A new notion of convergence defined by weak Fibonacci lacunary statistical convergence in normed spaces Ibrahim S. Ibrahim, María C. Listán-García, Rifat Colak Journal of Applied Analysis, 2025 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.
New Perspectives on Generalised Lacunary Statistical Convergence of Multiset Sequences María C. Listán-García, Ömer Kişi, Mehmet Gürdal Mathematics, 2025 This paper explores the concepts of J-lacunary statistical limit points, J-lacunary statistical cluster points, and J-lacunary statistical Cauchy multiset sequences. Building upon previous work in the field, we investigate the relationships between J-lacunary statistical convergence and J*-lacunary statistical convergence in multiset sequences. The findings contribute to a deeper understanding of the convergence behaviour of multiset sequences and provide new insights into the application of ideal convergence in this context.
A New Notion of Convergence Defined by The Fibonacci Sequence: A Novel Framework and Its Tauberian Conditions Ibrahim S. Ibrahim, María C. Listán-García Mathematics, 2024 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.
On statistical convergence and strong Cesàro convergence by moduli for double sequences Fernando León-Saavedra, María del Carmen Listán-García, María del Pilar Romero de la Rosa Journal of Inequalities and Applications, 2022 A remarkable result on summability states that the statistical convergence and the strong Cesàro convergence are closely connected. Given a modulus function f, we will establish that a double sequence that is f-strong Cesàro convergent is always f-statistically convergent. The converse, in general, is false even for bounded sequences. However, we will characterize analytically the modulus functions f for which the converse of this result remains true. The results of this paper adapt to several variables the results obtained in (León-Saavedra et al. in J. Inequal. Appl. 12:298, 2019).
General methods of convergence and summability Francisco Javier García-Pacheco, Ramazan Kama, María del Carmen Listán-García Journal of Inequalities and Applications, 2021 This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\\ell _{\\infty }(X)$ ℓ ∞ ( X ) is a space of convergence associated with any free ultrafilter of $\\mathbb{N} $ N ; and that if X is not complete, then $\\ell _{\\infty }(X)$ ℓ ∞ ( X ) is never the space of convergence associated with any free filter of $\\mathbb{N} $ N . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\\ell _{\\infty }(X)$ ℓ ∞ ( X ) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$ c ( X ) is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$ c ( X ) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set $\\mathcal{HB}(\\lim ):= \\{T\\in \\mathcal{B} (\\ell _{\\infty }(X),X): T|_{c(X)}= \\lim \\text{ and }\\|T\\|=1\\}$ HB ( lim ) : = { T ∈ B ( ℓ ∞ ( X ) , X ) : T | c ( X ) = lim and ∥ T ∥ = 1 } and prove that $\\mathcal{HB}(\\lim )$ HB ( lim ) is a face of $\\mathsf{B} _{\\mathcal{L}_{X}^{0}}$ B L X 0 if X has the Bade property, where $\\mathcal{L}_{X}^{0}:= \\{ T\\in \\mathcal{B} (\\ell _{\\infty }(X),X): c_{0}(X) \\subseteq \\ker (T) \\} $ L X 0 : = { T ∈ B ( ℓ ∞ ( X ) , X ) : c 0 ( X ) ⊆ ker ( T ) } . Finally, we study the multipliers associated with series for the above methods of convergence.
