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λ-Almost Statistical Convergence of Order α

λ-Almost Statistical Convergence of Order α In this paper, we introduce the concept . Also some relations between ^ and strong (V , )-almost summability of order are given. Furthermore some relations ^ ^ between the space [Vp , , f ] and S are examined. Mathematics Subject Classification 2010: 40A05, 40C05, 46A45. Key words: statistical convergence, almost convergence, Ces`ro summability. a 1. Introduction The idea of statistical convergence was given by Zygmund [27] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [26] and Fast [7] and later reintroduced by Schoenberg [25] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy [8], Connor [5], Savas [24], Mur¸ saleen [17], Mursaleen and Alotaibi [18], Miller and Orhan [16], Rath and Tripathy [21], Salat [23] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability (for details, see [20]). 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 linear functional L on is said to be a Banach limit (see [1]) if it has the properties: i) L (x) 0 if x 0 (i.e. xn 0 for all n); ii) L (e) = 1, where e = (1, 1, . . .); where D is the shift operator defined by (Dxn ) = (xn+1 ) . Let B be the set of all Banach limits on . A bounded sequence x is said to be almost convergent to a number L if L (x) = L for all L B. The set of all almost convergent sequences will be denoted by c. Lorentz [12] ^ 1 proved that x = (xk ) c if and only if limn n n xk+m exists, uniformly ^ k=1 in m. Several authors including Lorentz [12], Duran [6], g [10], Colak ¸ and Cakar [4] have studied almost convergent sequences. Maddox [13] ¸ has defined x to be strongly almost convergent to a number L if lim iii) L (Dx) = L (x) , 1 n n k=1 |xk+m - L| = 0, uniformly in m. It can be shown that the sequence x = (1, 0, 1, 0, 1, 0, ...) is strongly 1 almost convergent to 2 . By [^] we denote the space of all strongly almost c convergent sequences. It is easy to see that c [^] c and the c ^ inclusions are strict, for example the sequence x = (xk ) = ((-1)k ) almost convergent but not strongly almost convergent. A sequence x = (xk ) is said to be statistically convergent to the number 1 L if for every > 0, limn n |{k n : |xk - L| }| = 0, where the vertical bars indicate the number of elements in the enclosed set. In this case, we write S - lim x = L or xk L(S) and S denotes the set of all statistically convergent sequences. It is note that statistically convergent a sequence need not be bounded. For this, consider the sequence x = (xk ) defined by k, k = n2 , n = 1, 2, ... xk = . 3, k = n2 Then S - lim x = 3, but x . / The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in [9] and after then statistical convergence of order and strong p-Ces`ro summability of order studied by Colak [3]. a ¸ The statistical convergence of order is defined as follows. Let 0 < 1 be given. The sequence (xk ) is said to be statistically convergent of order if there is a complex number L such that limn n1 |{k n : |xk - L| }| = 0, for every > 0, in which case we say that x is statistically convergent of order , to L. In this case we write S - lim xk = L. The set of all statistically convergent sequences of order will be denoted by S . It is easy to see that every convergent sequence is statistically convergent of order , that is c S for each 0 < 1, but converse does not hold. For example the sequence x = (xk ) defined by 1 , k = n2 , n = 1, 2, ... xk = k 1, k = n2 1 is statistically convergent of order with S - lim xk = 1 for > 3 , but is not convergent. The generalized de la Vall´e-Poussin mean is defined by e 1 tn (x) = xk , n is (V, )-summable to 0 for = (n) . We write [C, 1] = x = (xk ) : lim 1 n n 1 where = (n ) is a non-decreasing sequence of positive numbers such that n+1 n + 1, 1 = 1, n as n and In = [n - n + 1, n] . A sequence x = (xk ) is said to be (V, ) -summable to a number L (see [11]) if tn (x) L as n . For example the sequence x = (xk ) defined by 1, if k is odd xk = -1, if k is even n k=1 |xk - L| = 0 for some L , |xk - L| = 0 for some L , [V, ] = x = (xk ) : lim n n for the sets of sequences x = (xk ) which are strongly Ces`ro summable and a strongly (V, )-summable respectively. The notion of -statistical convergence was introduced by Mursaleen [17] and the concept of almost -statistical convergence was studied by Savas [24]. Recently -statistical convergence was generalized by Colak ¸ ¸ and Bektas [2]. ¸ 2. Main results In this section we give the main results of the paper. In Theorem 2.4 we give the inclusion relations between the sets of -almost statistical convergent sequences of order for different ' s, and so that the inclusion relations between the set of -almost statistical convergent sequences of order and the set of -almost statistical convergent sequences. In Theorem 2.8 we give ^ the relationship between the strong almost [Vp , ]-summability of order ^ and the strong almost [Vp , ]-summability of order . In Theorem 2.11 ^ we give the relationship between the strong almost [Vp , ]-summability of order and the - almost statistical convergence of order . Definition 2.1. Let the sequence = (n ) of real numbers be defined as above and 0 < 1 be given. The sequence x = (xk ) w is said to be -almost statistically convergent of order if there is a complex number L such thatlimn 1 |{k In : |xk+m - L| }| = 0, uniformly in m, where n In = [n - n + 1, n] and denote the th power (n ) of n , that is n ^ = ( ) = ( , , ..., , ...) . In this case we write S - lim xk = L. n n 1 2 The set of all - almost statistically convergent sequences of order will be ^ denoted by S . For example the sequence x = (xk ) defined by xk+m = k + m, k + m = n2 , 0, k + m = n2 n = 1, 2, ... is -almost statistically convergent sequences of order to 0, for 1 < 1 2 and = (n) . The is same with the ^ ^ almost statistical convergence, that is S = S for = 1. The is well defined for 0 < 1. But it is not well defined for > 1 in general. For this let x = (xk ) be defined as 5 follows: 1, 0, if k + m even . if k + m odd xk+m = Then both n n lim |{k In : |xk+m - 1| }| = lim n 2 n = 0, uniformly in m and 1 n |{k In : |xk+m - 0| }| = lim = 0, uniformly in m, n n n 2 n lim for > 1, so that x = (xk ) , -almost statistically converges of order , ^ ^ both to 1 and 0, i.e S - lim xk = 1 and S - lim xk = 0. But this is impossible. Theorem 2.2. Let 0 < 1 and x = (xk ), y = (yk ) be sequences of complex numbers. ^ ^ (i) If S - lim xk = x0 and c C, then S - lim(cxk ) = cx0 ; ^ ^ ^ (ii) If S - lim xk = x0 and S - lim yk = y0 , then S - lim(xk + yk ) = x0 + y 0 . Proof. (i) It is clear in case c = 0. Suppose that c = 0 then the proof of (i) follows from 1 1 |{k In : |cxk+m - cx0 | }| = {k In : |xk+m - x0 | } n |c| n and that of (ii) follows from 1 |{k In : |xk+m + yk+m - (x0 + y0 )| }| n 1 1 + k In : |yk+m - y0 | k In : |xk+m - x0 | n 2 n 2 ^ If we take n = n for 0 < < 1 in the definition of S then we obtain ^ S . That is why S is different from S defined in [24] in general. ^ ^ ^ S ^ ^ ^ If n = n with = 1, that is n = n then S = S = S, that is the almost statistical convergence of order , -almost statistical convergence and almost statistical convergence are coincide in case n = n with =1. Definition 2.3. Let the sequence = (n ) be as above, (0, 1] be any real number and let p be a positive real number. A sequence x is said ^ to be strongly (V , )-almost summable of order , if there is a complex number L such that 1 n n lim |xk+m - L|p = 0 uniformly in m, where In = [n - n + 1, n] . The strong (V, )- almost summability of order ^ reduces to the strong (V , )- summability for = 1. The set of all ^ ^ strongly (V , )- summable sequences of order will be denoted by [Vp , ]. ^ ^ Theorem 2.4. If 0 < 1 then S S . Proof. If 0 < 1 then we may write 1 n |{k In : |xk+m - L| }| 1 |{k In : |xk+m - L| }| , n ^ ^ for every > 0 and this gives that S S . If we take = 1 in Theorem 2.4 then we obtain the following result. Corollary 2.5. If a sequence is -almost statistically convergent of ^ order , to L, then it is -almost statistically convergent to L, that is S ^ S for each (0, 1]. From Theorem 2.4 we have the following results and the proof is easy. ^ ^ Corollary 2.6. (i) S = S = . ^ ^ (ii) S = S = 1. ^ ^ Theorem 2.7. S S if (1) lim inf n > 0. n Proof. For given > 0 we have {k n : |xk+m - L| } {k In : |xk+m - L| }. Therefore we may write 1 1 |{k n : |xk+m - L| }| |{k In : |xk+m - L| }| n n 1 n . |{k In : |xk+m - L| }| . n n ^ Tag the limit as n and using (1), we get xk L(S) = xk ^ ). L(S Theorem 2.8. Let 0 < 1 and p be a positive real number. ^ ^ Then [Vp , ] [Vp , ]. ^ Proof. Let x = (xk ) [Vp , ]. Then given and such that 0 < 1 and a positive real number p, we may write 1 1 |xk+m - L|p |xk+m - L|p n n ^ ^ and this gives that [Vp , ] [Vp , ]. The following result is a consequence of Theorem 2.8. ^ ^ Corollary 2.9. For each (0, 1] and 0<p<, we have [Vp , ][Vp , ]. ^ Theorem 2.10. Let 0 < 1 and 0 < p < q < . Then [Vq , ] ^ [Vp , ]. Proof is seen from H¨lder inequality. o Theorem 2.11. Let and be fixed real numbers such that 0 < ^ 1 and 0 < p < . If a sequence is strongly (V , )- almost summable of order , to L, then it is -almost statistically convergent of order , to ^ ^ L, i.e [Vp , ] S . Proof. For any sequence x = (xk ) and > 0, we have |xk+m - L|p = |xk+m - L|p + < |xk+m - L|p |xk+m - L|p |{k In : |xk+m - L| }| .p and so that 1 n |xk+m - L|p 1 |{k In : |xk+m - L| }| .p n 1 n |{k In : |xk+m - L| }| .p . ^ From this it follows that if x = (xk ) is strongly (V , )-almost summable of order , to L, then it is -almost statistically convergent of order , to L. If we take = in Theorem 2.11 we obtain the following result. Corollary 2.12. Let be a fixed real number such that 0 < 1 ^ and 0 < p < . If a sequence is strongly (V , )- almost summable of order , to L, then it is -almost statistically convergent of order , to L, i.e ^ ^ [Vp , ] S . Corollary 2.13. Let 0 < 1 and p be a positive real number. Then ^ ^ [Vp , ] S and the inclusion is strict if 0 < < 1. ^ ^ Proof. From Corollary 2.5 and Corollary 2.12 we have [Vp , ] S . 3. Modulus function and statistical convergence The notion of a modulus was introduced by Nakano [19]. 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. It follows that f must be continuous everywhere on [0, ). A modulus may be bounded or unbounded. For example, f (x) = xp , (0 < p 1) is x unbounded and f (x) = 1+x is bounded. Maddox [14], Ruckle [22], Malkowsky and Savas [15] used a modulus function to construct some ¸ sequence spaces. Definition 3.1. Let f be a modulus function, p = (pk ) be a sequence of strictly positive real numbers and let (0, 1] be any real number. Now we define ^ [Vp , , f ] = {x = (xk ) : lim 1 n [f (|xk+m - L|)]pk = 0 uniformly in m, for some L}. In the following theorems we shall assume that the sequence p = (pk ) is bounded and 0 < h = inf k pk pk supk pk = H < . Theorem 3.2. Let , (0, 1] be any real numbers such that ^ ^ and f be a modulus function. Then [Vp , , f ] S . ^ Proof. Let x [Vp , , f ] and let > 0 be given. Then since n n for each n we may write 1 n [f (|xk+m - L|)]pk [f (|xk+m - L|)]pk + [f (|xk+m - L|)]pk [f (|xk+m - L|)]pk 1 n 1 < [f (|xk+m - L|)]pk + 1 n < n 1 n [f ()]pk min([f ()]h , [f ()]H ) |{k In : |xk+m - L| }| min([f ()]h , [f ()]H ). ^ Since x [Vp , , f ], the left hand side of the above inequality tends to zero as n uniformly in m. Since min([f ()]h , [f ()]H ) > 0 the right hand side of above inequality tends to zero as n uniformly in m. This ^ ^ implies that [Vp , , f ] S . Theorem 3.3. Let , (0, 1] be any real numbers such that . ^ ^ If the modulus f is bounded then S [Vp , , f ]. ^ Proof. Let x S and suppose that f is bounded. Let > 0 be given and 1 and 2 denote the sums over k In , |xk+m - L| and k In , |xk+m - L| < respectively. Since f is bounded there exists an integer K such that f (x) K, for all x 0. Then since for each n we may n n write 1 n 1 n 1 n = [f (|xk+m - L|)]pk 1 n [f (|xk+m - L|)]pk [f (|xk+m - L|)]pk + 1 n [f (|xk+m - L|)]pk [f ()]pk max K h , K H + max K h , K H 1 |{k n : |xk+m - L| }| + max f ()h , f ()H . n ^ Since x S the first term of the right hand side of above inequality tends to zero as n uniformly in m, and the second term can be made as small as desired. Therefore the left hand side of above inequality tends to ^ zero as n uniformly in m. Hence x [Vp , , f ]. ^ ^ Theorem 3.4. Let (0, 1] be any real number. Then S = [Vp , , f ] if and only if f is bounded. Proof. Let f be bounded. By Theorem 3.2 and Theorem 3.3 we have ^ ^ S = [Vp , , f ]. Conversely assume that f is unbounded. Using the same technique as ^ ^ in proof of Theorem 3 in [15] we may find a sequence x S - [Vp , , f ]. This completes the proof. If we take = 1 in Theorem 3.4 we obtain Theorem 3 in [15] as the following. ^ ^ Corollary 3.5. The S = [Vp , , f ] equality holds if and only if f is bounded. Acknowledgment. The authors wish to thank the referee for their careful reading of the manuscript and valuable of the manuscript and valuable suggestions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

λ-Almost Statistical Convergence of Order α

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de Gruyter
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Copyright © 2014 by the
ISSN
1221-8421
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1221-8421
DOI
10.2478/aicu-2013-0041
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Abstract

In this paper, we introduce the concept . Also some relations between ^ and strong (V , )-almost summability of order are given. Furthermore some relations ^ ^ between the space [Vp , , f ] and S are examined. Mathematics Subject Classification 2010: 40A05, 40C05, 46A45. Key words: statistical convergence, almost convergence, Ces`ro summability. a 1. Introduction The idea of statistical convergence was given by Zygmund [27] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [26] and Fast [7] and later reintroduced by Schoenberg [25] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy [8], Connor [5], Savas [24], Mur¸ saleen [17], Mursaleen and Alotaibi [18], Miller and Orhan [16], Rath and Tripathy [21], Salat [23] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability (for details, see [20]). 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 linear functional L on is said to be a Banach limit (see [1]) if it has the properties: i) L (x) 0 if x 0 (i.e. xn 0 for all n); ii) L (e) = 1, where e = (1, 1, . . .); where D is the shift operator defined by (Dxn ) = (xn+1 ) . Let B be the set of all Banach limits on . A bounded sequence x is said to be almost convergent to a number L if L (x) = L for all L B. The set of all almost convergent sequences will be denoted by c. Lorentz [12] ^ 1 proved that x = (xk ) c if and only if limn n n xk+m exists, uniformly ^ k=1 in m. Several authors including Lorentz [12], Duran [6], g [10], Colak ¸ and Cakar [4] have studied almost convergent sequences. Maddox [13] ¸ has defined x to be strongly almost convergent to a number L if lim iii) L (Dx) = L (x) , 1 n n k=1 |xk+m - L| = 0, uniformly in m. It can be shown that the sequence x = (1, 0, 1, 0, 1, 0, ...) is strongly 1 almost convergent to 2 . By [^] we denote the space of all strongly almost c convergent sequences. It is easy to see that c [^] c and the c ^ inclusions are strict, for example the sequence x = (xk ) = ((-1)k ) almost convergent but not strongly almost convergent. A sequence x = (xk ) is said to be statistically convergent to the number 1 L if for every > 0, limn n |{k n : |xk - L| }| = 0, where the vertical bars indicate the number of elements in the enclosed set. In this case, we write S - lim x = L or xk L(S) and S denotes the set of all statistically convergent sequences. It is note that statistically convergent a sequence need not be bounded. For this, consider the sequence x = (xk ) defined by k, k = n2 , n = 1, 2, ... xk = . 3, k = n2 Then S - lim x = 3, but x . / The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in [9] and after then statistical convergence of order and strong p-Ces`ro summability of order studied by Colak [3]. a ¸ The statistical convergence of order is defined as follows. Let 0 < 1 be given. The sequence (xk ) is said to be statistically convergent of order if there is a complex number L such that limn n1 |{k n : |xk - L| }| = 0, for every > 0, in which case we say that x is statistically convergent of order , to L. In this case we write S - lim xk = L. The set of all statistically convergent sequences of order will be denoted by S . It is easy to see that every convergent sequence is statistically convergent of order , that is c S for each 0 < 1, but converse does not hold. For example the sequence x = (xk ) defined by 1 , k = n2 , n = 1, 2, ... xk = k 1, k = n2 1 is statistically convergent of order with S - lim xk = 1 for > 3 , but is not convergent. The generalized de la Vall´e-Poussin mean is defined by e 1 tn (x) = xk , n is (V, )-summable to 0 for = (n) . We write [C, 1] = x = (xk ) : lim 1 n n 1 where = (n ) is a non-decreasing sequence of positive numbers such that n+1 n + 1, 1 = 1, n as n and In = [n - n + 1, n] . A sequence x = (xk ) is said to be (V, ) -summable to a number L (see [11]) if tn (x) L as n . For example the sequence x = (xk ) defined by 1, if k is odd xk = -1, if k is even n k=1 |xk - L| = 0 for some L , |xk - L| = 0 for some L , [V, ] = x = (xk ) : lim n n for the sets of sequences x = (xk ) which are strongly Ces`ro summable and a strongly (V, )-summable respectively. The notion of -statistical convergence was introduced by Mursaleen [17] and the concept of almost -statistical convergence was studied by Savas [24]. Recently -statistical convergence was generalized by Colak ¸ ¸ and Bektas [2]. ¸ 2. Main results In this section we give the main results of the paper. In Theorem 2.4 we give the inclusion relations between the sets of -almost statistical convergent sequences of order for different ' s, and so that the inclusion relations between the set of -almost statistical convergent sequences of order and the set of -almost statistical convergent sequences. In Theorem 2.8 we give ^ the relationship between the strong almost [Vp , ]-summability of order ^ and the strong almost [Vp , ]-summability of order . In Theorem 2.11 ^ we give the relationship between the strong almost [Vp , ]-summability of order and the - almost statistical convergence of order . Definition 2.1. Let the sequence = (n ) of real numbers be defined as above and 0 < 1 be given. The sequence x = (xk ) w is said to be -almost statistically convergent of order if there is a complex number L such thatlimn 1 |{k In : |xk+m - L| }| = 0, uniformly in m, where n In = [n - n + 1, n] and denote the th power (n ) of n , that is n ^ = ( ) = ( , , ..., , ...) . In this case we write S - lim xk = L. n n 1 2 The set of all - almost statistically convergent sequences of order will be ^ denoted by S . For example the sequence x = (xk ) defined by xk+m = k + m, k + m = n2 , 0, k + m = n2 n = 1, 2, ... is -almost statistically convergent sequences of order to 0, for 1 < 1 2 and = (n) . The is same with the ^ ^ almost statistical convergence, that is S = S for = 1. The is well defined for 0 < 1. But it is not well defined for > 1 in general. For this let x = (xk ) be defined as 5 follows: 1, 0, if k + m even . if k + m odd xk+m = Then both n n lim |{k In : |xk+m - 1| }| = lim n 2 n = 0, uniformly in m and 1 n |{k In : |xk+m - 0| }| = lim = 0, uniformly in m, n n n 2 n lim for > 1, so that x = (xk ) , -almost statistically converges of order , ^ ^ both to 1 and 0, i.e S - lim xk = 1 and S - lim xk = 0. But this is impossible. Theorem 2.2. Let 0 < 1 and x = (xk ), y = (yk ) be sequences of complex numbers. ^ ^ (i) If S - lim xk = x0 and c C, then S - lim(cxk ) = cx0 ; ^ ^ ^ (ii) If S - lim xk = x0 and S - lim yk = y0 , then S - lim(xk + yk ) = x0 + y 0 . Proof. (i) It is clear in case c = 0. Suppose that c = 0 then the proof of (i) follows from 1 1 |{k In : |cxk+m - cx0 | }| = {k In : |xk+m - x0 | } n |c| n and that of (ii) follows from 1 |{k In : |xk+m + yk+m - (x0 + y0 )| }| n 1 1 + k In : |yk+m - y0 | k In : |xk+m - x0 | n 2 n 2 ^ If we take n = n for 0 < < 1 in the definition of S then we obtain ^ S . That is why S is different from S defined in [24] in general. ^ ^ ^ S ^ ^ ^ If n = n with = 1, that is n = n then S = S = S, that is the almost statistical convergence of order , -almost statistical convergence and almost statistical convergence are coincide in case n = n with =1. Definition 2.3. Let the sequence = (n ) be as above, (0, 1] be any real number and let p be a positive real number. A sequence x is said ^ to be strongly (V , )-almost summable of order , if there is a complex number L such that 1 n n lim |xk+m - L|p = 0 uniformly in m, where In = [n - n + 1, n] . The strong (V, )- almost summability of order ^ reduces to the strong (V , )- summability for = 1. The set of all ^ ^ strongly (V , )- summable sequences of order will be denoted by [Vp , ]. ^ ^ Theorem 2.4. If 0 < 1 then S S . Proof. If 0 < 1 then we may write 1 n |{k In : |xk+m - L| }| 1 |{k In : |xk+m - L| }| , n ^ ^ for every > 0 and this gives that S S . If we take = 1 in Theorem 2.4 then we obtain the following result. Corollary 2.5. If a sequence is -almost statistically convergent of ^ order , to L, then it is -almost statistically convergent to L, that is S ^ S for each (0, 1]. From Theorem 2.4 we have the following results and the proof is easy. ^ ^ Corollary 2.6. (i) S = S = . ^ ^ (ii) S = S = 1. ^ ^ Theorem 2.7. S S if (1) lim inf n > 0. n Proof. For given > 0 we have {k n : |xk+m - L| } {k In : |xk+m - L| }. Therefore we may write 1 1 |{k n : |xk+m - L| }| |{k In : |xk+m - L| }| n n 1 n . |{k In : |xk+m - L| }| . n n ^ Tag the limit as n and using (1), we get xk L(S) = xk ^ ). L(S Theorem 2.8. Let 0 < 1 and p be a positive real number. ^ ^ Then [Vp , ] [Vp , ]. ^ Proof. Let x = (xk ) [Vp , ]. Then given and such that 0 < 1 and a positive real number p, we may write 1 1 |xk+m - L|p |xk+m - L|p n n ^ ^ and this gives that [Vp , ] [Vp , ]. The following result is a consequence of Theorem 2.8. ^ ^ Corollary 2.9. For each (0, 1] and 0<p<, we have [Vp , ][Vp , ]. ^ Theorem 2.10. Let 0 < 1 and 0 < p < q < . Then [Vq , ] ^ [Vp , ]. Proof is seen from H¨lder inequality. o Theorem 2.11. Let and be fixed real numbers such that 0 < ^ 1 and 0 < p < . If a sequence is strongly (V , )- almost summable of order , to L, then it is -almost statistically convergent of order , to ^ ^ L, i.e [Vp , ] S . Proof. For any sequence x = (xk ) and > 0, we have |xk+m - L|p = |xk+m - L|p + < |xk+m - L|p |xk+m - L|p |{k In : |xk+m - L| }| .p and so that 1 n |xk+m - L|p 1 |{k In : |xk+m - L| }| .p n 1 n |{k In : |xk+m - L| }| .p . ^ From this it follows that if x = (xk ) is strongly (V , )-almost summable of order , to L, then it is -almost statistically convergent of order , to L. If we take = in Theorem 2.11 we obtain the following result. Corollary 2.12. Let be a fixed real number such that 0 < 1 ^ and 0 < p < . If a sequence is strongly (V , )- almost summable of order , to L, then it is -almost statistically convergent of order , to L, i.e ^ ^ [Vp , ] S . Corollary 2.13. Let 0 < 1 and p be a positive real number. Then ^ ^ [Vp , ] S and the inclusion is strict if 0 < < 1. ^ ^ Proof. From Corollary 2.5 and Corollary 2.12 we have [Vp , ] S . 3. Modulus function and statistical convergence The notion of a modulus was introduced by Nakano [19]. 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. It follows that f must be continuous everywhere on [0, ). A modulus may be bounded or unbounded. For example, f (x) = xp , (0 < p 1) is x unbounded and f (x) = 1+x is bounded. Maddox [14], Ruckle [22], Malkowsky and Savas [15] used a modulus function to construct some ¸ sequence spaces. Definition 3.1. Let f be a modulus function, p = (pk ) be a sequence of strictly positive real numbers and let (0, 1] be any real number. Now we define ^ [Vp , , f ] = {x = (xk ) : lim 1 n [f (|xk+m - L|)]pk = 0 uniformly in m, for some L}. In the following theorems we shall assume that the sequence p = (pk ) is bounded and 0 < h = inf k pk pk supk pk = H < . Theorem 3.2. Let , (0, 1] be any real numbers such that ^ ^ and f be a modulus function. Then [Vp , , f ] S . ^ Proof. Let x [Vp , , f ] and let > 0 be given. Then since n n for each n we may write 1 n [f (|xk+m - L|)]pk [f (|xk+m - L|)]pk + [f (|xk+m - L|)]pk [f (|xk+m - L|)]pk 1 n 1 < [f (|xk+m - L|)]pk + 1 n < n 1 n [f ()]pk min([f ()]h , [f ()]H ) |{k In : |xk+m - L| }| min([f ()]h , [f ()]H ). ^ Since x [Vp , , f ], the left hand side of the above inequality tends to zero as n uniformly in m. Since min([f ()]h , [f ()]H ) > 0 the right hand side of above inequality tends to zero as n uniformly in m. This ^ ^ implies that [Vp , , f ] S . Theorem 3.3. Let , (0, 1] be any real numbers such that . ^ ^ If the modulus f is bounded then S [Vp , , f ]. ^ Proof. Let x S and suppose that f is bounded. Let > 0 be given and 1 and 2 denote the sums over k In , |xk+m - L| and k In , |xk+m - L| < respectively. Since f is bounded there exists an integer K such that f (x) K, for all x 0. Then since for each n we may n n write 1 n 1 n 1 n = [f (|xk+m - L|)]pk 1 n [f (|xk+m - L|)]pk [f (|xk+m - L|)]pk + 1 n [f (|xk+m - L|)]pk [f ()]pk max K h , K H + max K h , K H 1 |{k n : |xk+m - L| }| + max f ()h , f ()H . n ^ Since x S the first term of the right hand side of above inequality tends to zero as n uniformly in m, and the second term can be made as small as desired. Therefore the left hand side of above inequality tends to ^ zero as n uniformly in m. Hence x [Vp , , f ]. ^ ^ Theorem 3.4. Let (0, 1] be any real number. Then S = [Vp , , f ] if and only if f is bounded. Proof. Let f be bounded. By Theorem 3.2 and Theorem 3.3 we have ^ ^ S = [Vp , , f ]. Conversely assume that f is unbounded. Using the same technique as ^ ^ in proof of Theorem 3 in [15] we may find a sequence x S - [Vp , , f ]. This completes the proof. If we take = 1 in Theorem 3.4 we obtain Theorem 3 in [15] as the following. ^ ^ Corollary 3.5. The S = [Vp , , f ] equality holds if and only if f is bounded. Acknowledgment. The authors wish to thank the referee for their careful reading of the manuscript and valuable of the manuscript and valuable suggestions.

Journal

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Nov 24, 2014

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