Báo cáo toán học: " Strongly Almost Summable Diﬀerence Sequences"

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Nhập văn bản hoặc địa chỉ trang web hoặc dịch tài liệu. Hủy Bản dịch từ Tiếng Anh sang Tiếng Việt ý tưởng của không gian chuỗi sự khác biệt đã được giới thiệu bởi Kızmaz [12] và tổng quát của Et và Colak [6]. Trong bài báo này chúng tôi giới thiệu, kiểm tra ¸ một số thuộc tính của ba không gian trình tự quy định bằng cách sử dụng một chức năng mô đun và cung cấp cho các thuộc tính khác nhau và quan hệ bao gồm các không gian này....

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Vietnam Journal of Mathematics 34:3 (2006) 331–339 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 9$ 67 Strongly Almost Summable Diﬀerence Sequences Hifsi Altinok, Mikail Et, and Yavuz Altin Department of Mathematics, Firat University, 23119, Elazı˘-Turkey g Received November 28, 2005 Revised Ferbuary 14, 2006 Abstract. The idea of diﬀerence sequence space was introduced by Kızmaz [12] and was generalized by Et and Colak [6]. In this paper we introduce and examine ¸ some properties of three sequence spaces deﬁned by using a modulus function and give various properties and inclusion relations on these spaces. 2000 Mathematics Subject Classiﬁcation: 40A05, 40C05, 46A45. Keywords: Diﬀerence sequence, statistical convergence, modulus function. 1. Introduction Let w be the set of all sequences of real numbers and ∞ , c and c0 be respectively the Banach spaces of bounded, convergent and null sequences x = (xk ) with the usual norm x = sup |xk|, where k ∈ N = {1, 2, . . . }, the set of positive integers. A sequence x ∈ ∞ is said to be almost convergent [14] if all Banach limits of x coincide. Lorentz [14] deﬁned that n 1 c = x : lim ˆ xk + m exists, uniformly in m . nn k =1 Several authors including Lorentz [14], Duran [2] and King [11] have studied almost convergent sequences. Maddox ( [16, 17]) has deﬁned x to be strongly almost convergent to a number L if n 1 lim |xk+m − L| = 0, uniformly in m. n n k =1
332 Hifsi Altinok, Mikail Et, and Yavuz Altin By [ˆ] we denote the space of all strongly almost convergent sequences. It is easy c to see that c ⊂ [ˆ] ⊂ ˆ ⊂ ∞. c c The space of strongly almost convergent sequences was generalized by Nanda ([20, 21]). Let p = (pk ) be a sequence of strictly positive real numbers. Nanda [20] deﬁned n 1 |xk+m − L|pk = 0, [ˆ, p] = x = (xk ) : lim c uniformly in m , n n k =1 n 1 |xk+m|pk = 0, [ˆ, p]0 = x = (xk ) : lim c uniformly in m , n n k =1 n 1 | x k + m | pk < ∞ . [ˆ, p]∞ = x = (xk ) : sup c n n,m k =1 Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 1. The generalized de la Vall´e-Pousin mean is deﬁned by e 1 t n ( x) = xk , λn k ∈I n where In = [n − λn + 1, n] for n = 1, 2, .... A sequence x = (xk ) is said to be (V, λ)−summable to a number L [13] if tn (x) → L as n → ∞. If λn = n, then (V, λ)−summability and strongly (V, λ)−summability are reduced to (C, 1) −summability and [C, 1] −summability, respectively. The idea of diﬀerence sequence spaces was introduced by Kızmaz [12]. In 1981, Kızmaz[12] deﬁned the sequence spaces X (∆) = {x = (xk ) : ∆x ∈ X } for X = ∞ , c and c0 , where ∆x = (xk − xk+1) . Then Et and Colak [6] generalized the above sequence spaces to the sequence ¸ spaces X (∆r ) = x = (xk ) : ∆r x ∈ X c and c0 , where r ∈ N, ∆0x = (xk ) , ∆x = (xk − xk+1) , for X = ∞, r r (−1)v ∆r x = ∆r−1 xk − ∆r−1xk+1 , and so ∆r xk = xk + v . v v =0 Recently Et and Ba¸arır [5] extended the above sequence spaces to the sequence s spaces X (∆r ) for X = ∞ (p), c(p), c0 (p), [ˆ, p] , [ˆ, p]0 and [ˆ, p]∞ . c c c We recall that a modulus f is a function from [0,∞) to [0,∞) such that i) f (x) = 0 if and only if x = 0, ii) f (x + y) ≤ f (x) + f (y) for x, y ≥ 0, iii) f is increasing, iv) f is continuous from the right at 0.
Strongly Almost Summable Diﬀerence Sequences 333 It follows that f must be continuous everywhere on [0, ∞). A modulus may be unbounded or bounded. Ruckle [23] and Maddox [15] used a modulus f to construct some sequence spaces. Subsequently modulus function has been discussed in ([3, 4, 19, 22, 26]). Let X, Y ⊂ w. Then we shall write x−1 ∗ Y = a ∈ w : ax ∈ Y M (X, Y ) = for all x ∈ X [27]. x∈X The set X α = M (X, 1 ) is called the K¨the-Toeplitz dual space or α−dual of X. o Let X be a sequence space. Then X is called i) Solid (or normal) if (αk xk ) ∈ X whenever, (xk ) ∈ X for all sequences (αk ) of scalars with |αk | ≤ 1 for all k ∈ N. ii) Symmetric if (xk ) ∈ X implies (xπ(k)) ∈ X, where π(k) is a permutation of N. iii) Perfect if X = X αα . iv) A sequence algebra if x.y ∈ X, whenever x, y ∈ X. It is well known that if X is perfect then X is normal [10]. The following inequality will be used throughout this paper. pk pk p + |bk | k } , |ak + bk | ≤ C {|ak | (1) where ak , bk ∈ C, 0 < pk ≤ supk pk = H, C = max 1, 2H −1 [18]. 2. Main Results ˆ In this section we prove some results involving the sequence spaces V , ∆r , λ, f, p , 0 ˆ ˆ r r V , ∆ , λ, f, p and V , ∆ , λ, f, p . 1 ∞ Deﬁnition 1. Let f be a modulus function and p = (pk ) be any sequence of strictly positive real numbers. We deﬁne the following sequence sets 1 pk ˆ V , ∆r , λ, f, p [f (|∆r xk+m − L|)] = x = (xk ) : lim = 0, λn n 1 k ∈I n uniformly in m, for some L > 0 , 1 pk ˆ V , ∆r , λ, f, p [f (|∆r xk+m |)] = x = (xk ) : lim = 0, uniformly in m , 0 λn n k ∈I n 1 pk ˆ V , ∆r , λ, f, p [f (|∆r xk+m |)] = x = (xk ) : sup
334 Hifsi Altinok, Mikail Et, and Yavuz Altin ˆ In the case f (x) = x and pk = 1 for all k ∈ N, we shall write V , ∆r , λ Z ˆ ˆ ˆ and V , ∆r , λ, f instead of V , ∆r , λ, f, p . If x ∈ V , ∆r , λ then we say Z Z 1 r that x is ∆λ −strongly almost convergent to L. The proofs of the following theorems are obtained by using the known stan- dard techniques, therefore we give them without proofs (For detail see [3, 22]). ˆ Theorem 2.1. Let (pk ) be bounded. Then the spaces V , ∆r , λ, f, p are linear Z spaces over the set of complex numbers C. Theorem 2.2. Let the sequence p = (pk ) be bounded and f be a modulus function , then ˆ ˆ ˆ V , ∆r , λ, f, p ⊂ V , ∆r , λ, f, p ⊂ V , ∆r , λ, f, p . 0 1 ∞ ˆ ˆ Theorem 2.3. If r ≥ 1, then the inclusion V , ∆r−1, λ, f ⊂ V , ∆r , λ, f is Z Z ˆ ˆ strict. In general V , ∆i, λ, f ⊂ V , ∆r , λ, f for all i = 1, 2, . . . , r − 1 and Z Z the inclusion is strict. Proof. We give the proof for Z = ∞ only. It can be proved in a similar way for ˆ Z = 0, 1. Let x ∈ V , ∆r−1, λ, f . Then we have ∞ 1 f ∆r−1xk+m < ∞. sup λn m,n k ∈I n By deﬁnition of f, we have 1 1 1 f (|∆r xk+m |) ≤ ∆r−1xk+m + ∆r−1xk+m+1 f f < ∞. λn λn λn k ∈I n k ∈I n k ∈I n ˆ ˆ Thus V , ∆r−1, λ, f ⊂ V , ∆r , λ, f . Proceeding in this way one will have ∞ ∞ ˆ ˆ V , ∆i, λ, f ⊂ V , ∆r , λ, f for i = 1, 2, . . . , r − 1. Let λn = n for all n ∈ N, ∞ ∞ ˆ then the sequence x = (kr ) , for example, belongs to V , ∆r , λ, f , but does ∞ ˆ not belong to V , ∆r−1, λ, f for f (x) = x. (If x = (kr ), then ∆r xk = (−1)r r! ∞ + (r−1) ) and ∆r−1xk = (−1)r+1 r!(k for all k ∈ N). 2 The proof of the following result is a routine work. ˆ ˆ Proposition 2.4. V , ∆r−1, λ, f ⊂ V , ∆r , λ, f . 1 0 Theorem 2.5. Let f1 , f2 be modulus functions. Then we have ˆ ˆ i) V , ∆r , λ, f1 ⊂ V , ∆r , λ, f1 ◦ f2 , Z Z
Strongly Almost Summable Diﬀerence Sequences 335 ˆ ˆ ˆ ii) V , ∆r , λ, f1, p ∩ V , ∆r , λ, f2, p ⊂ V , ∆r , λ, f1 + f2 , p . Z Z Z Proof. Omitted. The following result is a consequence of Theorem 2.5 (i). ˆ ˆ Proposition 2.6. Let f be a modulus function. Then[V , ∆r , λ]Z ⊂ [V , ∆r , λ, f ]Z . ˆ ˆ ˆ Theorem 2.7. The sequence spaces [V , ∆r , λ, f, p]0, [V , ∆r , λ, f, p]1 and V , r ∆ , λ, f, p ∞ are not solid for r ≥ 1. Proof. Let pk = 1 for all k, f (x) = x and λn = n for all n ∈ N. Then when αk = (−1)k ˆ /ˆ (xk ) = (kr ) ∈ V , ∆r , λ, f, p but (αk xk ) ∈ V , ∆r , λ, f, p ∞ ∞ ˆ for all k ∈ N. Hence V , ∆r , λ, f, p is not solid. The other cases can be proved ∞ by considering similar examples. From the above theorem we may give the following corollary. ˆ ˆ V , ∆r , λ, f, p V , ∆r , λ, f, p Corollary 2.8. The sequence spaces , and 0 1 ˆ V , ∆r , λ, f, p are not perfect for r ≥ 1. ∞ ˆ ˆ Theorem 2.9. The sequence spaces V , ∆r , λ, f, p and V , ∆r , λ, f, p are 1 ∞ not symmetric for r ≥ 1. Proof. Let pk = 1 for all k, f (x) = x and λn = n for all n ∈ N. Then ˆ (xk ) = (kr ) ∈ [V , ∆r , λ, f, p]∞. Let (yk ) be a rearrangement of (xk ), which is deﬁned as follows: (yk ) = {x1 , x2, x4, x3, x9, x5, x16, x6, x25, x7, x36, x8, x49, x10,...} . Then (yk ) ∈ / ˆ [V , ∆r , λ, f, p]∞. ˆ Remark. The space [V , ∆r , λ, f, p]0 is not symmetric for r ≥ 2. ˆ Theorem 2.10. The sequence spaces [V , ∆r , λ, f, p]Z are not sequence algebras. Proof. Let pk = 1 for all k ∈ N, f (x) = x and λn = n for all n ∈ N. Then ˆ ˆ x = (kr−2 ), y = (kr−2 ) ∈ [V , ∆r , λ, f, p]Z , but x.y ∈ [V , ∆r , λ, f, p]Z . 3. Statistical Convergence The notion of statistical convergence was introduced by Fast [7] and studied by various authors ([1, 9, 24, 25]). In this section we deﬁne ∆r −almost statistically convergent sequences and λ ˆ give some inclusion relations between s(∆r ) and V , ∆r , λ, f, p . ˆ λ 1
336 Hifsi Altinok, Mikail Et, and Yavuz Altin Deﬁnition 2. A sequence x = (xk ) is said to be ∆r −almost statistically con- λ vergent to the number L if for every ε > 0, 1 |{k ∈ In : |∆r xk+m − L| ≥ ε}| = 0, uniformly in m. lim n λn In this case we write s(∆r ) − lim x = L or xk → Ls(∆r ). ˆλ ˆλ In the case λn = n we shall write s(∆r ) instead of s(∆r ). ˆ ˆλ The proof of the following theorem is easily obtained by using the same techniques of Theorem 2 in Savas [25], therefore we give it without proof. Theorem 3.1. Let λ = (λn ) be the same as in Sec. 1, then ˆ i) If xk → L V , ∆r , λ ⇒ xk → Ls(∆r ), ˆλ 1 ˆ r ) and xk → Ls(∆r ), then xk → L V , ∆r , λ , ii) If x ∈ ∞ (∆ ˆλ 1 ˆ s(∆r ) r r r iii) ˆλ ∩ ∞ (∆ ) = V ,∆ ,λ ∩ ∞ (∆ ). 1 λn Theorem 3.2. s(∆r ) ⊆ s(∆r ) if and only if lim inf n ˆ ˆλ > 0. n Proof. The suﬃciency part of the proof can be obtained using the same technique as the suﬃciency part of the proof of Theorem 3 in Savas [25]. For the necessity suppose that lim inf n λn = 0. As in ([8], p.510) we can n λ choose a subsequence (n (j )) such that nn(jj)) < 1 . Deﬁne x = (xi) such that ( j 1, if i ∈ In (j ) , j = 1, 2, ... ∆r xi = 0, otherwise. /ˆ Then x ∈ [ˆ] (∆r ) and by [4, Theorem 3.1 (i)], x ∈ s(∆r ). But x ∈ V , ∆r , λ c ˆ 1 and Theorem 3.1 (ii) implies that x ∈ s(∆r ). This completes the proof. /ˆ λ ˆ Theorem 3.3. Let f be a modulus function and supk pk = H . Then [V , ∆r , λ, r f, p]1 ⊂ s(∆λ). ˆ ˆ Proof. Let x ∈ [V , ∆r , λ, f, p]1 and ε > 0 be given. Let Σ1 denote the sum over k ≤ n such that |∆r xk+m − L| ≥ ε and Σ2 denote the sum over k ≤ n such that |∆r xk+m − L| < ε. Then
Strongly Almost Summable Diﬀerence Sequences 337 1 pk [f (|∆r xk+m − L|)] λn k ∈I n 1 1 pk pk [f (|∆r xk+m − L|)] [f (|∆r xk+m − L|)] = + λn λn 1 2 1 pk r ≥ [f (|∆ xk+m − L|)] λn 1 1 pk ≥ [f (ε)] λn 1 1 inf pk H ≥ min [f (ε)] , [f (ε)] λn 1 1 inf p H |{k ∈ In : |∆r xk+m − L| ≥ ε}| min [f (ε)] k , [f (ε)] . ≥ λn Hence x ∈ s(∆r ). ˆλ Theorem 3.4. Let f be bounded and 0 < h = inf k pk ≤ pk ≤ supk pk = H < ∞. ˆ Then s(∆r ) ⊂ V , ∆r , λ, f, p . ˆλ 1 Proof. Suppose that f is bounded. Let ε > 0 and Σ1 and Σ2 be denoted in the previous theorem. Since f is bounded there exists an integer K such that f (x) < K, for all x ≥ 0. Then 1 pk [f (|∆r xk+m − L|)] λn k ∈I n 1 1 pk pk [f (|∆r xk+m − L|)] [f (|∆r xk+m − L|)] = + λn λn 1 2 1 1 pk h H ≤ max K , K + [f (ε)] λn λn 1 2 1 ≤ max K h , K H |{k ∈ In : |∆r xk+m − L| ≥ ε}| λn + max f (ε)h , f (ε)H . ˆ Hence x ∈ V , ∆r , λ, f, p . 1 Theorem 3.5. Let f be bounded and 0 < h = inf k pk ≤ pk ≤ supk pk = H < ∞. ˆ We have s(∆r ) = [V , ∆r , λ, f, p]1 if and only if f is bounded . ˆλ Proof. Let f be bounded. By the Theorem 3.3 and Theorem 3.4 we have ˆ s(∆r ) = [V , ∆r , λ, f, p]1. ˆλ Conversely, suppose that f is unbounded. Then there exists a positive se- quence (tk ) with f (tk ) = k2 , for k = 1, 2, .... If we choose
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