Vietnam Journal of Mathematics 34:3 (2006) 331–339 Strongly Almost Summable Difference Sequences Hifsi Altinok, Mikail Et, and Yavuz Altin Department of Mathematics, Firat University, 23119, Elazı˘g-Turkey 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 intro duce 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 ∞ ,cand c 0 be respectively the Banach spaces of bounded, convergent and null sequences x =(x k ) with the usual norm x = sup |x k |, 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 ˆc = x : lim n 1 n n k=1 x k+m exists, uniformly in m . 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 numb er L if lim n 1 n n k=1 |x k+m − L| =0, uniformly in m. 332 Hifsi Altinok, Mikail Et, and Yavuz Altin By [ˆc] we denote the space of all strongly almost convergent sequences. It is easy to see that c ⊂ [ˆc] ⊂ ˆc ⊂ ∞. The space of strongly almost convergent sequences was generalized by Nanda ([20, 21]). Let p =(p k ) be a sequence of strictly positive real numbers. Nanda [20] defined [ˆc,p]= x =(x k ) : lim n 1 n n k=1 |x k+m − L| p k =0, uniformly in m , [ˆc, p] 0 = x =(x k ) : lim n 1 n n k=1 |x k+m | p k =0, uniformly in m , [ˆc,p] ∞ = x =(x k ) : sup n,m 1 n n k=1 |x k+m | p k < ∞ . Let λ =(λ n ) be a non-decreasing sequence of positive numbers tending to ∞ such that λ n+1 ≤ λ n +1,λ 1 =1. The generalized de la Vall´ee-Pousin mean is defined by t n (x)= 1 λ n k∈I n x k , where I n =[n − λ n +1,n] for n =1, 2, A sequence x =(x k ) is said to be (V, λ)−summable to a number L [13] if t n (x) → L as n →∞. If λ n = 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 =(x k ):∆x ∈ X} for X = ∞ ,cand c 0 , where ∆x =(x k − x k+1 ) . Then Et and ¸Colak [6] generalized the above sequence spaces to the sequence spaces X(∆ r )= x =(x k ):∆ r x ∈ X for X = ∞ ,cand c 0 , where r ∈ N, ∆ 0 x =(x k ) , ∆x =(x k − x k+1 ) , ∆ r x = ∆ r−1 x k − ∆ r−1 x k+1 , and so ∆ r x k = r v=0 (−1) v r v x k+v . Recently Et and Ba¸sarır [5] extended the above sequence spaces to the sequence spaces X (∆ r ) for X = ∞ (p),c(p), c 0 (p), [ˆc, p] , [ˆc, p] 0 and [ˆc, p] ∞ . 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 M(X,Y )= x∈X x −1 ∗ Y = a ∈ w : ax ∈ Y for all x ∈ X [27]. The set X α = M(X, 1 ) is called the K¨othe-Toeplitz dual space or α−dual of X. Let X be a sequence space. Then X is called i) Solid (or normal)if(α k x k ) ∈ X whenever, (x k ) ∈ X for all sequences (α k ) of scalars with |α k |≤1 for all k ∈ N. ii) Symmetric if (x k ) ∈ X implies (x π (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. |a k + b k | p k ≤ C {|a k | p k + |b k | p k } , (1) where a k ,b k ∈ C, 0 <p k ≤ sup k p k = H, C = max 1, 2 H−1 [18]. 2. Main Results In this section we prove some results involving the sequence spaces ˆ V,∆ r ,λ,f,p 0 , ˆ V,∆ r ,λ,f,p 1 and ˆ V,∆ r ,λ,f,p ∞ . Definition 1. Let f be a modulus function and p =(p k ) be any sequence of strictly positive real numbers. We define the following sequence sets ˆ V,∆ r ,λ,f,p 1 = x =(x k ) : lim n 1 λ n k∈I n [f (|∆ r x k+m − L|)] p k =0, uniformly in m, for some L>0 , ˆ V,∆ r ,λ,f,p 0 = x =(x k ) : lim n 1 λ n k∈I n [f (|∆ r x k+m |)] p k =0, uniformly in m , ˆ V,∆ r ,λ,f,p ∞ = x =(x k ) : sup n,m 1 λ n k∈I n [f (|∆ r x k+m |)] p k < ∞ . If x ∈ ˆ V,∆ r ,λ,f,p 1 then we shall write x k → L ˆ V,∆ r ,λ,f,p 1 and L will be called λ−strongly almost difference limit of x with respect to the modulus f. Throughout the paper Z will denote any one of the notation 0, 1, or ∞. 334 Hifsi Altinok, Mikail Et, and Yavuz Altin In the case f (x)=x and p k = 1 for all k ∈ N, we shall write ˆ V,∆ r ,λ Z and ˆ V,∆ r ,λ,f Z instead of ˆ V,∆ r ,λ,f,p Z . If x ∈ ˆ V,∆ r ,λ 1 then we say that x is ∆ r 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]). Theorem 2.1. Let (p k ) be bounded. Then the spaces ˆ V,∆ r ,λ,f,p Z are linear spaces over the set of complex numbers C. Theorem 2.2. Let the sequence p =(p k ) be bounded and f be a modulus function , then ˆ V,∆ r ,λ,f,p 0 ⊂ ˆ V,∆ r ,λ,f,p 1 ⊂ ˆ V,∆ r ,λ,f,p ∞ . Theorem 2.3. If r ≥ 1, then the inclusion ˆ V,∆ r−1 ,λ,f Z ⊂ ˆ V,∆ r ,λ,f Z is strict. In general ˆ V,∆ i ,λ,f Z ⊂ ˆ V,∆ r ,λ,f Z for all i =1, 2, ,r− 1 and the inclusion is strict. Proof. We give the proof for Z = ∞ only. It can be proved in a similar way for Z =0, 1. Let x ∈ ˆ V,∆ r−1 ,λ,f ∞ . Then we have sup m,n 1 λ n k∈I n f ∆ r−1 x k+m < ∞. By definition of f, we have 1 λ n k∈I n f (|∆ r x k+m |) ≤ 1 λ n k∈I n f ∆ r−1 x k+m + 1 λ n k∈I n f ∆ r−1 x k+m+1 < ∞. Thus ˆ V,∆ r−1 ,λ,f ∞ ⊂ ˆ V,∆ r ,λ,f ∞ . Proceeding in this way one will have ˆ V,∆ i ,λ,f ∞ ⊂ ˆ V,∆ r ,λ,f ∞ for i =1, 2, ,r− 1. Let λ n = n for all n ∈ N, then the sequence x =(k r ) , for example, belongs to ˆ V,∆ r ,λ,f ∞ , but does not belong to ˆ V,∆ r−1 ,λ,f ∞ for f(x)=x. (If x =(k r ), then ∆ r x k =(−1) r r! and ∆ r−1 x k =(−1) r+1 r!(k + (r−1) 2 ) for all k ∈ N). The proof of the following result is a routine work. Proposition 2.4. ˆ V,∆ r−1 ,λ,f 1 ⊂ ˆ V,∆ r ,λ,f 0 . Theorem 2.5. Let f 1 ,f 2 be modulus functions. Then we have i) ˆ V,∆ r ,λ,f 1 Z ⊂ ˆ V,∆ r ,λ,f 1 ◦ f 2 Z , Strongly Almost Summable Difference Sequences 335 ii) ˆ V,∆ r ,λ,f 1 ,p Z ∩ ˆ V,∆ r ,λ,f 2 ,p Z ⊂ ˆ V,∆ r ,λ,f 1 + f 2 ,p Z . Proof. Omitted. The following result is a consequence of Theorem 2.5 (i). Proposition 2.6. Let f be a modulus function. Then[ ˆ V,∆ r ,λ] Z ⊂[ ˆ V,∆ r ,λ,f] Z . Theorem 2.7. The sequence spaces [ ˆ V,∆ r ,λ,f,p] 0 , [ ˆ V,∆ r ,λ,f,p] 1 and ˆ V, ∆ r , λ,f,p ∞ are not solid for r ≥ 1. Proof. Let p k = 1 for all k, f(x)=x and λ n = n for all n ∈ N. Then (x k )=(k r ) ∈ ˆ V,∆ r ,λ,f,p ∞ but (α k x k ) /∈ ˆ V,∆ r ,λ,f,p ∞ when α k =(−1) k for all k ∈ N. Hence ˆ 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. Corollary 2.8. The sequence spaces ˆ V,∆ r ,λ,f,p 0 , ˆ V,∆ r ,λ,f,p 1 and ˆ V,∆ r ,λ,f,p ∞ are not perfect for r ≥ 1. Theorem 2.9. The sequence spaces ˆ V,∆ r ,λ,f,p 1 and ˆ V,∆ r ,λ,f,p ∞ are not symmetric for r ≥ 1. Proof. Let p k = 1 for all k, f(x)=x and λ n = n for all n ∈ N. Then (x k )=(k r ) ∈ [ ˆ V,∆ r ,λ,f,p] ∞ . Let (y k ) be a rearrangement of (x k ), which is defined as follows: (y k )={x 1 ,x 2 ,x 4 ,x 3 ,x 9 ,x 5 ,x 16 ,x 6 ,x 25 ,x 7 ,x 36 ,x 8 ,x 49 ,x 10, }. Then (y k ) /∈ [ ˆ V,∆ r ,λ,f,p] ∞ . Remark. The space [ ˆ V,∆ r ,λ,f,p] 0 is not symmetric for r ≥ 2. Theorem 2.10. The sequence spaces [ ˆ V,∆ r ,λ,f,p] Z are not sequence algebras. Proof. Let p k = 1 for all k ∈ N, f(x)=x and λ n = n for all n ∈ N. Then x =(k r−2 ),y=(k r−2 ) ∈ [ ˆ V,∆ r ,λ,f,p] Z , but x.y ∈ [ ˆ 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 λ −almost statistically convergent sequences and give some inclusion relations between ˆs(∆ r λ ) and ˆ V,∆ r ,λ,f,p 1 . 336 Hifsi Altinok, Mikail Et, and Yavuz Altin Definition 2. A sequence x =(x k ) is said to be ∆ r λ −almost statistically con- vergent to the number L if for every ε>0, lim n 1 λ n |{k ∈ I n : |∆ r x k+m − L|≥ε}| =0, uniformly in m. In this case we write ˆs(∆ r λ ) − lim x = L or x k → Lˆs(∆ r λ ). In the case λ n = n we shall write ˆs(∆ r ) instead of ˆs(∆ r λ ). 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 =(λ n ) be the same as in Sec. 1, then i) If x k → L ˆ V,∆ r ,λ 1 ⇒ x k → Lˆs(∆ r λ ), ii) If x ∈ ∞ (∆ r ) and x k → Lˆs(∆ r λ ), then x k → L ˆ V,∆ r ,λ 1 , iii) ˆs(∆ r λ ) ∩ ∞ (∆ r )= ˆ V,∆ r ,λ 1 ∩ ∞ (∆ r ). Theorem 3.2. ˆs(∆ r ) ⊆ ˆs(∆ r λ ) if and only if lim inf n λ n n > 0. 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 λ n n =0. As in ([8], p.510) we can choose a subsequence (n (j)) such that λ n(j) n(j) < 1 j . Define x =(x i ) such that ∆ r x i = 1, if i ∈ I n (j) ,j=1, 2, 0, otherwise. Then x ∈ [ˆc](∆ r ) and by [4, Theorem 3.1 (i)], x ∈ ˆs(∆ r ). But x/∈ ˆ V,∆ r ,λ 1 and Theorem 3.1 (ii) implies that x/∈ ˆs(∆ r λ ). This completes the proof. Theorem 3.3. Let f be a modulus function and sup k p k = H. Then [ ˆ V,∆ r ,λ, f,p] 1 ⊂ ˆs(∆ r λ ). Proof. Let x ∈ [ ˆ V,∆ r ,λ,f,p] 1 and ε>0 be given. Let Σ 1 denote the sum over k ≤ n such that |∆ r x k+m − L|≥ε and Σ 2 denote the sum over k ≤ n such that |∆ r x k+m − L| <ε.Then Strongly Almost Summable Difference Sequences 337 1 λ n k∈I n [f (|∆ r x k+m − L|)] p k = 1 λ n 1 [f (|∆ r x k+m − L|)] p k + 1 λ n 2 [f (|∆ r x k+m − L|)] p k ≥ 1 λ n 1 [f (|∆ r x k+m − L|)] p k ≥ 1 λ n 1 [f (ε)] p k ≥ 1 λ n 1 min [f (ε)] inf p k , [f (ε)] H ≥ 1 λ n |{k ∈ I n : |∆ r x k+m − L|≥ε}| min [f (ε)] inf p k , [f (ε)] H . Hence x ∈ ˆs(∆ r λ ). Theorem 3.4. Let f be bounded and 0 <h= inf k p k ≤ p k ≤ sup k p k = H<∞. Then ˆs(∆ r λ ) ⊂ ˆ V,∆ r ,λ,f,p 1 . Proof. Suppose that f is bounded. Let ε>0 and Σ 1 and Σ 2 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 λ n k∈I n [f (|∆ r x k+m − L|)] p k = 1 λ n 1 [f (|∆ r x k+m − L|)] p k + 1 λ n 2 [f (|∆ r x k+m − L|)] p k ≤ 1 λ n 1 max K h ,K H + 1 λ n 2 [f (ε)] p k ≤ max K h ,K H 1 λ n |{k ∈ I n : |∆ r x k+m − L|≥ε}| + max f(ε) h ,f(ε) H . Hence x ∈ ˆ V,∆ r ,λ,f,p 1 . Theorem 3.5. Let f be bounded and 0 <h= inf k p k ≤ p k ≤ sup k p k = H<∞. We have ˆs(∆ r λ )=[ ˆ V,∆ r ,λ,f,p] 1 if and only if f is bounded . Proof. Let f be bounded. By the Theorem 3.3 and Theorem 3.4 we have ˆs(∆ r λ )=[ ˆ V,∆ r ,λ,f,p] 1 . Conversely, suppose that f is unbounded. Then there exists a positive se- quence (t k ) with f( t k )=k 2 , for k =1, 2, If we choose 338 Hifsi Altinok, Mikail Et, and Yavuz Altin ∆ r x i = t k ,i= k 2 ,i=1, 2, 0, otherwise . Then we have 1 λ n |{k ∈ I n : |∆ r x k+m |≥ε}| ≤ λ n−1 λ n for all n and m and so x ∈ ˆs(∆ r λ ), but x/∈ ˆ V,∆ r ,λ,f,p 1 . This contradicts to ˆs(∆ r λ )= ˆ V,∆ r ,λ,f,p . References 1. J. S. 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