Δm-ideal convergence

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Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. pp. 101-110, 2011 Applied Mathematics

∆m_{-Ideal Convergence} Hafize Gok Gumus, Fatih Nuray

Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon, Turkiye

e-mail: hafi ze_ 1409@ hotm ail.com,f_ nuray@ hotm ail.com

Received Date: April 23, 2011 Accepted Date: May 23, 2011

Abstract. Statistical convergence has several applications in diﬀerent fields of Mathematics: Number theory, trigonometric series, summability theory, prob-ability theory, measure theory, optimization and approximation theory. The notion of ideal convergence corresponds to a generalization of the statistical convergence. In this paper, we define the ∆m_(c

I) spaces by using generalized sequence spaces and ideal convergence. Furthermore we establish some topo-logical results and give inclusion relation between ∆m_(wT

p)−convergence and ∆m_{−ideal convergence.}

Key words: Diﬀerence Sequence Spaces, I-Convergence 2000 Mathematics Subject Classification: 40G15, 40A35. 1. Introduction

Let l_∞, c and c0be the linear spaces of bounded, convergent and null sequences, respectively normed by

kxk∞= sup k

xk

where k ∈ N = {1, 2, ...} , the positive integers. In 1981 the idea of diﬀerence sequence spaces was introduced by Kızmaz.(see [10]) He defined the sequence spaces

l_∞(∆) = _{{x = (x}k) : ∆x ∈ l∞} c(∆) = _{{x = (x}k) : ∆x ∈ c} c0(∆) = {x = (xk) : ∆x ∈ c0}

where ∆x = (∆xk) = (xk − xk+1) and showed that these spaces are Banach spaces with the norm

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Then Et and Çolak generalized the concept when they defined the sequence spaces l_∞(∆m₎ ₌ _{{x = (x} k) : ∆mx ∈ l∞} c(∆m₎ ₌ _{{x = (x} k) : ∆mx ∈ c} c0(∆m) = {x = (xk) : ∆mx ∈ c0} where m ∈ N and ∆m_{x = (∆}m_x k) = (∆m−1xk − ∆m−1xk+1). They showed that spaces are Banach spaces with the norm

kxk∆= m P i=1|x

i| + k∆mxk_∞.

(see [4] and [5]). Recently Et and Nuray defined a general sequence spaces ∆m_{(X) = {x = (x}

k) : (∆mxk) ∈ X} where X is any sequence space and ∆mx = (∆m_x

k) = (∆m−1xk − ∆m−1xk+1) for m ∈ N. They established some inclu-sion relations and topological results. They also studied ∆m_{−statistical} conver-gence.(see [6]).

The notion of statistical convergence of sequences of real numbers was intro-duced by H. Fast (see [8]) and it is based on the notion of asymptotic density of a set A ⊆ N.

Definition 1. _{For A ⊆ N and n ∈ N we define} dn(A) = 1 n n P k=1 χ_A(k)

where χ_A is the characteristic function of A. The numbers d(A) = lim inf

n→∞dn(A), d(A) = lim supn→∞ dn(A)

are called, respectively, lower and upper asymptotic density of A. If we have that the limit

d(A) = lim n→∞dn(A)

exists, then d(A) is called the asymptotic density of A.(see [11])

Definition 2. A sequence x = (xk) of real numbers is said to be statistically convergent to L ∈ R provided that for each ε > 0 we have d(Aε) = 0 where

Aε= {k ∈ N : |xk− L| ≥ ε} .

The set of all statistically convergent sequences is denoted by S. (see [11]) Definition 3. A sequence x = (xk) of real numbers is said to be ∆m−statistically convergent to L ∈ R provided that for each ε > 0,

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The set of all ∆m_{−statistically convergent sequences is denoted by ∆}m(S) . (see [6])

Definition 4. _{A family of sets I ⊂ 2}N_{is an ideal in N if and only if} (i) _{∅ ∈ I}

(ii) For each A, B ∈ I we have A ∪ B ∈ I

(iii) For each A ∈ I and each B ⊆ A we have B ∈ I

An ideal is called non-trivial if N /∈ I and a non-trivial ideal is called admissible if {n} ∈ I for each n ∈ N.(see [12])

Definition 5. _{A family of sets F ⊂ 2}N_{is a filter in N if and only if} (i) _{∅ /}_{∈ F}

(ii) For each A, B ∈ F we have A ∩ B ∈ F

(iii) For each A ∈ F and each B ⊇ A we have B ∈ F (see [12]) Proposition 1. _{I is a non-trivial ideal in N if and only if}

F = F (I) = {M = N\A : A ∈ I} is a filter in N. (see [12])

In 2000, Kostyrko, ˘Salát and Wilczy´nski introduced and studied the concept of I−convergence of sequences in metric spaces where I is an ideal (see [11]). Definition 6. A real sequence x = (xk) is said to be I−convergent to L ∈ R if and only if for each ε > 0 the set

{k ∈ N : |xk− L| ≥ ε}

belongs to I. The number L is called I−limit of the sequence x. (see [11]) The set of all I−convergent sequences is denoted by cI.

2. Main Results

Definition 7. _{Let I ⊆ 2}N_{be a non-trivial ideal in N. The sequence x = (x} k) of real numbers is said to be ∆m_{I−convergent to L ∈ R if for each ε > 0 the set}

{k ∈ N : |∆mxk− L| ≥ ε}

belongs to I. In this case we write xk → L(∆m(cI)). The space of all ∆m_{I−convergent sequences is denoted by ∆}m_(c

I).

Example 1. Define If = {A ⊆ N : A is finite} . If is an admissible ideal in N and

∆m(cIf) = ∆

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Example 2. Define the non-trivial ideal Id to be Id= {A ⊆ N : d(A) = 0} . In this case

∆m(cId) = ∆

m_(S).

Theorem 1. Let I be an admissible ideal, and let x = (xk) and y = (yk) be real valued sequences.

(a) If xk → L1(∆m(cI)) and yk → L2(∆m(cI)) then xk + yk → L1 + L2(∆m(cI)).

(b) If xk → L1(∆m(cI)) and λ ∈ R then λxk→ λL1(∆m(cI)). Proof. (a) Let ε > 0 be given. If xk → L1(∆m(cI)) we can write

A1= n k ∈ N : |∆m_x k− L1| < ε 2 o ∈ F (I) and if yk→ L2(∆m(cI)) we can write

A2= n k ∈ N : |∆myk− L2| < ε 2 o ∈ F (I).

In this case A1∩ A2 ∈ F (I) and since F (I) is a filter, A1∩ A26= ∅.. For each k ∈ A1∩ A2, |∆m_(x k+ yk) − (L1+ L2)| = |(∆mxk− L1) + (∆myk− L2)| ≤ |∆m_x k− L1| + |∆myk− L2| < ε₂+ε₂ = ε

(b) Let xk→ L1(∆m(cI)). Then, for each k ∈ A1, |∆m_(λx

k) − λL1| = |λ(∆mxk− L1)| ≤ |λ| |∆m_x

k− L1| < λε₂

As ε > 0 was arbitrary, it follows that {k : |∆m_(λx

k) − λL1| < η} ∈ F (I) for any η > 0.

Definition 8. _{Let I ⊆ 2}N_{be an ideal in N. If} {k + 1 : k ∈ A} ∈ I

for any A ∈ I, then I is said to be a translation invariant ideal. (see [3]) Corollary 1. If I is translation invariant and (xk) ∈ cI then (xk+1) ∈ cI.

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Example 3. Id is a translation invariant ideal.

Proposition 2. Suppose that I ⊆ 2N _{is an admissible translation invariant} ideal and m ∈ N. Then

∆m−1(cI) ⊆ ∆m(cI). (see [3])

Proof. _{Let x ∈ ∆}m−1_(c_I_{) then}¡_∆m−1_x_k¢_{∈ c}_I_{. Since I is translation invariant,} ¡ ∆m−1_x k+1 ¢ ∈ cI. We know that (∆mxk) =¡∆m−1xk− ∆m−1xk+1¢ and hence x ∈ ∆m_(c I) .

Definition 9. _{Let I ⊆ 2}N_{be a non-trivial ideal in N. If for each ε > 0 there is} a number N (= N (ε)) such that

{k ∈ N : |∆mxk− ∆mxN| ≥ ε} ∈ I then x is called ∆m_{I−Cauchy sequence.}

Theorem 2. If x = (xk) is a ∆mI−convergent sequence then x is ∆mI−Cauchy sequence.

Proof. Suppose that xk → L(∆m(cI)) and ε > 0. Then A1= n k ∈ N : |∆mxk− L| < ε 2 o

< ε 2+

ε 2 = ε Hence x is a ∆m_{I−Cauchy sequence.}

Theorem 3. Let I be a non-trivial ideal in N and x be a sequence. If there is a ∆m_{I−convergent y such that}

{k ∈ N : ∆mxk6= ∆myk} ∈ I then x is also ∆m_{I−convergent.}

Proof. _{Assume that {k ∈ N : ∆}m_x

k 6= ∆myk} ∈ I and y is ∆mI−convergent to L ∈ R. For each ε > 0

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As the right-hand side of the inclusion is in I, we have that {k ∈ N : |∆mxk− L| ≥ ε} ∈ I.

Proposition 3. _{Let I ⊆ 2}N _{be a non-trivial ideal. Then ∆}m_{(c) ⊆ ∆}m_(c I). Proof. _{We know that c ⊆ c}I and for any X and Y spaces, if X ⊆ Y then ∆m_{(X) ⊆ ∆}m_{(Y ). (see [6] ). Hence it is easy to see that ∆}m_{(c) ⊆ ∆}m_(c

I). Remark 1. The converse of (i) is not always true.

Example 4. Define the sequence x such that ∆mxk = ( k12 if k is a square 0 otherwise then x ∈ ∆m_(c Id) but x /∈ ∆ m_{(c) .} Example 5. Define the ideal I such that

A ∈ I ⇐⇒ A eventually only contains even natural numbers then I is a non-trivial ideal in N. When

∆mx = (∆mxk) = (1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, ...) we have

A_ε0 = {k ∈ N : ∆mx_k 6= 0}

= _{{1, 2, 3, 5, 6, 7, 10, 12, 14, 16, 18, ...}} and x is ∆m_{I−convergent to zero i.e. x}

k → 0(∆m(cI)). Hence Aε0 ∈ I and ∆m_{x ∈ c}

I. Now lets look at the statistical convergence of the sequence. limn→∞n1|Aε0| = limn→∞n1 ¯ ¯(B) +n 2 ¯ ¯ = 1 2

where B is a finite number and |Aε0| is the cardinality of Aε0. Hence ∆mx /∈ S. Definition 10. Let T = (tnk)n,k∈N be a non-negative regular matrix, A ⊆ N and χA be the characteristic function of A. For each n ∈ N let

d(n)_T (A) = P∞ k=1

tnkχA(k). If we have the limit

lim n→∞d

(n)

T (A) = dT(A) then this limit is called T −density of A. The ideal

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is a non-trivial admissible ideal in N. When we take tnk=

( 1

n, k ≤ n 0, k > n

then we have ∆−statistical convergence, which is the case when m = 1 in ∆m_{−statistical convergence.}

Definition 11. Let T = (tnk)n,k∈Nbe a non-negative regular matrix and define the space ∆m_(wT p) to be ∆m(w_pT) = ½ x = (xk) : lim n→∞ ∞ P k=1 tnk|∆mxk− L|p= 0, p > 0 for some L ¾ .

In this case we write xk→ L(∆m(w

T

p)).

Theorem 4. _{Let p ∈ R, 0 < p < ∞, T = (t}nk) be a non-negative regular matrix and A ⊆ N. If xk → L(∆m(wTp)) then xk → L(∆m(cI_dT)). If x ∈ ∆m(l∞) and xk → L(∆m(cIdT)) then xk→ L(∆

m_(wT p)). Proof. Assume that xk → L(∆m(wTp)) then

lim n→∞ ∞ P k=1 tnk|∆mxk− L|p= 0 for some L. Let ε > 0 be given and define

A∆m_ε= {k ∈ N : |∆mx_k− L| ≥ ε} and write ∞ P k=1 tnk|∆mxk− L|p = P k∈A∆mε tnk|∆mxk− L|p+ P k /∈A∆mε tnk|∆mxk− L|p ≥ Ã P k∈A∆mε tnk ! εp Then we have xk→ L(∆m(cI_dT)).

Now suppose that x ∈ ∆m_(l

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where Ac is complement of A. If we consider regularity of T, Ac _{∈ I}dT and

boundedness of (∆m_x

k) right side tends to zero. Hence left side also tends to zero.

3. Matrix Transformations

Kızmaz obtained some matrix transformations between c and c(∆) spaces (see [10]) and Et and Çolak generalized them for c and ∆m_{(c). (see [4]) Gok, studied} about some matrix transformations related with summability methods and also investigated matrix transformations by using I−convergence and she obtained characterizations of matrix transformations between cI and ∆ (cI)∩∆ (l∞). (see [2] , see [3]). In this part of the paper we generalize these results for ∆m_(c

I) ∩ ∆m_(l

∞).

Theorem 5 _{Let T be a non-negative matrix, T ∈ (l}_∞, l_∞) and I be an ad-missible ideal in N. T ∈ (cI ∩ l∞, cI) and I − lim x = I − lim(T x) if and only if (i) ½ n : ¯ ¯ ¯ ¯_kP_∈Atnk ¯ ¯ ¯ ¯ ≥ ε ¾

∈ Ifor all A ∈ I and ε > 0.

(ii) I − lim P∞ k=1

tnk= 1

(see [2] and [3])

Theorem 6. Let T = (tnk) be a non-negative regular matrix, T ∈ (l∞, l∞), and I be an admissible translation invariant ideal in N. Then T ∈ (∆ (cI) ∩ ∆ (l_∞) , cI) if and only if (i) ( n ∈ N : ¯ ¯ ¯ ¯ ¯ P k∈A Ã ∞ P j=k tnj !¯¯ ¯ ¯ ¯≥ ε )

∈ I for all A ∈ I and ε > 0.

(ii) I − limn µ_P_∞ k=1 ktnk ¶ = 1 (see [2] and [3])

Theorem 7. Let T = (tnk) be a nonnegative regular matrix, T ∈ (l∞, l∞) and I be an admissible translation invariant ideal in N. Then, for each m ∈ N, T ∈ (∆m_(c I) ∩ ∆m(l∞) , cI) if and only if (i) ( n ∈ N : ¯ ¯ ¯ ¯ ¯ P k∈A Ã ∞ P j=k (j−(k−1))(j−(k−2))...(j−(k−(m−1))) (m−1)! tnk !¯¯_¯ ¯ ¯≥ ε ) ∈ I for all A ∈ I and ε > 0. (ii) I − limn µ_∞ P k=1 k(k+1)(k+2)...(k+(m−1) m! ¶ = 1

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Poof. _{Suppose that T ∈ (∆}m(cI) ∩ ∆m(l∞) , cI) and define the matrix Smby Sm= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 0 0 ... m 1 0 0 0 ... m(m+1) 2! m 1 0 0 ... m(m+1)(m+2) 3! m(m+1) 2! m 1 0 ... m(m+1)(m+2)(m+3) 4! m(m+1)(m+2) 3! m(m+1) 2! m 1 ... .. . ... ... ... ... ... ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Then for each x ∈ cI ∩ l∞ we have that ∆m(Smx) =

(

−xn+m , if m is odd xn+m , if m is even By the translation invariance of the ideal ∆m_(Sm_{x) ∈ c}

I ∩ l∞ and hence (Smx)_n _{∈ c}I(∆m)∩l∞(∆m). By the definition of (Sm)−1, for each y ∈ (∆m(cI)∩ ∆m_(l

∞)) we have (Sm)−1(y) ∈ cI∩l∞. Then the matrix R = T Smmaps cI∩l∞ into cI. If we apply theorem 5 to this matrix, then we have that T satisfies (i) and (ii). In order to prove the converse, note that (i) and (ii) yield that R is a matrix transformation between cI∩ l∞and cI. For each x ∈ ∆m(cI) ∩ ∆m(l∞)

³

R¡S−1¢m´=³(T Sm)¡S−1¢m´(x) = T³Sm¡S−1¢m´(x) = T (x) and, as x is arbitrary, we have T ∈ (∆m_(c

I) ∩ ∆m(l∞) , cI). References

1. Bekta¸s,Ç.A., Et, M.and Çolak R., 2004, Generalized diﬀerence sequence spaces and their dual spaces, J.Math.Anal.Appl. 292, 423-432

2. Gok, H., 2005, Some summability methods and matrix transformations, Master’ Thesis, Afyon Kocatepe University, Science Institute, Afyonkarahisar

3. Gok Gumus, H., _I−convergence and asymptotic _I−equivalence of diﬀerence se-quences, PhD. thesis, Afyon Kocatepe University, Science Institute, Afyonkarahisar, to appear

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