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A Framework for I*-Statistical Convergence of Fuzzy Numbers
(Mdpi, 2024) Tanushri; Ahmad, Ayaz; Esi, Ayhan
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*-lim in f.
A new notion of convergence in gradual normed linear spaces
(Institute of Mathematics, 2024) Choudhury, Chiranjib; Debnath, Shyamal; Esi, Ayhan
In this paper we introduce the notion of Iλ −statistical convergence of sequences as one of the extensions of I− statistical convergence in the gradual normed linear spaces. We investigate some fundamental properties of the newly introduced notion and its relationship with I− statistical convergence. In the end, we introduce and investigate the concept of Iλ −statistical limit points, cluster points and establish some implication relations. © 2024, Institute of Mathematics. All rights reserved.
A new type of g-neutrosophic metric spaces with order n
(Springer Heidelberg, 2025) Khan, Vakeel A.; Kamran, Mohd; Arshad, Mohammad; Esi, Ayhan
In this paper, we define the concept of the g-neutrosophic metric spaces of order n as a generalization of neutrosophic metric space. We described some properties of this novel space and construct examples based on it, then we propose the concept of statistical convergence. Also, we define the statistical convergence and statistical Cauchy sequences in g-neutrosophic metric spaces. Lastly, we offer an example to demonstrate our theorem.
A study of triple sequence spaces in fuzzy anti-normed linear spaces
(Univ Nis, Fac Sci Math, 2023) Pandit, Shailendra; Ahmad, Ayaz; Esi, Ayhan
The proposed article aims to study some topological properties of triple real sequence spaces with respect to fuzzy-anti-normed linear space (FANLS). In this work the algebra of fuzzy I lambda-limit and fuzzy I lambda-anti limit, where 0 < lambda < 1 and I is an ideal on N3, of triple real sequences along with an interesting example have been studied. Furthermore, the completeness of a special kind of sequence space with respect to fuzzy anti-norm has been examined.
A Study of Triple Sequences Statistical Convergence in Neutrosophic Normed Spaces
(Univ Craiova, 2024) Shankar, Hari; Ahmad, Ayaz; Esi, Ayhan
Neutrosophic logic, probability, and sets are all included in this discipline. The generalization of conventional sets, fuzzy sets, intuitionistic fuzzy sets, and other related ideas is the neutrosophic set theory. It is a mathematical concept that deals with situations involving inconsistent, ambiguous, and imprecise data. To fully understand sequence spaces, statistical convergence is a crucial concept. In this particular scientific work, we introduce the notion of statistical convergence for triple sequences in the neutrosophic normed space. In this neutrosophic normed space, we also explore the statistical properties of completeness and triple Cauchy sequences.
Advances in rough ideal convergence within L-fuzzy normed spaces
(University of Nis, 2026) Hossain, Nesar; Tripathy, Binod Chandra; Esi, Ayhan
This study introduces and develops the concept of rough ideal convergence for sequences in L-fuzzy normed spaces. Within this framework, it defines the notion of the rough I-limit set and investigates its topological and geometrical properties. The concept of I-boundedness for sequences is also introduced, and its relationship with the rough I-limit set is thoroughly examined. Furthermore, the study defines the rough I-cluster point of a sequence and derives several notable results concerning closed balls and their connection to the rough I-limit set in the context of L-fuzzy normed spaces. © 2026, University of Nis. All rights reserved.
ANALYZING LACUNARY △m-STATISTICAL CONVERGENCE IN NEUTROSOPHIC NORMED SPACES
(Amer Inst Mathematical Sciences-Aims, 2026) Shankar, Hari; Ahmad, Ayaz; Chaurasiya, Chandan; Esi, Ayhan
. This paper explores the notions of lacunary triangle m-statistical convergence, lacunary triangle m-statistical Cauchy sequences, and lacunary strongly triangle m-convergence within the framework of neutrosophic normed linear spaces (NNLS). We establish the uniqueness of limit and certain algebraic properties of lacunary triangle m-statistical convergence in NNLS, offering a rigorous foundation for these concepts. Additionally, we introduce the concept of strong Ces & aacute;ro summability in NNLS and conduct an in-depth analysis of its properties. The connections between lacunary strong triangle m-convergence and Ces & aacute;ro summability are also examined, revealing significant interrelationships between these two convergence methods.
Assessing the Fuzzy n-Inner Product Spaces and Its Impact on Linear Operators
(Soc Paranaense Matematica, 2026) Ahmad, Md Arman; Ahmad, Ayaz; Esi, Ayhan
Fuzzy n-inner product space (f-n-IPS) and its concept induced fuzzy n-normed linear space (f-n-NLS), which generalizes the classical n-inner product spaces to the fuzzy setting, and investigates what effects these generalized fuzzy structures have on the properties of linear operators. We construct a specific f-n-IPS and obtain some fundamental properties that relate the fuzzy inner product to the fuzzy norm. Then, the paper characterizes fuzzy bounded and fuzzy continuous linear operators T : (X, N-1) -> (Y, N-2) between these fuzzy spaces. One of the surprising findings is that, in this fuzzy context, a fuzzy bounded linear operator is always fuzzy continuous; however, the converse may not be true in general, which represents a critical difference from the classic theory of operators. Nevertheless, we prove that in the case when the space X is finite-dimensional, fuzzy continuity yields fuzzy boundedness. Finally, this study generalizes basic concepts from classical operator theory, such as adjoint, self-adjoint, normal, and unitary operators, to this new fuzzy context. With this work, a solid framework for the study of linear operators in imprecise contexts is laid, which provides the possibility for investigation into areas such as fuzzy differential equations and signal processing under uncertainty. Through a rigorous investigation of operator behavior in these uncertain situations, it represents a significant contribution to the growing field of fuzzy mathematics. We provide a basis for possible applications in fuzzy modeling, uncertain signal processing, and multidimensional decision-making systems.
Asymptotically double λ2-statistically equivalent sequences of interval numbers
(Editura Academiei Române, 2020) Esi, Ayhan; Debnath, Shyamal; Saha, Subrata
In this paper we have introduced the concept of ?2-asymptotically double statistical equivalent of interval numbers and strong ?2-asymptotically double statistical equivalent of interval numbers. We have investigated the rela-tions related to these spaces. © 2020, Publishing House of the Romanian Academy. All rights reserved.
AUTOMATED ASSESSMENT AND OPTIMIZATION OF FUZZY LOGIC IN METRIC SPACE APPLICATIONS
(Editura Bibliotheca-Bibliotheca Publ House, 2024) Ahmad, Arman; Ahmad, Ayaz; Esi, Ayhan
. In this paper, for fuzzy metric spaces we will attempt to demonstrate some novel fixed-point theorems that is common. To support our findings with evidence, we employ the concept of compatibility. The previous review found several useful mutual fixedpoint theorems for metric and fuzzy metric spaces. Our results greatly expand and strengthen these theorems. Reconsideration of fuzzy metric spaces is done by this study so that fuzzy scalars are utilized to construct fuzzy metric spaces rather than real numbers or fuzzy numbers. In contrast to the preceding work, which defined fuzzy metric spaces using fuzzy or real numbers, this definition uses fuzzy numbers. It has been demonstrated that any complete regular metric space can, under certain conditions, complete fuzzy metric space are given. In addition, we provide evidence that the fuzzy metric spaces generate fuzzy topology that is specified in this study and is consistent with previously presented fuzzy topologies. In the coming years, these findings will pave the way for additional research into imperfect optimization and pattern recognition.
Continuous and bounded linear operators in neutrosophic normed spaces
(IOS Press, 2021) Khan, Vakeel A.; Esi, Ayhan; Ahmad, Mobeen; Daud Khan, Mohammad
In this article, we show that the addition and scalar multiplication in neutrosophic normed spaces are continuous. The neutrosophic boundedness and continuity of linear operators between neutrosophic normed spaces are examined. Moreover, we analyzed that the set of all neutrosophic continuous linear operators and the set of all neutrosophic bounded linear operators from neutrosophic normed spaces into another are vector spaces.
Deferred Riesz Mean in Complex Uncertain Sequences
(CRC Press, 2025) Sharma, Sonali; Raj, Kuldip; Esi, Ayhan
[Abstract Not Available]
DEFERRED STATISTICAL CONVERGENCE IN NEUTROSOPHIC 2–NORMED SPACES
(Ele-Math, 2024) Hossain, Nesar; Esi, Ayhan
In this research paper, we present a circumstantial investigation of deferred statistical convergence and prove some fundamental results within the framework of a neutrosophic 2normed space. Furthermore, we define and provide a comprehensive exploration of deferred statistical Cauchy sequence and establish that every neutrosophic 2-normed space is deferred statistically complete within this specific framework. © D l,Zagreb Paper JCA-25-03.
DEFERRED STATISTICAL CONVERGENCE IN NEUTROSOPHIC 2–NORMED SPACES [2]
(Ele-Math, 2025) Hossain, Nesar; Esi, Ayhan
In this research paper, we present a circumstantial investigation of deferred statistical convergence and prove some fundamental results within the framework of a neutrosophic 2-normed space. Furthermore, we define and provide a comprehensive exploration of deferred statistical Cauchy sequence and establish that every neutrosophic 2-normed space is deferred statistically complete within this specific framework. © EEMEN, Zagreb.
Editorial Conclusion for the Special Issue Applications of Symmetric Functions Theory to Certain Fields
(Mdpi, 2023) Araci, Serkan; Esi, Ayhan
[Abstract Not Available]
Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective
(Mdpi, 2023) Shaba, Timilehin Gideon; Araci, Serkan; Adebesin, Babatunde Olufemi; Esi, Ayhan
This research article introduces a novel operator termed q-convolution, strategically integrated with foundational principles of q-calculus. Leveraging this innovative operator alongside q-Bernoulli polynomials, a distinctive class of functions emerges, characterized by both analyticity and bi-univalence. The determination of initial coefficients within the Taylor-Maclaurin series for this function class is accomplished, showcasing precise bounds. Additionally, explicit computation of the second Hankel determinant is provided. These pivotal findings, accompanied by their corollaries and implications, not only enrich but also extend previously published results.
Exploring Bounded Linear Operators in Neutrosophic Normed Linear Spaces
(Tsinghua Univ Press, 2025) Chaurasiya, Chandan; Ahmad, Ayaz; Shankar, Hari; Esi, Ayhan
This study focuses on analyzing the convergence of the product of sequences and investigating the Cauchy sequences under specific conditions within neutrosophic normed linear spaces (NNLS). Additionally, we define various types of bounded linear operators (both strong and weak) that constitute a linear space, and we explore the concept of neutrosophic compact linear operators.
Further results on I-deferred statistical convergence
(Univ Nis, Fac Sci Math, 2024) Choudhury, Chiranjib; Debnath, Shyamal; Esi, Ayhan
For a non-empty set X, an ideal I represents a family of subsets of X that is closed under taking finite unions and subsets of its elements. Considering X = N, in the present study, we set forth with the new notion of I-deferred statistical limit point, I-deferred statistical cluster point and study various properties of the newly introduced notion. For a real valued sequence x = (xn), we prove that every I-deferred statistical limit point is an I-deferred statistical cluster point. Moreover, the collection of all I-deferred statistical cluster points of x is a closed subset of R. We also introduce the notion of I-deferred statistical limit superior and inferior for real valued sequences and prove several interesting properties. In the end, we establish a necessary and sufficient condition under which a I-deferred statistically bounded real valued sequence is I-deferred statistically convergent.
GEOMETRIC PROPERTIES AND COMPACT OPERATOR ON FRACTIONAL RIESZ DIFFERENCE SPACE
(Univ Kragujevac, Fac Science, 2023) Yaying, Taja; Hazarika, Bipan; Esi, Ayhan
In this article we introduce the Riesz difference sequence space r(p)(q)(Delta(B alpha)) of fractional order a, defined by the composition of fractional backward difference operator Delta(B alpha) given by (Delta(B alpha)upsilon)(k) = Sigma(8)(i=0)(-1)(i)/Gamma(alpha+1) /iota!Gamma(alpha-i+1) upsilon(k- i) and the Riesz matrix Rq. We give some topological properties, obtain the Schauder basis and determine the alpha-, beta- and gamma- duals and investigate certain geometric properties of the space r(p)(q)(Delta(B alpha)). Finally, we characterize certain classes of compact operators on the space r(p)(q)(Delta(B alpha)) using Hausdorff measure of non-compactness.
IDEAL CONVERGENCE IN NEUTROSOPHIC n-NORMED LINEAR SPACES
(Palestine Polytechnic University, 2025) Hossain, Nesar; Esi, Ayhan
In this research paper, we present a thorough exploration of I-convergence and I-Cauchy sequence in neutrosophic n-normed linear spaces. Furthermore, we prove some interest-ing results associated with this notion. Additionally, we define the concept of I∗-convergence and I∗-Cauchy sequence and establish the connection between these notions within this specific framework. © Palestine Polytechnic University-PPU 2025.

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