Pratulananda Das

I and I*-convergence of double sequences

The idea of I-convergence was introduced by Kostyrko et al (2001) and also independently by Nuray and Ruckle (2000) (who called it generalized statistical convergence) as a generalization of statistical convergence (Fast (1951), Schoenberg(1959)). For the last few years, study of these convergences of sequences has become one of the most active areas of research in classical Analysis. In 2003 Muresaleen and Edely introduced the concept of statistical convergence of double sequences. In this paper we consider the notions of I and I*-convergence of double sequences in real line as well as in general metric spaces. We primarily study the inter-relationship between these two types of convergence and then investigate the category and porosity position of bounded I and I*-convergent double sequences.

A generalized statistical convergence via ideals

Abstract In this paper we make a new approach to the notions of [ V , λ ] -summability and λ -statistical convergence by using ideals and introduce new notions, namely, I − [ V , λ ] -summability and I − λ -statistical convergence. We mainly examine the relation between these two new methods as also the relation between I − λ -statistical convergence and I -statistical convergence introduced by the authors recently. We carry out the whole investigation in normed linear spaces.

On generalizations of certain summability methods using ideals

In this paper, following the line of Savas and Das (2011) [10], we provide a new approach to two well-known summability methods by using ideals, introduce new notions, namely, I-statistical convergence and I-lacunary statistical convergence, investigate their relationship, and make some observations about these classes.

Fuzzy compactness via summability

Abstract A fuzzy subset α of a fuzzy topological space is fuzzy sequentially compact if any sequence x = ( λ a n x n ) of fuzzy points in α has a generalized fuzzy subsequence which is fuzzy convergent in α . More generally we say that a fuzzy subset α of a fuzzy topological space X is G -fuzzy sequentially compact if any sequence x = ( λ a n x n ) of fuzzy points in α has a generalized fuzzy subsequence y such that G ( y ) ∈ α where G is a function from a suitable subset of the set of all sequences of fuzzy points in X . We investigate some of the basic properties of this compactness and suggest some open problems.

Some further results on I-Cauchy sequences and condition (AP)

In this paper we provide answers to two important questions regarding I and I^*-Cauchy sequences introduced and studied by Nabiev et al. (2007) [9] which were left unanswered. We then introduce the ideas of I and I^*-divergent sequences in a metric space and study their certain properties. Our investigation strengthens and reconfirms importance of condition (AP) in the study of summability through ideals.

Fixed point of contractive mappings in generalized metric spaces

We prove a fixed point theorem for contractive mappings of Boyd and Wong type in generalized metric spaces, a concept recently introduced in [BRANCIARI, A.: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31–37].

On I-convergence of nets in locally solid Riesz spaces

In this paper, following the line of (13) and (6), we introduce the ideas of Iτ-convergence, Iτ- boundedness and Iτ-Cauchy condition of nets in a locally solid Riesz space endowed with a topology τ and investigate some of its consequences.

Statistical convergence of order α in probability

Abstract In this paper the ideas of different types of convergence of a sequence of random variables in probability, namely, statistical convergence of order α in probability, strong p -Ces a ro summability of order α in probability, lacunary statistical convergence or S θ -convergence of order α in probability, and N θ -convergence of order α in probability have been introduced and their certain basic properties have been studied.

Well-posedness and porosity of a certain class of operators

We prove that several fixed point problems are well-posed and study the porosity behaviour of a certain class of operators.

On certain generalized matrix methods of convergence in (ℓ)-groups

Abstract We introduce a convergence of weight g: ℕ → [0, ∞) where g(n) → ∞ and n/g(n) ↛ 0 with respect to a summability matrix method A for sequences (which generalizes the notion of A-convergence of order α, 0 < α ≤ 1 [BOCCUTO, A.—DAS, P.: On matrix methods of convergence of order (α) in (ℓ)-groups, Filomat 29 (2015), 2069–2077] in a (ℓ)-group. We prove some basic results including a Cauchy-type criterion. Finally a closedness result for the space of such convergent sequences is proved.