ON $A^{\mathcal{I^{K}}}$-SUMMABILITY Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 8, No. 1, 2022, pp. 13-22

DOI: 10.15826/umj.2022.1.002

ON AIK-SUMMABILITY

Chiranjib Choudhury^, Shyamal Debnathtt

Tripura University (A Central University), Suryamaninagar-799022, Agartala, India

tchiranjibchoudhury123@gmail.com, chiranjib.mathematics@tripurauniv.in ttshyamalnitamath@gmail.com

Abstract: In this paper, we introduce and investigate the concept of A1 -summability as an extension of A1 -summability which was recently (2021) introduced by O.H.H. Edely, where A = (ankis a nonnegative regular matrix and I and K represent two non-trivial admissible ideals in N. We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that AK-summability always implies A1 -summability whereas A1 -summability not necessarily implies

iK

A1 -summability. Finally, we give a condition namely AP(I, K) (which is a natural generalization of the condition AP) under which A1 -summability implies A1 -summability.

Keywords: Ideal, Filter, I-convergence, IK-convergence, A1 -summa-bility, -summability.

1. Introduction

In 2000, Kostrkyo and Salat [12] introduced the notion of ideal convergence. They studied several fundamental properties of I and I*-convergence and showed that their idea was the extended version of so many known convergence methods. Based on I-convergence several generalizations were made by researchers and several analytical and topological properties have been investigated (see [1, 9, 11, 15-19, 21, 22] where many more references can be found) and this area becomes one of the most focused areas of research.

In 2011, M. Macaj and M. Sleziak [13] generalized the idea of I*-convergence to IK-convergence by involving two ideals I and K. In the case of IK-convergence, the convergence along the large set is taken with regard to another ideal K instead of considering ordinary convergence. So from that point of view the concept of IK-convergence being an extension of I*-convergence shows a strong analogy for further investigation. Recent developments in the direction of IK-convergence from topological aspects can be found from the works of Das et al. [4, 5], Banerjee and Paul [2, 3] and many others.

If x = (xk) be a real-valued sequence and A = (ank)^Dk=1 be an infinite matrix, then Ax is the sequence having nth term An(x) = ^^=1 ankxk. A sequence x = (xk) is said to be A-summable to L, if lim An(x) = L. A matrix A = (ank)^°k=1 is said to be regular if it maps a convergent

n—^^o ,

sequence into a convergent sequence keeping the same limit i.e., A € (c, c)reg if A € (c, c) and lim An(x) = lim xk. Here c, (c, c), and (c, c)reg denote the collection of all real-valued convergent

n—TO k—TO

sequences, collection of all matrices which maps an element of c to an element of c, and the collection of all regular matrices which maps an element of c to an element of c, respectively. The necessary and sufficient Silverman-Toeplitz conditions for an infinite matrix A = (ank)TO°k=1 to be regular are as follows:

(i) sup££=1 |ank | < to;

n

(ii) For any k € N, lim ank = 0;

(iii) lim £ank = 1-

n—y^o

In 2008, Edely and Mursaleen [7] generalized the notion of A-summability to statistical A-summability by using the concept of natural density. Recently, Edely [6] further extended the notion of statistical A-summability to Ax-summability, where I represents an ideal in N. In this paper we intend to introduce the notion of A1 -summability which is a natural generalization of A1 *

-summability. For more details regarding summability theory, one may refer to [8, 10, 14, 20].

Throughout the paper, we will use (yn) to denote the image (An(x)) of the sequence x = (xk) under the transformation of the non-negative regular infinite matrix A.

2. Definitions and preliminaries

Definition 1. A collection I containing subsets of a nonempty set X is called an ideal in X if and only if (i) 0 € I, (ii) P, Q € I implies P U Q € I (Additive), and (iii) P € I, Q C P implies Q € I (Hereditary).

If for any x € X {{x}} C I then it is said that I satisfies the admissibility property or simply is called admissible. Also I is called non-trivial if X / I and I = {0}.

Some standard examples of ideal are given below:

(i) The set If consisting of all subsets of N having finite cardinality is an admissible ideal in N.

(ii) The set Id of all subsets of natural numbers having natural density 0 is an ideal in N which is also admissible.

(iii) The

set Ic = {A C N : ^a 1 < to} is an ideal in N which also has the so called admissibility property.

to

(iv) Suppose N = (J Dp, where Dp C N for any p € N and for i = j, Di n Dj = 0. Then, the set

p= 1

I of all subsets of N which intersects finitely many Dp's forms an ideal in N.

More important examples can be found in [9] and [11].

Definition 2. A collection F containing subsets of a nonempty set X is called a filter in X if and only if (i) 0 / F (ii) M,N € F implies M n N € F and (iii) M € F, N D M implies N € F.

If I is a proper non-trivial ideal in X, then the collection F (I) = {M C X : 3 P € I suchthat M = X \ P} forms a filter in X. It is known as the filter associated with the ideal I.

Definition 3 [12]. Let I be an ideal in N which satisfies the admissibility property. A real-valued sequence x = (xk) is called I-convergent to l if for every e > 0 the set {k € N : |xk — l| > e} is contained in I. The number l is called the I-limit of the sequence x = (xk). Symbolically, I — lim x = l.

Definition 4 [12]. Let I be an ideal in N which satisfies the admissibility property. A sequence x = (xk) is called I*-convergent to l, if there exists a set M = {m1 <m2 < ... < mk < ...} in the associated filter F(I), for which lim xmk = l holds.

Definition 5 [13]. Let I, K denote two ideals in N. A sequence x = (xk) is called IK-convergent to l if the associated filter F(I) contains a set M such that the sequence y = (yk) defined by

= j xk, k € M, yk [ l, k/M

is K-convergent to l.

If we consider K = If then IK-convergence concept coincides with I*-convergence [12].

Definition 6 [13]. Let K be an ideal in N. Then, P CK Q denotes the property P \ Q € K. Also P CK Q and Q CK P together implies P Q. Thus P Q if and only if PAQ € K. A set P is said to be K-pseudointersection of a system {Pi : i € N} if for every i € N P CK Pi holds.

Definition 7 [13]. Let I and K be two ideals on N. Then I is said to have the additive property with respect to K or the condition AP(I, K) holds if every sequence (Fn)neN of sets from F(I) has K-pseudointersection in F(I).

Definition 8 [6]. A real-valued sequence x = (xk) is said to be A1 -summable to a real number L, if the transformed sequence (An(x)) is I-convergent to L. Symbolically, it is written as A1 — lim xk = L.

Definition 9 [6]. A real-valued sequence x = (xk) is said to be A1* -summable to a real number L, if there exists a set M = {m1 <m2 < ... < mi < ...} € F(I) such that

lim V" amikxk = lim ymi = L.

i—^^ z—* i—>-TO

3. Main results

Throughout the section, for a sequence x = (xk) we will use y = (yn) to denote the transformed sequence (An(x)) where A,n(x) = J2fc=1 ankxk.

Definition 10. Let A = (ank)TOk=1 be a non-negative regular matrix and suppose I, K be two admissible ideals in N. A real-valued sequence x = (xk) is said to be AlK-summable to L € R, if there exists a set M € F(I) such that the sequence z = (zk) defined by

= I yk, k€M, Zk {L, k/M

is K-convergent to L, where the sequence y = (yn) is defined as

to

yn = An (x) y ] ankxk.

k=1

In this case we write, AlK — lim xk = L.

Example 1. Consider the decomposition of N given by

TO

N = [J Di, Di = {2i-1 (2s — 1) : s = 1, 2, 3,...}.

i=1

Let I denotes the ideal consisting of all subsets of N which intersects finitely many of Di's. Consider the sequence x = (xk) defined by xk = 1/i if k € D-i and the infinite matrix A = (ank)TOk=1 as

I 1, k = n + 2, |0, otherwise.

tK

Then, the sequence is AI -summable to 0 for K = I. Justification: Clearly,

TO 1

yn = Y" ank,xk = n + 2 € A-^^ i k=1

Let M = N \ D1. Then, M € F(I) and it is easy to verify that the sequence z = (zk) defined by

z =1 yk, k € M, zk \0, k/M

is I-convergent to 0. Hence, A11 — lim xk = 0.

Theorem 1. Let A1* — lim xk = L t^en A1 K — lim xk = L.

P r o o f. Let A1 * — lim xk = L. Then, there exists a set

M = {m1 < m2 < ... < mk < ...} € F(I)

such that lim ymi = L. This implies that the sequence z = (zk) defined as

i

= Iyk, k € M, zk \ L, k/M

is usual convergent to L. Now by Theorem 2.1 of [11], we can say that for any ideal K, the sequence

tK

z = (zk) is K-convergent to L. Hence, A1 — lim xk = L. □

Theorem 2. Let AK — lim xk = L t^en AiK — lim xk = L.

Proof. Since AK — lim x = L, so for every e > 0,

{k € N : |yk — L| > e} € K. (3.1)

Choose M = N from F(I). Consider the sequence z = (zk) defined by zk = yk, k € M. Then, using (3.1), we get for every e > 0,

{k € N : |zk — L| > e} € K i.e. z = (zk) is K-convergent to L. Hence AI — lim xk = L. □

Remark 1. Converse of the above theorem is not necessarily true.

Example 2. Consider the ideals

Ic = {B C N : ^ b-1 < to}, Id = {B C N : d(B) = 0}

and the infinite matrix A = (ank)TOk=1 defined by

{1, k = n, 0, otherwise.

Let x = (xk) be the sequence defined as

{1, k is prime, 0, k is not prime.

Then, there exists set M of all non prime numbers € F(Id) such that the sequence z = (zk) defined

as

z f yk, k € M,

zk = 0, k / M

is Ic-converegnt to 0. Hence, AIdIc — lim xk = 0. But we claim that AIc — lim xk = 0. Because if AIc — lim xk = 0, then for any particular e with 0 < e < 1, we have the set

{k € N : |yk — 0| > e} = set of all prime numbers € Ic,

it is a contradiction.

The next theorem gives the condition under which A1 K-summability implies AK-summability.

K

Theorem 3. Let I and K be two admissible ideals in N. If I C K then A1 — lim xk = L implies AK — lim xk = L.

P r o o f. Let I C K. Then, AI — lim xk = L gives the assurance of the existence of a set M € F(I) such that the sequence z = (zk) defined as

z = I yk, k € M,

zk = L, k / M

is K-convergent to L and subsequently, we have

Ve > 0, {k € M : |yk — L| > e} € K. (3.2)

Now as the inclusion

{k € N : |yk — L| > e} C {k € M : |yfc — L| > e} U (N \ M) holds and by our assumption, N \ M € I C K, from (3.2) we have

{k € N : |yk — L| > e} € K. Hence, AK — lim xk = L. □

tK TK

Theorem 4. If every subsequence of x = (xk) is A1 -summable to L, then x is A1 -summable to L.

Proof. If possible let us assume the contrary. Then, for every M € F(I), the sequence z = (zk) defined as

z f yk, k€M,

zk = L, k / M

is not K-convergent to L. In other words, for any M € F(I), there exists an eu > 0 such that

B = M n {k € N : |yk — L| > eu} / K.

Since K is admissible, so B is infinite. Let B = {b1 < b2 < ... < bk < ...}. Construct a subsequence w = (wk) defined as wk = ybk for k € N. Then, A1 — lim wk = L, we get a contradiction to the hypothesis. □

tK

Theorem 5. Let x = (xk) be a sequence such that A1 — lim xk = L. Then, every subsequence

jK

of x is A1 -summable to L if and only if both I and K does not contain infinite sets. Proof. There are two possible cases.

Case I. Let K contain an infinite set. Suppose C be an infinite set and C € K. Then, N \ C € F(K) and N \ C is also infinite. Let e > 0 be arbitrary. Choose Li € R such that Li = L. Consider the infinite matrix A = (ank)TOk=1, defined as

ank —

and the sequence x = (xk) such that

xk =

Then,

1, k = n, 0, otherwise,

Li, k C, L, k N \ C.

{k € N : |yk — L| > e} C C € K.

This means that x is -summable to L. Therefore by Theorem 2, x is A1 -summable to L. But clearly the subsequence (xk)keC of x is A1 -summable to L1 and not to L.

tK

Case II. Let K does not contain an infinite set. Then K = If and A1 -summability concept coincides with A1 -summability.

Subcase I: if I contains an infinite set. Let B be any infinite set such that B € I. Then, N \ B € F(I) and N \ B is also infinite. Define a sequence x = (xk) as

it, k € B,

xk =

k L, k N \ B,

where t(= L) € R. Clearly x is

A1 * -summable to L for the infinite matrix considered in Case I. But clearly the subsequence (xk)keB of x is not A1*-summable to L.

Subcase II: if I does not contain an infinite set. In this subcase, we have I = K = If and therefore A1K-summability concept coincides with ordinary summability ([10]) so any subsequence of x is ordinary summable to L. □

Remark 2. If a sequence is

AIK

-summable then it may not be A1 -summable. Example 3. Let us consider the ideal I which is defined in Example 1 and the ideal

Ic = {A C N : Y^ a-1 < to}.

aeA

Let M = {k € N : k = 2p for some non-negative integer p}. Then, for the regular matrix A = (ank)TOk=1 defined as

{1, k = n, 0, otherwise,

the sequence x = (xk) defined by

(1, k € M, xk |0, k/M

is

-summable to 0 but x is not AI-summable to 0.

Theorem 6. Let I and K be two ideals in N. Let x = (xk) be any real-valued sequence. Then,

AIK

— lim xk = L implies AI — lim xk = L if and only if K C I.

P r o o f. Let K C I and suppose AI — lim xk = L. Then, the result follows directly from the following inclusion

{k € N : |yk — L| > e} C {k € M : |yfc — L| > e} U (N \ M).

For the converse part, we assume the contrary. Then, there exists a set say C € K \ I. Let L1 and L2 be two real numbers such that L1 = L2. Define a sequence x = (xk) as

L1, k € C,

xk =

k [L2, k € N \ C and the regular matrix A = (ank)nTOk=1 as

{1, k = n, 0, otherwise.

Then, for any e > 0 we have,

{k € N : |yk — L21 > e} C C € K

which means that x is

AK

-summable to L2. Therefore by Theorem 2, x is

AIK

-summable to L2.

By hypothesis x is A1 -summable to L2. Therefore for e = |L1 — L2|,

{k € N : |yk — L2|>|L1 — L2|} = C €I, it is a contradiction. Hence we must have K C I. □

Remark 3. If a sequence is AI-summable then it may not be

AIK

-summable. Consider the

ideal I and the sequence x = (xk) defined in Example 1. Then, proceeding as Example 1 of [6], we can prove that A1 f — lim xk = 0 although A1 — lim xk = 0.

Theorem 7. Let I and K be two admissible ideals of N such that the condition AP(I, K) holds. Then, for a sequence x = (xk), A1 -summability implies A1 -summability to the same limit.

Proof. Let A1 — lim xk = L. Choose a sequence of rationales (si)i&N. Then, for every i,

Mt = {k € N : lyk — L| <ej €F(I).

Thus by Definition 7, there exists a set M € F (I) such that for any i € N, M \ Mi € K. Consider the sequence z = (zk)keN defined by

z = i yk, k€M,

Zk {L, k / M.

To complete the proof, it is sufficient to show that the sequence z = (zk) is K-convergent to L. Now,

{k € N : |zk — L| < ei} = {k € M : |zk — L| < £i} U {k € N \ M : |zk — L| < e^}

= (N \ M) U {k € M : |zk — L| < ei} = (N \ M) U (Mi n M) = N \ (M \ Mi).

Now as M \ Mi € K, so N \ (M \ Mi) € F(K) and consequently we have

{k € N : |zk — L| < ei} € F(K)

tK

i.e. K — lim zk = L. Hence, A1 — lim Xk = L. This completes the proof. □

Theorem 8. Let I, I1, I2, K, K1, K2 be admissible ideals in N satisfying I1 C I2 and K1 C K2. Then,

(i) A1k — lim xk = L implies A1k — lim xk = L;

(ii) A1^1 — lim xk = L implies A1^2 — lim xk = L.

Proof. (i) Suppose A1k — lim xk = L. Then, by Definition 10, there exists M € F(I1) such that the sequence z = (zk) defined as

z f yk, k€M, Zk \ L, k/M

is K-convergent to L. Now since M € F(I1), we have N \ M € I1 and therefore by hypothesis

tK

N \ M € I2, which again implies M € F(I2). Hence we must have that A12 — lim xk = L.

1K1

(ii) Suppose A1 — lim xk = L. Then, by Definition 10, there exists M € F(I1) such that the sequence z = (zk) defined as,

z f yk, k€M, Zk \ l, k/M

satisfies the following property Ve > 0,

{k € N : |zk — l| > e} € Kx. Now by hypothesis the inclusion K1 C K2 holds, so we must have for Ve > 0,

{k € N : |zk — l| > e} € K2. Hence A1^2 — lim xk = L. □

4. Conclusion

Summability plays an important role in mathematics, particularly in mathematical analysis. In this paper, we introduce and investigate a few properties of A1 -summability. We generate a few examples and counterexamples in order to study some inclusion relationships with some known methods of summability. But the main focus was to link A1 and A1 * -summability with A1 -summability. We prove that the condition AP(I, K) plays a crucial role in this regard. In the future, this idea can be utilized by the researchers to develop some other forms of summability.

Acknowledgements

The authors thank the anonymous referees for their constructive suggestions to improve the quality of the paper. The first author is grateful to the University Grants Commission, India for their fellowships funding under the UGC-JRF scheme (F. No. 16-6(DEC. 2018)/2019(NET/CSIR)) during the preparation of this paper.

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