International Journal of Analysis and Applications Volume 16, Number 2 (2018), 222-231 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-222 QUASI-ALMOST LACUNARY STATISTICAL CONVERGENCE OF SEQUENCES OF SETS ESRA GÜLLE∗, UĞUR ULUSU Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University, 03200, Afyonkarahisar, Turkey ∗Corresponding author: egulle@aku.edu.tr Abstract. 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 con- vergence. 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. 1. 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 = (xk) is statistically convergent to the number L if for every ε > 0, lim n→∞ 1 n ∣∣∣{k ≤ n : |xk −L| ≥ 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 |σ1| and the strongly lacunary summable sequences space Nθ. Received 2017-10-26; accepted 2017-12-20; published 2018-03-07. 2010 Mathematics Subject Classification. 40A05, 40A35. Key words and phrases. almost convergence; quasi-almost convergence; quasi-almost statistical convergence; lacunary se- quence; sequences of sets; Wijsman convergence. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 222 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-222 Int. J. Anal. Appl. 16 (2) (2018) 223 By a lacunary sequence we mean an increasing integer sequence θ = {kr} such that k0 = 0 and hr = kr − kr−1 → ∞ as r → ∞. Throughout this study the intervals determined by θ will be denoted by Ir = (kr−1,kr] and ratio kr kr−1 will be abbreviated by qr. The concept of lacunary statistical convergence was introduced by Fridy and Orhan [13]. Let θ = {kr} be a lacunary sequence. A sequence x = (xk) is lacunary statistically convergent to L if for every ε > 0, lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |xk −L| ≥ 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 f : N → P(X) is defined by f(k) = Ak ∈ P(X) for each k ∈ N, where P(X) is power set of X. The sequence {Ak} = (A1,A2, . . .), 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 d(x,A) = inf a∈A ρ(x,a). Throughout the paper we take (X,ρ) as a metric space and A,Ak as any non-empty closed subsets of X. There are different convergence notions for sequence of sets. One of them handled in this paper is the concept of Wijsman convergence (see, [6–8, 14–16]). A sequence {Ak} is said to be Wijsman convergent to A if for each x ∈ X lim k→∞ d(x,Ak) = d(x,A) and denoted by Ak W→ A or W − lim Ak = A. A sequence {Ak} is said to be bounded if for each x ∈ X sup k { d(x,Ak) } < ∞. 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 {Ak} is Wijsman statistically convergent to A if for each x ∈ X and every ε > 0 lim n→∞ 1 n ∣∣∣{k ≤ n : |d(x,Ak) −d(x,A)| ≥ ε}∣∣∣ = 0 and it is denoted by st− limW Ak = A. The concepts of Wijsman lacunary summability ( (WNθ), [WN]θ, [WN] p θ ) and concept of Wijsman lacu- nary statistical convergence ( (WSθ) ) were introduced by Ulusu and Nuray [20, 21]. Int. J. Anal. Appl. 16 (2) (2018) 224 Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman lacunary summable to A if for each x ∈ X, lim r→∞ 1 hr ∑ k∈Ir d(x,Ak) = d(x,A) and it is denoted by Ak (WNθ)−→ A. Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman strongly lacunary summable to A if for each x ∈ X, lim r→∞ 1 hr ∑ k∈Ir |d(x,Ak) −d(x,A)| = 0 and it is denoted by Ak [WN]θ−→ A. Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman p-strongly lacunary summable to A if for each x ∈ X and 0 < p < ∞, lim r→∞ 1 hr ∑ k∈Ir |d(x,Ak) −d(x,A)| p = 0 and it is denoted by Ak [WN] p θ−→ A. A sequence {Ak} is Wijsman lacunary statistically convergent to A if for every ε > 0 and each x ∈ X, lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |d(x,Ak) −d(x,A)| ≥ ε}∣∣∣ = 0 and it is denoted by Ak (WSθ)−→ A. Also the concepts of Wijsman almost lacunary convergence and Wijsman almost lacunary statistical convergence were introduced by Ulusu [18, 19], too. Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman almost lacunary convergent to A if for each x ∈ X, lim r→∞ 1 hr ∑ k∈Ir d(x,Ak+i) = d(x,A) uniformly in i = 0, 1, 2, . . .. Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman strongly almost lacunary convergent to A if for each x ∈ X, lim r→∞ 1 hr ∑ k∈Ir |d(x,Ak+i) −d(x,A)| = 0 uniformly in i. Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman strongly p-almost lacunary convergent to A if for each x ∈ X and 0 < p < ∞, lim r→∞ 1 hr ∑ k∈Ir |d(x,Ak+i) −d(x,A)|p = 0 uniformly in i. Int. J. Anal. Appl. 16 (2) (2018) 225 A sequence {Ak} is Wijsman almost lacunary statistically convergent to A if for every ε > 0 and each x ∈ X, lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |d(x,Ak+i) −d(x,A)| ≥ ε}∣∣∣ = 0 uniformly in i. 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 con- vergence were studied by Gülle and Ulusu [4]. A sequence {Ak}∈ L∞ is Wijsman quasi-strongly almost convergent to A if for each x ∈ X, 1 p np+p−1∑ k=np ∣∣dx(Ak) −dx(A)∣∣ → 0 (as p →∞) uniformly in n and it is denoted by Ak [WQF] −→ A. A sequence {Ak} is Wijsman quasi-almost statistically convergent to A if for each x ∈ X and every ε > 0 lim p→∞ 1 p ∣∣∣{k ≤ p : |dx(Ak+np) −dx(A)| ≥ ε}∣∣∣ = 0, uniformly in n and it is denoted by Ak WQS−→ A. The set of all Wijsman quasi-almost statistically convergence sequences will be denoted by { WQS } . 2. MAIN RESULTS In this section, 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 relation- ships among these concepts. Furthermore, we investigate relationship between these concepts and some convergences types given earlier for sequences of sets, too. Definition 2.1. Let θ = {kr} be a lacunary sequence. A sequence {Ak} ∈ L∞ is Wijsman quasi-almost lacunary convergent to A if for each x ∈ X,∣∣∣∣∣ 1hr ∑ k∈Ir dx(Ak+nr) −dx(A) ∣∣∣∣∣ −→ 0 (as r →∞), (2.1) uniformly in n = 0, 1, 2, . . . where dx(Ak+nr) = d(x,Ak+nr) and dx(A) = d(x,A). In this case, we will write (WQF)θ − lim Ak = A or Ak (WQF)θ−→ A. Int. J. Anal. Appl. 16 (2) (2018) 226 Example 2.1. Let we define a sequence {Ak} as follows: Ak :=   {x ∈ R : 2 ≤ x ≤ kr −kr−1} , if k ≥ 2 and k is square integer, {1} , otherwise. This sequence is not Wijsman lacunary summable. But, since for each x ∈ X lim r→∞ ∣∣∣∣∣ 1hr ∑ k∈Ir dx(Ak+nr) −dx({1}) ∣∣∣∣∣ = 0 uniformly n, this sequence is Wijsman quasi-almost lacunary convergent to the set A = {1}. Theorem 2.1. If a sequence {Ak}∈ L∞ is Wijsman almost lacunary convergent to A, then {Ak} is Wijsman quasi-almost lacunary convergent to A. Proof. Suppose that the sequence {Ak} is Wijsman almost lacunary convergent to A. Then, for each x ∈ X and every ε > 0 there exists an integer r0 > 0 such that for all r > r0∣∣∣∣∣ 1hr ∑ k∈Ir dx(Ak+i) −dx(A) ∣∣∣∣∣ < ε, uniformly in i. If i is taken as i = nr, then we have∣∣∣∣∣ 1hr ∑ k∈Ir dx(Ak+nr) −dx(A) ∣∣∣∣∣ < ε, uniformly in n. Since ε > 0 is an arbitrary, the limit is taken for r →∞ we can write∣∣∣∣∣ 1hr ∑ k∈Ir dx(Ak+nr) −dx(A) ∣∣∣∣∣ −→ 0 uniformly in n. That is, the sequence {Ak} is Wijsman quasi-almost lacunary convergent to A. � Theorem 2.2. If a sequence {Ak} ∈ L∞ is Wijsman quasi-almost lacunary convergent to A, then {Ak} is Wijsman lacunary summable to A. Proof. Assume that the sequence {Ak} ∈ 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,∣∣∣∣∣ 1hr ∑ k∈Ir dx(Ak) −dx(A) ∣∣∣∣∣ −→ 0 (as r →∞); so, {Ak} is Wijsman lacunary summable to A. � Definition 2.2. Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman quasi-almost lacunary statistically convergent to A if for each x ∈ X and every ε > 0 lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣ = 0, Int. J. Anal. Appl. 16 (2) (2018) 227 uniformly in n. In this case, we will write (WQS)θ − lim Ak = A or Ak (WQS)θ−→ A. The set of all Wijsman quasi-almost lacunary statistically convergence sequences will be denoted by{ WQSθ } : { WQSθ } = { {Ak} : lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣ = 0} . Theorem 2.3. If a sequence {Ak} is Wijsman almost lacunary statistically convergent to A, then {Ak} is Wijsman quasi-almost lacunary statistically convergent to A. Proof. Suppose that the sequence {Ak} is Wijsman almost lacunary statistically convergent to A. Then, for every ε,δ > 0 and for each x ∈ X there exists an integer r0 > 0 such that for all r > r0 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+i) −dx(A)| ≥ ε}∣∣∣ < δ, uniformly in i. If i is taken as i = nr, then we have 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣ < δ, uniformly in n. Since δ > 0 is an arbitrary, we have lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣ = 0, uniformly in n which means that Ak (WQS)θ−→ A. � Theorem 2.4. For any lacunary sequence θ = {kr}; if lim infr qr > 1, then { WQS } ⊂ { WQSθ } . Proof. Suppose that lim infr qr > 1. Then for each r ≥ 1, there is a number δ ≥ 0 such that qr ≥ 1 + δ. Since qr ≥ 1 + δ and hr = kr −kr−1, we have hr kr ≥ δ 1 + δ . Assume that Ak WQS−→ A. For each x ∈ X, we can write 1 kr ∣∣∣{k ≤ kr : |dx(Ak+nkr ) −dx(A)| ≥ ε}∣∣∣ ≥ 1kr ∣∣∣{k ∈ Ir : |dx(Ak+nkr ) −dx(A)| ≥ ε}∣∣∣ = hr kr ( 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nkr ) −dx(A)| ≥ ε}∣∣∣) ≥ δ 1 + δ ( 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nkr ) −dx(A)| ≥ ε}∣∣∣) that is, 1 kr ∣∣∣{k ≤ kr : |dx(Ak+nkr ) −dx(A)| ≥ ε}∣∣∣ ≥ δ1 + δ ( 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nkr ) −dx(A)| ≥ ε}∣∣∣) Int. J. Anal. Appl. 16 (2) (2018) 228 uniformly in n. If the limit is taken for the above inequality; since Ak WQS−→ A, we have 0 ≥ δ 1 + δ · lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nkr ) −dx(A)| ≥ ε}∣∣∣. By the definition of lacunary sequence, we can write r instead of kr. Hence, for each x ∈ X we have lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣ = 0, uniformly in n, that is Ak (WQS)θ−→ A. � Definition 2.3. Let θ = {kr} be a lacunary sequence. A sequence {Ak} ∈ L∞ is Wijsman quasi-strongly almost lacunary convergent to A if for each x ∈ X, 1 hr ∑ k∈Ir |dx(Ak+nr) −dx(A)| −→ 0 (as r →∞); uniformly in n. In this case, we will write [WQF]θ − lim Ak = A or Ak [WQF]θ−→ A. Theorem 2.5. For any lacunary sequence θ = {kr}; if lim infr qr > 1, then Ak [WQF] −→ A ⇒ Ak [WQF]θ−→ A. Proof. Let lim infr qr > 1. Then for each r ≥ 1, there is a number δ ≥ 0 such that qr ≥ 1 + δ. Since qr ≥ 1 + δ and hr = kr −kr−1, we have kr hr ≤ 1 + δ δ and kr−1 hr ≤ 1 δ · (2.2) Assume that Ak [WQF] −→ A. For each x ∈ X, we can write 1 hr ∑ k∈Ir |dx (Ak+nr) −dx (A)| = 1 hr kr∑ i=1 |dx (Ai+nr) −dx (A)|− 1 hr kr−1∑ i=1 |dx (Ai+nr) −dx (A)| = kr hr ( 1 kr kr∑ i=1 |dx (Ai+nr) −dx (A)| ) − kr−1 hr ( 1 kr−1 kr−1∑ i=1 |dx (Ai+nr) −dx (A)| ) . Hence, for each x ∈ X we have lim r→∞ 1 hr ∑ k∈Ir |dx (Ak+nr) −dx (A)| = lim r→∞ kr hr ( 1 kr kr∑ i=1 |dx (Ai+nr) −dx (A)| ) − lim r→∞ kr−1 hr ( 1 kr−1 kr−1∑ i=1 |dx (Ai+nr) −dx (A)| ) uniformly in n. Since Ak [WQF] −→ A, for each x ∈ X we have 1 kr kr∑ i=1 |dx (Ai+nr) −dx (A)|→ 0 and 1 kr−1 kr−1∑ i=1 |dx (Ai+nr) −dx (A)|→ 0 (2.3) Int. J. Anal. Appl. 16 (2) (2018) 229 uniformly in n. By using the Inequalities (2.2) and the Status (2.3), we handle lim r→∞ 1 hr ∑ k∈Ir |dx (Ak+nr) −dx (A)| = 0. The proof of theorem is completed. � Definition 2.4. Let θ = {kr} be a lacunary sequence. A sequence {Ak}∈ L∞ is Wijsman quasi q-strongly almost lacunary convergent to A if for each x ∈ X and 0 < q < ∞, 1 hr ∑ k∈Ir |dx(Ak+nr) −dx(A)|q −→ 0 (as r →∞); (2.4) uniformly in n. In this case, we will write [WQF] q θ − lim Ak = A or Ak [WQF] q θ−→ A. Theorem 2.6. Let 0 < q < ∞. Then, we have following assertions: i. If a sequence {Ak} is Wijsman quasi q-strongly almost lacunary convergent to A, then the sequence {Ak} is Wijsman quasi-almost lacunary statistically convergent to A. ii. If a sequence {Ak}∈ L∞ and Wijsman quasi-almost lacunary statistically convergent to A, then the sequence {Ak} 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∑ k∈Ir |dx(Ak+nr) −dx(A)|q ≥ εq ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣, (2.5) uniformly in n. Since the sequence {Ak} is Wijsman quasi q-strongly almost lacunary convergent to A; if the both side of Inequality (2.5) are multipled by 1 hr and after that the limit is taken for r →∞, then we have lim r→∞ 1 hr ∑ k∈Ir |dx(Ak+nr) −dx(A)|q ≥ εq lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣ 0 ≥ εq lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣. Hence, we handle lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ ε}∣∣∣ = 0, uniformly in n. That is Ak (WQS)θ−→ A. (ii) Since {Ak} is bounded, we can write sup k { dx(Ak) } + dx(A) = M, (0 < M < ∞), for each x ∈ X. Int. J. Anal. Appl. 16 (2) (2018) 230 If {Ak} is Wijsman quasi-almost lacunary statistically convergent to A, then for a given ε > 0 a number Nε ∈ N can be chosen such that for all r > Nε and each x ∈ X 1 hr ∣∣∣∣ { k ∈ Ir : |dx(Ak+nr) −dx(A)| ≥ (ε 2 )1/q}∣∣∣∣ < ε2Mq uniformly in n. Let take the set Lp = { k ≤ p : |dx(Ak+nr) −dx(A)| ≥ (ε 2 )1/q} . Thus, for each x ∈ X we have 1 hr ∑ k∈Ir ∣∣dx(Ak+nr) −dx(A)∣∣q = 1 hr ( ∑ k∈Ir k∈Lp |dx(Ak+nr) −dx(A)|q + ∑ k∈Ir k/∈Lp |dx(Ak+nr) −dx(A)|q ) < 1 hr hr ε 2Mq Mq + 1 hr hr ε 2 = ε 2 + ε 2 = ε uniformly in n. So, the proof is completed. � Theorem 2.7. If the sequence {Ak} is Wijsman quasi q-strongly almost lacunary convergence to A, then {Ak} is Wijsman q-strongly lacunary summable to A. Proof. Suppose that the sequence {Ak}∈ 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 hr ∑ k∈Ir |dx(Ak) −dx(A)|q −→ 0 (as r →∞); so, {Ak} is Wijsman q-strongly lacunary summable to A. � Theorem 2.8. If a sequence {Ak} is Wijsman quasi q-strongly almost lacunary convergence to A, then the sequence {Ak} is Wijsman lacunary statistically convergent to A. Proof. Assume that the sequence {Ak} is Wijsman quasi q-strongly almost lacunary convergence to A. Then, by Theorem 2.7, the sequence {Ak} is Wijsman q-strongly lacunary summable to A. For each x ∈ X and every ε > 0, we can write∑ k∈Ir |dx(Ak) −dx(A)|q ≥ εq ∣∣∣{k ∈ Ir : |dx(Ak) −dx(A)| ≥ ε}∣∣∣. (2.6) Since the sequence {Ak} is Wijsman q-strongly lacunary summable to A; if the both side of Inequality (2.6) are multipled by 1 hr and after that the limit is taken for r →∞, left side of the Inequality (2.6) is equal to 0. Hence, we handle lim r→∞ 1 hr ∣∣∣{k ∈ Ir : |dx(Ak) −dx(A)| ≥ ε}∣∣∣ = 0. Int. J. Anal. 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