ASYMPTOTICALLY I 2 -LACUNARY STATISTICAL EQUIVALENCE OF DOUBLE SEQUENCES OF SETS

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1 Journal of Inequalities and Special Functions ISSN: , URL: Volume 7 Issue 2(206, Pages ASYMPTOTICALLY I 2 -LACUNARY STATISTICAL EQUIVALENCE OF DOUBLE SEQUENCES OF SETS UǦUR ULUSU, ERDİNÇ DÜNDAR Abstract. In this paper, we introduce the concepts of Wijsman asymptotically I 2 -statistical equivalence, Wijsman strongly asymptotically I 2 -lacunary equivalence and Wijsman asymptotically I 2 -lacunary statistical equivalence of double sequences of sets and investigate the relationship between them.. introduction, definitions and notations Throughout the paper N denotes the set of all positive integers and R the set of all real numbers. The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [] and Schoenberg [30]. This concept was extended to the double sequences by Mursaleen and Edely [9]. Çakan and Altay [6] presented multidimensional analogues of the results presented by Fridy and Orhan [3]. The idea of I-convergence was introduced by Kostyrko et al. [7] as a generalization of statistical convergence which is based on the structure of the idea I of subset of the set of natural numbers. Recently, Das et al. [7] introduced new notions, namely I-statistical convergence and I-lacunary statistical convergence by using ideal. Das, Kostyrko, Wilczyński and Malik [8] introduced the concept of I-convergence of double sequences in a metric space and studied some properties of this convergence. The concept of convergence of sequences of numbers has been extended by several authors to convergence of sequences of sets (see, [3 5, 34, 35]. Nuray and Rhoades [20] extended the notion of convergence of set sequences to statistical convergence and gave some basic theorems. Ulusu and Nuray [32] defined the Wijsman lacunary statistical convergence of sequence of sets and considered its relation with Wijsman statistical convergence, which was defined by Nuray and Rhoades. Kişi and Nuray [5] introduced a new convergence notion, for sequences of sets, which is called Wijsman I-convergence by using ideal. Recently, Ulusu and Dündar [3] studied the concepts of Wijsman I-statistical convergence, Wijsman I-lacunary statistical convergence and Wijsman strongly I-lacunary convergence of sequences of sets. 200 Mathematics Subject Classification. 34C4, 40A05, 40A35. Key words and phrases. Statistical convergence, lacunary sequence, I 2 -convergence, asymptotically equivalence, double sequence of sets, Wijsman convergence. c 206 Ilirias Publications, Prishtinë, Kosovë. Submitted January 5, 206. Published June 27, 206. This study supported by Afyon Kocatepe University Scientific Research Coordination Unit with the project number 5.HIZ.DES

2 ASYMPTOTICALLY I 2-LACUNARY STAT. EQU. OF DOUBLE SEQ. OF SETS 45 Nuray et al. [22] studied Wijsman statistical convergence, Hausdorff statistical convergence and Wijsman statistical Cauchy double sequences of sets and investigate the relationship between them. Nuray et al. [2] studied the concepts of Wijsman I 2, I2 -convergence and Wijsman I 2, I2 -Cauchy double sequences of sets. Dündar et al. [0] introduced the concepts of the Wijsman I 2 -statistical convergence, Wijsman I 2 -lacunary statistical convergence and Wijsman strongly I 2 -lacunary convergence of double sequences of sets. Marouf [8] peresented definitions for asymptotically equivalent and asymptotic regular matrices. Patterson [26] extend these concepts by presenting an asymptotically statistical equivalent analog of these definitions. Patterson and Savaş [27] extend the definitions presented in [26] to lacunary sequences. In addition to these definitions, natural inclusion theorems were presented. Recently, Savaş [28] presented the concept of I-asymptotically lacunary statistically equivalence which is a natural combination of the definitions for asymptotically equivalence and I-lacunary statistical convergence. The concept of asymptotically equivalence of sequences of real numbers which is defined by Marouf [8] has been extended by Ulusu and Nuray [33] to concept of Wijsman asymptotically equivalence of set sequences. In addition to these definitions, natural inclusion theorems are presented. Kişi et al. [6] introduced the concept of Wijsman I-asymptotically equivalence of sequences of sets. Now, we recall the basic definitions and concepts (See [ 3, 8 0, 2 4, 7, 8, 2 25, 29]. Two nonnegative sequences x = (x k and y = (y k are said to be asymptotically equivalent if x k lim =. k y k It is denoted by x y. 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 ρ(x, a. a A Let (X, ρ be a metric space and A, A k be any non-empty closed subsets of X. The sequence A k is Wijsman convergent to A if for each x X, lim d(x, A k = d(x, A. k A family of sets I 2 N is called an ideal if and only if (i I, (ii For each A, B I we have A B I, (iii For each A I and each B A we have B I. An ideal is called non-trivial if N / I and non-trivial ideal is called admissible if n I for each n N. Throughout the paper we take I 2 as an admissible ideal in N N. A non-trivial ideal I 2 of N N is called strongly admissible if i N and N i belongs to I 2 for each i N. It is evident that a strongly admissible ideal is admissible also. I 0 2 = A N N : ( m(a N(i, j m(a (i, j A. Then I 0 2 is a strongly admissible ideal and clearly an ideal I 2 is strongly admissible if and only if I 0 2 I 2.

3 46 U. ULUSU, E. DÜNDAR The class of all A N N which has natural density zero denoted by I f 2. Then I f 2 is strongly admissible ideal. A family of sets F 2 N is called a filter if and only if (i / F, (ii For each A, B F we have A B F, (iii For each A F and each B A we have B F. I is a non-trivial ideal in N if and only if F (I = M N : ( A I(M = N\A is a filter in N. An admissible ideal I 2 2 N N satisfies the property (AP2 if for every countable family of mutually disjoint sets A, A 2,... belonging to I 2, there exists a countable family of sets B, B 2,... such that A j B j I2, 0 i.e., A j B j is included in the finite union of rows and columns in N N for each j N and B = j= B j I 2 (hence B j I 2 for each j N. A double sequence x = (x kj k,j N of real numbers is said to be convergent to L R in Pringsheim s sense if for any ε > 0, there exists N ε N such that x kj L < ε, whenever k, j > N ε. In this case, we write P lim x kj = L or lim x kj = L. k,j k,j Throughout the paper, we let (X, ρ be a separable metric space, I 2 2 N N be a strongly admissible ideal and A, A kj be any non-empty closed subsets of X. The double sequence A kj is Wijsman convergent to A if for each x X, P lim d(x, A kj = d(x, A or k,j lim d(x, A kj = d(x, A. k,j The double sequence A kj is Wijsman statistically convergent to A if for every ε > 0 and for each x X, lim m,n mn k m, j n : d(x, A kj d(x, A ε = 0, that is, d(x, A kj d(x, A < ε for almost every (k, j. 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 as r. The double sequence θ = (k r, j s is called double lacunary sequence if there exist two increasing sequence of integers such that and k 0 = 0, h r = k r k r as r j 0 = 0, hu = j u j u as u. We use following notations in the sequel: k ru = k r j u, h ru = h r hu, I ru = (k, j : k r < k k r and j u < j j u, q r = k r k r and q u = j u j u. Let θ be a double lacunary sequence. The double sequence A kj is Wijsman strongly lacunary convergent to A if for each x X, lim r,u h r hu k r j u k=k r + j=j u + d(x, A kj d(x, A = 0.

4 ASYMPTOTICALLY I 2-LACUNARY STAT. EQU. OF DOUBLE SEQ. OF SETS 47 Let θ be a double lacunary sequence. The double sequence A kj is Wijsman lacunary statistically convergent to A, if for every ε > 0 and for each x X, lim r,u h r hu (k, j I ru : d(x, A kj d(x, A ε = 0. The double sequence of sets A kj is Wijsman I 2 -convergent to A, if for every ε > 0 and for each x X, (k, j N N : d(x, A kj d(x, A ε I 2. In this case we write I W2 lim A kj = A. k,j We define d(x; A kj, B kj as follows: d(x, A kj, x A kj B kj d(x, B d(x; A kj, B kj = kj L, x A kj B kj. The double sequences A kj and B kj are Wijsman asymptotically equivalent of multiple L if for each x X, lim d(x; A kj, B kj = L. k,j The double sequences A kj and B kj are Wijsman asymptotically statistical equivalent of multiple L if for every ε > 0 and for each x X, lim k m, j n : d(x; A kj, B kj L ε = 0. m,n mn Let θ be a double lacunary sequence. The double sequences A kj and B kj are Wijsman strongly asymptotically lacunary equivalent of multiple L if for each x X, lim r,u d(x; A kj, B kj L = 0. k,j I ru Let θ be a double lacunary sequence. The double sequences A kj and B kj are Wijsman asymptotically lacunary statistical equivalent of multiple L if for every ε > 0 and each x X, lim r,u (k, j Iru : d(x; A kj, B kj L ε = 0. The sequence A kj is Wijsman I 2 -statistical convergent to A or S (I W2 -convergent to A if for every ε > 0, δ > 0 and for each x X, (m, n N N : mn k m, j n : d(x, A kj d(x, A ε δ I 2. In this case, we write A kj A (S (I W2. Let θ be a double lacunary sequence. The sequence A kj is said to be Wijsman strongly I 2 -lacunary convergent to A or N θ [I W2 ]-convergent to A if for every ε > 0 and for each x X, (r, u N N : In this case, we write A kj A (N θ [I W2 ]. d(x, A kj d(x, A ε I 2. (k,j I ru

5 48 U. ULUSU, E. DÜNDAR Let θ be a double lacunary sequence. The sequence A kj is Wijsman I 2 - lacunary statistical convergent to A or S θ (I W2 -convergent to A if for every ε > 0, δ > 0 and for each x X, (r, u N N : (k, j I ru : d(x, A kj d(x, A ε δ I 2. In this case, we write A kj A (S θ (I W2. X R, f, g : X R functions and a point a X are given. If f(x = α(xg(x for x o U δ (a X, then for x X we write f = O(g as x a, where for any δ > 0, α : X R is bounded function on o U δ (a X. In this case, if there exists a c 0 such that f(x c g(x for x o U δ (a X, then for x X, f = O(g as x a. 2. main results In this section, we define the concepts of Wijsman asymptotically I 2 -statistical equivalence, Wijsman strongly asymptotically I 2 -lacunary equivalence and Wijsman asymptotically I 2 -lacunary statistical equivalence of double sequences of sets and investigate the relationship between them. Definition 2.. The double sequences A kj and B kj are Wijsman asymptotically I 2 -equivalent of multiple L if for every ε > 0 and each x X (k, j N N : d(x; A kj, B kj L ε I 2. I L W In this case, we write A 2 kj Bkj and simply Wijsman asymptotically I 2 -equivalent if L =. Definition 2.2. The double sequences A kj and B kj are Wijsman asymptotically I 2 -statistical equivalent of multiple L if for every ε > 0, δ > 0 and for each x X, (m, n N N : mn k m, j n : d(x; A kj, B kj L ε δ I 2. S(I L W In this case, we write A 2 kj B kj and simply Wijsman asymptotically I 2 - statistical equivalent if L =. The set of Wijsman asymptotically I 2 -statistical equivalent double sequences will be denoted by S ( IW L 2. For I 2 = I f 2, Wijsman asymptotically I 2-statistical equivalent of multiple L coincides with Wijsman asymptotically statistical equivalent of multiple L which is defined in [23]. As an example, consider the following double sequences; (x, y R A kj = 2 : x 2 + y 2 + kjy = 0, if k and j are a square integer, (,, otherwise. and B kj = (x, y R 2 : x 2 + y 2 kjy = 0, if k and j are a square integer, (,, otherwise.

6 ASYMPTOTICALLY I 2-LACUNARY STAT. EQU. OF DOUBLE SEQ. OF SETS 49 If we take I 2 = I f 2, since (m, n N N : mn k m, j n : d(x; A kj, B kj L ε δ I f 2, then the double sequences A kj and B kj are Wijsman asymptotically I 2 -statistical equivalent. Definition 2.3. Let θ be a double lacunary sequence. The double sequences A kj and B kj are Wijsman asymptotically I 2 -lacunary equivalent of multiple L if for every ε > 0 and for each x X, ( (r, u N N : d(x; A kj, B kj L ε I 2. (k,j I ru In this case, we write A kj N θ(i L W 2 B kj and simply Wijsman asymptotically I 2 - lacunary equivalent if L =. Definition 2.4. Let θ be a double lacunary sequence. The double sequences A kj and B kj are said to be Wijsman strongly asymptotically I 2 -lacunary equivalent of multiple L if for every ε > 0 and for each x X, (r, u N N : d(x; A kj, B kj L ε I 2. (k,j I ru N θ[i L W In this case, we write A 2 ] kj B kj and simply Wijsman strongly asymptotically I 2 -lacunary equivalent if L =. The set of Wijsman strongly asymptotically I 2 - lacunary equivalent double sequences will be denoted by [ ] N θ I L W2. As an example, consider the following double sequences; (x, y R A kj := 2 : (x kj 2 + y2 kj 2kj = k r < k < k r + [ h r ],, if j u < j < j u + [ h u ]. (,, otherwise. and (x, y R B kj := 2 : (x + kj 2 + y2 kj 2kj = k r < k < k r + [ h r ],, if j u < j < j u + [ h u ]. (,, otherwise. If we take I 2 = I f 2, since (r, u N N : d(x; A kj, B kj L ε If 2, (k,j I ru then the double sequences A kj and B kj are Wijsman strongly asymptotically I 2 -lacunary equivalent. Definition 2.5. Let θ be a double lacunary sequence. The double sequences A kj and B kj are Wijsman asymptotically I 2 -lacunary statistical equivalent of multiple L if for every ε > 0, δ > 0 and for each x X, (r, u N N : (k, j I ru : d(x; A kj, B kj L ε δ I 2.

7 50 U. ULUSU, E. DÜNDAR In this case, we write A kj S θ(i L W 2 B kj and simply Wijsman asymptotically I 2 - lacunary statistical equivalent if L =. The set of Wijsman asymptotically I 2 - lacunary statistical equivalent double sequences will be denoted by S θ ( I L W2. For I 2 = I f 2, Wijsman asymptotically I 2-lacunary statistical equivalent of multiple L coincides with Wijsman asymptotically lacunary statistical equivalent of multiple L which is defined in [23]. As an example, consider the following double sequences; A kj := (x, y R 2 : x 2 + (y 2 = kj, if (0, 0, otherwise. k r < k < k r + [ h r ], j u < j < j u + [ h u ] and k is a square integer, and (x, y R B kj := 2 : x 2 + (y + 2 = k r < k < k r + [ h r ],, if j kj u < j < j u + [ h u ] and k is a square integer, (0, 0, otherwise. If we take I 2 = I f 2, since (r, u N N : (k, j I ru : d(x; A kj, B kj L ε δ I f 2 h r h, u then the sequences A kj and B kj is Wijsman asymptotically I 2 -lacunary statistical equivalent. Theorem 2.6. Let θ be a double lacunary sequence. Then, A kj N θ[i L W 2 ] B kj A kj S θ(i L W 2 B kj. Proof. Suppose that A kj and B kj is Wijsman strongly asymptotically I 2 - lacunary equivalent of multiple L. Given ε > 0 and for each x X we can write d(x; A kj, B kj L d(x; A kj, B kj L (k,j I ru (k,j I ru d(x;a kj,b kj L ε ε. (k, j I ru : d(x; A kj, B kj L ε and so we get d(x; A kj, B kj L (k, j Iru : d(x; A kj, B kj L ε. ε h (k,j I r h u ru Hence, for each x X and for any δ > 0, we have (r, u N N : (k, j Iru : d(x; A kj, B kj L ε δ (r, u N N : d(x; A kj, B kj L ε δ I 2 (k,j I ru and so A kj S θ(i L W 2 B kj.

8 ASYMPTOTICALLY I 2-LACUNARY STAT. EQU. OF DOUBLE SEQ. OF SETS 5 Theorem 2.7. Let θ be a double lacunary sequence and d(x, A kj O ( d(x, B kj. Then, S θ(i L W A 2 N θ[i L W kj B kj A 2 ] kj B kj. Proof. Suppose that A kj and B kj is Wijsman asymptotically I 2 -lacunary statistical equivalent of multiple L and d(x, A kj O ( d(x, B kj. Then, there exists an M > 0 such that d(x; A kj, B kj L M, for each x X and all k, j N. Given ε > 0, for each x X we get d(x; A kj, B kj L (k,j I ru = (k,j I ru d(x;a kj,b kj L ε 2 d(x; A kj, B kj L + (k,j I ru d(x; A kj, B kj L d(x;a kj,b kj L < ε 2 M (k, j Iru : d(x; A kj, B kj L ε ε Hence, for each x X we have (r, u N N : d(x; A kj, B kj L ε (k,j I ru (r, u N N : (k, j Iru : d(x; A kj, B kj L ε 2 ε 2M I 2 and so A kj N θ[i L W 2 ] B kj. We have the following Theorem by Theorem 2.6 and Theorem 2.7. Theorem 2.8. Let θ be a double lacunary sequence. If d(x, A kj O ( d(x, B kj, then Sθ ( I L W2 = Nθ [ I L W2 ]. Theorem 2.9. Let θ be a double lacunary sequence. If lim inf r q r > and lim inf u q u > then, A kj S(I L W 2 B kj A kj S θ(i L W 2 B kj. Proof. Assume that lim inf r q r > and lim inf u q u >, then there exist λ, µ > 0 such that q r + λ and q u + µ for sufficiently large r, u which implies that k ru λµ ( + λ( + µ.

9 52 U. ULUSU, E. DÜNDAR If A kj and B kj is Wijsman asymptotically I 2 -statistical equivalent of multiple L, then for every ε > 0, for each x X and for sufficiently large r, u, we get k r j u k kr, j j u : d(x; A kj, B kj L ε k r j u (k, j Iru : d(x; A kj, B kj L ε ( λµ ( + λ( + µ. (k, j Iru : d(x; A kj, B kj L ε. Hence, for each x X and for any δ > 0 we have (r, u : (k, j Iru : d(x; A kj, B kj L ε δ (r, u : k r j u k kr, j j u : d(x; A kj, B kj L ε δλµ (+λ(+µ I 2 and so A kj S L θ (I W2 B kj. Theorem 2.0. Let θ be a double lacunary sequence. If lim sup r q r < and lim sup u q u <, then A kj S θ(i L W 2 B kj A kj S(I L W 2 B kj. Proof. If lim sup r q r < and lim sup u q u <, then there is an M, N > 0 such that q r < M and q u < N, for all r, u. Suppose that A kj and B kj is Wijsman asymptotically I 2 -lacunary statistical equivalent of multiple L and let U ru = U(r, u, x := (k, j I ru : d(x; A kj, B kj L ε. Since A kj and B kj is Wijsman asymptotically I 2 -lacunary statistical equivalent of multiple L, it follows that for every ε > 0 and δ > 0, for each x X, (r, u N N : (k, j Iru : d(x; A kj, B kj L ε δ Hence, we can choose a positive integers r 0, u 0 N such that Now let = U ru < δ, for all r > r 0, u > u 0. K := max U ru : r r 0, u u 0 U ru (r, u N N : δ I 2. and let t and v be any integers satisfying k r < t k r and j u < v j u.

10 ASYMPTOTICALLY I 2-LACUNARY STAT. EQU. OF DOUBLE SEQ. OF SETS 53 Then, we have k t, j v : d(x; A kj, B kj L ε tv = k r j u k kr, j j u : d(x; A kj, B kj L ε ( U + U 2 + U 2 + U U r0u k r j U ru u K k r j u r 0 u 0 ( U r0,u + h r0 h 0+ u0+ k r j u h r0 h u0+ r 0u 0 K k r j u + k r j u sup r>r 0 u>u 0 r 0u 0 K k r j u + ε (k r k r0 (j u j u0 k r j u r 0u 0 K k r j u + ε q r q u r 0u 0 K k r j u + ε M N. U r0+,u + h 0 U ru r0+h u0 + + h r0+h u0 U ru ( h r0 h u0+ + h r0+h u0 + + Since k r j u as t, v, it follows that k t, j v : d(x; A kj, B kj L ε 0 tv and consequently, for any δ > 0 the set (t, v N N : k t, j v : d(x; A kj, B kj L ε δ I 2. tv This shows that A kj and B kj is Wijsman asymptotically I 2 -statistical equivalent of multiple L. We have the following Theorem by Theorem 2.9 and Theorem 2.0. Theorem 2.. Let θ be a double lacunary sequence. If then < lim inf r q r lim sup r q r < and < lim inf u Sθ (I L W 2 = S(I L W 2. q u lim sup q u <, u Theorem 2.2. Let I 2 2 N N be a strongly admissible ideal satisfying property (AP 2 and θ F(I 2. If A kj, B kj S(I L W 2 S θ (I L2 W 2, then L = L 2. S L (IW2 Proof. Assume that A kj S L2 θ (I W2 B kj, A kj B kj and L L 2. Let 0 < ε < 2 L L 2.

11 54 U. ULUSU, E. DÜNDAR Since I 2 satisfies the property (AP 2, there exists M F(I 2 (i.e., N N\M I 2 such that for each x X and for (m, n M, lim k m, j n : d(x; A kj, B kj L ε = 0. m,n mn Let the sets and P = k m, j n : d(x; A kj, B kj L ε R = k m, j n : d(x; A kj, B kj L 2 ε. Then, mn = P R P + R. This implies that Since so we must have R mn and P mn + R mn. lim P m,n mn = 0, R lim m,n mn =. Let M = M θ F(I 2. Then, for (k l, j t M the k l j t th term of the statistical limit expression k m, j n : d(x; A kj, B kj L 2 ε mn is (k, j k l j t where l,t r,u=, I ru : d(x; A kj, B kj L 2 ε = l,t r,u=, v ru = (k, j Iru : d(x; A kj, B kj L 2 ε I 2 0 l,t r,u=, v ru, (2. because A kj and B kj is Wijsman asymptotically I 2 -lacunary statistical equivalent of multiple L 2. Since θ is a double lacunary sequence, (2. is a regular weighted mean transform of v ru s and therefore it is also I 2 -convergent to 0 as l, t, and so it has a subsequence which is convergent to 0 since I 2 satisfies property (AP 2. But since this is a subsequence of k m, j n : d(x; A kj, B kj L 2 ε, mn (m,n M we infer that k m, j n : d(x; A kj, B kj L 2 ε mn (m,n M is not convergent to. This is a contradiction. Hence, the proof is completed.

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13 56 U. ULUSU, E. DÜNDAR [3] U. Ulusu, E. Dündar, I-Lacunary statistical convergence of sequences of sets, Filomat 28(8 (204, [32] U. Ulusu, F. Nuray, Lacunary statistical convergence of sequence of sets, Progress in Applied Mathematics 4(2 (202, [33] U. Ulusu, F. Nuray, On Asymptotically Lacunary Statistical Equivalent Set Sequences, Journal of Mathematics 203 (203, 5 pages. [34] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc. 70( (964, [35] R. A. Wijsman, Convergence of Sequences of Convex sets, Cones and Functions II, Trans. Amer. Math. Soc. 23( (966, Uǧur Ulusu department of mathematics, faculty of science and literature, afyon kocatepe university, afyonkarahisar, turkey address: ulusu@aku.edu.tr Erdinç Dündar department of mathematics, faculty of science and literature, afyon kocatepe university, afyonkarahisar, turkey address: edundar@aku.edu.tr