1. Introduction EKREM SAVAŞ

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1 Matematiqki Bilten ISSN X Vol.39 (LXV) No (9 28) UDC:55.65:[57.52: Skopje, Makedonija I θ -STATISTICALLY CONVERGENT SEQUENCES IN TOPOLOGICAL GROUPS EKREM SAVAŞ Abstract. Recently, Das, Savaş and Ghosal [4], defined the lacunary statistical analogue for the sequence x = (x k ) as follows: A real sequence x = (x k ) is said to be I-lacunary statistically convergent to L or Sθ I -convergent to L if for any ɛ > 0 and δ > 0 {r N : {k I r : x k L ɛ} δ} I. In this case write S I θ lim x = L or x k L(S I θ ). In this paper, we introduce and study I-lacunary statistical convergence for sequence in topological groups and we shall also present some inclusion theorems.. Introduction We begin this section by giving some preliminaries. The notion of statistical convergence, which is an extension of the idea of usual convergence, was introduced by Fast [7] and Schoenberg [26] and its topological consequences were studied first by Fridy [8] and Šalát [20]. Di Maio and Kočinac [6] introduced the concept of statistical convergence in topological spaces and statistical Cauchy condition in uniform spaces and established the topological nature of this convergence. The notion has also been defined and studied in different steps, for example, in the locally convex space [5]; in intuitionistic fuzzy normed spaces[8]. In [] Albayrak and Pehlivan studied this notion in locally solid Riesz spaces. Recently, in [7] Mohiuddine and et.al introduced the concept of lacunary statistical convergence, lacunary statistically bounded and lacunary statistically Cauchy in the framework of locally solid Riesz spaces. Quite recently, Das and Savas [5] introduced the ideas of I τ -convergence, I τ -boundedness and I τ -Cauchy condition of nets in a locally solid Riesz space Mathematics Subject Classification. Primary 42B5; Secondary 40C05. Key words and phrases. double lacunary, statistical convergence, topological groups. 9

2 20 EKREM SAVAŞ The notion of statistical convergence depends on the density of subsets of N. A subset E of N is said to have density δ (E) if n δ (E) = lim χ E (k) exists. n n k= Note that if K N is a finite set, then δ(k) = 0, and for any set K N, δ(k C ) = δ(k). Definition.. A sequence x = (x k ) is said to be statistically convergent to l if for every ε > 0 δ ({k N : x k l ε}) = 0. In this case, we write st lim k x k = L and we denote the set of all statistical convergent sequences by st. By a lacunary sequence, we mean an increasing sequence θ = (k r ) of positive integers such that k 0 = 0 and : k r k r as r. Throughout this paper, the intervals determined by θ will be denoted by I r = (k r, k r ), and the ratio (k r ) (k r ) will be abbreviated by q r. In another direction, in [9], a new type of convergence called lacunary statistical convergence was introduced a follows: A sequence (x k ) of real numbers is said to be lacunary statistically convergent to L (or, S θ -convergent to L ) if for any ɛ > 0, lim r {k I r : x k L ɛ} = 0 where A denotes the cardinality of A N. In [9] the relation between lacunary statistical convergence and statistical convergence was established among other things. In [8], Mursaleen and Mohiuddine extended the idea of lacunary statistical convergence witespect to the intuitionistic fuzzy normed space. In [6], P. Kostyrko et al. introduced the concept of I-convergence of sequences in a metric space and studied some properties of such convergence. Note that I-convergence is an interesting generalization of statistical convergence. More investigations in this direction and more applications of ideals can be found in [4, 6,, 4, 2, 22, 23, 24, 25]. The purpose of this paper is to study I-lacunary statistical convergence of sequences in topological groups and to give some important inclusion theorems. 2. Definitions and Notations The following definitions and notions will be needed. Definition 2.. A family I 2 N is said to be an ideal of N if the following conditions hold:

3 I θ -STATISTICAL CONVERGENCE 2 (a) A, B I implies A B I, (b) A I, B A implies B I. Definition 2.2. A non-empty family F 2 N is said to be an ideal of N if the following conditions hold: (a) φ / F, (b) A, B F implies A B F, (c) A F, A B implies B F. If I is a proper ideal of N (i.e., N / I), then the family of sets F (I) = {M N : A I : M = N \ A} is a filter of N. It is called the filter associated with the ideal. Definition 2.3. A proper ideal I is said to be admissible if {n} I for each n N. Throughout the paper, I will stand for a proper admissible ideal of N and by sequence we always mean sequences of real numbers. Definition 2.4 (See [6]). Let I 2 N be a proper admissible ideal in N. (i) The sequence {x k } of elements of R is said to be I-convergent to L R if for each ɛ > 0 the set A(ɛ) = {n N : x k L ɛ} I. (ii) The sequence {x k } of elements of R is said to be I -convergent to L R if there exists M F (I) such that {x k } k M converges to L. In [4], Das, Savaş and Ghosal defined I-statistical convergence and I-lacunary statistical convergence as follows: Definition 2.5. A sequence x = {x k } is said to be I-statistically convergent to L or S(I)-convergent to L if for each ɛ > 0 and δ > 0 {n N : n {k n : x k L ɛ} δ} I. In this case we write x k L(S(I)). The class of all I-statically convergent sequences will be denoted by simply S(I). Definition 2.6. Let θ be a lacunary sequence. A sequence x = {x k } is said to be I-lacunary statistically convergent to L or Sθ I -convergent to L if for any ɛ > 0 and δ > 0 {r N : {k I r : x k L ɛ} δ} I. In this case we write x k L(Sθ I ). The class of all I-lacunary statistically convergent sequences will be denoted by Sθ I.

4 22 EKREM SAVAŞ By X, we will denote an abelian topological Hausdorff group, written additively, which satisfies the first axiom of countability. For a subset A of X, s(a) will denote the set of all sequences (x k ) such that x k is in A for k =, 2,... In [2], a sequence (x k ) in X is called to be statistically convergent to an element L of X if for each neighbourhood U of 0, lim n n {k n : x k L / U} = 0, where the vertical bars indicate the number of elements in the enclosed set. The set of all statistically convergent sequences in X is denoted by st(x). Also, Çakalli [3] defined lacunary statistical convergence in topological groups as follows: A sequence (x k ) is said to be S θ -convergent to L (or lacunary statistically convergent to L) if for each neighborhood U of 0, In this case, we write and define S θ (X) = lim r ( ) k I r : x k L / U} = 0. S θ lim k x k = L or x k L(S θ ) { } (x k ) : for some L, S θ lim x k = L. k Now we are ready to give the main definitions of I-statistical convergence and I-lacunary statistical convergence in topological groups as follows: Definition 2.7. A sequence x = (x k ) is said to be S I -statistically convergent to L or S I -convergent to L if for each neighborhood U of 0 and δ > 0 {n N : n {k n : x k L / U} δ} I. In this case we write x k L(S I ). The class of all S I -statically convergent sequences will be denoted by simply S I (X). Remark 2.8. For I = I fin, I- statistical convergence becomes statistical convergence in topological groups which is studied by Çakalli [2]. Definition 2.9. A sequences x = (x k ) is said to be I-lacunary statistically convergent to L or Sθ I -convergent to L if for each neighborhood U of 0 and δ > 0 {r N : {k I r : x k L / U} δ} I.

5 I θ -STATISTICAL CONVERGENCE 23 In this case, we write and define and in particular, S I θ (X) = S I θ lim k x k = L or x k L(S I θ ) { } (x k ) : for some L, Sθ I lim x k = L k S I θ (X) 0 = { } (x k ) : Sθ I lim x k = 0. k Remark 2.0. For I = I fin, I-lacunary statistical convergence becomes lacunary statistical convergence in topological groups which is studied by Çakalli [3]. It is obvious that every I-lacunary statistically convergent sequence has only one limit, that is, if a sequence is I-lacunary statistically convergent to L and L 2, then L = L Inclusion Theorems In this section, we prove the following theorems. Theorem 3.. For any double lacunary sequence θ = {k r }, S I (X) Sθ I (X) if and only if lim inf q r >. Proof. Sufficiency. Suppose that lim inf q r >, lim inf q r = α, say. Write β = (α )/2. Then there exists a positive integer r 0 such that q r + β for r r 0. Hence for r r 0, (k r ) = ( (k r )(k r ) ) = ( (q r ) ) ( ( + β) ) = (β( + β ) ). Take any (x k ) S I (X) and S I lim (k) x k = L, say. We prove that S I θ lim k x k = L. Let us take any neighborhood U of 0. Then for r r 0, we have (k r ) {k k r : x k L / U} (k r ) {k I r : x(k) L / U} {k I r : x k L / U}. Then for any δ > 0, we get = (k r ) ( ) {k I r : x k L / U} β( + β) ( ) {r N : ( ) {k I r : x k L / U} δ} {r N : (k r ) {k k r : x k L / U} δβ( + β) } I.

6 24 EKREM SAVAŞ This proves the sufficiency. Necessity. Suppose that lim inf r q r =. Then we can choose a subsequence {(k r (j) of the lacunary sequence θ r = {k r } such that k r(j) (k r(j) ) < + j, k r(j) (k r(j ) ) > j where r(j) > r(j ) + 2. Take an element x of X different from 0. Define a sequence (x k ) by x k = x if k {I r(j) } for some j =, 2,..., n,..., and x k = 0 otherwise. Then (x k ) S I (X) (in fact (x k ) S0 I (X)). To see this take any neighborhood U of 0. Then we may choose a neighborhood W of 0 such that W U and x / W. On the other hand, for each m we can find a positive number j m such that k r(jm) < m k r(jm)+. Then (m) {k m : x k / U} k r(j m) {k m : x k / W } < { k { r(j m) k kr(jm) : x k / W } + { k r(jm) < k m : x(k) / W } } { k k r(jm) : x k / W } + k ) k r(j m) ( (jm + ) + + j m ( r(j kr(jm)+ m) k r(jm) ) < ( (j m + ) + jm ) for each m. Therefore (x k ) S0 I (X). Now let us see that (x(k)) / S0 I (X). Since X is a Hausdorff space, there exists a symmetric neighborhood V of 0 such that x / V. Hence ( ) { lim hr(j) kr(j) < k k r(j) : x k / V } j and = lim j = lim j ( hr(j) ) ( kr(j) k r(j) ) ( hr(j) ) hr(j) =, lim h r r {k r < k k r : x k x / V } = 0. r r(j) j=,2,... Hence neither x nor 0 can be I-lacunary statistical limit of (x k ). No other point of X can be I-lacunary statistical limit of the sequence as well. Thus (x k ) / Sθ I (X). This completes the proof. For the next result we assume that the lacunary sequence θ satisfies the condition that for any set C F (I), {n : k r < n < k r, r C} F ( I ). Theorem 3.2. If lim sup r q r <, then for any lacunary sequence θ = {k r }, S I θ (X) SI (X).

7 I θ -STATISTICAL CONVERGENCE 25 Proof. If lim supq r <, then without any loss of generality, we can assume that r there exists a 0 < B < such that q r < B for all r. Suppose that x k L(S I θ ).Take any neighborhood U of 0. For ɛ, δ, δ > 0 define the sets and C = {r N : {k I r : x k L / U} < δ} T = {n N : n {k n : x k L / U} < δ }. It is obvious from our assumption that C F (I), the filter associated with the ideal I. Further observe that A j = h j {k I j : x k L / U} < δ for all j C. Let n N be such that k r < n < k r for some r C. Now n {k n : x k L / U} {k k r : x k L / U} k r = k r {k I : x k L / U} + + k r {k I r : x k L / U} = k k r h {k I : x k L / U} + k 2 k k r h 2 {k I 2 : x k L / U} k r k r k r {k I r : x(k) L / U} = k A + k 2 k A k r k r A r k r k r k r k r supa j. < Bδ. j C k r Choosing δ = δ B and in view of the fact that {n : k r < n < k r, r C} T where C F (I) it follows from our assumption on θ that the set T also belongs to F (I) and this completes the proof of the theorem. Problem. When I-lacunary statistical convergence implies I-statistical convergence? Corollary 3.3. Let θ = {(k r )} be a lacunary sequence, then S I (X) = Sθ I (X) if and only if < lim inf r q r lim sup q r <. r

8 26 EKREM SAVAŞ Recall [0, 2] that an admissible ideal I is said to satisfy condition (AP) if for any mutually disjoint sequence of sets {A i } i N in I, there exists a sequence {B i } i N in I such that A i B i is finite for all i N and i NB i I. It was observed in [0] ( also [2] ) that for a sequence {x n } n N I-convergence is equivalent to I -convergence if and only if the ideal I satisfies the condition (AP ). More facts about condition (AP) and its importance can be seen from []. We are now ready to prove our next result. Of course, the S θ limit is unique as shown in section 2. It is possible, however, for a sequence to have different S θ limits for different θ s.the following theorem shows that this situation cannot occur if x S I (X). Theorem 3.4. If (x(k)) belongs to both S I (X) and Sθ I (X), then S I lim k x k = S I θ lim k x k. Proof. Take any (x k ) S I (X) Sθ I(X) and suppose SI lim x = L and Sθ I lim x = L 2 and L L 2. Let 0 < ɛ < 2 L L 2. Since X is a Hausdorff space, there exists a symmetric neighborhood U of 0 such that L L 2 / U. Then we may choose a symmetric neighborhood W of 0 such that W + W U. Since I satisfies the condition (AP ), so there exists M F (I) (i.e. N \ M I ) such that lim m k m {k k m : L x k / W } = 0, where M = {k, k 2, k 3,...}. Let A = {k k m : L x k / W } and B = {k k m : L 2 x k / W }. Then k m = A B A + B. This implies A + B. Since B A and lim = 0 so we must have k m k m k m m k m B lim =. m k m Let M = {k l, k l2, k l3,...} = M θ F (I). Then k lp th term of the statistical limit expression k m {k k m : L 2 x k / W } : k lp {k l p i= I i : L 2 x k / W } = l p h i i= l p i= t i h i () Where t i = h i {k I i : L 2 x k / W } I 0 because x k L 2 (S I θ ). Since θ is a lacunary sequence, () is a regular weighted mean transform of t i s and therefore it is also I-convergent to zero as p and so it has a subsequence which is convergent to zero since I satisfies condition (AP ). But since this is a subsequence

9 I θ -STATISTICAL CONVERGENCE 27 of {n {k n : L 2 x k / W } } n M, we infer that {n {k n : L 2 x k / W } } n M is not convergent to which is a contradiction. This completes the proof of the theorem. References [] H. Albayrak, S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topology Appl., 59 (202), [2] H. Çakalli, On Statistical Convergence in topological groups, Pure and Appl. Math. Sci. 43, No.-2 (996), [3] H. Çakalli, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26 (2)(995), 3 9. [4] P. Das, E. Savaş, S. K. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Letters, 24 (20), [5] P. Das, E. Savaş, On I-convergence of nets in locally solid Riesz spaces, Filomat 27: (203), [6] K. Dems, On I-Cauchy sequences, Real Anal. Exchance, 30 ( ), [7] H. Fast, Sur la convergence statistique, Colloq Math., 2 (95), [8] J. A. Fridy, On ststistical convergence, Analysis, 5 (985), [9] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math., 60 (993), [0] P. Kostyrko, T. Šalát, W.Wilczynki, I-convergence, Real Anal. Exchange, 26 (2) (2000/200), [] P. Kostyroko, M. Macaj, T. Šalát, and M. Sleziak, I-convergence and extremal I-limit points, Math. Slovaca, 55 (2005), [2] B. K. Lahiri, Pratulananda Das, I and I -convergence in topological spaces, Math. Bohem., 30 (2005), [3] J. Li, Lacunary Statistical convergence and Inclusion properties between Lacunary Methods, Internat. J. Math. & Math. Sci. 23(3)(2000), [4] B. K. Lahiri, P. Das, On I and I convergence of nets, Real Anal.Exchange, 33 (2) ( ), [5] I. J. Maddox, Statistical convergence in locally convex spaces, Math Proc. Camb. Phil. Soc., 04 (988), [6] G. D. Maio, L. D. R. Kočinac, Statistical convergence in topology, Topology Appl., 56 (2008), [7] S. A. Mohiuddine, M. A. Alghamdi, Statistical summability through lacunary sequence in locally solid Riesz spaces, Journal of Inequalities and Applications (202), 202:225. [8] M. Mursaleen, S.A. Mohiuddine, On lacunary statistical convergence witespect to the intuitionistic fuzzy normed space, Journal of Computational and Applied Mathematics, 233(2) (2009), [9] R. F. Patterson, Analogues of some fundamental theorems of summability theory, Internat. J. Math. & Math. Sci. 23() (2000), 9. [20] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (980),

10 28 EKREM SAVAŞ [2] E. Savaş, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24(20), [22] E. Savaş, m -strongly summable sequence spaces in 2-normed spaces defined by ideal convergence and an Orlicz function, Appl. Math. Comput., 27 (200), [23] E. Savaş, A-sequence spaces in 2-normed space defined by ideal convergence and an Orlicz function, Abst. Appl. Analy., Vol. 20 (20), Article ID [24] E. Savaş, On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function, J. Ineq. Appl. (200) Article Number: , DOI: 0.55/200/ [25] E. Savaş, On generalized double statistical convergence via ideals, The Fifth Saudi Science Conference, 6 8 April, 202. [26] I.J Schoenberg, The integrability methods, Amer. Math. Monthly, 66 (959), İstanbul Commerce University, Department of Mathematics, Uskudar, İstanbul, Turkey address: ekremsavas@yahoo.com