Statistical Convergence of Order α

Filomat 32:16 (2018), 5565–5572 https://doi.org/10.2298/FIL1816565K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat ∆m p −Statistical Convergence of Order α Abdulkadir Karakaşa , Yavuz Altınb , Mikail Etb a Department b Department of Mathematics, Siirt University, Siirt, Turkey of Mathematics, Fırat University, Elazig, Turkey Abstract. In this work, we generalize the concepts of statistically convergent sequence of order α and statistical Cauchy sequence of order α by using the generalized difference operator ∆m . We prove that a m sequence is ∆m p −statistically convergent of order α if and only if it is ∆p −statistically Cauchy of order α. 1. Introduction Throughout we denote the space of all complex sequences by w and ℓ∞ , c and c0 be the linear spaces of bounded, convergent and null sequences x = (xk ) with complex terms, respectively normed by kxk∞ = supk |xk |, where k ∈ N = {1, 2, 3, ...}, the set of positive integers. In 1981, the difference sequence spaces X (∆) were introduced by Kızmaz [16] for X = ℓ∞ , c and c0 and the notion was generalized by Et and Çolak [10]. Out of these, using the generalized difference operator ∆m , Ioan [17] introduced and discussed the concept of p−convex sequences. Later on, Karakaş and Altin [15] defined and studied some basic topological and algebraic properties of the sequence spaces X ∆m p m P m m v m m−v for X = ℓ∞ , c, c0 , where p, m ∈ N, ∆p x = pxk − xk+1 , and ∆p x = ∆p xk = (−1) v p xk+v . In the case v=0 m m m x ∈ X ∆m p (for X = ℓ∞ , c and c0 ), we call ∆p − bounded, ∆p − conver1ent and ∆p − zero, respectively. Let X be any sequence space, if x ∈X (∆m ) then there exists one and only one y = (yk ) ∈ X such that xk = k−m X i=1 (−1)m ! ! k X k−i−1 m k+m−i−1 (−1) yi−m , yi = m−1 m−1 i=1 y1−m = y2−m = · · · = y0 = 0 (1) for sufficiently large k, for instance k > 2m. We use this fact to formulate (2), (3) and (4). Recently the difference sequence spaces have been studied by many researchers [1],[2],[8],[15],[19],[26]. The idea of statistical convergence goes back to the first edition of monograph of Zygmund [27]. This notion has firstly been defined for real and complex sequences by Steinhaus [23] and Fast [12]. Schoenberg [21] has defined from a sequence- to- sequence summability method called D−convergence which, implies 2010 Mathematics Subject Classification. Primary 40A05; Secondary 40D255 Keywords. Statistical convergence, difference sequence Received: 30 May 2017; Revised: 16 October 2017; Accepted: 06 November 2017 Communicated by Ljubiša D.R. Kočinac Email addresses: [email protected] (Abdulkadir Karakaş), [email protected] (Yavuz Altın), [email protected] (Mikail Et) A. Karakaş et al. / Filomat 32:16 (2018), 5565–5572 5566 statistical convergence. Later on, it has been studied by Bhuania et al. [3], Connor [4], Çolak [5], Çolak and Altin [6], Et et al. [9, 11, 22], Fridy [13], Gadjiev and Orhan [14], Moricz [18], Šalát [20], Tripathy [25], Dutta and Tripathy [7], and many others. The concept of statistical convergence depends on the density of subsets of the set N. The natural density of a subset A of N is defined by δ (A) = limn n1 |{k ≤ n : k ∈ A}| , if the limit exists, where |.| is cardinality of set A. A sequence x = (xk ) of complex numbers is said to be statistically convergent to some number L if, for every positive number ε, δ ({k ∈ N: |xk − L| ≥ ε}) has natural density zero. The number L is called the statistical limit of (xk ) and written as S − lim xk = L. We denote the space of all statistically convergent sequences by S. 2. Some Properties of ∆m p (X) In this section, we give some topological properties of ∆m p (X) and some inclusion relations. m m Theorem 2.1. The sequence spaces ℓ∞ ∆m p , c ∆p and c0 ∆p are BK−spaces with norm kxk1 = m X xi + ∆m px i=1 ∞ . Proof. The proof is similar to the proof of Theorem 1.1 of Et and Çolak [10]. m Theorem 2.2. Let X be a vector space and let A ⊂ X. If A is a convex set, then ∆m p (A) is a convex set in ∆p (X) . Proof. Can be established using standard techniques, so omitted. Theorem 2.3. The following statements hold: i) ℓ∞ ⊂ ℓ∞ ∆m p and the inclusion is strict, m and the inclusion is strict, ii) c ∆p ⊂ ℓ∞ ∆m p m iii) c (∆) ⊂ c ∆p and the inclusion is strict, m iv) The sequence space ℓ∞ (∆) is different from the sequence space ℓ∞ ∆m p and ℓ∞ (∆) ∩ ℓ∞ ∆p , ∅. Proof. i) Let x ∈ ℓ∞ . Then ! m m−v p xk+v v v=0 ! ! ! ! m m m m−1 m m−2 m ≤ p |xk | + p p p |xk+v | < M |xk+1 | + |xk+2 | + ... 0 1 2 m−1 m m for some M > 0; i.e. , ∆m p xk ∈ ℓ∞ and so x ∈ ℓ∞ (∆p ). Hence ℓ∞ ⊂ ℓ∞ ∆p . ∆m px = m X (−1)v k P To show that the inclusion is strict, let us consider the sequence x = (xk ) with xk = pk − pi so that i=1 m−1 , p(p − 1)m−1 , p(p − 1)m−1 , ... . Then we obtain ∆m ∆m p xk ∈ ℓ∞ but (xk ) < ℓ∞ . p x = p(p − 1) m m m m ii) Let x ∈ c ∆m p . Then, we have ∆p x ∈ c ⊂ ℓ∞ , that is, x ∈ ℓ∞ ∆p . Therefore, c ∆p ⊂ ℓ∞ ∆p . To show that the inclusion is strict, define a sequence x = (xk ) such that xk = 0, p, 0, p, 0, ... , m then x ∈ ℓ∞ ∆m p \c ∆p . 5567 iii) If we choose (xk ) = p, 2p, 3p, 4p, ... , then we obtain x ∈ c (∆) but x < c ∆m p . m iv) If we choose (xk ) = (1, 2, 3, ...) , then x ∈ ℓ∞ (∆) , but x < ℓ∞ ∆p . Let us take the sequence x = (xk ) such k P that xk = pk − pi . Then, we get x < ℓ∞ (∆) but x ∈ ℓ∞ ∆m p . Since all constant sequences belong to both i=1 m ℓ∞ (∆) and ℓ∞ ∆m p , the spaces ℓ∞ (∆) and ℓ∞ ∆p are overlapping. A. Karakaş et al. / Filomat 32:16 (2018), 5565–5572 3. Main Results In this section, we introduce and examine the concepts of ∆m p −statistically convergent sequence of order −statistically Cauchy sequence of order α. α and ∆m p Definition 3.1. Let x = (xk ) ∈ w and 0 < α ≤ 1 be given. The sequence x = (xk ) is said to be ∆m p −statistically convergent of order α if there exists a complex number L such that o 1 n m =0 k ≤ n : ∆ x − L ≥ ε k p n→∞ nα lim m for every ε > 0. In this case we write stat(α) − lim ∆m p xk → L. The set of ∆p −statistically convergent k→∞ α m sequences of order α will be denoted by Sα ∆m p . In case of L = 0, we shall write S0 ∆p . m Theorem 3.2. Let 0 < α ≤ 1. If a sequence x = (xk ) is ∆m p −statistically convergent of order α, then stat(α)− lim ∆p xk k→∞ is unique. m Proof. Suppose that stat(α) − lim ∆m p xk = L1 and stat(α) − lim ∆p xk = L2 . Given ε ≥ 0, consider the following sets: k→∞ k→∞ ε K1 (ε) = k ∈ N : ∆m p xk − L1 ≥ 2 and ε ≥ x − L K2 (ε) = k ∈ N : ∆m . 2 p k 2 α m Therefore, we obtain δα (K1 (ε)) = 0 since stat(α) − lim ∆m p xk = L1 and δ (K2 (ε)) = 0 since stat(α) − lim ∆p xk = k→∞ k→∞ L2 . Now, let K (ε) = K1 (ε) ∪ K2 (ε) . Thus, we get δα (K (ε)) = 0 which implies N/δα (K (ε)) = 0. Now let Kc (ε) = N/K (ε) , then we get m |L1 − L2 | ≤ L1 − ∆m p xk + ∆p xk − L2 ε ε < + = ε. 2 2 Therefore, we have |L1 − L2 | = 0, i.e. L1 = L2 . From Theorem 3.2 we see that the ∆m p −statistical convergence of order α is well defined for 0 < α ≤ 1. However, for α > 1 it is not well defined, since stat(α) − lim ∆m p xk is not uniquely defined. To show it, let k→∞ x = (xk ) be defined as ( 1, k = 2n ( n = 1, 2, 3...) . xk = 0, k , 2n otherwise Then we have ( p, k = 2n ( n = 1, 2, 3...) ∆p xk = 0, k , 2n A. Karakaş et al. / Filomat 32:16 (2018), 5565–5572 5568 for m = 1. Then both o n n lim k ≤ n : ∆m =0 p xk − p ≥ ε ≤ lim n 2nα n→∞ and o n 1 n m ≤ lim α = 0 ≥ ε ∆ x − 0 k ≤ n : k p n 2n n→∞ nα lim for α > 1, so that x = (xk ) is ∆m p −statistically convergent of order α both to p and 0. m Since the α−density of a finite set is zero, every ∆m p −convergent sequence is ∆p −statistically convergent of order α, but the converse is not true in general as can be seen in the following example. Let x = (xk ) be defined as ( p, k = n2 ( n = 1, 2, 3...) . xk = 0, otherwise Then we obtain  2  p,    −p, ∆p xk =     0, k = n2 ( n = 1, 2, 3...) k + 1 = n2 , otherwise Then we have  2  p,    −p, ∆p xk =     0, k = n3 ( n = 1, 2, 3...) k + 1 = n3 .. otherwise for m = 1. It is easy to see that x = (xk ) is ∆p −statistically convergent of order α for α > 21 , but is not convergent. m β Theorem 3.3. Let 0 < α ≤ β ≤ 1. Then Sα ∆m p ⊆ S ∆p and the inclusion is strict for at least those α and β for 1 which there is a k ∈ N such that α < k < β. m β Proof. The inclusion part of proof is trivial. To show the inclusion Sα ∆m p ⊆ S ∆p is strict choose m = 1 and define a sequence x = (xk ) by ( p, k = n3 ( n = 1, 2, 3...) xk = 0, k , n3 and so √ o 2 3n 1 n m k ≤ n : ≤ lim ∆ x − 0 =0 ≥ ε p k n→∞ nβ n nβ 1 1 m β m α hence stat(β) − lim ∆m p xk = 0, i.e x ∈ S ∆p for 3 < β ≤ 1, but x < S ∆p for 0 < α ≤ 3 so that the inclusion k→∞ 1 1 m β Sα ∆m p ⊂ S ∆p is strict. This holds for 3 = α < β < 2 for example, but there is no a number k ∈ N such that α < 1k < β. Therefore, the condition α < 1k < β is sufficient but not necessary for strictness of inclusion m β Sα ∆m p ⊂ S ∆p . lim Corollary 3.4. If a sequence is ∆m p −statistically convergent of order α to L, for some 0 < α ≤ 1, then it is 1 m m ∆p −statistically convergent to L, that is Sα ∆m p ⊆ S ∆p and inclusion is strict at least for 0 < α < 2 . 5569 A. Karakaş et al. / Filomat 32:16 (2018), 5565–5572 We state the following theorems without proof, since these can be established using standard techniques. Theorem 3.5. Let α ∈ (0, 1] and x = (xk ), y = (yk ) be sequences of real numbers. Then m i) If stat(α) − lim ∆m p xk = L1 and c ∈ C, then stat(α) − lim c∆p xk = cL1 , k→∞ k→∞ m m m ii) If stat(α) − lim ∆m p xk = L1 and stat(α) − lim ∆p yk = L2 , then stat(α) − lim ∆p xk + ∆p yk = L1 + L2 . k→∞ k→∞ k→∞ m m Theorem 3.6. Let x = (xk ) , y = yk and z = (zk ) be real sequences such that ∆m p xk ≤ ∆p yk ≤ ∆p zk . If m m m stat(α) − lim ∆p xk = L = stat(α) − lim ∆p zk , then stat(α) − lim ∆p yk = L. k→∞ k→∞ k→∞ m ∩ ℓ ∆ is a closed subset of ℓ ∆m Theorem 3.7. Let α ∈ (0, 1] be arbitrary real number, then Sα ∆m ∞ ∞ p p p . m m Theorem 3.8. The set Sα ∆m p ∩ ℓ∞ ∆p is nowhere dense in ℓ∞ ∆p . Proof. Since every closed linear subspace of an arbitrary linear normed E is a nowhere space Edifferent from m m dense set in E, by Theorem 3.7 we only need to show that Sα ∆m ∩ ℓ ∆ , ℓ ∆ . For this, choose ∞ ∞ p p p p = 1 and consider a sequence x = (xk ) defined by  √  k, k = n2    m ∆ xk =  (2) n = 1, 2, 3, ... ,    0, k , n2 m then x ∈ Sα ∆m p , but x < ℓ∞ ∆p by (1). Definition 3.9. Let α ∈ (0, 1] be arbitrary real number and q be a positive real number. A sequence x ∈ w is m α said to be wq ∆p −summable of order α (or wq ∆m p −summable) if there exists a real number L such that n 1 X m q lim ∆p xk − L = 0, where p, m ∈ N. n→∞ nα k=1 m In this case we write xk → L(wq ∆m p ). The set of all wq ∆p −summable sequences of order α to L will be denoted by wαq ∆m p . Theorem 3.10. Let αo ∈ (0, 1] and qo be a positive real number. The sequence space wαq00 ∆m p is a Banach space for 1 ≤ qo < ∞ normed by kxk2 = m X i=1  q1  n 0  1 X q 0  ∆m x xi + sup  α  k p n 0 n k=1 and a complete q−normed space for 0 < qo < 1 by kxk3 = m X i=1 q xi + sup n n 1 X m q0 ∆p xk nα k=1 Proof. The proof has been omitted. In the next theorem, we give the relationship between ∆m p −statistically convergent of order α and m wq ∆p −summable sequences of order α. Theorem 3.11. Let α, β be fixed real numbers such that 0 < α ≤ β ≤ 1, p, m ∈ N and let q be a positive real number, m β then wαq ∆m p ⊂ S ∆p and the inclusion is strict. 5570 Proof. The inclusion part of proof is easy. Taking p = 1 we show the strictness of the inclusion wαq ∆m p ⊂ m β S ∆p for a special case. For this, choose p = 1 and consider the sequence x = (xk ) defined by A. Karakaş et al. / Filomat 32:16 (2018), 5565–5572 m ∆ xk = ( 1, 0, if k = n2 if k , n2 n = 1, 2, .... (3) 1 , 1 we have 2 √ n 1 1 m |{k ≤ n : |∆ xk − 0| ≥ ε}| ≤ α = α− 12 nα n n 1 1 so xk → 0 (Sα (∆m )) for α ∈ , 1 by (1). On the other hand for α ∈ 0, we have 2 2 √ n−1 1 P m p 1 P m ≤ α |∆ xk | = α |∆ xk − 0|p , nα n k∈In λn k∈In For every ε > 0 and α ∈ and so xk 9 0 wαq (∆m ) by (1). m Corollary 3.12. If a sequence x = (xk ) is wq ∆m p −summable of order α to L, then it is ∆p −statistically convergent of order α to L. Even if x = (xk ) is a ∆m p −bounded sequence, the converse of Theorem 3.11 and Corollary 3.12 do not m hold, in general. To show this we must find a sequence that is ∆m p −bounded ( that is x ∈ ℓ∞ ∆p ) and m ∆m p −statistically convergent of order β, but need not to be wq ∆p −summable of order α, for some real numbers α and β such that 0 < α ≤ β ≤ 1. For this, choose p = 1 and consider a sequence x = (xk ) defined by   1    √ , k , n3 m ∆ xk =  n = 1, 2, ... . (4) k    0, k = n3 1 1 m α α m Then x ∈ ℓ∞ ∆m p and x ∈ S ∆p for α ∈ ( , 1], but x < wq ∆p for α ∈ (0, ) by (1). 3 2 Definition 3.13. Let α ∈ (0, 1] . A sequence x = (xk ) is said to be ∆m p −statistically Cauchy of order α if for every ε ≥ 0 there exists a number N = N (ε) ∈ N such that o 1 n m m =0 ≥ ε ∆ x − ∆ x k ≤ n : N k p p n→∞ nα n o m that is; the set k ≤ n : ∆m p xk − ∆p xN ≥ ε has α−density zero. lim We establish the following theorem with help of the method used by Fridy [13] and Tabib [24]. Theorem 3.14. A real sequence x = (xk ) is ∆m p −statistically convergent of order α if and only if x = (xk ) is ∆m p −statistically Cauchy of order α. Proof. Let α ∈ (0, 1] be given. Suppose that the sequence x = (xk ) is ∆m p −statistically convergent of order α to L. Then for every ε > 0 the set ε A(ε) = k ≤ n, ∆m p xk − L ≥ 2 A. Karakaş et al. / Filomat 32:16 (2018), 5565–5572 5571 has α−density zero. Choose positive integer number N such that ∆m p xN − L ≥ ε. Now let us take the sets ε m , Bε = k ≤ n, |∆m p xk − ∆p xN | ≥ 2 ε , Cε = k ≤ n, ∆m p xk − L ≥ 2 ε Dε = N ≤ n, |∆m . p xN − L| ≥ 2 Then Bε ⊆ Cε ∪ Dε and therefore δα (Bε ) ≤ δα (Cε ) + δα (Dε ) = 0. Hence x = (xk ) is ∆m p −statistically Cauchy of order α. Conversely let x = (xk ) be a ∆m p −statistically Cauchy sequence of order α, then for every ε > 0, there exists N0 ∈ N such that o n δα k ≤ n : ∆m p xk − L < ε = 1. Hence, we obtain o n m δα k ≤ n : ∆m p xk < ∆p xN0 + ε = 1 and o n m δα k ≤ n : ∆m = 1. p xN0 − ε < ∆p xk We define the following sets: n o o n A = a ∈ R : δα k ≤ n : ∆m p xk < a = 1 , and n o o n B = b ∈ R : δα k ≤ n : ∆m p xk > b = 1 , m then ∆m p xN0 + ε ∈ A and ∆p xN0 − ε ∈ B. Let a ∈ A and b ∈ B, then we have o o n n m δα k ≤ n : ∆m p xk < a = 1 and δα k ≤ n : ∆p xk > b = 1. Therefore, we get o n δα k ≤ n : b < ∆m p xk < a = 1. This implies b < a . We have m ∆m p xN0 − ε ≤ sup B ≤ inf A ≤ ∆p xN0 + ε. Since ε was arbitrary positive number, we get sup B = inf A and sup B = inf A = L. Let ε > 0 be given and there exists a ∈ A and b ∈ B such that L − ε < b < a < L + ε. The definitions of A and B imply o n δα k ≤ n : L − ε < ∆m p xk < L + ε = 1, we obtain o o n n m δα k ≤ n : ∆m p xk − L < ε = 1 or δα k ≤ n : ∆p xk − L ≥ ε = 0. Therefore, x = (xk ) is ∆m p −statistically convergent of order α. Theorem 3.15. If x = (xk ) is a sequence for which there exists a ∆m p −statistically convergent of order α sequence y m m (α) such that ∆m x = ∆ y for almost all k . Then, x is ∆ −statistically convergent sequence of order α. p k p k p Proof. The proof has been omitted. A. Karakaş et al. / Filomat 32:16 (2018), 5565–5572 5572 References [1] Y. Altın, Properties of some sets of sequences defined by a modulus function, Acta Math. Sci. Ser. B Engl. Ed. 29 (2009) 427–434. [2] Y. Altın,M. Et, R. 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