-metric spaces, we introduce the concept of a generalized Ćirić-type contraction and obtain several fixed-point theorems for this contraction. First, we present a significant lemma, which is used to ensure that the Picard sequence is a Cauchy sequence. Using this lemma, we establish three fixed-point theorems satisfying different conditions. Second, we construct new examples to illustrate our results. As applications, we deduce the famous Ćirić fixed-point theorem in terms of b-metric spaces using our results. In addition, we obtain Reich-type contraction fixedpoint theorems in such a space using the aforementioned lemma. Our results improve and complement many recent findings. In particular, we substantially enlarge the range of the contraction constant in our results. Finally, we consider the existence and uniqueness of solutions for integral equation applying our new results. [ABSTRACT FROM AUTHOR]

We present and explore the concept of a quaternion-valued g-metric space with specific foundational properties related to convergence in this study. The main goal of this study is to introduce γ-statistically convergent sequences and γ-statistically g-Cauchy sequences, examining their properties within quaternion-valued g-metric spaces. The research also includes discussions on natural inclusion theorems relevant to these concepts. [ABSTRACT FROM AUTHOR]

In this paper, a novel of convergence named the fuzzy u-uniform convergence is defined on the basis of the fuzzy order convergence in fuzzy Riesz spaces. The relation between the fuzzy u-uniform convergence and the fuzzy order convergence in fuzzy Archimedean Riesz spaces is studied. We also discuss the existence of the fuzzy u-uniform limit for the fuzzy u-uniform Cauchy sequence in fuzzy Archimedean Riesz Spaces. We give the characterization of fuzzy order completeness in fuzzy Riesz spaces. [ABSTRACT FROM AUTHOR]

In this work, the main discussion centres on avoiding the use of triangle inequality while proving Cauchy sequence to establish best proximity point theorems for single valued as well as multivalued non-self mappings. We prove such best proximity point theorems in the setting of non-triangular metric spaces and elaborate through examples. In this process host of the existing best proximity results are generalized and improved. To arouse further interest in the subject, we connect this work in solving a specific type of partial differential equation problem. [ABSTRACT FROM AUTHOR]

In this study, we present statistical convergence, statistical limit points, and statistical cluster points of double sequences in neutrosophic fuzzy G-metric space with order q, extending the notion of neutrosophic fuzzy metric space. We support our assertions with relevant theorems and elucidate them through illustrative examples. Following the establishment of statistical convergence and the scrutiny of its properties within these spaces, we explore the concepts of lacunary statistical convergence and strongly lacunary convergence of double sequences, while also investigating the relationships among them. [ABSTRACT FROM AUTHOR]

The primary aim of this article is to introduce the concept of ideal convergence for double sequences in neutrosophic fuzzy G-metric spaces (NFGMS) with order q. We explore the fundamental properties of this concept, analyze I2-Cauchy double sequences, and examine I2 completeness in NFGMS. We provide examples of ideal convergent NFGMS and discuss their algebraic and topological properties. Additionally, we investigate the I2-limit and I2-cluster points of sequences. We also present I2-statistical convergence, I2-lacunary statistical convergence, strong I2-Cesàro summability, and strong I2-lacunary summability with order q in NFGMS, along with their associated properties. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of I and I*- convergence of sequences of bi-complex number. We study some basic properties and implication relations of the newly defined convergence concepts. We introduce the notions of I and I*- Cauchy sequences of bi-complex number and investigate the relations between them. Also, we introduce the notion of I-limit point, I-cluster point and prove certain properties of both the notions. [ABSTRACT FROM AUTHOR]

In this paper, by using the notions of λ-density, (V , λ) -summability and Orlicz function ϕ, we introduce a new concept of convergence, namely λ-statistically ϕ-convergence, as a generalization of the λ-statistical convergence and statistically ϕ-convergence. Based on this concept, we introduce a new sequence space S λ - ϕ and investigate some of its properties. Also, we find its relations with statistically ϕ-convergence, [ C , 1 ] ϕ -summability and [ V , λ ] ϕ -summability. Finally, we introduce and investigate the concept of S λ - ϕ Cauchy sequences. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the p -generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions : c × ⋯ × c ︸ p (total distance) : = ∑ ♣ 1 , ⋯ , ♣ p ∈ Z ν ♣ 1 + ⋯ + ♣ p = total distance ∏ j = 1 p c (♣ j). In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for p = 2 [11] , and the combinatorics become harder for larger values of p. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, p = 3. Next, we prove existence and uniqueness in the general case p ≥ 2 , which then covers the remaining cases p ≥ 4. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general p. [ABSTRACT FROM AUTHOR]

This paper's main purpose is to present λ²-statistically convergent and condition of being λ²-statistically Cauchy for double sequences in fuzzy nnormed spaces. At the same time, in fuzzy n-normed space, we have introduced the concept of λ²-summability and Cesaro summability for double sequences in fuzzy n normed spaces. Then, we studied the relation between these concepts and statistical convergence in fuzzy n-normed space. [ABSTRACT FROM AUTHOR]

In present paper, we aim to define a new summability method using σ-mean called σ-statistical summability in an intuitionistic fuzzy 2-normed space (briefly IF 2N S). We also define σ-statistical Cauchy sequence and Sσ-completeness in an IF 2NS and study some of their properties. We display an example that shows our method of summability is more stronger in these spaces. [ABSTRACT FROM AUTHOR]

In this paper, within framework credibility theory, we examine several notions of convergence and almost convergence of fuzzy variable sequences. We investigate relations between these notions. Utilizing fuzzy variables, the almost convergence with regards to (w.r.t.) almost surely, credibility, mean, distribution, and uniformly almost surely are investigated. We also examine almost Cauchy sequence types in credibility theory and obtain significant results. [ABSTRACT FROM AUTHOR]

In this study, we introduced the concepts of rough convergence, rough Cauchy double sequence, and the set of rough limit points of a double sequence, as well as the rough convergence criteria associated with this set in n-normed spaces. Later, we proved that this set is both closed and convex. Finally, we presented the relationships between rough convergence and rough Cauchy double sequence in n-normed spaces. [ABSTRACT FROM AUTHOR]

In this paper, the semi-local convergence of the Cordero's sixth-order iterative method in Banach space was proved by the method of recursion relation. In the process of proving, the auxiliary sequence and three increasing scalar functions can be derived using Lipschitz conditions on the first-order derivatives. By using the properties of auxiliary sequence and scalar function, it was proved that the iterative sequence obtained by the iterative method was a Cauchy sequence, then the convergence radius was obtained and its uniqueness was proven. Compared with Cordero's process of proving convergence, this paper does not need to ensure that G (s) is continuously differentiable in higher order, and only the first-order Fréchet derivative was used to prove semi-local convergence. Finally, the numerical results showed that the recursion relationship is reasonable. [ABSTRACT FROM AUTHOR]

In this current paper, our objective is to establish neutrosophic n-normed linear space and introduce the convergence structure within these spaces. We also define Cauchy sequences, and obtained the relations between these notions.

This paper is devoted to showing the relevance of the notion of completeness used to establish a fixed point theorem in fuzzy metric spaces introduced by Kramosil and Michalek. Specifically, we show that demanding a stronger notion of completeness, called p-completeness, it is possible to relax some extra conditions on the space to obtain a fixed point theorem in this framework. To this end, we focus on a fixed point result, proved by Mihet for complete non-Archimedean fuzzy metric spaces (Theorem 1). So, we define a weaker concept than the non-Archimedean fuzzy metric, called t-strong, and we establish an alternative version of Miheţ's theorem for p-complete t-strong fuzzy metrics (Theorem 2). In addition, an example of t-strong fuzzy metric spaces that are not non-Archimedean is provided. [ABSTRACT FROM AUTHOR]

In this study, we have defined the concepts of invariant continuity, invariant compactness, invariant boundedness and invariant Cauchy sequence in normed linear spaces. In general, there is no relation between continuity and invariant continuity. We have proved that if f is a linear map, then continuity of f implies invariant continuity of f. Additionally, we have shown that continuity of f and invariant continuity of f is equal under a condition. Also, we have proved that every invariant convergent sequence is invariant Cauchy. Finally, we have proved that invariant continuous image of an invariant compact space is invariant compact. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to define λ-statistical summability, Vλ-statistical summability in neutrosophic-2-normed spaces (briefly called N - 2 - NS) and study some relationships among these notions. We give an example that shows λ-statistical summability is stronger method in neutrosophic-2-normed spaces. Finally, we define λ-statistically Cauchy sequence and λ-statistically completeness in neutrosophic-2-normed spaces and obtain the Cauchy convergence criteria in these spaces. [ABSTRACT FROM AUTHOR]

Investigating the concepts of N örlund I2-lacunary statistical convergence and N örlund I2--lacunary statistical convergence of double sequence in intuitionistic fuzzy normed spaces (IFNS) is the goal of this article. Then, we introduce the concepts of N örlund strongly I2 -lacunary convergence, N örlund strongly I2-lacunary Cauchy, and N örlund strongly I2 -lacunary Cauchy double sequences in IFNS and get astounding results. [ABSTRACT FROM AUTHOR]

In this study, we define lacunary Δm-statistical convergence in the framework of intuitionistic fuzzy normed spaces (IFNS) for triple sequences. We prove several results for lacunary Δm-statistical convergence of triple sequence in IFNS. We further established lacunary Δm-statistical Cauchy sequences and provided the Cauchy convergence criterion for this novel idea of convergence. [ABSTRACT FROM AUTHOR]

In this paper the definition of fuzzy anti-norm is modified the authors have studied some properties of convergence and continuity in fuzzy anti-normed spaces. Firstly, the authors have given some definitions and theorems, such as fuzzy anti-normed, fuzzy anti–continuous, and fuzzy anti-convergent sequence and Cauchy sequence on fuzzy anti-normed space and definitions of open ball and closed ball. The authors have presented some examples by using these definitions. Additionally, the authors have given a definition, sequentially fuzzy anticontinuous. and some of their properties are studied. [ABSTRACT FROM AUTHOR]

The purpose of this work is to define the generalized b-Menger spaces and prove a fixed point theorem in this new setting. As application, we establish the existence and uniqueness of a solution for Volterra type integral equation. Our results extend and generalize the existing results in literature. [ABSTRACT FROM AUTHOR]

The factor group of a Hausdorff topological Abelian group by a discrete subgroup is complete iff the topological group is complete and the factor ring of a Hausdorff topological ring by a discrete ideal is complete iff the topological ring is complete. [ABSTRACT FROM AUTHOR]

Phu [16] introduced the generalized version of the concept of ordinary convergence of sequences as rough convergence of sequences in a normed linear space. Here using the idea of Phu we have introduced the idea of rough convergence of sequences in an S-metric space and discussed some of the fundamental properties of rough limit set for some degree of roughness. It has been shown that for a certain degree of roughness, a sequence that is rough convergent may not necessarily be convergent. [ABSTRACT FROM AUTHOR]

The existence of mild solutions and approximate controllability for Riemann-Liouville fractional neutral evolution systems with nonlocal conditions of a fractional order is investigated. The Laplace transform and semigroup theory are the tools used to prove the existence. In turn, approximate controllability is proved on the basis of a Nemytskii operator, a Mittag-Leffler function and certain hypotheses using fixed point theorems, as well as the construction of a Cauchy sequence. An example is provided to highlight the main results. [ABSTRACT FROM AUTHOR]

In this current paper, our objective is to establish neutrosophic n-normed linear space(briefly called N -n-NLS) and introduce the convergence structure within these spaces. We also define Cauchy sequences, completeness in N - n - NLS and obtained the relations between these notions. [ABSTRACT FROM AUTHOR]

In this article, we have defined the ideal Cauchy and ideal bounded double sequences on L-fuzzy normed spaces, which are generalizations of normed spaces, fuzzy normed spaces and intuitionistic fuzzy normed spaces, with the help of ideal convergence of double sequences, which can be accepted as a generalization of known convergence types, and examined their relationships. [ABSTRACT FROM AUTHOR]

In the study conducted here, we have given some new concepts in summability theory. In this sense, firstly, using the lacunary sequence we have given the concept of strongly Iθ2*-convergence and we have examined the relations between Iθ2*-convergence and strongly Iθ2*-convergence and also between strongly Iθ2-convergence and strongly Iθ2*-convergence. Also, using the lacunary sequence we have given the concept of strongly Iθ2*-Cauchy sequence and examined the relations between strongly Iθ2-Cauchy sequence and strongly Iθ2*-Cauchy sequence. [ABSTRACT FROM AUTHOR]

In the study conducted here, we have given some new concepts in summability. In this sense, firstly, we have given the concept of lacunary I2 -convergence and we have investigated the relations between lacunary I2-convergence and lacunary I2 -convergence. Also, we have given the concept of lacunary I2-Cauchy sequence and investigated the relations between lacunary I2-Cauchy sequence and lacunary I2-Cauchy sequence. [ABSTRACT FROM AUTHOR]

The goal of this research is to create a new general family of Topp-Leone distributions called the Topp-Leone Cauchy Family (TLC), which is exceedingly versatile and results from a careful merging of the Topp-Leone and Cauchy distribution families. Some of the new family's theoretical properties are investigated using specific results on stochastic functions, quantile functions and associated measures, generic moments, probability weighted moments, and Shannon entropy. A parametric statistical model is built from a specific member of the family. The maximum likelihood technique is used to estimate the model's unknown parameters. Furthermore, to emphasize the new family's practical potential, we applied our model to two real-world data sets and compared it to existing rival models. [ABSTRACT FROM AUTHOR]

The article deals with the approximate controllability of Atangana–Baleanu semilinear control systems. The outcomes are derived by applying Gronwall's inequality and Cauchy sequence, and avoid the use of the fixed point theorem. We have also included an example for the validation of theoretical results. [ABSTRACT FROM AUTHOR]

We show ideal convergence (I-convergence), ideal Cauchy (I-Cauchy) sequences, I*- convergence and I*-Cauchy sequences for double sequences in fuzzy metric spaces. We define the I-limit and I-cluster points of a double sequence in these spaces. Afterward, we provide certain fundamental properties of the aspects. Lastly, we discuss whether the phenomena should be further investigated. [ABSTRACT FROM AUTHOR]

The aim of this paper was to develop the summability literature by introducing the concept of I p -convergence in the partial metric space (X, p). First, we give some properties of I p -convergence. Also, we introduce the concept of I p ∗ -convergence in the partial metric space (X, p) and examine relations between newly defined concepts. Then, we present the concepts of I p -Cauchy and I p ∗ -Cauchy sequence in the partial metric space (X, p) and investigate relations between these Cauchy sequences. [ABSTRACT FROM AUTHOR]