In this paper, we introduce the concepts of $\mathcal{I}$ and $\mathcal{I^{*}}-$convergence of sequences in gradual normed linear spaces. We study some basic properties and implication relations of the newly defined convergence concepts. Also, we introduce the notions of $\mathcal{I}$ and $\mathcal{I^{*}}-$Cauchy sequences in the gradual normed linear space and investigate the relations between them.
In this paper, we present the notion of quasi-statistical convergence of triple sequences in the neutrosophic normed spaces mainly as a generalization of statistical convergence of triple sequences. We investigate a few principal properties of the newly presented notion and investigate the relationship with statistical convergence of triple sequences in the neutrosophic normed spaces. In the end, we introduce the concept of quasi-statistical Cauchy sequence of triple sequences and show that quasi-statistical Cauchy sequences for triple sequences are equivalent to quasi-statistical convergent triple sequences in the neutrosophic normed spaces.
In this paper, we introduce and investigate the concept of \(A^{\mathcal{I^{K}}}\)-summability as an extension of \(A^{\mathcal{I^{*}}}\)-summability which was recently (2021) introduced by O.H.H.~Edely, where \(A=(a_{nk})_{n,k=1}^{\infty}\) is a non-negative regular matrix and \(\mathcal{I}\) and \(\mathcal{K}\) represent two non-trivial admissible ideals in \(\mathbb{N}\). We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that \(A^{\mathcal{K}}\)-summability always implies \(A^{\mathcal{I^{K}}}\)-summability whereas \(A^{\mathcal{I}}\)-summability not necessarily implies \(A^{\mathcal{I^{K}}}\)-summability. Finally, we give a condition namely \(AP(\mathcal{I},\mathcal{K})\) (which is a natural generalization of the condition \(AP\)) under which \(A^{\mathcal{I}}\)-summability implies \(A^{\mathcal{I^{K}}}\)-summability.
In this paper, we introduce and investigate the notion of lacunary statistical convergence of sequences in gradual normed linear spaces. We study some of its basic properties and some inclusion relations. In the end, we introduce the notion of lacunary statistical Cauchy sequences and prove that it is equivalent to the notion of lacunary statistical convergence.
In this paper we investigate the notion of I-statistical ϕ-convergence and introduce IS-ϕ limit points and IS-ϕ cluster points of real number sequence and also studied some of its basic properties.
Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of
ℐ
{\mathcal{I}}
-statistical supremum and
ℐ
{\mathcal{I}}
-statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of
ℐ
{\mathcal{I}}
-statistical monotonicity and establish the condition under which an
ℐ
{\mathcal{I}}
-statistical monotonic sequence is
ℐ
{\mathcal{I}}
-statistical convergent. We end up giving a necessary and a sufficient condition for the
ℐ
{\mathcal{I}}
-statistical convergence of a real-valued sequence.
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