compiles/77965cce38181ced4e0ea8972acce766/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 2, 2013, 137-146 ISSN 1307-5543 – www.ejpam.com Statistically Almost λ-convergence of Sequences of Sets Bipan Hazarika1, Ayhan Esi 2,∗ 1 Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791 112, Arunachal Pradesh, India 2 Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman, Turkey Abstract. The concept of Wijsman statistical convergence was defined by Nuray and Rhoades [9]. In this paper we define statistically almost λ- convergence for sequences for sets in sense of Wijsman and study some properties of this concept. 2010 Mathematics Subject Classifications: 40A05, 40A35, 40G15, 46E25 Key Words and Phrases: Statistical convergence, λ−sequence, almost convergence, Wijsman conver- gence 1. Introduction The concept of statistical convergence play a vital role not only in pure mathematics but also in other branches of science involving mathematics, especially in information theory, computer science, biological science, dynamical systems, geographic information systems, population modeling, and motion planning in robotics. The concept of convergence of sequences of points has been extended by several authors to convergence of sequences of sets. The one of these such extensions considered in this paper is the concept of Wijsman convergence. We shall define Wijsman statistically almost λ-convergence for sequences of sets and establish some basic results regarding this notions. The idea of statistical convergence was formerly given under the name “almost conver- gence” by Zygmund in the first edition of his celebrated monograph published in Warsaw in 1935 [13]. The concept was formally introduced by Steinhaus [11] and Fast [2] and later was introduced by Schoenberg [10], and also independently by Buck [1]. A lot of developments have been made in this areas after the works of S̆alát [12] and Fridy [4]. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. In the recent years, generalization of statistical ∗Corresponding author. Email addresses: bh_rgu@yahoo.co.in (B. Hazarika), aesi23@hotmail.com (A. Esi) http://www.ejpam.com 137 c© 2013 EJPAM All rights reserved. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 138 convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on Stone-C̆ech compactification of the natural num- bers. A real or complex number sequence x = � xk � is said to be statistically convergent to L if for every ε > 0 lim n 1 n � � � ¦ k ≤ n : � �xk − L � �≥ ε © � � �= 0. In this case, we write S − lim x = L or xk → L(S) and S denotes the set of all statistically convergent sequences. The generalized de la Vallée-Poussin mean is defined by tn (x) = 1 λn ∑ k∈In xk where In = � n−λn+ 1, n � . A sequence x = � xk � is said to be (V,λ)−summable to number L [5] if tn (x)→ L as n→∞. If λn = n, then (V,λ)−summability reduces to (C , 1)-summability. Mursaleen [8] defined λ−statistically convergent sequence as follows: A sequence x = � xk � is said to be λ− statistically convergent to the number L if for every ε > 0 lim n→∞ 1 λn � � � ¦ k ∈ In : � �xk − L � �≥ ε © � � �= 0. Let Sλ denotes the set of all λ−statistically convergent sequences. If λn = n, then Sλ is the same as S. The idea of almost convergence of sequences of points was introduced by Lorentz [6]. A sequence x = (xk) is said to be almost convergent to L if lim n→∞ 1 n n ∑ k=1 xk+m = L uniformly in m. Maddox [7] and Freedman et al. [3] introduced the notion of strong almost convergence of sequences of points independently. A sequence x = (xk) is said to be strongly almost convergent to L if lim n→∞ 1 n n ∑ k=1 |xk+m− L|= 0 uniformly in m. Let `∞, c, ac and |ac| denote the sets of all bounded , convergent, almost convergent and strongly almost convergent sequences, respectively. It is known [7] that c ⊂ ac ⊂ |ac| ⊂ `∞. 2. Wijsman Convergence and Preliminaries Let (X ,ρ) be a metric space. For any point x ∈ X and any non-empty subset A ⊂ X , the distance from x to A is defined by d(x ,A) = inf y∈A ρ � x , y � . B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 139 Definition 1 ([9]). Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X (k ∈ N), we say that the sequence � Ak � is Wijsman convergent to A if limk d(x ,Ak) = d(x ,A) for each x ∈ X . In this case we write W − lim Ak = A. The concepts of Wijsman statistical convergence and boundedless for the sequence � Ak � were given by Nuray and Rhoades [9] as follows: Definition 2. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X (k ∈ N), we say that the sequence � Ak � is Wijsman statistical convergent to A if the sequence � d(x ,Ak) � is statistically convergent to d(x ,A), i.e., for ε > 0 and for each x ∈ X lim n 1 n � � � ¦ k ≤ n : � �d(x ,Ak)− d(x ,A) � �≥ ε © � � �= 0. In this case, we write st − limk Ak = A or Ak→ A(WS). The sequence � Ak � is bounded if supk d(x ,Ak)<∞ for each x ∈ X . The set of all bounded sequences of sets denoted by L∞. Definition 3 ([9]). Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman Cesaro summable to A if {d(x ,Ak)} is Cesaro summable to d(x ,A), i.e. for each x ∈ X , lim n→∞ 1 n n ∑ k=1 d(x ,Ak) = d(x ,A). Definition 4 ([9]). Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman strongly Cesaro summable to A if {d(x ,Ak)} is Cesaro summable to d(x ,A), i.e. for each x ∈ X , lim n→∞ 1 n n ∑ k=1 |d(x ,Ak)− d(x ,A)|= 0. Definition 5 ([9]). Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman almost convergent to A if for each x ∈ X , lim n→∞ 1 n n ∑ k=1 d(x ,Ak+m) = d(x ,A) uniformly in m. Definition 6 ([9]). Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman strongly almost convergent to A if for each x ∈ X , lim n→∞ 1 n n ∑ k=1 |d(x ,Ak+m)− d(x ,A)|= 0 uniformly in m. Let L∞, C ,AC and |AC | denote the sets of all bounded , Wijsman convergent, Wijsman almost convergent and Wijsman strongly almost convergent sequences, respectively. It is known [9] that C ⊂ AC ⊂ |AC | ⊂ L∞. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 140 Definition 7 ([9]). Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman almost statistically convergent to A if for each ε > 0 and for each x ∈ X , lim n→∞ 1 n |{k ≤ n : |d(x ,Ak+m)− d(x ,A)| ≥ ε}|= 0 uniformly in m. 3. Wijsman Statistically Almost λ-convergence In this section, we will define Wijsman strongly λ-summable and Wijsman statistically al- most λ-convergence of sequences of sets and will give the relations between Wijsman strongly λ-summable and Wisjman statistically almost λ− convergence of sequences of sets. Let λ = � λn � be a non-decreasing sequence of positive numbers such that λn+1 ≤ λn+ 1,λ1 = 1,λn→∞ as n→∞ and In = � n−λn+ 1, n � . Definition 8. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman λ-summable to A if for each x ∈ X , lim n→∞ 1 λn ∑ k∈In d(x ,Ak) = d(x ,A). If λn = n, then Wijsman λ-summable reduces to Wijsman Cesaro summable. Definition 9. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman strongly λ-summable to A if for each x ∈ X , lim n→∞ 1 λn ∑ k∈In |d(x ,Ak)− d(x ,A)|= 0. In this case, we write wW λ − limk Ak = A or Ak→ A � wW λ � and wW λ =    � Ak � : lim n 1 λn ∑ k∈In � �d(x ,Ak)− d(x ,A) � �= 0    . If λn = n, then Wijsman strongly λ-summable reduces to Wijsman strongly Cesaro summable, i.e. wW = ( � Ak � : lim n 1 n ∑ k∈N � �d(x ,Ak)− d(x ,A) � �= 0 ) . Definition 10. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman almost λ-convergent to A if for each x ∈ X , lim n→∞ 1 λn ∑ k∈In d(x ,Ak+m) = d(x ,A) uniformly in m. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 141 If λn = n, then Wijsman almost λ-convergent reduces to Wijsman almost convergent. In special case m= 0, then Wijsman almost λ-convergent reduces to Wijsman λ-summable. Definition 11. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman strongly almost λ-convergent to A if for each x ∈ X , lim n→∞ 1 λn ∑ k∈In |d(x ,Ak+m)− d(x ,A)|= 0 uniformly in m. In this case, we write wW λ − limk Ak = A or Ak→ A � wW λ � . If λn = n, then Wijsman strongly almost λ-convergent reduces to Wijsman strongly almost convergent. In special case m = 0, then Wijsman strongly almost λ-convergent reduces to Wijsman strongly λ-summable. Definition 12. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X (k ∈ N), we say that the sequence � Ak � is Wijsman statistically λ-convergent to A if the sequence � d(x ,Ak) � is statistically λ− convergent to d(x ,A), i.e., for ε > 0 and for each x ∈ X lim n 1 λn � � � ¦ k ∈ In : � �d(x ,Ak)− d(x ,A) � �≥ ε © � � �= 0. In this case, we write sW λ − limk Ak = A or Ak→ A � sW λ � . If λn = n, then Wijsman statistical λ-convergent reduces to Wijsman statistical convergent. Definition 13. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman almost statistically λ-convergent to A if for each ε > 0 and for each x ∈ X , lim n→∞ 1 λn |{k ∈ In : |d(x ,Ak+m)− d(x ,A)| ≥ ε}|= 0 uniformly in m. In this case, we write sW λ − limk Ak = A or Ak→ A � sW λ � . If λn = n, then Wijsman almost statistically λ-convergent reduces to Wijsman almost statistically convergent. In special case m= 0, then Wijsman almost statistically λ-convergent reduces to Wijsman statistically λ-convergent. Example 1. Let X = R2 and the sequence � Ak � is defined as follows: Ak = ( ¦ � x , y � : x2+ � y − 1 �2 = k−1 © , if n− �� �λn � � � + 1≤ k ≤ n, k is square integer {(0,0)} , otherwise . Then the sequence � Ak � is Wijsman λ−statistical convergent to A= {(0,0)} since lim n 1 λn � � � ¦ k ∈ In : � �d(x ,Ak)− d(x , {(0,0)}) � �≥ ε © � � �= 0. But it is not Wijsman convergent. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 142 Theorem 1. Let (X ,ρ) be a metric space and A,Ak ⊂ X (k ∈ N) be non-empty closed subsets of X . Then a) wW λ ⊂ sW λ and the inclusion is proper. b) Let � Ak � ∈ L∞, then sW λ ⊂ wW λ . c) sW λ ∩ L∞ = wW λ ∩ L∞, where L∞ = {(Ak) : sup k,m |d(x ,Ak+m)− d(x ,A)|<∞}. Proof. a) Let ε > 0 and � Ak � ∈ wW λ . Then for all m ∈ N we can write ∑ k∈In � �d(x ,Ak+m)− d(x ,A) � �≥ ∑ k∈In |d(x ,Ak+m)−d(x ,A)|≥ε � �d(x ,Ak+m)− d(x ,A) � � ≥ε � � � ¦ k ∈ In : � �d(x ,Ak+m)− d(x ,A) � �≥ ε © � � � which gives the result. To show that the inclusion is strict, we define the sequence � Ak � as follows: Ak = ( {k} , if n− �� �λn � � � + 1≤ k ≤ n; {0} , otherwise It is clear that � Ak � /∈ L∞ and for ε > 0, lim n 1 λn � � � ¦ k ∈ In : � �d(x ,Ak+m)− d(x , {0}) � �≥ ε © � � �= lim n 1 λn �� �λn � � � = 0. So � Ak � ∈ sW λ , but lim n 1 λn ∑ k∈In � �d(x ,Ak+m)− d(x , {0}) � �= lim n 1 λn ��� �λn � � ���� �λn � � � + 1 �� 2 = 1 2 6= 0. Therefore � Ak � /∈ wW λ . This completes the proof of (a). b) Suppose that � Ak � ∈ sW λ and � Ak � ∈ L∞, say � �d(x ,Ak+m)− d(x ,A) � � ≤ M for each x ∈ X and for all k, m ∈ N. Given ε > 0, we get 1 λn ∑ k∈In � �d(x ,Ak+m)− d(x ,A) � �= 1 λn ∑ k∈In |d(x ,Ak+m)−d(x ,A)|≥ε � �d(x ,Ak+m)− d(x ,A) � � B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 143 + 1 λn ∑ k∈In |d(x ,Ak+m)−d(x ,A)|<ε � �d(x ,Ak+m)− d(x ,A) � � ≤ M λn � � � ¦ k ∈ In : � �d(x ,Ak+m)− d(x ,A) � �≥ ε © � � �+ ε from which the result follows. c) It follows from (a) and (b). If we let λn = n in Theorem 1, then we have the following corollary. Corollary 1. Let (X ,ρ) be a metric space and A,Ak ⊂ X (k ∈ N) be non-empty closed subsets of X . Then a) wW ⊂ sW and the inclusion is proper. b) Let � Ak � ∈ L∞, then sW ⊂ wW . c) sW ∩ L∞ = wW ∩ L∞. Theorem 2. sW ⊂ sW λ if and only if lim inf λn n > 0. Proof. Suppose that lim inf λn n > 0. For given ε > 0, for all m ∈ N, we have ¦ k ≤ n : � �d(x ,Akm)− d(x ,A) � �≥ ε © ⊃ ¦ k ∈ In : � �d(x ,Ak+m)− d(x ,A) � �≥ ε © . Therefore 1 n � � � ¦ k ≤ n : � �d(x ,Ak+m)− d(x ,A) � �≥ ε © � � �≥ 1 n � � � ¦ k ∈ In : � �d(x ,Ak+m)− d(x ,A) � �≥ ε © � � � ≥ λn n . 1 λn � � � ¦ k ∈ In : � �d(x ,Ak+m)− d(x ,A) � �≥ ε © � � � . Taking the limit as n→∞ and using lim inf λn n > 0, we get the desired result. Conversely, suppose that lim infn λn n = 0. Then we can select a subsequence (n(i))∞i=1 such that λn(i) n(i) < 1 i . We define a sequence (Ak) as follows: Ak = ( {1}, if n(i)− �� �λn(i) � � � + 1≤ k ≤ n(i), i = 1,2,3, . . . ; {0}, otherwise . Then (Ak) is Wijsman-statistically convergent, so (Ak) ∈ sW . But (Ak) /∈ wW λ . Therefore the Theorem 1 (b) implies that (Ak) /∈ sW λ . This completes the proof. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 144 Theorem 3. sW λ ⊂ sW if lim inf λn n = 1. Proof. Since limn λn n = 1, then for ε > 0, for all m ∈ N, we observe that 1 n |{k ≤ n : |d(x ,Ak+m)− d(x ,A)| ≥ ε}| ≤ 1 n |{k ≤ n−λn : |d(x ,Ak+m)− d(x ,A)| ≥ ε}| + 1 n |{k ∈ In : |d(x ,Ak+m)− d(x ,A)| ≥ ε}| ≤ n−λn n + 1 n |{k ∈ In : |d(x ,Ak+m− d(x ,A)| ≥ ε}| = n−λn n + λn n 1 λn |{k ∈ In : |d(x ,Ak+m− d(x ,A)| ≥ ε}|. This implies that (Ak) Wijsman almost statistically convergent, if (Ak) is Wijsman almost sta- tistically λ-convergent. Thus sW λ ⊂ sW . Remark 1. Since limn λn n = 1, implies that lim infn λn n > 0, then from Theorem 2, we have sW ⊂ sW λ . Hence sW λ = sW . Definition 14 ([9]). Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman strongly p-almost convergent to A if for each x ∈ X , p ∈ (0,∞), lim n→∞ 1 n n ∑ k=1 |d(x ,Ak+m)− d(x ,A)|p = 0 uniformly in m. We introduced the following definition. Definition 15. Let (X ,ρ) be a metric space. For any non-empty closed subsets A,Ak ⊂ X , we say {Ak} is Wijsman strongly almost λp-summable to A if for each x ∈ X , p ∈ (0,∞), lim n→∞ 1 λn ∑ k∈In |d(x ,Ak+m)− d(x ,A)|p = 0 uniformly in m. If λn = n, Wijsman strongly almost λp-summable reduces to Wijsman strongly almost p-Cesaro summable defined as follows: lim n→∞ 1 n n ∑ k=1 |d(x ,Ak+m)− d(x ,A)|p = 0 uniformly in m. Theorem 4. Let (X ,ρ) be a metric space and A,Ak ⊂ X (k ∈ N) be non-empty closed subsets of X . If (Ak) is Wijsman strongly almost λp-summable to A, then it is Wijsman statistically almost λ-convergent to A. B. Hazarika, A. Esi / Eur. J. Pure Appl. Math, 6 (2013), 137-146 145 Proof. For any (Ak, ) fix an ε > 0 and for all m ∈ N, we have ∑ k∈In |d(x ,Ak+m)− d(x ,A)|p ≥ ε|{k ∈ In : |d(x ,Ak+m)− d(x ,A)|p ≥ ε}|, and it follows that if (Ak) is Wijsman strongly almost λp-summable to A, then it is Wijsman statistically almost λ-convergent to A. Theorem 5. Let (X ,ρ) be a metric space and A,Ak ⊂ X (k ∈ N) be non-empty closed subsets of X . If (Ak) is bounded and Wijsman statistically almost λ-convergent to A, then it is Wijsman strongly almost λp-summable to A and hence (Ak) is Wijsman strongly almost p-Cesaro summable to A. Proof. Let (Ak) is bounded and Wijsman statistically almost λ-convergent to A. Since (Ak) is bounded,then there exists M > 0 such that |d(x ,Ak+m)− d(x ,A)| ≤ M for all k, m ∈ N. Let ε > 0 be given and for all m ∈ N, we select n0 = n0(ε) such that 1 λn � � � � � ¨ k ∈ In : |d(x ,Ak+m)− d(x ,A)|p ≥ �ε 2 � 1 p « � � � � � < ε 2M p for all n> n0. We put K(ε) = ¨ k ∈ In : |d(x ,Ak+m)− d(x ,A)| ≥ �ε 2 � 1 p « . For all m ∈ N, we have 1 λn ∑ k∈In |d(x ,Ak+m)− d(x ,A)|p = 1 λn ∑ k∈In,k∈K(ε) |d(x ,Ak+m)− d(x ,A)|p + 1 λn ∑ k∈In,k/∈K(ε) |d(x ,Ak+m)− d(x ,A)|p = T1+ T2 where T1 = 1 λn ∑ k∈In,k∈K(ε) |d(x ,Ak+m)− d(x ,A)|p and T2 = 1 λn ∑ k∈In,k/∈K(ε) |d(x ,Ak+m)− d(x ,A)|p. If k ∈ K(ε), then T2 < ε 2 . 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