Lacunary ideal convergence of multiple sequences in probabilistic normed spaces

Applied Mathematics and Computation 279 (2016) 139–153

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Lacunary ideal convergence of multiple sequences in probabilistic normed spaces Bipan Hazarika∗ Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, India

a r t i c l e

i n f o

MSC: 40G15 46S70 54E70 Keywords: Ideal convergence Double sequence Probabilistic normed space Double lacunary sequence θ -convergence

a b s t r a c t An ideal I is a family of subsets of positive integers N × N which is closed under finite unions and subsets of its elements. The aim of this paper is to study the notion of lacunary I-convergence of double sequences in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary I-limit points and lacunary I-cluster points have been defined and the relation between them has been established. Furthermore, lacunary-Cauchy and lacunary I-Cauchy, lacunary I ∗ -Cauchy, lacunary I ∗ -convergent double sequences are introduced and studied in probabilistic normed spaces. Finally, we provided example which shows that our method of convergence in probabilistic normed space is more general. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Menger [40] proposed the probabilistic concept of the distance by replacing the number d(p, q) as the distance between points p, q by a probability distribution function Fp, q (x). He interpreted Fp, q (x) as the probability that the distance between ˘ p and q is less than x. This led to the development of the area now called probabilistic metric spaces. Serstnev [60] who first used this idea of Menger to introduce the concept of a PN space. In 1993, Alsina et al. [2] presented a new definition of ˘ probabilistic normed space which includes the definition of Serstnev as a special case. For an extensive view on this subject, we refer [1,3,13,21,22,29,34,36,44,49,58,59]. Steinhaus [61] and Fast [16] independently introduced the notion of statistical convergence for sequences of real numbers. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from various points of view. For example, statis˘ tical convergence has been investigated in summability theory by (Connor [11], Fridy [18], Salát [55]), number theory and mathematical analysis by (Buck [4], Mitrinovic´ et al., [42]), topological groups (Çakalli [5,6]), topological spaces (Di Maio and Ko˘cinac [39]), function spaces (Caserta and Ko˘cinac [8]), locally convex spaces (Maddox [38]), measure theory (Cheng et al., [9], Connor and Swardson [12], Miller [41]). Fridy and Orhan [19] introduced the concept of lacunary statistical convergence. Some work on lacunary statistical convergence can be found in [20,25,37,50,52]. Kostyrko et al., [31] introduced the notion of I-convergence as a generalization of statistical convergence which is based on the structure of an admissible ideal I of subset of natural numbers N. Kostyrko et al., [32] gave some of basic properties of I-convergence and dealt with extremal I-limit points. Tripathy and Tripathy [62] introduced the concept of I-convergence of double sequences of real numbers and studied some basic properties of this notion. Mursaleen and Mohiuddine [47] ∗

Tel.: +91 3602278512; fax: +91 3602277881. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.amc.2015.12.048 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

and Rahmat [54] studied the ideal convergence in probabilistic normed spaces and Kumar and Kumar [33] studied ICauchy and I∗ -Cauchy sequences in probabilistic normed spaces. Kumar and Guillén [34] introduced ideal convergence of double sequences in probabilistic normed spaces and proved some interesting results. The notion of lacunary ideal convergence of real sequences was introduced in [10,63] and Hazarika [23,24], introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some properties. Debnath [14] introduced the notion lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. Yamanci and Gürdal [65] introduced the notion lacunary ideal convergence in random n-normed space. Recently, in [28] Hazarika introduced the lacunary ideal convergent double sequences of fuzzy real numbers and studied some interesting properties of this notion. Further details on ideal convergence we refer to [7,15,26,27,35,43,45,48,51,56,64], and many others. By a lacunary sequence θ = (kr ), where k0 = 0, we shall mean an increasing sequence of non-negative integers with kr − kr−1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Jr = (kr−1 , kr ] and we let hr := kr − kr−1 . The space of lacunary strongly convergent sequences Nθ was defined by Freedman et al., [17] as follows.



Nθ = x = (xk ) : lim r

 1  | xk − L| = 0, for some L . k∈Jr hr

By the convergence of a double sequence we mean the convergence in the Pringsheim’s sense [53]. A double sequence x = (xk,l ) has a Pringsheim limit L (denoted by P − lim x = L) provided that given an ε > 0 there exists an n ∈ N such that |xk,l − L| < ε whenever k, l > n. We shall describe such an x = (xk,l ) more briefly as “P − convergent”. In [46] Mursaleen and Edely introduced the two dimensional analogue of natural (or asymptotic) density. Let K ⊂ N × N and K(m, n) denotes the number of (i, j) in K such that i ≤ m and j ≤ n. Then the lower natural density of K is defined m,n ) m,n ) . In case, the sequence ( K (mn ) has a limit in Pringsheim’s sense, then we say that K has a by δ 2 (K ) = lim infm,n→∞ K (mn

m,n ) double natural density and is defined by P − limm,n→∞ K (mn = δ2 ( K ) .

The double sequence θ = θr,s = {(kr , ls )} is called double lacunary sequence if there exist two increasing of integers such that (see [57])

ko = 0, hr = kr − kr−1 → ∞ as r → ∞ and

lo = 0, hs = ls − ls−1 → ∞ as s → ∞. We denote kr,s = kr ls , hr,s = hr hs and θ r, s is determined by

Jr,s = {(k, l ) : kr−1 < k ≤ kr qr =

kr , kr−1

qs =

ls ls−1

and

and

ls−1 < l ≤ ls },

qr,s = qr qs .

In this paper we study the concept of lacunary I-convergence of double sequences in probabilistic normed spaces. We also define lacunary I-limit points and lacunary I-cluster points in probabilistic normed space and prove some interesting results. 2. Basic definitions and notations Now we recall some notations and basic definitions that we are going to use in this paper. The notion of statistical convergence depends on the density (asymptotic or natural) of subsets of N. Definition 2.1. A subset E of N is said to have natural density δ (E) if

1 n

δ (E ) = n→∞ lim |{k ≤ n : k ∈ E }| exists. Definition 2.2. A sequence x = (xk ) is said to be statistically convergent to  if for every ε > 0

δ ({k ∈ N : |xk − | ≥ ε} ) = 0. In this case, we write S − lim x =  or xk → (S) and S denotes the set of all statistically convergent sequences. Definition 2.3. A family of subsets of N, positive integers, i.e. I ⊂ 2N is an ideal on N if and only if (i) φ ∈ I, (ii) A ∪ B ∈ I for each A, B ∈ I, (iii) each subset of an element of I is an element of I. Definition 2.4. A non-empty family of sets F ⊂ 2N is a filter on N if and only if (a) φ ∈ F (b) A ∩ B ∈ F for each A, B ∈ F, (c) any superset of an element of F is in F.

B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

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Definition 2.5. An ideal I is called non-trivial if I = φ and N ∈ / I. Clearly I is a non-trivial ideal if and only if F = F (I ) = {N − A : A ∈ I} is a filter in N, called the filter associated with the ideal I. Definition 2.6. A non-trivial ideal I is called admissible if and only if {{n} : n ∈ N} ⊂ I. Definition 2.7. A non-trivial ideal I is maximal if there cannot exists any non-trivial ideal J = I containing I as a subset. Definition 2.8 [31]. A sequence x = (xk ) of points in R is said to be I-convergent to a real number  if {k ∈ N : |xk − | ≥ ε} ∈ I for every ε > 0. In this case we write I − lim xk = . Definition 2.9 [10,63]. Let I ⊂ 2N be a non-trivial ideal. A real sequence x = (xk ) is said to be lacunary I-convergent or Iθ convergent to L ∈ R if for every ε > 0 the set



1  r∈N: |xk − L| ≥ ε hr



∈ I.

k∈Jr

L is called the Iθ -limit of the sequence x = (xk ), and we write Iθ − lim x = L. Definition 2.10 [31]. An admissible ideal I ⊂ 2N is said to satisfy the condition (AP) if for every countable family of mutually disjoints sets {A1 , A2 , . . .} belonging to I there exists a countable family of sets {B1 , B2 , . . .} such that Aj Bj is a finite set for  j ∈ N and B = ∞ j=1 B j ∈ I. Definition 2.11 [31]. A sequence x = (xk ) of points in R is said to be I∗ -convergent to a real number  if there exists a set M ∈ F(I) (i.e. N − M ∈ I), M = {km : k1 < k2 < · · · < km < · · · } such that limm xkm = . In this case we write I∗ − lim xk =  and  is called the I∗ -limit of x. Definition 2.12. A distribution function (briefly a d.f.) F is a function from the extended reals (−∞, +∞ ) into [0, 1] such that (a) it is non-decreasing ; (b) it is left-continuous on (−∞, +∞ ); (c) F (−∞ ) = 0 and F (+∞ ) = 1. The set of all d.f.’s will be denoted by . The subset of  consisting of proper d.üf’s, namely of those elements F such that + F (−∞ ) = F (−∞ ) = 0 and − F (+∞ ) = F (+∞ ) = 1 will be denoted by D. A distance distribution function (briefly, d.d.f.) is a d.f. F such that F (0 ) = 0. The set of all d.d.f.f’s will be denoted by + , while D+ := D ∩ + will denote the set of proper d.d.f.’s. Definition 2.13. A triangular norm or, briefly, a t-norm is a binary operation T: [0, 1] × [0, 1] → [0, 1] that satisfies the following conditions (see [30]): (T1) (T2) (T3) (T4)

T T T T

is commutative, i.e.,T (s, t ) = T (t, s ) for all s and t in [0, 1]; is associative, i.e., T (T (s, t ), u ) = T (s, T (t, u )) for all s, t and u in [0, 1]; is nondecreasing, i.e., T(s, t) ≤ T(s , t) for all t ∈ [0, 1] whenever s ≤ s ; satisfies the boundary condition T (1, t ) = t for every t ∈ [0, 1].

T∗ is a continuous t-conorm, namely, a continuous binary operation on [0, 1] that is related to a continuous t-norm through T ∗ (s, t ) = 1 − T (1 − s, 1 − t ). Notice that by virtue of its commutativity, any t-norm T is nondecreasing in each place. Some examples of t-norms T and its t-conorms T∗ are: M (x, y ) = min{x, y}, (x, y ) = x.y and M∗ (x, y ) = max{x, y}, ∗ (x, y ) = x + y − x.y. ˘ Using the definitions just given above Serstnev [60] defined a PN space as follows. Definition 2.14. A triplet (X, ν , T) is called a probabilistic normed space (in short PNS) if X is a real vector space, ν is a mapping from X into D and for x ∈ X, the d.f. ν (x) is denoted by ν x , ν x (t) is the value of ν x at t ∈ R and T is a t-norm. ν satisfies the following conditions : (i) νx (0 ) = 0; (ii) νx (t ) = 1 for  all t > 0 if and only if x = 0;

(iii) νax (t ) = νx |at | for all a ∈ R{0}; (iv) νx+y (s + t ) ≥ T (νx (s ), νy (t )) for all x, y ∈ X and s, t ∈ R+ . 0 Let (X, ||.||) be a normed space and μ ∈ D with μ(0 ) = 0 and μ = 0 , where



0 (t ) =

0, 1,

if t ≤ 0 if t > 0

For x ∈ X, t ∈ R, if we define

νx (t ) = μ

 t , x = 0, ||x||

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B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

then in [36], it is proved that (X, ν , T) is a PN space in the sense of Definition 2.14. Alsina et al. [2] gave new definition of a PN-space. Before giving this, we recall for the reader’s convenience the concept of a triangle function, that of a PN space from the point of view of the new definition. Definition 2.15. A triangle function is a mapping τ from + × + into + such that, for all F, G, H, K in + , (1) (2) (3) (4)

τ (F, 0 ) = F ; τ (F, G ) = τ (G, F ); τ (F, G) ≤ τ (H, K) whenever F ≤ H, G ≤ K; τ (τ (F, G ), H ) = τ (F, τ (G, H )).

Particular and relevant triangle functions are the functions τT , τT ∗ and those of the form T which, for any continuous t-norm T, and any x > 0, are given by

τT (F, G )(x ) = sup{T (F (u ), G(v )) : u + v = x}, τT ∗ (F, G )(x ) = inf{T ∗ (F (u ), G(v )) : u + v = x} and

T (F, G )(x ) = T (F (x ), G(x )). Definition 2.16 [2]. A probabilistic normed space is a quad-ruple (X, ν , τ , τ ∗ ), where X is a real linear space, τ and τ ∗ are continuous triangle functions such that τ ≤ τ ∗ and the mapping ν : X → + called the probabilistic norm, satisfies for all p and q in X, the conditions (PN1) (PN2) (PN3) (PN4)

ν p = 0 if and only if p = θ (θ is the null vector in X); ∀ p ∈ X, ν−p = ν p ; ν p+q ≥ τ (ν p , νq ); ∀a ∈ [0, 1], ν p ≤ τ ∗ (νap , ν(1−a) p ).

If a PN space (X, ν , τ , τ ∗ ), satisfies the following condition ˘ ˘ implies that the ˘ ∀ p ∈ X, ∀λ ∈ R{0}, ∀t > 0, νλ p (t ) = ν p ( t ), then it is called a Serstnev PN space; the condition (S) (S) |λ|

best-possible selection for τ ∗ is τ ∗ = τM , which satisfies a stricter version of (PN4), namely,

∀a ∈ [0, 1], ν p = τM (νap , ν(1−a) p ). Definition 2.17. A Menger PN space under T is a PN space (X, ν , τ , τ ∗ ) denoted by (X, ν , T), in which τ = τT and τ ∗ = τT ∗ , for some continuous t-norm T and its t-conorm T∗ . Lemma 2.18 [36]. The simple space generated by (X, ||.||) and by μ is a Menger PN space under M and also a S˘ erstnev PN space. Here M(x, y) := min {x, y}. For further study, by a PN space we mean a PN space in the sense of Definition 2.14. We now give a quick look on the characterization of convergence and Cauchy sequences on these spaces. Let (X, ν , T) be a PN space and x = (xk ) be a sequence in X. We say that (xk ) is convergent to  ∈ X with respect to the probabilistic norm ν if for each ε > 0 and α ∈ (0, 1) there exists a positive integer m such that νxk − (ε ) > 1 − α whenever ν

k ≥ m. The element  is called the ordinary limit of the sequence (xk ) and we shall write ν − lim xk =  or xk →  as k → ∞. A sequence (xk ) in X is said to be Cauchy with respect to the probabilistic norm ν if for each ε > 0 and α ∈ (0, 1) there exists a positive integer M = M (ε , α ) such that νxk −x p (ε ) > 1 − α whenever k, p ≥ M. Definition 2.19. Let (X, ν , T) be an probabilistic normed space, and let r ∈ (0, 1) and x ∈ X. The set

B(x, r; t ) = {y ∈ X : νy−x (t ) > 1 − r} is called open ball with center x and radius r with respect to t. Throughout the paper, we denote I is an admissible ideal of subsets of N × N and θ¯ = (kr,s ) a non increasing sequence of positive real numbers, respectively, unless otherwise stated. 3. Lacunary ideal convergence of multiple sequences We now obtain main results.





Definition 3.1. Let I ⊂ 2N×N and (X, ν , T) be an PNS. A double sequence x = xk,l in X is said to be Iθ¯ -convergent to L ∈ X with respect to the probabilistic norm ν if for every ε > 0 and α ∈ (0, 1) the set



1 (r, s ) ∈ N × N : hr,s

 (k,l )∈Jr,s



νxk,l −L (ε ) ≤ 1 − α ∈ I. 



L is called the Iθ¯ −limit of the sequence x = xk,l in X, and we write I ν¯ − lim x = L. θ

B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

143

Example 3.1. Let (R, |.| ) denote the space of all real numbers with the usual norm, and let T (a, b) = ab for all a, b ∈ [0, 1]. For all x ∈ R and every t > 0, consider νx (t ) = t+t|x| . Then (R, ν, T ) is an PNS. If we take I = {A ⊂ N × N : δ2 (A ) = 0}, where

 δ 2 (A) denotes the double natural density of the set A, then I is a non-trivial admissible ideal. Define a sequence x = xk,l as follows:



xk,l =

if k, l = i2 , i ∈ N

1, 0,

otherwise.

Then for every α ∈ (0, 1) and for any ε > 0,



1 (r, s ) ∈ N × N : hr,s

K=





νxk,l (ε ) ≤ 1 − α

(k,l )∈Jr,s

will be a finite set. Hence δ2 (K ) = 0 and consequently K ∈ I, i.e., I ν¯ − lim x = 0. θ

Lemma 3.1. Let (X, ν , T) be an PNS and x = (xk,l ) be a double sequence in X. Then for every ε > 0 and α ∈ (0, 1) the following statements are equivalent: (i) I ν¯ − lim x = L,

θ 

(r, s ) ∈ N × N : h1r,s (k,s)∈Jr,s νxk,l −L (ε ) ≤ 1 − α ∈ I.  

(iii) (r, s ) ∈ N × N : h1 ν ( ε ) > 1 − α ∈ F ( I ), x −L ( k,l ) ∈J r,s k,l r,s (iv) Iθ¯ − lim νxk,l −L (ε ) = 1. (ii)

Proof. (i) ⇒ (ii) Suppose that (i) holds. By definition for every ε > 0 and α ∈ (0, 1) we have



1 (r, s ) ∈ N × N : hr,s





νxk,l −L (ε ) ≤ 1 − α ∈ I

(k,l )∈Jr,s

which is the result (ii). (ii) ⇒ (iii) Suppose the result (ii) holds. Since



1 (r, s ) ∈ N × N : hr,s



 1 (r, s ) ∈ N × N : hr,s

νxk,l −L (ε ) ≤ 1 − α



1 (r, s ) ∈ N × N : hr,s



1 (r, s ) ∈ N × N : hr,s

νxk,l −L (ε ) > 1 − α c

νxk,l −L (ε ) ≤ 1 − α

∈ F ( I ).

(k,l )∈Jr,s



 νxk,l −L (ε ) > 1 − α ∈ F (I ).

(k,l )∈Jr,s

(iii ) ⇒ (iv ) Suppose





(k,l )∈Jr,s

Therefore we have



c

(k,l )∈Jr,s

1 (r, s ) ∈ N × N : hr,s

= and





 νxk,l −L (ε ) > 1 − α ∈ F (I ).

(k,l )∈Jr,s

Then for every (r, s ) ∈ N × N we have

1 hr,s



νxk,l −L (ε ) > 1 − α .

(k,l )∈Jr,s

Therefore Fθ¯ − lim νxk,l −L (ε ) = 0. i.e. Iθ¯ − lim νxk,l −L (ε ) = 1. (iv ) ⇒ (i ) Suppose that (iv) holds. Then for α ∈ (0, 1) we have



1 (r, s ) ∈ N × N : hr,s

 (k,l )∈Jr,s



νxk,l −L (ε ) > 1 − α ∈ F (I ).

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B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

But

 1 (r, s ) ∈ N × N : hr,s







νxk,l −L (ε ) ≤ 1 − α

(k,l )∈Jr,s

=

c



1 (r, s ) ∈ N × N : hr,s

νxk,l −L (ε ) > 1 − α

∈ I.

(k,l )∈Jr,s

Hence I ν¯ − lim x = L.  θ

Theorem 3.1. Let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is Iθ¯ -convergent to L ∈ X with respect to the probabilistic norm ν , then I ν¯ − lim x is unique. θ

Proof. Suppose that I ν¯ − lim x = L1 and I ν¯ − lim x = L2 (L1 = L2 ). Given α > 0 and choose β ∈ (0, 1) such that θ

θ

T (1 − β , 1 − β ) > 1 − α .

(3.1)

Then for ε > 0 define the following sets:



K1 =



1 (r, s ) ∈ N × N : hr,s 1 (r, s ) ∈ N × N : hr,s

K2 =



νxk,l −L1

ε 2

(k,l )∈Jr,s



νxk,l −L2

ε 2

(k,l )∈Jr,s

 ≤1−β

,

 ≤1−β

,

Since I ν¯ − lim x = L1 , using Lemma 3.1, we have K1 ∈ I. Also using I ν¯ − lim x = L2 we get K2 ∈ I. Now let θ

θ

K = K1 ∪ K2 . Then K ∈ I. This implies that its complement Kc is a non-empty set in F (I ). Now if (r, s) ∈ Kc , let us consider (r, s ) ∈ K1c ∩ K2c . Then we have

1 hr,s



νxk,l −L1

ε

(k,l )∈Jr,s

2

1 hr,s

> 1 − β and

 (k,l )∈Jr,s

νxk,l −L2

ε 2

> 1 − β.

Now we choose a (m, n ) ∈ N × N such that

νxm,n −L1 and

νxm,n −L2

ε 2

ε 2

>

>

1 hr,s

1 hr,s



νxk,l −L1

ε 2

(k,l )∈Jr,s



νxk,l −L2

ε 2

(k,l )∈Jr,s

>1−β

> 1 − β.

Then from (3.1) we have

 ε  ε νL1 −L2 (ε ) ≥ T νxm,n −L1 , νxm,n −L2 > T (1 − β , 1 − β ) > 1 − α . 2

2

Since α > 0 is arbitrary we have νL1 −L2 (ε ) = 1 for all ε > 0, which implies that L1 = L2 . Therefore we conclude that I ν¯ − θ

lim x is unique.  Now we introduce the notion of θ¯ -convergence in an PNS and discuss some properties.

Definition 3.2. Let (X, ν , T) be an PNS. A double sequence x = (xk,l ) in X is θ¯ -convergent to L ∈ X with respect to the probabilistic norm ν if for α ∈ (0, 1) and every ε > 0, there exists N ∈ N such that

1 hr,s



νxk,l −L (ε ) > 1 − α

(k,l )∈Jr,s

for all k, l ≥ N. In this case we write ν θ − lim x = L. ¯

Theorem 3.2. Let (X, ν , T) be an PNS and let x = (xk,l ) be a double sequence in X. If x = (xk,l ) is θ¯ -convergent with respect to the probabilistic norm ν , then ν θ − lim x is unique. ¯

B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

145

Proof. Suppose that ν θ − lim x = L1 and ν θ − lim x = L2 (L1 = L2 ). Given α ∈ (0, 1) and choose β ∈ (0, 1) such that T (1 − β , 1 − β ) > 1 − α . Then for any ε > 0 there exists (r1 , s1 ) ∈ N × N such that ¯

1 hr,s



¯

νxk,l −L1 (ε ) > 1 − α

(k,l )∈Jr,s

for all r ≥ r1 , s ≥ s1 . Also there exists (r2 , s2 ) ∈ N × N such that

1 hr,s



νxk,l −L2 (ε ) > 1 − α

(k,l )∈Jr,s

for all r ≥ r2 , s ≥ s2 . Now consider ro = max {r1 , r2 } and so = max{s1 , s2 }. Then for r ≥ ro , s ≥ so we will get a (m, n ) ∈ N × N such that

νxm,n −L1 and

νxm,n −L2

ε 2

ε 2

>

>

1 hr,s

1 hr,s



νxk,l −L1

ε 2

(k,l )∈Jr,s



νxk,l −L2

ε

(k,l )∈Jr,s

2

>1−β

> 1 − β.

Then we have

 ε  ε νL1 −L2 (ε ) ≥ T νxm,n −L1 , νxm,n −L2 > T (1 − β , 1 − β ) > 1 − α . 2

2

Since α > 0 is arbitrary, we have νL1 −L2 (ε ) = 1 for all ε > 0, which implies that L1 = L2 .  ¯ Theorem 3.3. Let (X, ν , T) be an PNS and let x = (xk,l ) be a double sequence in X. If ν θ − lim x = L, then I ν¯ − lim x = L.

θ

θ¯

Proof. Let ν − lim x = L, then for every ε > 0 and given α ∈ (0, 1), there exists (r0 , s0 ) ∈ N × N such that

1 hr,s



νxk,l −L (ε ) > 1 − α

(k,l )∈Jr,s

for all r ≥ r0 , s ≥ s0 . Therefore the set



B=

1 (r, s ) ∈ N × N : hr,s



 νxk,l −L (ε ) ≤ 1 − α ⊆ {(1, 1 ), (2, 2 ), . . . , (m0 − 1, n0 − 1 )}.

(k,l )∈Jr,s

Since I is an admissible ideal, we have B ∈ I. Hence I ν¯ − lim x = L.  θ

Remark 3.1. If I = I f = {A ⊂ N × N : A is Theorem 3.3 is true.

finite a subset} and I is an admissible ideal of N × N. Then the converse of

Theorem 3.4. Let (X, ν , T) be an PNS and x = (xk,l ), y = (yk,l ) be two double sequences in X. (i) If I ν¯ − lim xk,l = L1 and I ν¯ − lim yk,l = L2 , then I ν¯ − lim(xk,l ± yk,l ) = L1 ± L2 ; θ

θ

θ

(ii) If I ν¯ − lim xk,l = L and a be a non-zero real number, then I ν¯ − lim axk,l = aL. If a = 0, then result is true only if I is an θ

θ

admissible of N × N.

Theorem 3.5. Let (X, ν , T) be an PNS and let x = (xk,l ) be a double sequence in X. If ν θ − lim x = L, then there exists a subsequence (xmk ,nl ) of x = (xk,l ) such that ν − lim xmk ,nl = L. ¯

Proof. Let ν θ − lim x = L. Then for every ε > 0 and given α ∈ (0, 1) there exists (r0 , s0 ) ∈ N × N such that ¯

1 hr,s



νxk,l −L (ε ) > 1 − α

(k,l )∈Jr,s

for all r ≥ r0 , s ≥ s0 . Clearly, for each r ≥ r0 , s ≥ s0 we can select (mk , nl ) ∈ Jr, s such that

νxmk ,nl −L (ε ) >

1 hr,s



νxk,l −L (ε ) > 1 − α .

(k,l )∈Jr,s

It follows that ν − lim xmk ,nl = L.  Definition 3.3. Let (X, ν , T) be an PNS and let x = (xk,l ) be a double sequence in X. Then

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B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

(1) an element L ∈ X is said to be I ν¯ -limit point of x = (xk,l ) if there is a set M = {m1 < m2 < . . . < mk < . . . ; n1 < n2 < θ

. . . < nl < . . .} ⊂ N × N such that the set Mı = {(r, s ) ∈ N × N : (mk , nl ) ∈ Jr,s } ∈ / I and ν θ − lim xmk ,nl = L. (2) an element L ∈ X is said to be I ν¯ -cluster point of x = (xk,l ) if for every ε > 0 and α ∈ (0, 1) we have θ



I ν¯ θ (x )





1 (r, s ) ∈ N × N : hr,s Let 

¯

νxk,l −L (ε ) > 1 − α ∈/ I.

(k,l )∈Jr,s ν

I denote the set of all I ν¯ -limit points and  θ¯ (x ) denote the set of all I ν¯ -cluster points in X, respectively.

θ

θ

Theorem 3.6. Let (X, ν , T) be an PNS. For each sequence x = (xk,l ) in X,

Iν  θ¯ (x )

⊂

Iν  θ¯ (x ).

Iν

Proof. Let L ∈  θ¯ (x ). Then there exists a set M ⊂ N × N such that Mı ∈ / I, where M and Mı are as in the Definition 3.3, ¯ satisfies ν θ − lim xmk ,nl = L. Thus for every ε > 0 and α ∈ (0, 1), there exists (r0 , s0 ) ∈ N × N such that



1 hr,s

(k,l )∈Jr,s

νxmk ,nl −L (ε ) > 1 − α

for all r ≥ r0 , s ≥ s0 . Therefore





1 (r, s ) ∈ N × N : hr,s

B=

 νxk,l −L (ε ) > 1 − α

(k,l )∈Jr,s

⊇ Mı \ {(m1 , n1 ), (m2 , n2 ), . . . , (mr0 , ns0 )}. Since I is admissible, we must have Mı \ {(m1 , n1 ), (m2 , n2 ), . . . , (mr0 , ns0 )} ∈ / I and as such B ∈ / I. Hence L ∈ 

I ν¯ θ ( x ).



The converse of Theorem 3.6 discuss in the following result. Theorem 3.6.A. Let (X, ν , T) be an PNS. Let x = (xk,l ) be sequence in X. If I is an admissible ideal of N × N and Iθ¯ − lim x = L. I ν¯ Iν θ (x ) =  θ¯ (x ) = {L}.

Then 

Proof. Suppose that Iθ¯ − lim x = L. Then for every ε > 0 and α ∈ (0, 1) we have



1 (r, s ) ∈ N × N : hr,s

A= i.e.



1 (r, s ) ∈ N × N : hr,s

which implies that L ∈ 

νxk,l −L (ε ) ≤ 1 − α ∈ I

(k,l )∈Jr,s

 Ac =





 νxk,l −L (ε ) > 1 − α ∈/ I

(k,l )∈Jr,s

I ν¯ θ ( x ).

We assume that there exists at least L0 ∈ 



1 (r, s ) ∈ N × N : hr,s



⊇ Since



1 (r, s ) ∈ N × N : hr,s

1 (r, s ) ∈ N × N : hr,s



νxk,l −L (ε ) ≤ 1 − α 

 νxk,l −L0 (ε ) > 1 − α .

(k,l )∈Jr,s





νxk,l −L (ε ) ≤ 1 − α ∈ I

(k,l )∈Jr,s

this implies that



such that L = L0 . Then there exists ε > 0 and α ∈ (0, 1) such that

(k,l )∈Jr,s

1 (r, s ) ∈ N × N : hr,s



I ν¯ θ (x )





νxk,l −L0 (ε ) > 1 − α ∈ I,

(k,l )∈Jr,s

which contradicts that L0 ( = L ) ∈ 

I ν¯ θ ( x ).

Hence 

I ν¯ θ (x )

= {L}.

B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153 Iν

147

Iν

Next we prove that  θ¯ (x ) = {L}. Suppose that L0 , L ∈  θ¯ (x ) such that L0

= L. Then by Definition 3.3 there exist two sets M = {m1 < m2 < · · · < mk < · · · ; n1 < n2 < · · · < nl < · · · } ⊂ N × N and P = { p1 < p2 < · · · < pk < · · · ; q1 < q2 < · · · < ql < · · · } ⊂ N × N such that Mı = {(r, s ) ∈ N × N : (mk , nl ) ∈ Jr,s } ∈/ I and / I we have ν θ − lim xmk ,nl = L and P ı = {(r, s ) ∈ N × N : ( pk , ql ) ∈ Jr,s } ∈ ¯

ν θ¯ − lim x pk ,ql = L0 .

(3.2)

By the relation (3.2) we have for every ε > 0 and α ∈ (0, 1) the set



1 (r, s ) ∈ N × N : hr,s

B=



 ( pk ,ql )∈Jr,s

νx pk ,ql −L0 (ε ) ≤ 1 − α

is a finite set and therefore belongs to I as I is an admissible ideal. Therefore



Bc =





1 (r, s ) ∈ N × N : hr,s

( pk ,ql )∈Jr,s

νx pk ,ql −L0 (ε ) > 1 − α ∈/ I.

(3.3)

We have Ac ∩ Bc = φ . This implies that Bc ⊂ A and since A ∈ I therefore Bc ∈ I which contradicts relation (3.3). Therefore Iν

Iν

Iν

 θ¯ (x ) = {L}. Hence  θ¯ (x ) =  θ¯ (x ) = {L}. 



Theorem 3.7. Let (X, ν , T) be an PNS. For each sequence x = xk,l



in X, the set 

I ν¯ θ (x )

is closed set in X with respect to the

usual topology induced by the probabilistic norm ν θ . ¯

Iν

Proof. Let y ∈  θ¯ (x ). Take ε > 0 and α ∈ (0, 1). Then there exists L0 ∈  ε) ⊂ B(y, α , ε). We have



G=



1 (r, s ) ∈ N × N : hr,s 1 (r, s ) ∈ N × N : hr,s

⊇



Choose δ > 0 such that B(L0 , δ ,



νxk,l −y (ε ) > 1 − α

(k,l )∈Jr,s



I ν¯ θ (x ) ∩ B (y, α , ε ).

 νxk,l −L0 (ε ) > 1 − δ = H.

(k,l )∈Jr,s

Thus H ∈ / I and so G ∈ / I. Hence y ∈ 

I ν¯ θ (x )

.







Theorem 3.8. Let (X, ν , T) be an PNS and let x = xk,l be a double sequence in X. Then the following statements are equivalent: (1) L is a I ν¯ −limit point of x, θ





¯ (2) There exist two sequences y and z in X such that x = y + z and ν θ − lim y = L and (r, s ) ∈ N × N : (k, l ) ∈ Jr,s , zk,l = 0 ∈

I, where 0 is the zero element of X. Proof. Suppose that (1) holds. Then there exist sets M and Mı as in Definition 3.3 such that Mı ∈ / I and ν θ − lim xmk ,nl = L. Define the sequences y and z as follows: ¯



yk,l = and

zk,l =

xk,l , L,

0, xk,l − L,

if (k, l ) ∈ Jr,s ; (r, s ) ∈ Mı , otherwise.

if (k, l ) ∈ Jr,s ; (r, s ) ∈ Mı , otherwise.

It suffices to consider the case (k, l) ∈ Jr, s such that (r, s ) ∈ N × NMı . Then for each α ∈ (0, 1) and ε > 0, we have

νyk,l −L (ε ) = 1 > 1 − α . Thus in this case 1 hr,s



νyk,l −L (ε ) = 1 > 1 − α .

(k,l )∈Jr,s

Hence ν θ − lim y = L. Now {(r, s ) ∈ N × N : (k, l ) ∈ Jr,s , zk,l = 0¯ } ⊂ NMı so that {(r, s ) ∈ N × N : (k, l ) ∈ Jr,s , zk,l = 0¯ } ∈ I. Next suppose that (2) holds. Let Mı = {(r, s ) ∈ N × N : (k, l ) ∈ Jr,s , zk,l = 0¯ }. Then clearly Mı ∈ F (I ) and so it is an infinite set. ¯

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B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

Construct the set M = {m1 < m2 < · · · < mk < · · · ; n1 < n2 < · · · < nl < · · · } ⊂ N × N such that (mk , nl ) ∈ Jr, s and zmk ,nl = 0.

Since xmk ,nl = ymk ,nl and ν θ − lim y = L we obtain ν θ − lim xmk ,nl = L. This completes the proof.  ¯

¯

Theorem 3.9. Let (X, ν , T) be an PNS and x = (xk,l ) be a double sequence in X. Let I be an admissible ideal in N × N. If there is a I ν¯ -convergent sequence y = (yk,l ) in X such that {(k, l ) ∈ N × N : yk,l = xk,l } ∈ I then x is also I ν¯ -convergent. θ

θ

Proof. Suppose that {(k, l ) ∈ N × N : yk,l = xk,l } ∈ I and I ν¯ − lim y = . Then for every θ



1 hr,s

(r, s ) ∈ N × N :





νyk,l −L (ε ) ≤ 1 − α ∈ I.

(k,l )∈Jr,s

For every α ∈ (0, 1) and ε > 0 we have



1 (r, s ) ∈ N × N : hr,s

α ∈ (0, 1) and ε > 0, the set





νxk,l −L (ε ) ≤ 1 − α

(k,l )∈Jr,s



⊆ {(k, l ∈ N × N : yk,l = xk,l } ∪

1 (r, s ) ∈ N × N : hr,s



 νyk,l −L (ε ) ≤ 1 − α .

(3.4)

(k,l )∈Jr,s

As both the sets of right-hand side of (3.4) is in I, therefore we have that



1 (r, s ) ∈ N × N : hr,s





νxk,l −L (ε ) ≤ 1 − α ∈ I.

(k,l )∈Jr,s

This completes the proof of the theorem.



Now we define the notion I ν¯ ∗ -convergence in probabilistic normed space and obtain some relations between the notions θ ν ∗ I ¯ -convergence and I ν¯ -convergence. θ θ Definition 3.4. Let I be an admissible ideal of N × N. Let (X, ν , T) be an PNS. A double sequence x = (xk,l ) in X is said to be I ν¯ ∗ -convergent to L ∈ X with respect to the probabilistic norm ν if there is a set M = θ

{m1 < m2 < · · · < mk < · · · ; n1 < n2 < · · · < nl < · · · } ⊂ N × N such that Mı = {(r, s ) ∈ N × N : (mk , nl ) ∈ Jr,s } ∈ F (I ) and ν θ¯ − lim xmk ,nl = L. In this case we write I ν¯ ∗ − lim xk,l = L and L is called the I ν¯ ∗ -limit of the sequence x = (xk,l ). θ θ Theorem 3.10. Let (X, ν , T) be an PNS and x = (xk,l ) be a double sequence in X. Let I be an admissible ideal in N × N. If I ν¯ ∗ − lim xk,l = L, then I ν¯ − lim xk,l = L. θ

θ

Proof. Suppose that I ν¯ ∗ − lim xk,l = L. Then by Definition 3.4, there exists a set M = {m1 < m2 < · · · < mk < · · · ; n1 < n2 < θ

· · · < nl < · · · } ⊂ N × N such that Mı ∈ F (I ) and ν θ − lim xmk ,nl = L. ¯

Let ε > 0 and α ∈ (0, 1) be given. Since ν θ − lim xmk ,nl = L there exists N ∈ N such that ¯



1 hr,s

(k,l )∈Jr,s

Since

νxmk ,nl −L (ε ) > 1 − α for every k, l ≥ N.

 A=

1 (r, s ) ∈ N × N : hr,s

 (k,l )∈Jr,s

 νxmk ,nl −L (ε ) ≤ 1 − α

⊆ {(m1 , n1 ), (m2 , n2 ), . . . , (mN−1 , nN−1 )}. Also since I is an admissible ideal, we have A ∈ I. Hence



1 (r, s ) ∈ N × N : hr,s





νxk,l −L (ε ) ≤ 1 − α

(k,l )∈Jr,s

⊆ M ∪ {(m1 , n1 ), (m2 , n2 ), . . . , (mN−1 , nN−1 )} ∈ I, for every ε > 0 and α ∈ (0, 1). Therefore we conclude that I ν¯ − lim xk,l = L.  θ

Remark 3.2. The converse of this theorem need not be true. To prove it we consider the following example. Example 3.2. Let (R, |.| ) denote the space of all real numbers with the usual norm, and let T (a, b) = ab for all a, b ∈ [0, 1]. For all x ∈ R and every t > 0 consider νx (t ) = t+t|x| . Then (R, ν, T ) is an PNS. Consider a decomposition of N × N as

B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

149



N × N = i, j i, j such that for any m, n ∈ N each i, j contains infinitely many i, j where i ≥ m, j ≥ n and i, j ∩ m,n = φ for i = m, j = n. Let I be the class of all subsets of N × N which intersect almost a finite number of i, j ’s. Then I is an admissible ideal of N × N. We define a sequence (xm, n ) as xm,n = i1j , i, j = 1, 2, 3 . . . if m, n ∈ i, j . Then we have

νxm,n (t ) =

t → 1 as m, n → ∞. t + |xm,n |

Hence I ν¯ − lim xk,l = 0. θ

Now we show that I ν¯ ∗ − lim xk,l = 0. Suppose that I ν¯ ∗ − lim xk,l = 0. θ

θ

Then by Definition 3.4, there exists a subset M = {m1 < m2 < · · · < mk < · · · ; n1 < n2 < · · · < nl < · · · } ⊂ N × N such that

¯ Mı ∈ F (I ) and ν θ − lim xm p ,nq = 0. Since Mı ∈ F (I ), there exists K ∈ I such that M = N \ K. Then there exist positive integers p, q such that



K⊆

p 



m=1

∞ 



m,n





n=1

q 



n=1

∞ 



m,n

.

m=1

1 Thus  p+1,q+1 ⊂ M and so xmk ,nl = ( p+1 )( > 0 for infinitely many values (mi , nj )’s in M. This contradicts the assumption q+1 ) that I ν¯ ∗ − lim xm,n = 0. Hence I ν¯ ∗ − lim xm,n = 0. Hence the converse of the theorem need not be true.

θ

θ

The following theorem shows that the converse of the Theorem 3.10 holds if the ideal I satisfies condition (AP). Theorem 3.11. Let (X, ν , T) be an PNS and x = (xk,l ) be a double sequence in X. Let I be an admissible ideal satisfy the condition (AP). If I ν¯ − lim xk,l = L, then I ν¯ ∗ − lim xk,l = L. θ

θ

Proof. Since I ν¯ − lim xk,l = L, then for every ε > 0 and α ∈ (0, 1), the set θ



1 (r, s ) ∈ N × N : hr,s





νxk,l −L (ε ) ≤ 1 − α ∈ I.

(k,l )∈Jr,s

We define the set Ap for p ∈ N as



A p = (k, l ) ∈ N × N : 1 −



1 1 ≤ νxk,l −L < 1 − . p p+1

Then it is clear that {A1 , A2 , . . .} is a countable family of mutually disjoint sets belonging to I and so by the condition (AP)  there is a countable family of sets {B1 , B2 , . . .} ∈ I such that Ai Bi is a finite set for each i ∈ N and B = ∞ i=1 Bi ∈ I. Since B ∈ I, there is a set M ∈ F (I ) such that M = N × N \ B. Now to prove the result it is sufficient to show that νθ¯ − limk,l∈M xk,l = L. Let η ∈ (0, 1) and ε > 0. Choose a positive integer q such that



(r, s ) ∈ N × N :



1 hr,s



< η. Then we have

νxk,l −L (ε ) ≤ 1 − η

(k,l )∈Jr,s

1 (r, s ) ∈ N × N : hr,s

⊂



1 q

 (k,l )∈Jr,s

 1 νxk,l −L (ε ) ≤ 1 − q



q+1

⊂

Ai .

i=1

Since Ai Bi is a finite set for each i = 1, 2, 3, . . . q + 1, there exists (m0 , n0 ) ∈ N × N such that







q+1

Bi





{(k, l ) : k ≥ m0 and l ≥ n0 } =

i=1





q+1

Ai



{(k, l ) : k ≥ m0 and l ≥ n0 }.

i=1

If k ≥ m0 , l ≥ n0 and (k, l) ∈ M then (k, l) ∈ B. This implies that (k, l ) ∈ / m0 , l ≥ n0 and (k, l) ∈ M we have

1 hr,s



q+1 i=1

Bi and therefore (k, l ) ∈ /

q+1 i=1

Ai . Hence for k ≥

νxk,l −L (ε ) > 1 − η.

(k,l )∈Jr,s

Since this holds for every ε > 0 and η ∈ (0, 1), we have I ν¯ ∗ − lim xk,l = L. This completes the proof of the theorem. θ



Theorem 3.12. Let (X, ν , T) be an PNS and let I be an admissible ideal. Let x = xk,l following statements are equivalent:





be a double sequence in X. Then the

(1) I ν¯ ∗ − lim xk,l = L, θ

(2) There exist two double sequences y and z in X such that x = y + z and ν θ − lim y = L and {(r, s ) ∈ N × N : (k, l ) ∈ Jr,s , zk,l = 0} ∈ I, where 0 is the zero element of X. ¯

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B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

Proof. The proof of the theorem follows from the definitions.  Definition 3.5. Let (X, ν , T) be an PNS. A sequence x = (xk,l ) in X is said to be θ¯ -Cauchy sequence with respect to the probabilistic norm ν if for every ε > 0 and α ∈ (0, 1), there exist r0 , s0 , m, n ∈ N satisfying

1 hr,s



νxk,l −xm,n (ε ) > 1 − α

(k,l )∈Jr,s

for all r ≥ r0 , s ≥ s0 . Definition 3.6. Let I be an admissible ideal of N × N. Let (X, ν , T) be an PNS. A double sequence x = (xk,l ) in X is said to be I ν¯ -Cauchy sequence with respect to the probabilistic norm ν if for every ε > 0 and α ∈ (0, 1), there exists m, n ∈ N θ

satisfying



1 (r, s ) ∈ N × N : hr,s



 νxk,l −xm,n (ε ) > 1 − α ∈ F (I ).

(k,l )∈Jr,s

Definition 3.7. Let I be an admissible ideal of N × N. Let (X, ν , T) be an PNS. A double sequence x = (xk,l ) in X is said to be I ν¯ ∗ -Cauchy sequence with respect to the probabilistic norm ν if there exists a set M = {m1 < m2 < · · · < mk < · · · ; n1 < θ

n2 < · · · < nl < · · · } ⊂ N × N such that the set Mı = {(r, s ) ∈ N × N : (mk , nl ) ∈ Jr,s } ∈ F (I ) and the subsequence (xmk ,nl ) of x = (xk,l ) is a θ¯ -Cauchy sequence with respect to the probabilistic norm ν .

Theorem 3.13. Let I be an admissible ideal of N × N. Let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is θ¯ -Cauchy sequence with respect to the probabilistic norm ν , then it is I ν¯ -Cauchy sequence with respect to the same norm. θ

Proof. The proof of the theorem follows from Theorem 3.3 .



Theorem 3.14. Let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is θ¯ -Cauchy sequence with respect to the probabilistic norm ν , then there is a subsequence of x = (xk,l ) which is ordinary Cauchy sequence with respect to the same norm. Proof. The proof of the theorem follows from Theorem 3.5 .



Theorem 3.15. Let I be an admissible ideal of N × N and let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is Iθν ∗ Cauchy sequence, then it is Iθν -Cauchy sequence as well. Proof. Let (xk, l ) be a Iθν ∗ -Cauchy sequence. Then for every ε > 0 and α ∈ (0, 1), there exists M ∈ F (I ) and M1 , N1 ∈ N such that

1 hr,s

 (k,l )∈Jr,s

νxmk ,nl −xm p ,nq (ε ) > 1 − α for every k, p ≥ M1 and l, q ≥ N1 .

Now fix p = mM1 +1 , q = nN1 +1 . Then for every ε > 0 and α ∈ (0, 1) we have

1 hr,s

 (k,l )∈Jr,s

νxmk ,nl −x p,q (ε ) > 1 − α for every k ≥ M1 , l ≥ N1 .

Let K = N × N \ M. It is clear that K ∈ I and



A (ε , α ) =

1 (r, s ) ∈ N × N : hr,s



 νxk,l −x p,q (ε ) ≤ 1 − α

(k,l )∈Jr,s

⊂ K ∪ {m1 < m2 < · · · < mM1 ; n1 < n2 < · · · < nN1 } ∈ I. Therefore for every ε > 0 and α ∈ (0, 1) we can find p, q ∈ N such that A(ε , α ) ∈ I, i.e. (xk, l ) is a Iθν -Cauchy sequence.  Theorem 3.16. Let sequence, then it is

I satisfy the condition (AP) and let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is Iθν -Cauchy Iθν ∗ -Cauchy sequence as well.

Proof. The proof of the theorem follows from Theorem 3.11 and Theorem 3.15 .



Theorem 3.17. Let I be an admissible ideal of N × N and let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is Iθν ∗ convergent, then it is Iθν -Cauchy sequence. Proof. Suppose that Iθν ∗ − limk,l xk,l = L. By Definition 3.4 there exists a set M = {m1 < m2 < · · · < mk < · · · ; n1 < n2 < · · · < nl < · · · } ⊂ N × N such that Mı = {(r, s ) ∈ N × N : (mk , nl ) ∈ Jr,s } ∈ F (I ) and ν θ − lim xmk ,nl = L. ¯

B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

151

¯ For every α ∈ (0, 1) and choose β ∈ (0, 1) such that T (1 − β , 1 − β ) > (1 − α ). Since ν θ − lim xmk ,nl = L. Then for β ∈ (0, 1) we have

νxmk ,nl −L

ε 2

>

1 hr,s

 (mk ,nl )∈Jr,s

νxmk ,nl −L

ε

Also for any (mp , nq ) ∈ Jr, s we have

νxm p ,nq −L

ε 2

>

1 hr,s



νxm p ,nq −L

2

> 1 − β.

ε

(m p ,nq )∈Jr,s

2

> 1 − β.

Then for every ε > 0, and α ∈ (0, 1) we have

 ε  ε νxmk ,nl −xm p ,nq (ε ) ≥ T νxmk ,nl −L , νxm p ,nq −L 2 2 > T (1 − β , 1 − β ) > 1 − α

for every k, p ≥ M, l, q ≥ N for M, N ∈ N. Therefore we have

1 hr,s



(k,l )∈Jr,s

νxmk ,nl −xm p ,nq (ε ) > 1 − α for every k, p ≥ M; l, q ≥ N.

i.e. (xk, l ) is an Iθν ∗ -Cauchy sequence in X. Then by Theorem 3.15, we have (xk, l ) is an Iθν -Cauchy sequence in X.



Theorem 3.18. Let I be an admissible ideal of N × N and let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is Iθν convergent then it is an Iθν -Cauchy sequence. Proof. Suppose that Iθν − limk,l xk,l = L. Let ε > 0 and α ∈ (0, 1). Choose β ∈ (0, 1) such that T (1 − β , 1 − β ) > (1 − α ). Since Iθν − limk,l xk,l = L. Then for every ε > 0 and β ∈ (0, 1), we have



1 (r, s ) ∈ N × N : hr,s

A=





νxk,l −L (ε ) ≤ 1 − β ∈ I.

(k,l )∈Jr,s

This implies that



φ = A = c

1 (r, s ) ∈ N × N : hr,s



 νxk,l −L (ε ) > 1 − β ∈ F (I ).

(k,l )∈Jr,s

Let (p, q) ∈ Ac . Then we have

1 hr,s If we take

B=



νx p,q −L (ε ) > 1 − β .

( p,q )∈Jr,s

 1 (r, s ) ∈ N × N : hr,s



 νxk,l −x p,q (ε ) ≤ 1 − α ,

(k,l )∈Jr,s

then it is sufficient to show that B ⊂ A. Let (k, l) ∈ B. Then we have

1 hr,s



νxk,l −x p,q (ε ) ≤ 1 − α .

(k,l )∈Jr,s

Furthermore we have either

1 hr,s or

1 hr,s If

1 hr,s



νxk,l −L

ε 2

(k,l )∈Jr,s



νxk,l −L

ε 2

(k,l )∈Jr,s



νxk,l −L

ε

(k,l )∈Jr,s

then we have (k, l) ∈ A.

2

≤1−β

> 1 − β.

≤1−β

152

B. Hazarika / Applied Mathematics and Computation 279 (2016) 139–153

Suppose that

1 hr,s



νxk,l −L

(k,l )∈Jr,s

ε 2

> 1 − β.

Then we have

νxk,l −L (ε ) >

1 hr,s



νxk,l −L

(k,l )∈Jr,s

ε 2

and

νxm,n −L (ε ) >

1 hr,s

 (m,n )∈Jr,s

νxm,n −L

>1−β

ε 2

> 1 − β.

Therefore we get

1−α ≥

 ε  ε νxk,l −x p,q (ε ) ≥ T νxk,l −L , νxm,n −L

2 > T (1 − β , 1 − β ) > 1 − α

2

which is is not possible. Hence B ⊂ A and therefore B ∈ I. This shows that (xk, l ) is an Iθν -Cauchy sequence.  Theorem 3.19. Let I be an admissible ideal of N × N and let (X, ν , T) be an PNS. If a double sequence x = (xk,l ) in X is Iθν ∗ convergent then it is an Iθν ∗ -Cauchy sequence. Proof. The proof of the theorem follows from Theorem 3.16 and Theorem 3.18.  References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

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