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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