pure and applied mathematics letters

2 downloads 0 Views 1MB Size Report
Pure and Applied Mathematics Letters 1(2013)25-33. A characterization of weak statistical convergence via lacunary sequences and ideals. Meenakshi1, M. S.Β ...
P A M L December 2013

PURE AND APPLIED MATHEMATICS LETTERS An International Journal A characterization of weak statistical convergence via lacunary sequences and ideals

HCTM Technical Campus P A M L December 2 0 1 3

Pure and Applied Mathematics Letters 1(2013)25-33

Contents lists available at HCTM, Technical Campus

Pure and Applied Mathematics Letters Journal homepage: http:pamletters.org

A characterization of weak statistical convergence via lacunary sequences and ideals Meenakshi1, M. S. Saroa1 1

Department of Mathematics, Maharishi Markandeshwar University, Mullana Ambala, Haryana, India.

Abstract For any lacunary sequence πœƒ = (π‘˜π‘Ÿ ) and an ideal 𝐼 βŠ‚ 𝑃(β„•), we intend in present paper to define and study certain new generalizations of weak convergence which we call π‘Šπ‘†(𝐼) βˆ’convergence and π‘Šπ‘†πœƒ (𝐼) βˆ’convergence on a Banach space 𝑋. Later, notions of π‘Šπ‘†πœƒ (𝐼) βˆ’limit points and π‘Šπ‘†πœƒ (𝐼) βˆ’cluster points are also defined and some specific relations are obtained.

Keywords: weak convergence, statistical convergence, lacunary statistical convergence, ideal convergence.

AMS subject classification: 40A05; 40A35; 46B10; 46B99. Article Information: Received 07.07.2013

Revised 28.12.2013

Accepted 30.12.2013

Communicated by Prof. S. S. Bhatia

1. Introduction and Background Fast [8] and Schonenberg [23] independently introduced statistical convergence in order to assign limit to those sequences which are not convergent in the usual sense with the help of asymptotic density of subsets of β„•, the set of positive integers. Definition 1.1 [8] A sequence π‘₯ = π‘₯π‘˜ of numbers is said to be statistically convergent to a number 𝐿 provided that, for every πœ€ > 0, 1 lim π‘˜ ≀ 𝑛: π‘₯π‘˜ βˆ’ 𝐿 β‰₯ πœ€ = 0, 𝑛 𝑛 where {β‹…} denotes the cardinality of the enclosed set. In this case, we write 𝑆 βˆ’ limπ‘˜β†’βˆž π‘₯π‘˜ = 𝐿. In past years, many problems arising naturally in analysis have been solved by use of statistical convergence. But the quick process of change came after the works of Ε alΓ‘t [22], Fridy [9] and Connor [3]. This concept has been discussed in many fields. For instance in trigonometric series [24], summability theory [12], intuitionistic fuzzy normed spaces [16], probabilistic normed spaces [15], locally convex spaces [20] and Banach spaces [17]. In [11], Fridy and Orhan generalized statistical convergence to lacunary statistical convergence with the help of a lacunary sequence. By a lacunary sequence, we mean an increasing sequence πœƒ = (π‘˜π‘Ÿ ) of positive integers such that π‘˜0 = 0 and β„Žπ‘Ÿ = π‘˜π‘Ÿ βˆ’ π‘˜π‘Ÿβˆ’1 β†’ ∞ as π‘Ÿ β†’ ∞. The intervals determined by πœƒ will be denoted by πΌπ‘Ÿ = (π‘˜π‘Ÿβˆ’1 , π‘˜π‘Ÿ ] and the ratio π‘˜π‘Ÿ /π‘˜π‘Ÿβˆ’1 is denoted by π‘žπ‘Ÿ . Definition 1.2 [11] For a lacunary sequence πœƒ = π‘˜π‘Ÿ , the sequence π‘₯ = π‘₯π‘˜ of numbers is said to be lacunary statistically convergent to a number 𝐿 provided that, for every πœ€ > 0, π›Ώπœƒ π‘˜ ∈ πΌπ‘Ÿ : π‘₯π‘˜ βˆ’ 𝐿 β‰₯ πœ€ where π›Ώπœƒ (𝐾) = limπ‘Ÿ

1 β„Žπ‘Ÿ

= 0,

{π‘˜ ∈ πΌπ‘Ÿ : π‘˜ ∈ 𝐾} . In this case, we write π‘†πœƒ βˆ’ limπ‘˜ β†’βˆž π‘₯π‘˜ = 𝐿.

Definition 1.3 [11] A sequence π‘₯ = (π‘₯π‘˜ ) of numbers is said to be π‘πœƒ βˆ’convergent to 𝐿 provided that, for each πœ€ > 0, lim π‘Ÿ

1 β„Žπ‘Ÿ

‍|π‘₯π‘˜ βˆ’ 𝐿| = 0. π‘˜βˆˆπΌπ‘Ÿ

In this case, we write π‘πœƒ βˆ’ limπ‘˜β†’βˆž π‘₯π‘˜ = 𝐿.

______________________ Corresponding author: Meenakshi, Mob. No. +91 8950651607, E-mail addresses: [email protected] 25

Connor 𝑒𝑑 π‘Žπ‘™. [5], have generalized statistical convergence to weak statistical convergence and given the description of Banach spaces with separable duals using weak statistical convergence. Let 𝑋 be a Banach space and (π‘₯π‘˜ ) be a sequence in 𝑋. Definition 1.4 [5] A sequence (π‘₯π‘˜ ) is said to be weak statistically convergent to π‘₯ ∈ 𝑋 provided that, for each 𝑓 ∈ 𝑋 βˆ— ,(𝑋 βˆ— is the continuous dual of X), each πœ€ > 0, 𝛿 {π‘˜ ≀ 𝑛: |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€} = 0. In this case, we write π‘Šπ‘† βˆ’ limπ‘˜β†’βˆž π‘₯π‘˜ = π‘₯. Although, Bhardwaj and Bala [2] have introduced weak statistical Cauchy sequences in a normed space and weak statistical convergence in 𝑙𝑝 spaces. Bala [1], has generalized weak statistical convergence to weak* statistical convergence. In 2010, F. Nuray [21], has generalized weak statistical convergence to lacunary weak statistical convergence as follows. Definition 1.5 [21] A sequence (π‘₯π‘˜ ) is said to be weak lacunary statistically convergent to π‘₯ ∈ 𝑋 provided that, for each 𝑓 ∈ 𝑋 βˆ— , each πœ€ > 0, π›Ώπœƒ {π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€} = 0. In this case, we write π‘Šπ‘†πœƒ βˆ’ limπ‘˜β†’βˆž π‘₯π‘˜ = π‘₯. Definition 1.6 [21] A sequence

π‘₯π‘˜

is said to be weak π‘πœƒ βˆ’convergent to π‘₯ ∈ 𝑋 provided that, for each 𝑓 ∈ 𝑋 βˆ— , each πœ€ > 0, 1 β„Žπ‘Ÿ

lim π‘Ÿ

‍ 𝑓 π‘₯π‘˜ βˆ’ π‘₯

=0

π‘˜βˆˆπΌπ‘Ÿ

We next give a brief description of 𝐼 βˆ’convergence. Kostyrko, Ε alΓ‘t and Wilczynski [18] have defined 𝐼 βˆ’convergence as a natural generalization of statistical convergence. For the set of positive integers β„•, let 𝑃(β„•) denotes the power set of β„•. Definition 1.7 A family of sets 𝐼 βŠ‚ 𝑃(β„•) is called an ideal in β„• if and only if (i) (ii) (iii)

βˆ… ∈ 𝐼; For each 𝐴, 𝐡 ∈ 𝐼 we have 𝐴 βˆͺ 𝐡 ∈ 𝐼; For 𝐴 ∈ 𝐼 and 𝐡 βŠ† 𝐴 we have 𝐡 ∈ 𝐼.

Definition 1.8 A non-empty family of sets 𝐹 βŠ‚ 𝑃(β„•) is called a filter on β„• if and only if (i) (ii) (iii)

βˆ… βˆ‰ 𝐹; For each 𝐴, 𝐡 ∈ 𝐹 we have 𝐴 ∩ 𝐡 ∈ 𝐹; For 𝐴 ∈ 𝐹 and 𝐡 βŠ‡ 𝐴 we have 𝐡 ∈ 𝐹.

An ideal 𝐼 is called non-trivial if 𝐼 β‰  𝑃(β„•). It immediately implies that 𝐼 βŠ‚ 𝑃(β„•) is a non-trivial ideal if and only if the class 𝐹 = 𝐹(𝐼) = {β„• βˆ’ 𝐴: 𝐴 ∈ 𝐼} is a filter on β„•. The filter 𝐹 = 𝐹(𝐼) is called the filter associated with the ideal 𝐼. A non-trivial ideal 𝐼 βŠ‚ 𝑃(β„•) is called an admissible ideal in β„• if and only if it contains all singletons i.e., if it contains {{π‘₯}: π‘₯ ∈ β„•}. An admissible ideal 𝐼 βŠ‚ 𝑃(β„•) is said to be satisfy the condition (𝐴𝑃) if for every countable family of mutually disjoint sets {𝐴1 , 𝐴2 . . . } belonging to 𝐼 there exists a countable family {𝐡1 , 𝐡2 . . . } in 𝐼 such that 𝐴𝑖 π›₯𝐡𝑖 is a finite set for each 𝑖 ∈ β„• and 𝐡 =βˆͺ∞ 𝑖=1 𝐡𝑖 ∈ 𝐼. Definition 1.9 A sequence π‘₯ = (π‘₯π‘˜ ) of numbers is said to be 𝐼 βˆ’convergent to a number 𝐿 provided that, for every πœ€ > 0, π‘˜ ∈ β„•: |π‘₯π‘˜ βˆ’ 𝐿| β‰₯ πœ€ ∈ 𝐼 Further, Das 𝑒𝑑 π‘Žπ‘™. [6] extended the notion of 𝐼 βˆ’convergence to 𝐼 βˆ’statistical convergence and 𝐼 βˆ’lacunary statistical convergence as follows. Definition 1.10 A sequence π‘₯ = (π‘₯π‘˜ ) of numbers is said to be 𝐼 βˆ’statistically convergent to 𝐿 provided that, for every πœ€ > 0 and each 𝛾 > 0, 𝑛 ∈ β„•:

1 {π‘˜ ≀ 𝑛: |π‘₯π‘˜ βˆ’ 𝐿| β‰₯ πœ€} β‰₯ 𝛾 ∈ 𝐼 𝑛

Moreover, above definition is also generalized for lacunary sequences as follows. Definition 1.11 Let πœƒ = (π‘˜π‘Ÿ ) be a lacunary sequence as defined above. A sequence π‘₯ = (π‘₯π‘˜ ) of numbers is said to be 𝐼 βˆ’lacunary statistically convergent to 𝐿 or π‘†πœƒ (𝐼) βˆ’convergent to 𝐿 provided that, for every πœ€ > 0 and each 𝛾 > 0, π‘Ÿ ∈ β„•:

1 {π‘˜ ∈ πΌπ‘Ÿ : |π‘₯π‘˜ βˆ’ 𝐿| β‰₯ πœ€} β‰₯ 𝛾 ∈ 𝐼. β„Žπ‘Ÿ

In this case we write π‘₯π‘˜ β†’ 𝐿(π‘†πœƒ (𝐼)). Definition1.12 A sequence π‘₯ = (π‘₯π‘˜ ) of numbers is said to be π‘πœƒ (𝐼) βˆ’ convergent to 𝐿 provided that, for every πœ€ > 0,

π‘Ÿ ∈ β„•:

1 {π‘˜ ∈ πΌπ‘Ÿ : β„Žπ‘Ÿ

‍|π‘₯π‘˜ βˆ’ 𝐿)| β‰₯ πœ€}

∈ 𝐼.

π‘˜βˆˆπΌπ‘Ÿ

In this case we write π‘₯π‘˜ β†’ 𝐿(π‘πœƒ (𝐼)). Some further applications of 𝐼 βˆ’convergence can be found in [13] ,[14] and [19]. Throughout this paper, 𝑋 will represent a Banach space, πœƒ = (π‘˜π‘Ÿ ) be a lacunary sequence as defined above, π‘₯ = (π‘₯π‘˜ ) be a sequence of elements of 𝑋 and 𝐼 βŠ† 𝑃(β„•) be a non trivial admissible ideal (as otherwise indicated). 26

2. Weak 𝑰 βˆ’ lacunary statistical convergence Nuray [21], defined weak lacunary statistical convergence and proved some relations between weak lacunary statistical convergence and weak statistical convergence. In this section, we generalized these notions and study relevent connections with the help of ideals. Definition 2.1 A sequence (π‘₯π‘˜ ) is said to be weak statistically convergent to π‘₯ with respect to 𝐼 in 𝑋 if for every πœ€ > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 βˆ— , where 𝑋 βˆ— is the continuous dual of 𝑋 we have 1 𝑛 ∈ β„•: |{π‘˜ ≀ 𝑛: |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€}| β‰₯ 𝛾 ∈ 𝐼. 𝑛 In this case we write π‘Šπ‘†(𝐼) βˆ’ limπ‘₯π‘˜ = π‘₯. We will denote the set of all weak 𝐼 βˆ’ statistically convergent sequences by π‘Šπ‘†(𝐼). For 𝐼 = 𝐼𝑓𝑖𝑛 , weak 𝐼 βˆ’statistical convergence coincides with weak statistical convergence of [2]. Definition 2.2 A sequence (π‘₯π‘˜ ) is said to be weak lacunary statistically convergent to π‘₯ with respect to 𝐼 in 𝑋 if for every πœ€ > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 βˆ— , where 𝑋 βˆ— is the continuous dual of 𝑋 we have π‘Ÿ ∈ β„•:

1 |{π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€}| β‰₯ 𝛾 ∈ 𝐼. β„Žπ‘Ÿ

In this case we write π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘₯π‘˜ = π‘₯. We will denote the set of all weak 𝐼 βˆ’ lacunary statistically convergent sequences by π‘Šπ‘†πœƒ (𝐼). For 𝐼 = 𝐼𝑓𝑖𝑛 , weak 𝐼 βˆ’lacunary statistical convergence coincides with weak lacunary statistical convergence of [21]. Definition 2.3 A sequence (π‘₯π‘˜ ) is said to be weak π‘πœƒ βˆ’convergent to π‘₯ with respect to 𝐼 in 𝑋 if for every πœ€ > 0 and each 𝑓 ∈ 𝑋 βˆ— , π‘Ÿ ∈ β„•:

1 β„Žπ‘Ÿ

‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€ ∈ 𝐼. π‘˜βˆˆπΌπ‘Ÿ

In this case we write π‘Šπ‘πœƒ (𝐼) βˆ’ limπ‘₯π‘˜ = π‘₯. We will denote the set of all weak π‘πœƒ (𝐼) βˆ’convergent sequences by π‘Šπ‘πœƒ (𝐼). For 𝐼 = 𝐼𝑓𝑖𝑛 , weak π‘πœƒ (𝐼) βˆ’convergence coincides with weak π‘πœƒ βˆ’convergence of [21]. Theorem 2.1 Let 𝐼 βŠ‚ 𝑃(β„•) be an admissible ideal. If π‘Šπ‘†πœƒ βˆ’ limπ‘˜ π‘₯π‘˜ = π‘₯, then π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘˜ π‘₯π‘˜ = π‘₯. Proof Suppose π‘Šπ‘†πœƒ βˆ’ limπ‘₯π‘˜ = π‘₯, then for each πœ€ > 0, each 𝑓 ∈ 𝑋 βˆ— , lim

π‘Ÿβ†’βˆž

1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓 π‘₯π‘˜ βˆ’ π‘₯ β„Žπ‘Ÿ

β‰₯ πœ€ = 0.

This immediately follows that for each 𝛾 > 0, there exists a positive integer π‘š such that 1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓 π‘₯π‘˜ βˆ’ π‘₯ β„Žπ‘Ÿ

β‰₯ πœ€ < 𝛾 for all 𝑛 β‰₯ π‘š

𝑖. 𝑒. π‘Ÿ ∈ β„•:

1 |{π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€}| β‰₯ 𝛾 βŠ† {1,2,3, . . . . , π‘š βˆ’ 1} β„Žπ‘Ÿ

Since 𝐼 is admissible, therefore π‘Ÿ ∈ β„•:

1 |{π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€}| β‰₯ 𝛾 ∈ 𝐼. β„Žπ‘Ÿ

This shows that π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘₯π‘˜ = π‘₯. ∎ The converse of above result is not true in general and can be seen from the following example. Example 2.1 Let 𝑋 = 𝑐00 be the normed linear space with ||. ||𝑝 (1 < 𝑝 < ∞), 𝐼 be an admissible ideal, 𝐴 ∈ 𝐼 is fixed and πœƒ = (2π‘Ÿ ) be a lacunary sequence as defined above. Define a sequence (π‘₯π‘˜ ) in 𝑐00 by π‘˜, (π‘˜) π‘₯𝑗

=

π‘˜, 0,

2π‘Ÿβˆ’1 + 1 ≀ π‘˜ ≀ 2π‘Ÿβˆ’1 + [ β„Žπ‘Ÿ ]

𝑖𝑓 𝑗 ≀ π‘˜, 𝑖𝑓 𝑗 ≀ π‘˜, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’,

2

π‘Ÿβˆ’1

π‘Žπ‘›π‘‘ π‘Ÿ βˆ‰ 𝐴;

π‘Ÿ

≀ π‘˜ ≀ 2 π‘Žπ‘›π‘‘ π‘Ÿ ∈ 𝐴;

where πΌπ‘Ÿ = (2π‘Ÿβˆ’1 , 2π‘Ÿ ]. For arbitrary 𝑓 ∈ 𝑋 βˆ— , there is unique 𝑦 ∈ π‘™π‘ž such that ∞ (π‘˜)

𝑓(π‘₯π‘˜ ) =

‍π‘₯𝑗

𝑦𝑗 ≀βˆ₯ π‘₯ βˆ₯𝑝 βˆ₯ 𝑦 βˆ₯π‘ž

(𝑏𝑦 π»π‘œπ‘™π‘‘π‘’π‘Ÿ π‘–π‘›π‘’π‘žπ‘’π‘Žπ‘™π‘–π‘‘π‘¦)

(1)

𝑗 =1

So, for each πœ€ > 0 (0 < πœ€ < 1) and each πœ‘ ∈ 𝑋 βˆ— , 1 π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ 0)| β‰₯ πœ€ β„Žπ‘Ÿ

≀

1 π‘˜ ∈ πΌπ‘Ÿ : βˆ₯ π‘₯ βˆ₯𝑝 βˆ₯ 𝑦 βˆ₯π‘ž β‰₯ πœ€ β„Žπ‘Ÿ 27

=

1 β„Žπ‘Ÿ

πœ€ βˆ₯ 𝑦 βˆ₯π‘ž

π‘˜ ∈ πΌπ‘Ÿ : βˆ₯ π‘₯ βˆ₯𝑝 β‰₯

=

1 β„Žπ‘Ÿ

(π‘˜)

π‘˜ ∈ πΌπ‘Ÿ : π‘₯𝑗

= π‘˜

=

[ β„Žπ‘Ÿ ] β†’ 0 as π‘Ÿ β†’ ∞ and π‘Ÿ βˆ‰ 𝐴 β„Žπ‘Ÿ

Thus, for each 𝛾 > 0,

π‘Ÿ ∈ β„•:

1 π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ 0)| β‰₯ πœ€ β„Žπ‘Ÿ

β‰₯ 𝛾 βŠ‚ 𝐴 βˆͺ {1,2, … , π‘š}

for some π‘š ∈ β„•. π‘Šπ‘†πœƒ (𝐼)

Since 𝐴 ∈ 𝐼 and 𝐼 is admissible, it gives that π‘₯π‘˜

β†’

0. Also by (1) we can see that,

∞ (π‘˜)

𝑓(π‘₯π‘˜ ) =

‍π‘₯𝑗

𝑦(𝑗) ≀βˆ₯ π‘₯ βˆ₯𝑝 βˆ₯ 𝑦 βˆ₯π‘ž

(by Holder inequality)

𝑗 =1

1 𝑝

∞

=

‍

‍ 𝑦(𝑗)

𝑗 =1

1 𝑝

‍

1 π‘ž

∞

(π‘˜) 𝑝 π‘₯𝑗

‍ 𝑦(π‘˜)

𝑗 =1

π‘ž

(by structure of the sequence)

π‘˜=1

1 𝑝

π‘˜

‍( π‘˜)𝑝

=

π‘ž

𝑗 =1

π‘˜

=

1 π‘ž

∞

(π‘˜) 𝑝 π‘₯𝑗

1

1 1 1 + 𝑝 π‘€π‘ž .

π‘€π‘ž

(for some positive constant 𝑀) = π‘˜ 2

𝑗 =1

This gives that, 1 β„Žπ‘Ÿ

‍|𝑓(π‘₯π‘˜ βˆ’ 0)| ≀ π‘˜βˆˆπΌπ‘Ÿ

π‘Šπ‘πœƒ

1 β„Žπ‘Ÿ

1 1 1 + 𝑝 π‘€π‘ž

β€π‘˜ 2

↛ 0 π‘Žπ‘  π‘Ÿ β†’ ∞,

π‘˜βˆˆπΌπ‘Ÿ

π‘Šπ‘†πœƒ

𝑖. 𝑒., π‘₯π‘˜ ↛ 0. Hence, by Theorem [4 ] of [21], we have π‘₯π‘˜ ↛ 0. ∎ Theorem 2.2 For a lacunary sequence πœƒ = (π‘˜π‘Ÿ ), the sequence (π‘₯π‘˜ ) is weak π‘πœƒ (𝐼) βˆ’convergent to π‘₯ if and only if (π‘₯π‘˜ ) is weak 𝐼 βˆ’lacunary statistically convergent to π‘₯. Proof Let (π‘₯π‘˜ ) is weak π‘πœƒ (𝐼) βˆ’convergent to π‘₯, then for every πœ€ > 0 and each 𝑓 ∈ 𝑋 βˆ— , we have π‘Ÿ ∈ β„•:

1 β„Žπ‘Ÿ

‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€ ∈ 𝐼. π‘˜βˆˆπΌπ‘Ÿ

Also for given πœ€ > 0, we have 1 β„Žπ‘Ÿ

‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ π‘˜βˆˆπΌπ‘Ÿ

β‰₯

1 β„Žπ‘Ÿ

‍ 𝑓(π‘₯π‘˜ βˆ’ π‘₯) π‘˜βˆˆπΌπ‘Ÿ 𝑓(π‘₯ π‘˜ βˆ’π‘₯) β‰₯πœ€

πœ€ {π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) β‰₯ πœ€} . β„Žπ‘Ÿ

Hence, for every πœ€ > 0, each 𝛾 > 0 and each 𝑓 ∈ 𝑋 βˆ— we have π‘Ÿ ∈ β„•:

1 1 |{π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€}| β‰₯ 𝛾 βŠ† π‘Ÿ ∈ β„•: β„Žπ‘Ÿ β„Žπ‘Ÿ

‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€π›Ύ ∈ 𝐼. π‘˜βˆˆπΌπ‘Ÿ

This immediately gives (π‘₯π‘˜ ) is weak 𝐼 βˆ’lacunary statistically convergent to π‘₯. Conversely, suppose that (π‘₯π‘˜ ) is weak 𝐼 βˆ’lacunary statistically convergent to π‘₯. Then for every πœ€ > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 βˆ— we have

π‘Ÿ ∈ β„•:

1 |{π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€}| β‰₯ 𝛾 ∈ 𝐼. β„Žπ‘Ÿ

Since 𝑓 ∈ 𝑋 βˆ— , therefore 𝑓 is bounded. Let |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| ≀ 𝑀 for all π‘˜. For given πœ€ > 0, we have 28

1 β„Žπ‘Ÿ

‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| = π‘˜βˆˆπΌπ‘Ÿ

≀

1 β„Žπ‘Ÿ

‍ 𝑓(π‘₯π‘˜ βˆ’ π‘₯) + π‘˜βˆˆπΌπ‘Ÿ 𝑓(π‘₯ π‘˜ βˆ’π‘₯) β‰₯πœ€

‍ 𝑓(π‘₯π‘˜ βˆ’ π‘₯) π‘˜βˆˆπΌπ‘Ÿ 𝑓(π‘₯ π‘˜ βˆ’π‘₯) 1. Proof Let liminfπ‘Ÿ π‘žπ‘Ÿ > 1, then there exists a 𝛿 > 0 such that π‘žπ‘Ÿ > 1 + 𝛿 for sufficiently large π‘Ÿ which implies that βˆ—

π‘˜π‘Ÿ β„Žπ‘Ÿ

≀

𝛿+1 𝛿

. Let π‘Šπ‘†(𝐼) βˆ’

limπ‘₯π‘˜ = π‘₯, then for every πœ€ > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 , we have 1 𝑛 ∈ β„•: |{π‘˜ ≀ 𝑛: |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€}| β‰₯ 𝛾 ∈ 𝐼. 𝑛 Also, one can easily observe that 1 1 {π‘˜ ≀ π‘˜π‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€} β‰₯ {π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€} π‘˜π‘Ÿ π‘˜π‘Ÿ β„Žπ‘Ÿ 1 {π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€} π‘˜π‘Ÿ β„Žπ‘Ÿ

= β‰₯

𝛿

1

𝛿+1 β„Ž π‘Ÿ

{π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€} ;

So, π‘Ÿ ∈ β„•:

1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓 π‘₯π‘˜ βˆ’ π‘₯ β„Žπ‘Ÿ

β‰₯πœ€

β‰₯ 𝛾 βŠ† π‘Ÿ ∈ β„•:

1 𝛾𝛿 {π‘˜ ≀ π‘˜π‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€} β‰₯ ∈ 𝐼, π‘˜π‘Ÿ 𝛿+1

which gives the result. Conversely, suppose that liminfπ‘Ÿ π‘žπ‘Ÿ = 1, then following as in [11], one can find a subsequence (π‘˜π‘Ÿ(𝑗 ) ) of πœƒ = (π‘˜π‘Ÿ ) such that π‘˜π‘Ÿ(𝑗 ) 1 < 1+ π‘˜π‘Ÿ(𝑗 )βˆ’1 𝑗

π‘Žπ‘›π‘‘

π‘˜π‘Ÿ(𝑗 )βˆ’1 >𝑗 π‘˜π‘Ÿ(𝑗 βˆ’1)

π‘€β„Žπ‘’π‘Ÿπ‘’

π‘Ÿ(𝑗) β‰₯ π‘Ÿ(𝑗 βˆ’ 1) + 2

Define a sequence (π‘₯π‘˜ ) as follows

π‘₯π‘˜ =

1, 0,

𝑖𝑓 π‘˜ ∈ πΌπ‘Ÿ (𝑗 ) , π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ .

Since 𝑓 ∈ 𝑋 βˆ— , therefore 𝑓 is bounded. Let |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| ≀ 𝑀 for all π‘˜. Thus for any 𝑓 ∈ 𝑋 βˆ— , π‘₯ ∈ 𝑋, 1 β„Žπ‘Ÿ(𝑗 )

‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| = π‘˜βˆˆπΌπ‘Ÿ(𝑗 )

1 β„Žπ‘Ÿ(𝑗 )

‍|𝑓(1 βˆ’ π‘₯)| ≀ 𝑀 π‘“π‘œπ‘Ÿ 𝑗 = 1,2,β‹…β‹…β‹… π‘˜βˆˆπΌπ‘Ÿ(𝑗 )

and 1 β„Žπ‘Ÿ

‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| = π‘˜βˆˆπΌπ‘Ÿ

1 β„Žπ‘Ÿ

‍|𝑓(βˆ’π‘₯)| ≀ 𝑀 π‘“π‘œπ‘Ÿ π‘Ÿ β‰  π‘Ÿπ‘— π‘˜βˆˆπΌπ‘Ÿ

which gives (π‘₯π‘˜ ) βˆ‰ π‘Šπ‘πœƒ (𝐼). By Theorem 2.2 we have (π‘₯π‘˜ ) βˆ‰ π‘Šπ‘†πœƒ (𝐼). Let π‘˜π‘Ÿ(𝑗 βˆ’1) ≀ 𝑛 ≀ π‘˜π‘Ÿ(𝑗 ) , then 1 𝑛

𝑛

π‘˜=1

1 ‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ 𝑛

β‰₯

𝑛

‍ 𝑓(π‘₯π‘˜ βˆ’ π‘₯) π‘˜=1 𝑓(π‘₯ π‘˜ βˆ’π‘₯) β‰₯πœ€

πœ€ {π‘˜ ≀ 𝑛: 𝑓(π‘₯π‘˜ βˆ’ π‘₯) β‰₯ πœ€} . 𝑛

We can write, 29

πœ€ 1 {π‘˜ ≀ 𝑛: 𝑓(π‘₯π‘˜ βˆ’ π‘₯) β‰₯ πœ€} ≀ 𝑛 𝑛

𝑛

π‘˜=1

1 1 2𝑀 ‍|𝑓(π‘₯π‘˜ βˆ’ π‘₯)| ≀ 𝑀( + ) = 𝑗 𝑗 𝑗

Thus, (π‘₯π‘˜ ) is weak 𝐼 βˆ’ statistically convergent to π‘₯. ∎ In [21] it has been shown that for any lacunary sequence πœƒ = (π‘˜π‘Ÿ ), π‘Šπ‘†πœƒ βŠ† π‘Šπ‘† if and only if limsupπ‘Ÿ π‘žπ‘Ÿ < ∞. However, for any admissible ideal 𝐼 this is not sure in general. Theorem 2.4 Let 𝐼 be an admissible ideal having condition (AP) and πœƒ ∈ 𝐹(𝐼). If π‘₯ ∈ π‘Šπ‘†(𝐼) ∩ π‘Šπ‘†πœƒ (𝐼), then π‘Šπ‘†(𝐼) βˆ’ limπ‘₯π‘˜ = π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘₯π‘˜ . Proof Suppose π‘Šπ‘†(𝐼) βˆ’ limπ‘₯π‘˜ = π‘₯ and π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘₯π‘˜ = 𝑦 and π‘₯ β‰  𝑦. Let |π‘₯ βˆ’ 𝑦| > 2πœ€, (πœ€ > 0). Since 𝐼 satisfy the condition (AP) so there exists 𝑀 ∈ 𝐹(𝐼) such that lim

π‘Ÿβ†’βˆž

1 {π‘˜ ≀ π‘šπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) β‰₯ πœ€} = 0, π‘šπ‘Ÿ

where 𝑀 = π‘š1 , π‘š2 , π‘š3 . . . . . Let, 𝐺 = π‘˜ ≀ π‘šπ‘Ÿ : 𝑓 π‘₯π‘˜ βˆ’ π‘₯

β‰₯ πœ€ and

𝐻 = π‘˜ ≀ π‘šπ‘Ÿ : 𝑓 π‘₯π‘˜ βˆ’ 𝑦 Since limπ‘Ÿβ†’βˆž

|𝐺| π‘šπ‘Ÿ

= 0, it follows that limπ‘Ÿβ†’βˆž

1 π‘šπ‘Ÿ

β‰₯πœ€ .

{π‘˜ ≀ π‘šπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ 𝑦) β‰₯ πœ€} = 1.

π‘‘β„Ž Let, 𝑀 βˆ— = π‘˜π‘™1 , π‘˜π‘™2 , π‘˜π‘™3 , … = 𝑀 ∩ πœƒ ∈ 𝐹(𝐼). Then the π‘˜π‘™π‘ term of the weak statistical limit expression

1 1 {π‘˜ ≀ π‘šπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ 𝑦) β‰₯ πœ€} : π‘šπ‘Ÿ π‘˜π‘™ 𝑝

𝑙𝑝

π‘˜βˆˆ

1 = π‘˜π‘™ 𝑝

β€πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ 𝑦) β‰₯ πœ€ π‘Ÿ=1

= where π‘‘π‘Ÿ =

1 β„Žπ‘Ÿ

𝑙𝑝

‍ {π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ 𝑦) β‰₯ πœ€} π‘Ÿ =1

1

𝑙𝑝

β€β„Žπ‘Ÿ π‘‘π‘Ÿ 𝑙𝑝 ‍ β„Ž π‘Ÿ π‘Ÿ=1 π‘Ÿ=1

(2)

{π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ 𝑦) β‰₯ πœ€} and π‘‘π‘Ÿ ⟢𝐼 0 as π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘₯π‘˜ = 𝑦. Since πœƒ is a lacunary sequence and the right side of above

expression is a regular weighted mean transform of the sequence 𝑑 = π‘‘π‘Ÿ , therefore it is also 𝐼-convergent to zero as 𝑝 β†’ ∞. Also 𝐼 satisfies 1 the condition (AP) which gives it has a subsequence converging to zero. Since this is a subsequence of {π‘˜ ≀ π‘šπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ 𝑦) β‰₯ πœ€} , we infer that

1 π‘šπ‘Ÿ

π‘šπ‘Ÿ

{π‘˜ ≀ π‘šπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ 𝑦) β‰₯ πœ€} is not convergent to 1 which is a contradiction. This completes the proof of the theorem. ∎

3. Weak 𝑰 βˆ’lacunary statistical limit and cluster point Fridy [10] introduced the notion of statistical limit point and statistical cluster point for the sequences of numbers. Connor 𝑒𝑑 π‘Žπ‘™. [4] generalised these notions to 𝐴 βˆ’statistical limit point and 𝐴 βˆ’statistical cluster point where 𝐴 is a nonnegative regular summability matrix. Demirci [7] generalised these notions to πœƒ βˆ’statistical limit point and πœƒ βˆ’statistical cluster point for sequences of numbers. GΓΌrdal 𝑒𝑑 π‘Žπ‘™. [14] have generalized the notion of 𝐴 βˆ’statistical limit point and 𝐴 βˆ’statistical cluster point to define 𝐴𝐼 βˆ’statistical limit point and 𝐴𝐼 βˆ’statistical cluster point where 𝐼 βŠ‚ 𝑃(β„•) is an ideal. Further these concepts has been defined in different spaces. Now we will introduce the notion of weak 𝐼 βˆ’lacunary statistical limit and cluster point in Banach spaces. Let πœƒ = π‘˜π‘Ÿ be a lacunary sequence. For a Banach space 𝑋, let π‘₯ = π‘₯π‘˜ be a sequence in 𝑋. Let π‘₯π‘˜ 𝐾 = π‘˜ 𝑗 : 𝑗 ∈ β„• , then we denote π‘₯π‘˜ 𝑗 by (π‘₯)𝐾 . If for every 𝛾 > 0, π‘Ÿ ∈ β„•:

𝑗

be a subsequence of π‘₯ and

1 π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 ∈ 𝐼; β„Žπ‘Ÿ

then (π‘₯)𝐾 is called πœƒ 𝐼 -thin subsequence. on the other hand (π‘₯)𝐾 is a πœƒ 𝐼 -nonthin subsequence of π‘₯ provided that π‘Ÿ ∈ β„•:

1 π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 βˆ‰ 𝐼. β„Žπ‘Ÿ

Definition 3.1 Let 𝑋 be a Banach space, πœƒ = (π‘˜π‘Ÿ ) be a lacunary sequence and 𝑓 ∈ 𝑋 βˆ— . An element π‘₯ ∈ 𝑋 is called a weak π‘†πœƒ (𝐼) βˆ’limit point of a sequence π‘₯ = (π‘₯π‘˜ ) in 𝑋 provided that there is a πœƒ(𝐼)-nonthin subsequence of (π‘₯π‘˜ ) that is weak convergent to π‘₯. Let Ξ›(π‘Šπ‘†πœƒ (𝐼), π‘₯) denotes the set of all π‘Šπ‘†πœƒ (𝐼)-limit points of the sequence π‘₯ = (π‘₯π‘˜ ). Definition 3.2 Let 𝑋 be a Banach space, πœƒ = π‘˜π‘Ÿ be a lacunary sequence and 𝑓 ∈ 𝑋 βˆ— . An element π‘₯ ∈ 𝑋 is called a weak π‘†πœƒ 𝐼 βˆ’cluster point of a sequence π‘₯ = π‘₯π‘˜ in 𝑋 provided that for every πœ€ > 0 and each 𝑓 ∈ 𝑋 βˆ— , we have π‘Ÿ ∈ β„•:

1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓 π‘₯π‘˜ βˆ’ π‘₯ β„Žπ‘Ÿ

0, π‘Ÿ ∈ β„•:

1 {π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„•} β‰₯ 𝛾 βˆ‰ 𝐼 β„Žπ‘Ÿ

(3)

Now, the containment π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) < πœ€ βŠ‡ {π‘˜(𝑗) ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜(𝑗 ) βˆ’ π‘₯) < πœ€} gives π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) < πœ€ βŠ‡ π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• βˆ’ π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ (𝑗 ) βˆ’ π‘₯) β‰₯ πœ€ ; which immediately implies 1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) < πœ€ β„Žπ‘Ÿ

β‰₯

1 1 π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• βˆ’ π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜(𝑗 ) βˆ’ π‘₯) β‰₯ πœ€ . β„Žπ‘Ÿ β„Žπ‘Ÿ

So, for each 𝛾 > 0 and πœ€ > 0, π‘Ÿ ∈ β„•:

1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) < πœ€ β„Žπ‘Ÿ

β‰₯ 𝛾 βŠ‡ π‘Ÿ ∈ β„•: π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 βˆ’ π‘Ÿ ∈ β„•: π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ (𝑗 ) βˆ’ π‘₯) β‰₯ πœ€

= π‘Ÿ ∈ β„•: π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 ∩ π‘Ÿ ∈ β„•: π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜(𝑗 ) βˆ’ π‘₯) β‰₯ πœ€

β‰₯𝛾

= π‘Ÿ ∈ β„•: π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 ∩ π‘Ÿ ∈ β„•: π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ (𝑗 ) βˆ’ π‘₯) β‰₯ πœ€

0 and each 𝑓 ∈ 𝑋 βˆ— , the set π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ (𝑗 ) βˆ’ π‘₯) β‰₯ πœ€ is finite for which we have limπ‘Ÿβ†’βˆž

1 π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜(𝑗 ) βˆ’ π‘₯) β‰₯ πœ€ β„Žπ‘Ÿ

=0,

𝑖. 𝑒. π‘Šπ‘†πœƒ βˆ’ limπ‘₯π‘˜ (𝑗 ) = π‘₯. Using Theorem 2.1, we get π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘₯π‘˜(𝑗 ) = π‘₯. Thus, for each πœ€ > 0, each 𝛾 > 0 and each 𝑓 ∈ 𝑋 βˆ— π‘Ÿ ∈ β„•:

1 |π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€| β‰₯ 𝛾 ∈ 𝐼 β„Žπ‘Ÿ

or

π‘Ÿ ∈ β„•:

1 |π‘˜ ∈ πΌπ‘Ÿ : |𝑓(π‘₯π‘˜ βˆ’ π‘₯)| β‰₯ πœ€| < 𝛾 ∈ 𝐹 𝐼 β„Žπ‘Ÿ

(5)

Using (3) and (5) in (4), we get π‘Ÿ ∈ β„•:

1 β„Žπ‘Ÿ

π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) < πœ€

β‰₯ 𝛾 ∈ 𝐹(𝐼).

𝑖. 𝑒. π‘Ÿ ∈ β„•:

1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯) < πœ€ β„Žπ‘Ÿ

β‰₯ 𝛾 βˆ‰ 𝐼.

This shows that π‘₯ ∈ Ξ“(π‘Šπ‘†πœƒ (𝐼), π‘₯) and therefore we have the containment Ξ›(π‘Šπ‘†πœƒ (𝐼), π‘₯) βŠ† Ξ“(π‘Šπ‘†πœƒ (𝐼), π‘₯). ∎ Theorem 3.2 Let πœƒ = π‘˜π‘Ÿ be a lacunary sequence. For any sequence π‘₯ = (π‘₯π‘˜ ) in a Banach space 𝑋, if π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘˜ π‘₯π‘˜ = π‘₯0 , then Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ = Ξ“ π‘Šπ‘†πœƒ (𝐼), π‘₯ = {π‘₯0 }. Proof. We first show that Ξ› π‘Šπ‘†πœƒ (𝐼) , π‘₯ = π‘₯0 . Let Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ = π‘₯0 , 𝑦0 such that π‘₯0 β‰  𝑦0 . By definition there exist two πœƒ(𝐼)-nonthin subsequences π‘₯π‘˜(𝑖) and π‘₯𝑙(𝑗 ) of the sequence π‘₯ = (π‘₯π‘˜ ) which are respectively weakly convergent to π‘₯0 and 𝑦0 . Choose πœ€ > 0 such that |𝑓(π‘₯ )βˆ’π‘“(𝑦 )|

0 0 0 0 and 𝛾 > 0, the set π‘Ÿ ∈ β„•: = π‘Ÿ ∈ β„•:

1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) < πœ€ β„Žπ‘Ÿ

β‰₯𝛾

1 1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 βˆ’ π‘Ÿ ∈ β„•: 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) β‰₯ πœ€ β„Žπ‘Ÿ β„Žπ‘Ÿ

β‰₯𝛾 31

= π‘Ÿ ∈ β„•:

1 1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 ∩ π‘Ÿ ∈ β„•: 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) β‰₯ πœ€ β„Žπ‘Ÿ β„Žπ‘Ÿ

𝑐

β‰₯𝛾 .

Since 𝑙(𝑗) is πœƒ(𝐼) βˆ’nonthin subsequence and π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘₯𝑙(𝑗 ) = 𝑦0 , therefore π‘Ÿ ∈ β„•:

1 1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„• β‰₯ 𝛾 ∩ π‘Ÿ ∈ β„•: 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) β‰₯ πœ€ β„Žπ‘Ÿ β„Žπ‘Ÿ

𝑐

β‰₯𝛾

∈ 𝐹(𝐼).

This gives, π‘Ÿ ∈ β„•:

1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) < πœ€ β„Žπ‘Ÿ

β‰₯𝛾 ∈𝐹 𝐼

(6)

Also π‘Šπ‘†πœƒ (𝐼) βˆ’ limπ‘˜ π‘₯π‘˜ = π‘₯0 , so we have π‘Ÿ ∈ β„•:

1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯0 ) β‰₯ πœ€ β„Žπ‘Ÿ

β‰₯𝛾 ∈𝐼

(7)

For 0 < 2πœ€ < 𝑓 π‘₯0 βˆ’ 𝑦0 , 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) < πœ€ ∩ π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯0 ) < πœ€ = βˆ… so we have, 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) < πœ€ βŠ† {π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯0 ) β‰₯ πœ€}; which implies immadetaly that 1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) < πœ€ β„Žπ‘Ÿ

≀

1 {π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯0 ) β‰₯ πœ€} . β„Žπ‘Ÿ

For πœ€ > 0, 𝛾 > 0 π‘Ÿ ∈ β„•:

1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) < πœ€ β„Žπ‘Ÿ

β‰₯ 𝛾 βŠ† π‘Ÿ ∈ β„•:

1 π‘˜ ∈ πΌπ‘Ÿ : 𝑓(π‘₯π‘˜ βˆ’ π‘₯0 ) β‰₯ πœ€} β‰₯ 𝛾 β„Žπ‘Ÿ

Using(7), we have π‘Ÿ ∈ β„•:

1 𝑙 𝑗 ∈ πΌπ‘Ÿ : 𝑓(π‘₯𝑙(𝑗 ) βˆ’ 𝑦0 ) < πœ€ β„Žπ‘Ÿ

β‰₯ 𝛾 ∈ 𝐼;

which contradict(6). Hence, Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ = π‘₯0 . Similarly, we can show that Ξ“ π‘Šπ‘†πœƒ (𝐼), π‘₯ = π‘₯0 . ∎ Theorem 3.3 If π‘₯ = π‘₯π‘˜

and 𝑦 = (π‘¦π‘˜ )

are two sequences in 𝑋 such that limπ‘Ÿ

1 β„Žπ‘Ÿ

π‘˜ ∈ πΌπ‘Ÿ : π‘₯π‘˜ β‰  π‘¦π‘˜

= 0, then Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ =

Ξ› π‘Šπ‘†πœƒ (𝐼), 𝑦 and Ξ“ π‘Šπ‘†πœƒ (𝐼), π‘₯ = Ξ“ π‘Šπ‘†πœƒ (𝐼), 𝑦 . Proof Suppose π‘₯ β€² ∈ Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ , then there exists a πœƒ(𝐼) βˆ’nonthin subsequence (π‘₯)𝐾 of the sequence π‘₯ = (π‘₯π‘˜ ) that weakly converges to π‘₯ β€² . 𝑖. 𝑒. If 𝐾 = π‘˜ 𝑗 : 𝑗 ∈ β„• , then for every 𝛾 > 0, π‘Ÿ ∈ β„•: Since, limπ‘Ÿ

1 β„Žπ‘Ÿ

1 {π‘˜ 𝑗 ∈ πΌπ‘Ÿ : 𝑗 ∈ β„•} β‰₯ 𝛾 βˆ‰ 𝐼 π‘Žπ‘›π‘‘ lim𝑓(π‘₯π‘˜(𝑗 ) βˆ’ π‘₯ β€² ) = 0 π‘˜ β„Žπ‘Ÿ

{π‘˜ ∈ πΌπ‘Ÿ : π‘˜ ∈ 𝐾, π‘₯π‘˜ β‰  π‘¦π‘˜ } = 0, it follows that limsup π‘Ÿ

1 {π‘˜ ∈ πΌπ‘Ÿ : π‘˜ ∈ 𝐾, π‘₯π‘˜ = π‘¦π‘˜ } > 0 β„Žπ‘Ÿ

(8)

Therefore, there exists a πœƒ(𝐼) βˆ’nonthin subsequence (𝑦)𝐾 of the sequence 𝑦 = (π‘¦π‘˜ ) that weakly converges to π‘₯ β€² . This shows that π‘₯ β€² ∈ Ξ› π‘Šπ‘†πœƒ (𝐼), 𝑦 and therefore Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ βŠ† Ξ› π‘Šπ‘†πœƒ (𝐼), 𝑦 . By symmetry we have Ξ› π‘Šπ‘†πœƒ (𝐼), 𝑦 βŠ† Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ . Hence, Ξ› π‘Šπ‘†πœƒ (𝐼), π‘₯ = Ξ› π‘Šπ‘†πœƒ (𝐼), 𝑦 . Similarly we can prove Ξ“ π‘Šπ‘†πœƒ (𝐼), π‘₯ = Ξ“ π‘Šπ‘†πœƒ (𝐼), 𝑦 . ∎

Acknowledgement The authors are thankful to the reviewers for their valuable comments and suggestions.

References [1]

Bala I., On weak* statistical convergence of sequences of functionals, International J. Pure Appl. Math., 70 (2011), pp. 647-653.

[2]

Bhardwaj V. K., Bala I., On weak statistical convergence, International J. Math. Math. Sc. (2007), Article ID 38530, 9 pages, doi:10.1155/2007/38530.

32

[3]

Connor J. S., The statistical and strong p-CesΓ ro convergence of sequences, Analysis, 8 (1988), pp. 47-63.

[4]

Connor J. A., Kline J., On statistical limit points and the consistency of statistical convergence, J. Math. Anal. Appl., 197 (1996), pp. 392–399.

[5]

Connor J., Ganichev M. and Kadets V., A characterization of Banach spaces with separable duals via weak statistical convergence, J. Math. Anal. Appl.,244 (2000), pp. 251261.

[6]

Das P., Savas E. and Ghosal K., On generalizations of certain summability methods using ideals, Appl. Math. Letters, 24(2011), pp. 1509-1514.

[7]

Demirci K., On lacunary statistical limit points, Demonstratio Mathematica, 35 (2002), pp. 93-101.

[8]

Fast H., Surla Convergence Statistique, Colloquium Mathematicum, 2 (1951), pp. 241-244.

[9]

Fridy J. A., On statistical convergence, Analysis, 5 (1985), pp. 301-313.

[10] Fridy J. A., Statistical limit points, Proceeding of American Mathematical Society, 118 (1993), pp. 1187-1192. [11] Fridy J. A. and Orhan C., Lacunary statistical convergence, Pacific Journal of Mathematics, 160 (1993), pp. 43-51. [12] Freedman A. R. and Sember J. J., Densities and summability, Pacific Journal of Mathematics, 95(1981), pp. 293–305. [13] GΓΌrdal M., 𝑆ahiner A., Extremal 𝐼 βˆ’limit points of double sequences, Appl. Math. E-Notes, 8 (2008), pp. 131–137. [14] GΓΌrdal M., Sari H., Extremal 𝐴 βˆ’statistical limit points via ideals, Journal of the Egyptian Mathematical Society, 22(1) (2013) , pp. 55–58. [15] Karakus S., Statistical convergence on probabilistic normed spaces, Math. Commun., 12 (2007), pp. 11-23. [16] Karakus S., Demirci K. and Duman O., Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 35 (2008), pp. 763-769. [17] Kolk E., The statistical convergence in Banach spaces, Acta et Comment. Univ. Tartu., 928 (1991), pp. 41-52. [18] Kostyrko P., Ε alΓ‘t T. and Wilczynski W., 𝐼 βˆ’Convergence, Real Anal. Exchange, 26(2000), pp. 669–686. [19] Kostyrko P., Macaj M., Ε alΓ‘t T. and Sleziak M., 𝐼 βˆ’Convergence and Extremal 𝐼 βˆ’Limit Points, Math. Slovaca, 55(2005), pp. 443–464. [20] Maddox I. J., Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc., 104(1)(1988), pp. 141-145. [21] Nuray F., Lacunary weak statistical convergence, Mathematica Bohemica, 136 (2011), pp. 259-268. [22] Ε alΓ‘t T., On statistically convergent sequences of real numbers, Mathematica Slovaca, 30 (2) (1980), pp. 139-150. [23] Schoenberg I. J., The integrability of certain functions and related summability methods, American Mathematical Monthly, 66 (1951), pp. 361-375. [24] Zygmund A., Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.

33