QUASI-ALMOST LACUNARY STATISTICAL CONVERGENCE OF SEQUENCES OF SETS

In this study, we defined concepts of Wijsman quasi-almost lacunary convergence, Wijsman quasi-strongly almost lacunary convergence and Wijsman quasi q-strongly almost lacunary convergence. Also we give the concept of Wijsman quasi-almost lacunary statistically convergence. Then, we study relationships among these concepts. Furthermore, we investigate relationship between these concepts and some convergences types given earlier for sequences of sets, too.


INTRODUCTION AND BACKGROUNDS
The concept of statistical convergence was first introduced by Fast [10].Also this concept was studied by Fridy [12], Šalát [17] and many others.
A sequence x = (x k ) is statistically convergent to the number L if for every ε > 0, where the vertical bars indicate the number of elements in the enclosed set.
Freedman et al. [1] established the connection between the strongly Cesàro summable sequences space |σ 1 | and the strongly lacunary summable sequences space N θ .
By a lacunary sequence we mean an increasing integer sequence θ = {k r } such that k 0 = 0 and h r = k r − k r−1 → ∞ as r → ∞.Throughout this study the intervals determined by θ will be denoted by I r = (k r−1 , k r ] and ratio kr kr−1 will be abbreviated by q r.The concept of lacunary statistical convergence was introduced by Fridy and Orhan [13]. Let θ = {k r } be a lacunary sequence.A sequence x = (x k ) is lacunary statistically convergent to L if for every ε > 0, The idea of almost convergence was introduced by Lorentz [9].Maddox [11] and (independently) Freedman [1] gave the concept of strong almost convergence.Similar concepts can be seen in [2].
Let X be any non-empty set and N be the set of natural numbers.The function , which is the range's elements of f , is said to be sequences of sets.
Let (X, ρ) be a metric space.For any point x ∈ X and any non-empty subset A of X, we define the distance from x to A by Throughout the paper we take (X, ρ) as a metric space and A, A k as any non-empty closed subsets of X.
A sequence The set of all bounded sequences of sets is denoted by L ∞ .
The concepts of Wijsman statistical convergence was introduced by Nuray and Rhoades [6].
A sequence {A k } is Wijsman statistically convergent to A if for each x ∈ X and every ε > 0 The concepts of Wijsman lacunary summability (W N θ ), [W N ] θ , [W N ] p θ and concept of Wijsman lacunary statistical convergence (W S θ ) were introduced by Ulusu and Nuray [20,21].
and it is denoted by A k Let θ = {k r } be a lacunary sequence.A sequence {A k } is Wijsman strongly lacunary summable to A if for each x ∈ X, Let θ = {k r } be a lacunary sequence.A sequence {A k } is Wijsman p-strongly lacunary summable to A if for each x ∈ X and 0 < p < ∞, A sequence {A k } is Wijsman lacunary statistically convergent to A if for every ε > 0 and each x ∈ X, Also the concepts of Wijsman almost lacunary convergence and Wijsman almost lacunary statistical convergence were introduced by Ulusu [18,19], too.
Let θ = {k r } be a lacunary sequence.A sequence A sequence {A k } is Wijsman almost lacunary statistically convergent to A if for every ε > 0 and each The idea of quasi-almost convergence in a normed space was introduced by Hajduković [3].Then, Nuray [5] studied concepts of quasi-invariant convergence and quasi-invariant statistical convergence in a normed space.
The concepts of Wijsman quasi-strongly almost convergence and Wijsman quasi-almost statistically convergence were studied by Gülle and Ulusu [4].
uniformly in n and it is denoted by A k uniformly in n and it is denoted by The set of all Wijsman quasi-almost statistically convergence sequences will be denoted by W QS .

MAIN RESULTS
In this section, we defined concepts of Wijsman quasi-almost lacunary convergence, Wijsman quasistrongly almost lacunary convergence and Wijsman quasi q-strongly almost lacunary convergence.Also we give the concept of Wijsman quasi-almost lacunary statistically convergence.Then, we study relationships among these concepts.Furthermore, we investigate relationship between these concepts and some convergences types given earlier for sequences of sets, too.
uniformly in n = 0, 1, 2, . . .where d x (A k+nr ) = d(x, A k+nr ) and d x (A) = d(x, A).In this case, we will write Example 2.1.Let we define a sequence {A k } as follows: This sequence is not Wijsman lacunary summable.But, since for each x ∈ X Proof.Suppose that the sequence {A k } is Wijsman almost lacunary convergent to A. Then, for each x ∈ X and every ε > 0 there exists an integer r 0 > 0 such that for all r > r 0 uniformly in n.Since ε > 0 is an arbitrary, the limit is taken for r → ∞ we can write Wijsman lacunary summable to A.
Proof.Assume that the sequence {A k } ∈ L ∞ is Wijsman quasi-almost lacunary convergent to A. Then, Equation (2.1) is true which for n = 0 implies for every ε > 0 and each x ∈ X, Definition 2.2.Let θ = {k r } be a lacunary sequence.A sequence {A k } is Wijsman quasi-almost lacunary statistically convergent to A if for each x ∈ X and every ε > 0 uniformly in n.In this case, we will write The set of all Wijsman quasi-almost lacunary statistically convergence sequences will be denoted by W QS θ : Wijsman quasi-almost lacunary statistically convergent to A.
Proof.Suppose that the sequence {A k } is Wijsman almost lacunary statistically convergent to A. Then, for every ε, δ > 0 and for each x ∈ X there exists an integer r 0 > 0 such that for all r > r 0 uniformly in n.Since δ > 0 is an arbitrary, we have Theorem 2.4.For any lacunary sequence θ = {k r }; if lim inf r q r > 1, then W QS ⊂ W QS θ .
Proof.Suppose that lim inf r q r > 1.Then for each r ≥ 1, there is a number δ ≥ 0 such that q r ≥ 1 + δ.
uniformly in n.If the limit is taken for the above inequality; since By the definition of lacunary sequence, we can write r instead of k r .Hence, for each x ∈ X we have uniformly in n.In this case, we will write Theorem 2.5.For any lacunary sequence θ = {k r }; if lim inf r q r > 1, then Proof.Let lim inf r q r > 1.Then for each r ≥ 1, there is a number δ ≥ 0 such that q r ≥ 1 + δ.Since The proof of theorem is completed.
Definition 2.4.Let θ = {k r } be a lacunary sequence.A sequence {A k } ∈ L ∞ is Wijsman quasi q-strongly almost lacunary convergent to A if for each x ∈ X and 0 < q < ∞, uniformly in n.In this case, we will write Theorem 2.6.Let 0 < q < ∞.Then, we have following assertions: i.If a sequence {A k } is Wijsman quasi q-strongly almost lacunary convergent to A, then the sequence ii.If a sequence {A k } ∈ L ∞ and Wijsman quasi-almost lacunary statistically convergent to A, then the sequence {A k } is Wijsman quasi q-strongly almost lacunary convergent to A.
Proof.(i) Let ε > 0 be given.Then, for each x ∈ X following inequality is proved uniformly in n.Since the sequence {A k } is Wijsman quasi q-strongly almost lacunary convergent to A; if the both side of Inequality (2.5) are multipled by 1 h r and after that the limit is taken for r → ∞, then we have lim Hence, we handle If {A k } is Wijsman quasi-almost lacunary statistically convergent to A, then for a given ε > 0 a number N ε ∈ N can be chosen such that for all r > N ε and each x ∈ X Thus, for each x ∈ X we have So, the proof is completed.
Theorem 2.7.If the sequence {A k } is Wijsman quasi q-strongly almost lacunary convergence to A, then {A k } is Wijsman q-strongly lacunary summable to A.
Proof.Suppose that the sequence {A k } ∈ L ∞ is Wijsman quasi q-strongly almost lacunary convergent to A.
Then, Equation (2.4) is true which for n = 0 implies for every ε > 0 and each x ∈ X, 1 h r k∈Ir |d x (A k ) − d x (A)| q −→ 0 (as r → ∞); so, {A k } is Wijsman q-strongly lacunary summable to A.
Theorem 2.8.If a sequence {A k } is Wijsman quasi q-strongly almost lacunary convergence to A, then the sequence {A k } is Wijsman lacunary statistically convergent to A.
Proof.Assume that the sequence {A k } is Wijsman quasi q-strongly almost lacunary convergence to A. Then, by Theorem 2.7, the sequence {A k } is Wijsman q-strongly lacunary summable to A. For each x ∈ X and every ε > 0, we can write Since the sequence {A k } is Wijsman q-strongly lacunary summable to A; if the both side of Inequality (2.6) are multipled by 1 h r and after that the limit is taken for r → ∞, left side of the Inequality (2.