Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 33 2 (2015): 59–67. ISSN-00378712 in press doi:10.5269/bspm.v33i2.21670 Statistical convergence of double sequences on probabilistic normed spaces defined by [V, λ, µ]-summability Pankaj Kumar, S.S. Bhatia and Vijay Kumar abstract: In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers λ = (λn ) and µ = (µn ) such that each tending to ∞, also λn+1 ≤ λn + 1, λ1 = 1, and µn+1 ≤ µn + 1, µ1 = 1. We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces. Key Words: Statistical convergence; λ-statistical convergence; Probabilistic normed spaces. Contents 1 Introduction 59 2 Background and preliminaries 60 3 Strong (λ, µ)-statistical convergence of double sequences on a PN-space 62 1. Introduction Before we go into the motivation for this paper and present main results, we move through the background of the topic. Menger [12] provoked a crucial generalization of a metric space and called it a probabilistic metric space. This concept was further developed by various authors [2,3,4], [6], [11] and [23,24]. Probabilistic normed space, which is an important family of probabilistic metric spaces, were firstly defined by Šternev [25]. Alsina et al. [1] gave a new definition of probabilistic normed space making Šternev definition a special case. As a result, a productive theory agreeable with ordinary normed spaces and probabilistic metric spaces originated. The notion of statistical convergence of sequence of numbers was introduced by Fast [5] and Schoenberg [22] independently in 1951 and discussed by [7], [13,14], [16,17,18,19,20,21], [26,27], [29] and [31]. During last few years, statistical convergence has been applied in various fields like fourier analysis, ergodic theory and number theory. Mursaleen [15] generalized the notion of statistical convergence with the help of a non-decreasing sequence λ = (λn ) of positive numbers tending to ∞ with λn+1 ≤ λn + 1, λ1 = 1 and called respectively λ-statistical convergence. Karakus extended the concept of the statistical convergence for single and double 2000 Mathematics Subject Classification: 40A05, 40C05, 46A45 59 Typeset by BSP style. M c Soc. Paran. de Mat. 60 Pankaj Kumar, S.S. Bhatia and Vijay Kumar sequences on probabilistic normed spaces in [8] and [9]. Tripathy et al. [28] discussed the double sequence spaces with the help of Orlicz function and in [30], they extended the concept to double sequence spaces of fuzzy numbers. Recently, Kumar and Mursaleen [10] defined (λ, µ)-statistical convergence of double sequences on intuitionistic fuzzy normed spaces. Following Kumar and Mursaleen [10], in this paper, we aim to define strongly (λ, µ)-statistical convergence of double sequences on probabilistic normed spaces. 2. Background and preliminaries First, We recall some notations and basic definitions those will be used in this paper. By a distribution function we mean a function F : R∪{−∞, +∞} → [0, 1] that is left-continuous and non-decreasing on R with F (−∞) = 0 and F (∞) = 1. We normalize all distribution functions to be left continuous on unextended real line R = (−∞, +∞). Moreover, for any a ≥ 0, εa is the distribution function defined by  0, x ≤ a εa (x) = 1, x > a. Let ∆ denotes the set of all the distribution functions, ∆+ = {F : F ∈ ∆ with F (0) = 0} and D+ ⊆ ∆+ is the set D+ = {F ∈ ∆+ : l− F (+∞) = 1} where l− f (x) = limt→x− f (t). For F, G ∈ ∆+ , F ≤ G iff F (x) ≤ G(x) for all x ∈ R and (∆+ , ≤) is a partially ordered set. The maximal element for ∆+ in this order is the d.f. given by  0, x ≤ 0 ε0 (x) = 1, x > 0. Definition 2.1. A triangle function is a mapping τ from ∆+ × ∆+ into ∆+ such that, for all F, G, H, K in ∆+ , (i) τ (F, ε0 ) = F ; (ii) τ (F, G) = τ (G, F ); (iii) τ (F, G) ≤ τ (H, K) whenever F ≤ H, G ≤ K; (iv) τ (τ (F, G), H) = τ (F, τ (G, H)). Particular and relevant triangle functions are the functions τ T , τ T ∗ and those of the form ΠT which, for any continuous t-norm T , and any x > 0, are given by τ T (F, G)(x) = sup{T (F (s), G(t)) : s + t = x} τ T ∗ (F, G)(x) = inf{T ∗ (F (s), G(t)) : s + t = x} and ΠT (F, G)(x) = T (F (x), G(x)). In 1993, using triangle functions, Alsina et al. [1] defined probabilistic normed spaces as follows: Statistical convergence of double sequences 61 Definition 2.2. [1] A probabilistic normed space, briefly PN-space, is a quadruple (V, v, τ , τ ∗) where V is a real linear space, τ and τ ∗ are continuous triangle functions with τ ≤ τ ∗ and v, the probabilistic norm, is a mapping from V into the space of distribution function ∆+ such that writing vp for v(p) for all p, q in V , the following conditions hold: (i) vp = ε0 if and only if p = θ, the null vector in V , (ii) v−p = vp , (iii) vp+q ≥ τ (vp , vq ), (iv) vp ≤ τ ∗ (vαp , v(1−α)p ) for every α ∈ [0, 1]. If, instead of (i), we only have vp = ε0 , then we shall speak of a probabilistic pseudo normed space, briefly a PPN-space. If the inequality (iv) is replaced by the equality vp = τ M (vαp , v(1−α)p ), then the PN-space is called a Šerstnev space, in this case, a condition stronger than (ii) holds, namely j ), ∀λ 6= 0 ∀p ∈ V, vλp = vp ( |λ| here j is the identity map on R. A Šerstnev space is denoted by (V, v, τ ). There is a natural topology in PN-space (V, v, τ , τ ∗ ), called the strong topology. It is defined, for t > 0, by the neighbourhoods Np (t) = {q ∈ V : dS (vq−p , ε0 ) < t} = {q ∈ V : vq−p (t) > 1 − t} S The strong neighbourhood system for V is the union p∈V Np λ where Np = {Np λ : λ > 0}. The strong neighbourhood system for V determines a Hausdroff topology for V . Definition 2.3. Let (V, v, τ , τ ∗ ) be a PN-space. A sequence (pn )n in V is said to be strongly convergent to p in V if for each λ > 0, there exists a positive integer N such that pn ∈ Np (λ), for n ≥ N . Definition 2.4. Let (V, v, τ , τ ∗ ) be a PN-space. A sequence (pn )n in V is called strongly Cauchy sequence if , for every λ > 0, there is a positive integer N such that vpn −pm (λ) > 1 − λ, whenever m, n > N . Definition 2.5. A PN-space (V, v, τ , τ ∗ ) is said to be strongly complete in the strong topology if and only if every strong Cauchy sequence in V is strongly convergent to a point in V . Lemma 2.6. If |α| ≤ |β|, then vβ p ≤ vαp for every p ∈ V . Definition 2.7. The natural density of a set K of positive integers is defined by δ(K) = lim 1 n→∞ n |{k ∈ K : k ≤ n}|. Where |{k ∈ K : k ≤ n}| denotes the number of elements of K not exceeding n. Definition 2.8. Let (V, v, τ , τ ∗ ) be a PN-space. A sequence (pn )n in V is said to be strongly statistical convergent to p in V if for each λ > 0, δ({n ∈ N : pn ∈ / Np (λ)}) = 0. 62 Pankaj Kumar, S.S. Bhatia and Vijay Kumar The element p is called the statistical (...truncated)