No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Concave modulars
In this paper, the concepts of Riesz lacunary statistical convergence, Riesz lacunary strong convergence, and Riesz lacunary uniform integrability of real sequences within the framework of power series are introduced and studied. Fundamental relationships among these notions are established, particularly in terms of uniform integrability. A characterization of Riesz lacunary uniform integrability is provided. Furthermore, these definitions by incorporating a modulus function are explored, and significant interrelations among these concepts are investigated.
The key ideas of summability theory have been the subject of extensive investigation in recent years in a variety of metric space extensions. Octonion-valued metric spaces are based on modifying the triangle inequality of a semi-metric space by multiplying one side of the inequality by a scalar b. This new generalisation of metric spaces is very interesting since octonions are not even a ring since they do not have the associative property of multiplication and the spaces do not satisfy the standard triangle inequality. We are prompted by this to study the notions of strong I-Cesàro summability, I-statistical convergence, I-lacunary statistical convergence, and similar notions that respect the modulus function in octonion valued b-metric spaces, an extension of metric spaces. We also examine the connections among these ideas.
In this study, the sequence sets ?p(Gk, v,?(r,s)) and ??(Gk, v,?(r,s)) which are depended on the Lucas band matrix and a sequence of modulus functions, are presented. After that, a few inclusion relationships of these sequence sets are given. Furthermore, a geometrical property such as the modulus of convexity of the sequence set ?p(Gk, v,?(r,s)) is discussed.
In this article we have introduced the Cartesian product sequence spaces , and related to cone metric space defined by the Orlicz function. We have established different properties like solidity, completeness etc. and also we obtain some inclusion results involving the spaces , and .
The paper aims to investigate λ -statistical convergence using modulus function and a generalized difference operator for double sequences of functions for order γ ∈ (0, 1]. Further, we prove that the statistical convergence in the newly formed sequence spaces is not well defined for γ > 1. Finally, we examine relevant inclusion relations concerning λ -statistical convergence and strongly λ -summable in the environment of the newly defined classes of double sequences of functions. Some interesting examples related to the examined results are also discussed in this paper.
Statistical convergence and some related types of convergence are a generalisation of topological convergence. Similarly, soft set theory, introduced by Molodtsov to deal with uncertainty in various scientific fields, is a generalisation of the classical concept of sets. Although both concepts have found extensive applications to various mathematical structures, the investigation of statistical convergence within soft topological spaces has not yet been undertaken. This study examines the lacunary statistical convergence of sequences of soft points in soft topological spaces, employing a density defined by an unbounded modulus function. Basic results and inclusion theorems concerning this convergence are presented.
Studies on difference sequences was introduced in the 1980s, and since then, many mathematicians have studied this kind of sequences and obtained some generalized difference sequence spaces. In this paper, using the generalized difference operator, we introduce the concept of the deferred f-statistical convergence of generalized difference sequences of the order α and give some inclusion relations between the deferred f-statistical convergence of generalized difference sequences and deferred f-statistical convergence of generalized difference sequences of the order α. Our results are more general than the corresponding results in the existing literature.
This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.
In this paper, the Fibonacci sequence, renowned for its significance across various fields, its ability to illuminate numerical concepts, and its role in uncovering patterns in mathematics and nature, forms the foundation of this research. This study introduces innovative concepts of weighted density, weighted statistical summability, weighted statistical convergence, and weighted statistical Cauchy, uniquely defined via the Fibonacci sequence and modulus functions. Key theorems, relationships, examples, and properties substantiate these novel principles, advancing our comprehension of sequence behavior. Additionally, we extend the notion of statistical cluster points within a broader framework, surpassing traditional definitions and offering deeper insights into convergence in a statistical context. Our findings in this paper open avenues for new applications and further exploration in various mathematical fields.
This paper introduces the neutrosophic -statistical convergent difference sequence spaces defined through a modulus function. Additionally, we establish new topological spaces and examine various topological properties within these neutrosophic -statistical convergent difference sequence spaces.
In this paper, deferred Riesz statistical convergence as well as -deferred Riesz statistical convergence in terms of power series method for real or complex sequences are introduced and studied. Their interconnection with Riesz statistical convergence is explored and illustrative examples in support of our results are presented. Applications of these convergences in the form of a Korovkin-type approximation theorem are established and illustrations demonstrating the superiority of our proven theorem over the classical Korovkin theorem are offered. Finally, the rate of convergence is computed.
The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.
The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces.
Modifying the definition of density functions is one method used to generalise statistical convergence. In the present study, we use sequences of modulus functions and order to introduce a new density. Based on this density framework, we define strong -lacunary summability of order and -lacunary statistical convergence of order for a sequence of modulus functions . This concept holds an intermediate position between the usual convergence and the statistical convergence for lacunary sequences. We also establish inclusion theorems and relations between these two concepts in the study.
Modulus statistical convergence has been studied in very different general settings such as topological spaces and uniform spaces. In this manuscript, modulus statistical convergence is defined and studied in partial metric spaces.
In this research paper, we introduce some concepts of λf-density in connection with modulus functions under certain conditions. Furthermore, we establish some relations between the sets of λf-statistically convergent and λf-statistically bounded sequences.
In this article, we first put forward the concept of deferred f-double natural density for double sequences, where f is an unbounded modulus. Then, we combine f-density with deferred statistical convergence for double sequences and investigate deferred f-statistical convergence and strongly deferred Cesàro summability with respect to modulus f. Moreover, we extend these concepts to deferred f-statistical convergence for double sequences of random variables in the Wijsman sense and prove some inclusions. Finally, we consider the concepts of deferred f-statistical convergence of order α \alpha and strongly deferred f-summability of order α \alpha for double sequences and obtain some conclusions.
The Orlicz function-defined sequence spaces of functions by relative uniform convergence of sequences related to p-absolutely summable spaces are a new concept that is introduced in this article. We look at its various attributes, such as solidity, completeness, and symmetry. We also look at a few insertional connections involving these spaces.
The central objective of this paper is to investigate the lacunary statistical convergence of order [Formula: see text] and strong lacunary summability of order [Formula: see text] for sequences of fuzzy functions by using the modulus function and a regular matrix. We examine the relevant inclusion relations between these spaces under specific conditions. Additionally, we study properties related to these spaces and establish several interesting results.
In the present paper, we introduce the concept of lacunary statistical boundedness of order \ with respect to a modulus function f for sequences of fuzzy numbers and give some relations between lacunary statistical boundedness of order \ and statistical boundedness with respect to a modulus function f with the help of many examples and figures. Furthermore, we study some properties like solidity, symmetricity, etc.
In this research paper, our aim is to introduce the concepts of -statistical convergence of order and -statistical boundedness of order by using unbounded modulus functions in metric spaces. Furthermore, we obtain some relations between these concepts, and we provide some counterexamples to support our results.
In this manuscript, we present the ideas of asymptotically -equivalence, asymptotically -equivalence, asymptotically -equivalence and asymptotically -equivalence for real sequences. In addition to, investigate some connections among these new ideas and we give some inclusion theorems about them.
In this study, we first give the definition of ρ-statistical convergence of order (α,β) for sequences of fuzzy numbers. We also define the strongly w(ρ,F,q)-summable of order (α,β) and the strongly w(ρ,F,q,f)-summable of order (α,β), defined by a modulus function f for sequences of fuzzy numbers. Later we give some coverage theorems between these sets and the set S_α^β (ρ,F).
Here, we continued the studies initiated by Vinod K. Bhardwaj and Shweta Dhawan which relate different convergence methods involving the classical statistical and the classical strong Cesàro convergences by means of lacunary sequences and measures of density in N modulated by a modulus function f. A method for constructing non-compatible modulus functions was also included, which is related to symmetries with respect to y=x.
In this study, we define a new type of statistical limit point using the notions of statistical convergence with respect to the power series method and then we present some examples to show the relations between these points and ordinary limit points. After that we also study statistical limit points of a sequence with the help of a modulus function in the sense of the power series method. Namely, we define -statistical limit and cluster points of the real sequences and compare the set of these limit points with the set of ordinary points.
Mursaleen introduced the concepts of statistical convergence in random 2-normed spaces. Recently Mohiuddine and Aiyup defined the notion of lacunary statistical convergence and lacunary statistical Cauchy in random 2-normed spaces. In this paper, we define and study the notion of lacunary statistical convergence and lacunary of statistical Cauchy sequences in random on χ 2 over p− metric spaces defined by Musielak and prove some theorems which generalizes Mohiuddine and Aiyup results.
In this paper, we introduced the space of all deferred -statistically convergent sequences of order in the normed space (X, q) with the help of generalized difference operator , unbounded modulus function f, and deferred sequences of non-negative integers with for any , . We also introduced the space of all strongly deferred -summable sequences of order in the normed space (X, q). Inclusion relations between spaces and are established under certain conditions.KeywordsStatistical convergenceModulus functionDifference sequence spaceDeferred sequences
In earlier works, F. León and coworkers discovered a remarkable structure between statistical convergence and strong Cesàro convergence, modulated by a function f (called a modulus function). Such nice structure pivots around the notion of compatible modulus function. In this paper, we will explore such a structure in the framework of lacunary statistical convergence for double sequences and discover that such structure remains true for lacunary compatible modulus functions. Thus, we continue the work of Hacer Şenül, Mikail Et and Yavuz Altin, and we fully solve some questions posed by them.
We introduce the notions of uniform f−statistical convergence of se- quences and uniform f−statistical Cauchy sequences of functions. The natural equivalence between these two notions, in Banach spaces, is proved. In doing so, we generalize and improve results of Gökhan and Güngör.
In this paper, by using unbounded modulus function the notion -density for measurable subsets of is defined and related notion -statistical convergence of measurable functions is investigated. In addition to the basic properties of -density and -statistical convergence, the definition of being -Cauchy is given and it is shown that these two concepts are equivalent for real valued measurable functions. Among others, the notion of f-strongly Cesaro summability is introduced. Finally, necessary and sufficient conditions between f-strongly Cesaro summability and -statistical convergence are given under explicated restrictions on measurable function and modulus function.
In this paper, a comprehensive literature review on the sequence of uncertain variables, complex uncertain variables defined by Orlicz function is realized. We procure different results and developments. In the recent years, ordinary sequences have been extended to new types and these extensions have been used in uncertain environment too, different type of convergence is one of the major developments in this directions. This literature review also analyzes the chronological development of these extensions. In the last section of the paper, we present our interpretations on the future of uncertainty theory.
In this paper, we examine and study statistical convergence of order of sequences of fuzzy numbers such that strong summability of order of sequences of fuzzy numbers such that and construct some interesting examples. We also give relations between these concepts. Furthermore, some relations between the space and are examined. The modulus function presents the containment connection.
The notion of fractional structures has been studied intensely in various fields. Using this concept, the main idea of this paper is to apply the Cesàro approach and introduce the new generalized Δ-structure of spaces on a fractional level. Also, the statistical notions will be studied using this new structure and some inclusion relations will be computed. In addition, the sequence space Wq(Δgκ,f) will be introduced, and some fundamental inclusion relations and topological properties concerning it will be given.
In this study, we discuss the idea of difference operators and examine some properties of these operators. We also describe the concepts of ordered statistical convergence and lacunary statistical by using the -difference operator. We examine some features of these sequence spaces and present some inclusion theorems. We obtain the Caputo fractional derivative in this work.
The classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the f -statistical convergence for f a modulus function. Here, we go several steps further and extend the f -statistical convergence to the scope of uniform spaces, obtaining particular cases of f -statistical convergence on pseudometric spaces and topological modules.
In this manuscript, we introduce the following novel concepts for real functions related to f -convergence and f -statistical convergence: f -statistical continuity, f -statistical derivative, and f -strongly Cesàro derivative. In the first subsection of original results, the f -statistical continuity is related to continuity. In the second subsection, the f -statistical derivative is related to the derivative. In the third and final subsection of results, the f -strongly Cesàro derivative is related to the strongly Cesàro derivative and to the f -statistical derivative. Under suitable conditions of the modulus f , several characterizations involving the previous concepts have been obtained.
We introduce the notion of generalized relative order convergence in (l)-groups by using generalized density. We prove some basic results including a Cauchy-type criterion. Furthermore we present an idea of ideal order convergence of sequences and study some of its properties by using the mathematical tools of the theory of (l)-group
In this paper we introduce the notion of f-partial statistical continuity of a function (where f is an unbounded modulus function) which is much weaker than continuity of a function. We give an example to show that f-partial statistical continuity is weaker than continuity. Then we apply unbounded modulus function to give some answers to farthest point problem in real normed linear space which improves the result in \cite{ni}. As an application, we provide a sufficient condition for the existence of an unique fixed point of a self mapping defined on a non-empty, closed, bounded uniquely remotal subset in Hilbert space. Lastly, we introduce the notion of f-statistically maximizing sequence of a non-empty bounded subset M of a normed linear space and show by an example that this notion is weaker than maximizing sequence.
ÖZET: Bu çalışmada invaryant yakınsaklık yardımı ile bazı dizi uzayları tanımlanarak bazı kapsamlar kuruldu. lσ ve lσσ uzaylarının lσ(p) ve lσσ(p) uzaylarına genelleştirildiği gibi [ωσ], ω̅σveω̿σuzaylarını da [ωσ(p)],ω̅σ(p)veω̿σ(p)uzaylarına genişletildi ve modülüs fonksiyonlar uygulandı.
Anahtar Kelimeler: İnvaryant yakınsaklık, dizi uzayları, modülüs fonsiyonu.
The Space of Modulus Functions Defined by Invariant Convergence
ABSTRACT: : In this study, some scopes are established by defining some sequence spaces with the help of invariant convergence. Just as the spaces lσ and lσσ are generalized to the spaceslσ(p) and lσσ(p), so do the spaces [ωσ], ω̅σ and ω̿σ it has been extended to the spaces [ωσ(p)], ω̅σ (p) ve ω̿σ (p) and modulus functions have been implemented.
Keywords: Invariant convergence, sequence spaces, module function.
In this paper, we introduce some new multiplier spaces related to a series ∑ixi in a normed space X through f–statistical summability and give some characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in X and X*, respectively. We also obtain a new version of the Orlicz-Pettis theorem within the frame of the f–statistical convergence.
This research paper focuses on defining the relationships between the sets of strongly lacunary summable and lacunary statistically convergent sequences of complex numbers by using different modulus functions f and g under certain conditions and different orders α, β ∈ (0, 1] such that α ≤ β. Furthermore, for some special modulus functions, we establish the relations between the sets of strongly f-lacunary summable sequences and strongly f-lacunary summable sequences of order α.
Bu çalışmada Modülüs fonksiyon yardımı ile tanımlanan invaryant yakınsak dizi uzayları tanımlanarak aralarında bazı kapsam bağıntıları kuruldu. [ω_σ (f)],ω ̅_(σ ) (f) ve ω ̿_(σ ) (f) uzayları [ω_σ (f)(p)],ω ̅_(σ ) (f)(p) ve ω ̿_(σ ) (f)(p) uzaylarına genişletildi. Genelleştirilen bu dizi uzaylarının topolojik özellikleri incelendi.
In this paper, we introduce the concepts of asymptotically f‐statistical equivalence, asymptotically f‐lacunary statistical equivalence, and strong asymptotically f‐lacunary equivalence for non‐negative two delta measurable real‐valued functions defined on time scales with the aid of modulus function f. Furthermore, the relationships between these new concepts are investigated. We also present some inclusion theorems.
In this paper, we first define a new density of a -measurable subset of a product time scale with respect to an unbounded modulus function. Then, by using this definition, we introduce the concepts of -statistical convergence and -statistical Cauchy for a -measurable real-valued function defined on product time scale and also obtain some results about these new concepts. Finally, we present the definition of strong -Cesaro summability on and investigate the connections between these new concepts.
Some algebraic properties of Cesáro ideal convergent sequence spaces, and , are studied in this article and some inclusion relations on these spaces are established.
1. Introduction
Consider the space of all real and complex sequences, where and are, respectively, the sets of all real and complex numbers.
Suppose that , c, and are the linear spaces of bounded, convergent, and null sequences, respectively, normed by
being the set of all natural numbers.
A sequence space of complex numbers is said to be (C, 1) summable to if for , limk = L. The sequence (C, 1) is also called Cesáro summable sequence of complex numbers over C. Let us denote by C1 the linear space of all (C, 1) summable sequences of complex numbers over C, i.e.,
Hardy and Littlewood [1] initiated the notion of strong Cesáro convergence for real numbers which is defined as follows.
A sequence () on a normed space (X, |) is said to be strongly Cesáro convergent to L if
In [2–6], the authors have extended the notion of strong Cesáro convergence in various fields. In 1951, Fast [7] introduced the term statistical convergence, while Steinhaus [8] independently introduced the term “ordinary and asymptotic convergences.”
Later on, Fridy [9, 10] also studied the statistical convergence and he linked it with the summability theory. Kostyrko et al. [11] gave the concept of ideal convergence (I- convergence) which was indeed a generalization of statistical convergence. Salat et al. [12] studied some properties of I-convergence, and further investigations in this field are done by Khan [13], Tripathy and Esi [14], Tripathy and Hazarika [15], and many others.
In this article, further interesting properties of Cesáro Ideal Convergent Sequences are established and a few inclusion relations are also proved.
2. Definitions of the Terms Used
Let us first present some definitions and notions that are required in the sequel.(1)A family of subsets I of is called an ideal set in (i)If I(ii)If the sets A, BI, then ABI(iii)If BA and AI, then BI(2)A nontrivial ideal set I is said to be admissible if {{n}: n }}I(3)A nonempty set F is known as a filter in if(a)(b)A, BF ABF(c)AF with AAB BF
Remark 1. For every ideal I, there is a filter F(I) (associated with I) defined as follows: A sequence ()X is said to be I-convergent to a number L if, for every >0, the set {x = () X: {k :|−L| ≥ } I}. In this case, we write I–lim = L. If L = 0, then it is called I-null. A sequence ω is said to be I-Cauchy if, for every > 0, there exists a number m = m() such that Let If be the class of all finite subsets of . If I = If, then I is admissible ideal set in . A sequence space X is said to be solid (normal) if ()X whenever ()X and () is a sequence of scalars with |αk| ≤ 1, for all k . A sequence space X is a Sequence Algebra if, for every (), () X, () X. Let K = and X be a sequence space. A K-step space of X is a sequence space . A canonical preimage of a sequence () is a sequence () defined by A sequence space is monotone if it contains the canonical preimages of its step spaces.
3. Result
A canonical preimage of a step space is a set of preimages of all elements in , i.e., y is in the canonical preimage of if and only if y is the canonical preimage of some x .
Let X and Y be two normed linear spaces. An operator T: X ⟶ Y is known as a compact linear operator if [16].
(a) T is linear
(b) If, for every bounded subset D of X, the image M(D) is relatively compact, i.e, the closure is compact
Lemma 1. (see [12]). Every solid space is monotone.
Lemma 2. (see [12]). Let KF(I) and M . If MI, then MK I.
Lemma 3. (see [11]). Let I and M . If MI, then MKI.
4. Main Results
Let us first define CI, the space of all Cesáro ideal convergent sequences and , the space of all Cesáro ideal null sequences which are given as follows:
Theorem 1. The sequence spaces CI and are linear.
Proof. Assume that x = (), y = () CI. Then, one hasLetLet and be some scalers.
By using the properties of norm, one can easily see that Then, from (9) and (10), we have for each ,Therefore, () CI, for all scalars α, β and (), ()CI.
Hence, CI is a linear space.
On the similar manner, one can prove that is also linear.
Theorem 2. Let x = () be any sequence Then, CI.
Proof. It can be easily observed.
Theorem 3. A sequence x = () CI is I-convergent if and only if, for every there exists l = l() such that
Proof. Suppose that x = () CI. Therefore, Then, for all >0 the setFix an l(). Then, we havewhich holds for all k. Hence,Conversely, suppose that, for all , the setThen, for every , we haveFor fixed one has as well as F(I).
Hence, .
This implies that , that is,That is diam P ≤ diam , where the diam P denotes the length of the interval of P.
In this way, by induction, one obtains the sequence of closed intervals:with the property that diam diam for k = 1, 2, 3, …, andfor k = 1,2,3, ….,. Then, there exists a such that showing that x = ()CI is I-convergent. Hence, the result holds.
Theorem 4. The space is solid and monotone.
Proof. Let () be any element. Then, one hasLet () be a sequence of scalars such that , for all k , and hence .
Then, the result (that is solid) follows from the above equation and inequality:for all k .
The space is monotone which follows from Lemma 1.
Hence, is solid and monotone.
Theorem 5. The space CI is neither solid nor monotone.
Proof. For this theorem, we provide a counter example for the proof.
5. Counter Example
Let I = If, and consider the k-step of defined as follows.
Let () and let () be such that
Let us consider the sequence () defined by for all k . Then, ()CI, but its K-step preimages do not belong to CI. Thus, () CI is not monotone.
Hence, () CI is not solid.
Theorem 6. Let x = () and y = () be two sequences in such a way that . Then, the space CI and are sequence algebra.
Proof. Let x = () and y = () be two elements of CI withFor every > 0 select β > 0 in such a way that < β, thenUsing the above and the property of norm, one obtainsTherefore, the setThus, ().() ICes. Hence, CI is a sequence algebra.
On the similar manner, one can prove that space is also sequence algebra.
Data Availability
The data used to support the findings of the study are obtained from the author upon request.
Conflicts of Interest
The author declares no conflicts of interest.
Authors’ Contributions
The author read and approved the final manuscript.
Acknowledgments
The research work was supported by Saudi Electronic University, Deanship of Scientific Research, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh, Kingdom of Saudi Arabia.
Our aim of this paper is to introduce some new techniques of spaces using modulus function. Some of basic inclusion properties will be taken care of.
In this study, we introduce the concepts of φλ,μ-double statistically convergence of order β in fuzzy sequences and strongly λ- double Cesaro summable of order β for sequences of fuzzy numbers. Also we give some inclusion theorems.
The concept of modulus function was introduced by Nakano [8] in 1953. The intention of this paper is initiate the concept of some new sequence spaces with respect to sequence of modulus functions F = (fk). Also, we discuss about some inclusion relations between these spaces.
The space $L^{P}$ with $0<p<1$, Bull. Amer
- M M Day
M. M. Day: The space $L^{P}$ with $0<p<1$, Bull. Amer. Math. Soc., 46 (1940), 816-
823. C. Sirvint: Espace des fonctionells lin\'eaires, Com. Rend. URSS., 26 (1940),
123-126.