lambda-STATISTICALLY CONVERGENT DOUBLE SEQUENCES IN PROBABILISTIC NORMED SPACES

Math. Slovaca 62 (2012), No. 1, 99–108

¯λ-STATISTICALLY CONVERGENT DOUBLE SEQUENCES

IN PROBABILISTIC NORMED SPACES

E. Savas¸* — S. A. Mohiuddine**

(Communicated by Michal Zajac )

ABSTRACT. The purpose of this paper is to introduce and study the concepts of double ¯λ-statistically convergent and double ¯λ-statistically Cauchy sequences in probabilistic normed space.

2012c Mathematical Institute Slovak Academy of Sciences

1. Introduction and preliminaries

The idea of statistical convergence for sequences of real number was first in- troduced by Fast [5] and Steinhauss [18] independently in the same year 1951 and since then several generalizations and applications of this concept have been investigated by various authors, namely ˇSal´at [14], Fridy [6], and many others.

The concept of statistical convergence for double sequences was studied by Mur- saleen and Edely [11] and further studied by Mursaleen and Mohiuddine [10].

The idea of λ-statistical convergence of single sequences x = (x_k) of real numbers has been studied by Mursaleen [12]; and for double sequences of fuzzy numbers by Savas [15]. Quite recently, Mohiuddine and Lohani [9] introduced the concept of λ-statistical convergence of single sequences x = (x_k) in intuitionistic fuzzy normed spaces.

An interesting and important generalization of the notion of metric space was introduced by Menger [8] under the name of statistical metric space, which is now called probabilistic metric space. The idea of Menger was to use distribution functions instead of nonnegative real numbers.

2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 42B15; Secondary 40C05.

K e y w o r d s: double sequences, t-norm, probabilistic normed space, statistical convergence,

¯λ-statistical convergence, ¯λ-statistical Cauchy.

In fact the probabilistic theory has become an area of active research for the last forty years. It has a wide range of applications in functional analysis [4]. An important family of probabilistic metric spaces are probabilistic normed spaces (briefly, PN-spaces). The notion of probabilistic normed spaces was introduced by Sherstnev [17] in 1963 and later on studied by various authors, see [2, 3, 7].

In [1], Alotaibi studied the notion of λ-statistical convergence for single se- quences in probabilistic normed spaces. In this paper, we study the concepts of double ¯λ-statistically convergence and double ¯λ-statistically Cauchy for se- quences in probabilistic normed spaces.

Let K be a subset of N, the set of natural numbers. Then the asymptotic density of K denoted by δ(K) (see [5, 18]), is defined as

δ(K) = lim

n

1

n|{k ∈ K : k ≤ n}|,

where the vertical bars denote the cardinality of the enclosed set.

1.1^ A number sequence x = (x_k) is said to be statistically conver- gent to the number  if for each ε > 0, the set

K(ε) =

k∈ N : |xk− | > ε has asymptotic density zero, i.e.

limn

1

n
k∈ N : k ≤ n, |xk− | > ε= 0.

In this case we write st- lim x = .

Notice that every convergent sequence is statistically convergent to the same limit, but its converse need not be true.

The following definitions were giving by Mursaleen [12].

 1.2^ Let λ = (λ_n) be a non-decreasing sequences of positive real numbers tending to∞ and such that

λ_n+1≤ λn+ 1, λ₁ = 0.

Let K be a subset ofN, the set of natural numbers. The number δ_λ(K) = lim

n

1

λ_n|{k ∈ K : n − λn+ 1≤ k ≤ n}|, is said to be the λ-density of K.

If λ_n= n for every n then every λ-density is reduced to asymptotic density.

 1.3^ A sequence x = (x_k) is said to be λ-statistically convergent to l if for every ε > 0, the set K(ε) has λ-density zero, i.e.,

limn

1

λ_n|Kn(ε)| = 0,

where

K_n(ε) =

k∈ In: |xk− l| > ε

and I_n= [n− λn+ 1, n].

In this case we write st_λ- lim x = l.

Firstly, we recall the following concepts for which we refer the readers to [7, 8, 16] for more details.

 1.4^ A triangular norm (t-norm) is a continuous mapping T : [0, 1]× [0, 1] → [0, 1] such that ([0, 1], T ) is an abelian monoid with unit one and for all a, b, c∈ [0, 1]:

(i) T (c, d)≥ T (a, b) if c ≥ a and d ≥ b;

1.5^ A function f :R → R⁺₀ is called a distribution function if it is non-decreasing and left-continuous with inf

t∈Rf (t) = 0 and sup

t∈Rf (t) = 1.

By D, we denote the set of all distribution functions.

 1.6^ Let X be a real linear space and ν : X → D. Then the probabilistic norm or ν-norm is a t-norm satisfying the following conditions:

(i) ν_x(0) = 0;

(ii) ν_x(t) = 1 for all t > 0 iff x = 0;

(iii) ν_αx(t) = ν_x _t

|α|

for all α∈ R \ {0};

(iv) ν_x+y(s + t)≥ T (νx(s), ν_y(t)) for all x, y∈ X and s, t ∈ R⁺₀. where ν_x means ν(x) and ν_x(t) is the value of ν_x at t∈ R.

Remark 1.1^ We can say that t-norm is a binary operation∗ given by T (a, b) = a∗ b.

Space (X, ν,∗) is called a probabilistic normed space (PN-space), and by a P N -space X we mean the triplet (X, ν,∗).

 1.7^ Let (X, ν,∗) be an P N-space. Then, a sequence x = (xk) is said to be convergent to  in probabilistic norm space X, that is, x_k →  if for^ν every ε > 0 and θ∈ (0, 1), there is a positive integer k0such that ν_xk−(ε) > 1−θ whenever k≥ k0. In this case we write ν- lim x = .

1.8^ Let (X, ν,∗) be an P N-space. Then, x = (xk) is called Cauchy sequence in probabilistic norm space X, if for every ε > 0 and θ∈ (0, 1), there is a positive integer k₀ such that ν_xk−xj(ε) > 1− θ for all j, k ≥ k0.

 1.9^ Let (X, ν,∗) be an P N-space. Then, x = (xk) is said to be bounded in probabilistic norm space X, if there is r∈ R⁺such that ν_x(r) > 1−θ, 0 < θ < 1.

We denote by ^ν_∞ the space of bounded sequences in P N -space.

2. Double ¯ λ-statistical convergence in PN-space

In this section we study the concept of ¯λ-statistically convergent sequences in probabilistic normed spaces. Before continuing with this paper we present the definition of density and related concepts which form the background of the present work.

By the convergence of a double sequence we mean the convergence on the Pringsheim sense that is, a double sequence x = (x_k,l) has Pringsheim limit L (denoted by P-lim x = L) provided that given ε > 0 there exists N ∈ N such that|xk,l− L| < ε whenever k, l > N ([13]).

We now recall the definition of density and related concepts which form the background of the present work.

Let K⊆ N×N be a two-dimensional set of positive integers and let K(m, n) be the numbers of (k, l) in K such that k≤ m and l ≤ n. Then the two-dimensional analogue of natural density can be defined as follows [11].

The lower asymptotic density of the set K⊆ N × N is defined as δ₂(K) = lim inf

m,n

K(m, n) mn .

In case the sequence (K(m, n)/mn) has a limit in Pringsheim’s sense then we say that K has a double natural density and is defined as

m,nlim

K(m, n)

mn = δ₂(K).

For example, let K =

(i², j²) : i, j∈ N . Then δ₂(K) = lim

m,n

K(m, n) mn ≤ lim

m,n

√m√ n mn = 0, i.e. the set K has double natural density zero, while the set

(i, 2j) : i, j ∈ N has double natural density 1/2.

 2.1^ A real double sequence x = (x_kl) is said to be double statis- tically convergent ([11]) to the number  if for each ε > 0, the set

(k, l) : k, l∈ N, k ≤ m, l ≤ n, |xkl− | ≥ ε has double natural density zero.

We denote the set of all double statistically convergent sequences by st₂. In this case we write st₂- lim

k,l x_kl= .

 2.2^ Let (X, ν,∗) be a P N-space. We say that a double sequence x = (x_kl) is said to be double statistically convergent to  in probabilistic norm space X (for short, S^{(P N)}-convergent), if for every ε > 0 and t∈ (0, 1),

δ

(k, l) : k, l ∈ N, k ≤ m, l ≤ n, νxkl−(ε)≤ 1 − t

= 0

or equivalently δ

(k, l) : k, l∈ N, k ≤ m l ≤ n, νxkl−(ε) > 1− t

= 1.

In this case we write x_kl →  or S^{ν (P N)}- lim x = , and denote the set of all S-convergent double sequences in probabilistic normed spaces by (S)_ν.

In this paper, we introduce the concept of double ¯λ-statistical convergence of sequences in PN-Spaces.

The idea of λ-statistical convergence of single sequences in PN-spaces was studied by Alotaibi [1].

First we define the concept of ¯λ-density:

Let λ = (λ_n) and µ = (µ_m) be two non-decreasing sequences of positive real numbers each tending to∞ and such that

λ_n+1≤ λn+ 1, λ₁ = 0 and

µ_m+1≤ µm+ 1, µ₁ = 0.

Let K ⊆ N × N. The number δ_¯λ(K) = P - lim

n,m

1

λ¯_nm
k∈ In, l∈ Jm : (k, l)∈ K

is said to be the ¯λ-density of K, provided the limit exists, where ¯λ_nm= λ_nµ_m We now ready to define the double ¯λ-statistical convergence.

 2.3^ A double sequence x = (x_kl) is said to be double ¯λ- statisti- cally convergent or S_¯λ-convergent to  if for every ε > 0,

P - lim

n,m

1

¯λnm
k∈ In, l∈ Jm: |xkl− | ≥ ε= 0.

In this case we write S_¯λ- lim x =  or x_jk → (S^P _¯λ) and we denote the set of all double S_¯λ-statistically convergent sequences by (S_¯λ).

If ¯λ_nm = nm, for all n, m, then the set of S_¯λ-convergent sequences reduces to the space st².

Now we define the S_¯λ-convergence in P N -space.

 2.4^ Let (X, ν,∗) be a P N-space. We say that a double sequence x = (x_kl) is said to be S_¯λ-convergent to  in probabilistic norm space X (for short, S_¯λ^{(P N)}-convergent), if for every ε > 0 and t∈ (0, 1),

δ_¯λ

k∈ In, l∈ Jm : ν_xkl−(ε)≤ 1 − t

= 0 or equivalently

δ_¯λ

k∈ In, l∈ Jm : ν_xkl−(ε) > 1− t

= 1.

In this case we write x_kl → (S^ν _¯λ) or S_¯λ^{(P N)}- lim x = , and denote the set of all S_¯λ-convergent double sequences in probabilistic normed spaces by (S_¯λ)_ν.



 2.1^ Let (X, ν,∗) be a P N-space. If a sequence x = (xkl) is a double

¯λ-statistically convergent in probabilistic normed space X, then S^{(P N)}_¯λ -limit is unique.

P r o o f. Suppose that S^{(P N)}_¯λ - lim x = ₁ and S^{(P N)}_¯λ - lim x = ₂. Let ε > 0 and t > 0. Choose s∈ (0, 1) such that (1 − s) ∗ (1 − s) ≥ 1 − t. Then we define the following sets as

K₁(s, ε) =

k∈ In, l∈ Jm: ν_xkl−1(ε/2)≤ 1 − s , K₂(s, ε) =

k∈ In, l∈ Jm: ν_xkl−2(ε/2)≤ 1 − s .

So that we have δ_¯λ(K₁(s, ε)) = 0 and δ_¯λ(K₂(s, ε)) = 0 for all ε > 0. Now let K₃(s, ε) = K₁(s, ε)∪ K2(s, ε).

It follows that δ_¯λ(K₃(s, ε)) = 0, which implies δ_¯λ(N × N \ K3(s, ε)) = 1.

If (k, l)∈ N × N \ K3(s, ε), we have

ν_1−2(ε) = ν_{(1−xkl)+(xkl−2)}(ε/2 + ε/2)

≥ νxkl−1(ε/2)∗ νxkl−2(ε/2)

> (1− s) ∗ (1 − s) ≥ 1 − t.

Since t > 0 was arbitrary, we get ν_1−2(ε) = 1 for all ε > 0, which gives ₁ = ₂. Hence S_¯λ^{(P N)} limit is unique.

This completes the proof of the theorem.



 2.2^ Let (X, ν,∗) be a P N-space. If a sequence x = (xkl) is a double statistically convergent to  in probabilistic normed space X, then S^{(P N)}_¯λ - lim x

=  if

P - lim inf

nm

λ¯_nm

nm > 0. (2.1)

P r o o f. For given ε > 0 and t∈ (0, 1),

k≤ n, l ≤ m : νxkl−(ε)≤ 1 − t

⊃

k∈ In, l∈ Jm : ν_xkl−(ε)≤ 1 − t

Therefore,

1 nm

k≤ n, l ≤ m : νxkl−(ε)≤ 1 − t

≥ 1 nm

k∈ In, l∈ Jm: ν_xkl−(ε)≤ 1 − t

≥ λ¯_nm nm

1 λ¯_nm

k∈ In, l∈ Jm: ν_xkl−1(ε)≤ 1 − t .

Taking the limit as n, m→ ∞ and using (2.1), we get S_¯λ^{(P N)}- lim x = .



2.3^ Let (X, ν,∗) be a P N-space. If ν- lim x =  then S^{(P N)}_¯λ - lim x = .

But converse need not be true.

P r o o f. Let ν- lim x = . Then for every t∈ (0, 1) and ε > 0, there is a couple (k₀, l₀)∈ N × N such that νxkl−(ε) > 1− t for all k ≥ k0, l≥ l0. Hence the set

k∈ In, l∈ Jm : ν_xkl−(ε)≤ 1 − t

has natural density zero and hence δ_¯λ

k∈ In, l∈ Jm : ν_xkl−(ε)≤ 1 − t

= 0, that is, S^{(P N)}_¯λ - lim x = .

For converse, we construct the following example:

Example 2.1. Let (R, |·|) denote the space of real numbers with the usual norm.

Let a∗ b = ab and νx(ε) = _ε+|x|^ε , where x ∈ X and ε ≥ 0. In this case, we observe that (R, ν, ∗) is a PN-space. Define a sequence x = (xkl) by

x_kl =

⎧⎪

⎨

⎪⎩

(k, l), for n−√ λ_n

+ 1≤ k ≤ n and m−√

µ_m

+ 1≤ l ≤ m (k, l ∈ N);

0, otherwise.

It is easy to see that, this sequence is S_¯λ-convergence to zero in P N -space, i.e., x_kl→ 0(S^ν _¯λ), while x_kl→ 0.^ν

This completes the proof of the theorem.



2.4^ Let (X, ν,∗) be a P N-space. Then, S^{(P N)}_¯λ - lim x =  if and only if there exists a subset K =

(k_n, l_n) : k₁ < k₂ < . . . ; l₁ < l₂ < . . .

⊆ N × N such that δ_¯λ(K) = 1 and ν- lim

n→∞x_knln = .

P r o o f.

Necessity. Suppose that S_¯λ^{(P N)}- lim x = . Then, for any ε > 0 and s∈ N, let K(s, ε) =

k∈ In, l∈ Jm: ν_xkl−(ε)≤ 1 − 1 s

,

and

M (s, ε) = 

k∈ In, l∈ Jm: ν_xkl−(ε) > 1− ¹_s . Then δ_¯λ(K(s, ε)) = 0 and

M (1, ε)⊃ M(2, ε) ⊃ · · · ⊃ M(i, ε) ⊃ M(i + 1, ε) ⊃ . . . (2.2) and

δ_¯λ(M (s, ε)) = 1, s = 1, 2, . . . .

Now we have to show for (k, l)∈ M(s, ε), x = (xkl) is ν-convergent to . Suppose that for some (k, l) ∈ M(s, ε), the double sequence x = (xkl) is not ν-con- vergent to . Therefore there is t > 0 and are positive integers l₀, k₀ such that ν_xkl−(ε)≤ 1 − t for all l ≥ l0, k ≥ k0. Let ν_xkl−(ε) > 1− t for all k ≤ k0, l≤ l0. Then

δ_¯λ

k∈ In, l∈ Jm : ν_xkl−(ε) > 1− t

= 0.

Since t > ¹_s, we have

δ_¯λ(M (s, ε)) = 0

which contradicts (2.2). Therefore x = (x_kl) is ν-convergent to .

Conversely, suppose that there exists a subset K =

(k_n, l_n) : k₁< k₂< . . . ; l₁ < l₂ < . . .

⊆ N × N such that δ_¯λ(K) = 1 and ν- lim

(k,l)∈Kx_kl = , there exists N∈ N such that for every t ∈ (0, 1) and ε > 0

ν_xkl−(ε) > 1− t, for all k, l≥ N.

Now

M (t, ε) =

k∈ In, l∈ Jm: ν_xkl−(ε)≤ 1 − t

⊆ N × N −

(k_N+1, l_N+1), (k_N+2, l_N+2), . . . . Therefore δ_¯λ(M (t, ε))≤ 1 − 1 = 0. Hence S^{(P N)}_¯λ - lim x = .

This completes the proof of the theorem.

In the next we now define double λ-statistically Cauchy sequence in proba- bilistic normed space.

2.5^ Let (X, ν,∗) be a P N-space. Then, a double sequence x=(xkl) is said to be S_¯λ-Cauchy in P N -space X if for every ε > 0 and t∈ (0, 1), there exist N = N (ε) and M = M (ε) such that

δ_¯λ

k∈ In, l∈ Jm: ν_xkl−xMN(ε)≤ 1 − t

= 0.



 2.5^ Let (X, ν,∗) be a P N-space and θrz be any double lacunary sequence. Then, a double sequence x = (x_kl) is S^{(P N)}_¯λ -convergent if and only if it is S_¯λ^{(P N)}-Cauchy in probabilistic norm space X.

P r o o f. Let x = (x_kl) be S^{(P N)}_¯λ -convergent to , i.e., S_¯λ^{(P N)}- lim x = . Then for a given ε > 0 and t∈ (0, 1), choose r > 0 such that (1 − t) ∗ (1 − t) > 1 − r.

Then, we have

δ_¯λ(A(t, ε)) = δ_¯λ

k∈ In, l∈ Jm: ν_xkl−(ε/2)≤ 1 − t

= 0 (2.3)

which implies that

δ_¯λ(A^C(t, ε)) = δ_¯λ

k ∈ In, l∈ Jm: ν_xkl−(ε/2) > 1− t

= 1.

Let (p, q)∈ A^C(t, ε). Then ν_xpq−(ε) > 1− t.

Now, let

B(t, ε) =

k∈ In, l∈ Jm: ν_xkl−xpq(ε)≤ 1 − r .

We need to show that B(t, ε)⊂ A(t, ε). Let (k, l) ∈ B(t, ε) \ A(t, ε). Then we have

ν_xkl−xpq(ε)≤ 1 − r and ν_xkl−(ε/2) > 1− t, in particular ν_xpq−(ε/2) > 1− t. Then

1− r ≥ νxkl−xpq(ε)≥ νxkl−(ε/2)∗ νxpq−(ε/2) > (1− t) ∗ (1 − t) > 1 − r, which is not possible. Hence B(t, ε)⊂ A(t, ε). Therefore, by (2.3), δ_¯λ(B(t, ε))

= 0. Hence x is S_¯λ^{(P N)}-Cauchy.

Conversely, let x = (x_kl) be S_¯λ^{(P N)}-Cauchy but not S_¯λ^{(P N)}-convergent. Now ν_xkl−xMN(ε)≥ νxkl−(ε/2)∗ νxMN−(ε/2) > (1− t) ∗ (1 − t) > 1 − r, since x is not S_¯λ^{(P N)}-convergent. Therefore δ_¯λ(B^C(t, ε)) = 0, where

B(t, ε) =

(k, l)∈ N × N : νxkl−xMN(ε)≤ 1 − r

and so δ_¯λ(B(t, ε)) = 1, which is contradiction, since x was S^{(P N)}_¯λ -Cauchy. Hence x must be S_¯λ^{(P N)}-convergent.

This completes the proof of the theorem.

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Received 8. 7. 2009 Accepted 16. 12. 2009

* Department of Mathematics Istanbul Ticaret University Usk¨¨ udar/Istanbul

TURKEY

E-mail : ekremsavas@yahoo.com

** Department of Mathematics Faculty of Science

King Abdulaziz University P.O. Box 80203

Jeddah 21589 SAUDI ARABIA

E-mail : mohiuddine@gmail.com