On Zweier I-convergent sequence spaces

Transcription

1 Proyecciones Journal of Mathematics Vol. 33, N o 3, pp , September Universidad Católica del Norte Antofagasta - Chile On Zweier I-convergent sequence spaces Vakeel A. Khan Khalid Ebadullah and Yasmeen Aligarh Muslim University, India Received : February Accepted : April 2014 Abstract In this article we introduce the Zweier I-convergent sequence spaces Z I, Z I 0 and ZI. We prove the decomposition theorem and study topological, algebraic properties and have established some inclusion relations of these spaces Mathematics Subject Classification : 40C05, 40J05, 46A45. Keywords and phrases : Ideal, filter, I-convergence field, monotone, solid, Lipschitz function, Zweier Space, Statistical convergence, Banach space.

2 260 Vakeel A. Khan, Khalid Ebadullah and Yasmeen 1. Introduction Let N,R and C be the sets of all natural, real and complex numbers respectively. We write ω = {x =(x k ):x k R or C }, the space of all real or complex sequences. Let,c and c 0 denote the Banach spaces of bounded,convergent and null sequences respectively normed by x =sup x k. k A sequence space λ with linear topology is called a K-space provided each of maps p i C defined by p i (x) =x i is continuous for all i N. AK-spaceλ is called an FK-space provided λ is a complete linear metric space. An FK-space whose topology is normable is called a BK-space. Let λ and µ be two sequence spaces and A =(a nk )beaninfinite matrix of real or complex numbers (a nk ), where n, k N. Then we say that A defines a matrix mapping from λ to µ, and we denote it by writting A : λ µ. If for every sequence x =(x k ) λ the sequence Ax = {(Ax) n },thea transform of x is in µ, where (1.1) (Ax) n = X k a nk x k, (n N). By (λ : µ), we denote the class of matrices A such that A : λ µ. Thus, A (λ : µ) if and only if series on the right side of (1) converges for each n N and every x λ. The approach of constructing new sequence spaces by means of the matrix domain of a particular limitation method have recently been employed by Altay,Başar and Mursaleen [1], Başar and Altay [2], Malkowsky [13], Ng and Lee [14], and Wang [21].

3 On Zweier I-convergent sequence spaces 261 Şengönül[18] defined the sequence y =(y i ) which is frequently used as the Z p transform of the sequence x =(x i )i.e., y i = px i +(1 p)x i 1 where x 1 =0,1<p< and Z p denotes the matrix Z p =(z ik )defined by p, (i = k), z ik = 1 p, (i 1=k); (i, k N), 0, otherwise. Following Başar and Altay[2], Şengönül[18] introduced the Zweier sequence spaces Z and Z 0 as follows : Z = {x =(x k ) ω : Z p x c} Z 0 = {x =(x k ) ω : Z p x c 0 }. Here we list below some of the results of Şengönül [18] which we will need as a reference in order to establish analogously some of the results of this article. Theorem 1.1. The sets Z and Z 0 are linear spaces with the co-ordinate wise addition and scalar multiplication which are the BK-spaces with the norm x Z = x Z0 = Z p x c [See (Theorem 2.1. [18])]. Theorem 1.2. The sequence spaces Z and Z 0 are linearly isomorphic to the spaces c and c 0 respectively, i.e Z = c and Z 0 = c0 [See (Theorem 2.2.[18])] Theorem 1.3. The inclusions Z 0 Z strictly hold for p 6= 1. [See(Theorem 2.3. [18])]. Theorem 1.4. Z 0 is solid.[see (Theorem 2.6.[18])]. Theorem 1.5. Z is not a solid sequence space.[see (Theorem 3.6. [18])]. The concept of statistical convergence was first introduced by Fast [7] and also independently by Buck [3] and Schoenberg [17] for real and complex sequences.further this concept was studied by Connor [4, 5], Connor,

4 262 Vakeel A. Khan, Khalid Ebadullah and Yasmeen Fridy and Kline [6] and many others. Statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence. A sequence x =(x k ) is said to be statistically convergent to L if for a given ε>0 1 lim k k {i : x i L ε, i k} =0. The notion of I-convergence generalizes and unifies different notions of convergence including the notion of statistical convergence. At the initial stage it was studied by Kostyrko, Šalát, Wilczyński [12]. Later on it was studied by Šalát, Tripathy, Ziman [15, 16]. Recentlly further it was studied by Tripathy [19, 20, 21, 22, 23, 24, 25, 26, 27], and V. A.Khan and Khalid Ebadullah [9-11]. Here we give some preliminaries about the notion of I-convergence. Let X be a non empty set. Then a family of sets I 2 X (2 X denoting thepowersetofx)issaidtobeanidealifiisadditivei.ea,b I A B I and hereditary i.e A I, B A B I. A non-empty family of sets (I) 2 X is said to be filter on X if and only if / (I), for A, B (I) wehavea B (I) andforeacha (I) and A B impliesb (I). An Ideal I 2 X is called non-trivial if I6= 2 X. A non-trivial ideal I 2 X is called admissible if {{x} : x X} I. A nontrivial ideal I is maximal if there cannot exist any non-trivial ideal J6=I containing I as a subset. For each ideal I, there is a filter (I) corresponding to I. i.e (I) ={K N : K c I}, where K c = N K. Definition 1.6. A sequence (x k ) ω is said to be I-convergent to a number L if for every ε>0. {k N : x k L ε} I. In this case we write I lim x k = L. The space c I of all I-convergent sequences to L is given by c I = {(x k ) ω : {k N : x k L ε} I,for some L C }.

5 On Zweier I-convergent sequence spaces 263 Definition 1.7. A sequence (x k ) ω is said to be I-null if L =0. Inthis case we write I lim x k =0. Definition 1.8. A sequence (x k ) ω is said to be I-Cauchy if for every ε>0 there exists a number m = m(ε) such that {k N : x k x m ε} I. Definition 1.9. A sequence (x k ) ω is said to be I-bounded if there exists M>0such that {k N : x k >M} I. Example Take for I the class I f of all finite subsets of N. Then I f is a non-trivial admissible ideal and I f convergence coincides with the usual convergence with respect to the metric in X. (see [12]). Definition For I = I δ and A N with δ(a) = 0 respectively. I δ is a non-trivial admissible ideal, I δ -convergence is said to be logarithmic statistical convergence(see[12]). Definition Amap h defined on a domain D X i.e h : D X R is said to satisfy Lipschitz condition if h(x) h(y) K x y, where K is known as the Lipschitz constant.the class of K-Lipschitz functions defined on D is denoted by h (D, K)(see[15,16]). Definition A convergence field of I-covergence is a set F (I) ={x =(x k ) l : there exists I lim x R}. The convergence field F (I) is a closed linear subspace of l with respect to the supremum norm, F (I) =l c I (See [15,16]). Define a function h : F (I) R such that h(x) =I lim x, forall x F (I), then the function h : F (I) R is a Lipschitz function. (see [15, 16]). Definition Let (x k ), (y k ) be two sequences. We say that (x k )=(y k ) for almost all k relative to I (a.a.k.r.i), if {k N : x k 6= y k } I(see[19, 20]).

6 264 Vakeel A. Khan, Khalid Ebadullah and Yasmeen The following Lemmas will be used for establishing some resultsofthisarticle: Lemma Let E be a sequence space. If E is solid then E is monotone.(see [8],page 53). Lemma If I 2 N and M N. If M / I, then M N / I. (see [19,20]). 2. Main Results In this section we introduce the following classes of sequence spaces : and Z I = {x =(x k ) ω : {k N : I lim Z p x = L, for some L C }}; Z I 0 = {x =(x k ) ω : {k N : I lim Z p x =0}}; Z I = {x =(x k ) ω : {k N :sup Z p x < }}. k We also denote by m I Z = Z Z I m I Z 0 = Z Z I 0. Throughout the article, for the sake of conveinence now we will denote by Z p (x k )=x /,Z p (y k )=y /,Z p (z k )=z / for x, y, z ω. Theorem 2.1. The classes of sequences Z I, Z0 I,mI Z and mi Z 0 spaces. are linear Proof. We shall prove the result for the space Z I. The proof for the other spaces will follow similarly. Let (x k ), (y k ) Z I and let α, β be scalars. Then I lim x / k L 1 =0, for some L 1 C;

7 On Zweier I-convergent sequence spaces 265 I lim y / k L 2 =0, for some L 2 C; That is for a given ε>0, we have A 1 = {k N : x / k L 1 > ε 2 } I, (2.1) A 2 = {k N : y / k L 2 > ε 2 } I. we have (αx / k + βy/ k ) (αl 1 + βl 2 ) α ( x / k L 1 ) + β ( y / k L 2 ) x / k L 1 + y / k L 2 Now, by (1) and (2), {k N: (αx / k +βy/ k ) (αl 1+βL 2 ) > } A 1 A 2. Therefore (αx k + βy k ) Z I Hence Z I is a linear space. Theorem 2.2. The spaces m I Z and mi Z 0 are normed linear spaces,normed by (2.2) x / k =sup Z p (x), k where x / k = Zp (x). Proof: It is clear from Theorem 2.1 that m I Z and mi Z 0 are linear spaces. It is easy to verify that (3) defines a norm on the spaces m I Z and mi Z 0. Theorem 2.3. A sequence x =(x k ) m I Z every >0 there exists N N such that I-converges if and only if for (2.3) {k N : x / k x/ N <ε} m I Z Proof. Suppose that L = I lim x /.Then B ε = {k N : x / k L < ε 2 } mi Z for all ε>0

8 266 Vakeel A. Khan, Khalid Ebadullah and Yasmeen.FixanN ε B ε.thenwehave x / N ε x / k x/ N ε L + L x / k < 2 + ε 2 = ε which holds for all k B ε. Hence {k N : x / k x/ N ε <ε} m I Z. Conversely, suppose that {k N : x / k x/ N ε <ε} m I Z. That is {k N : x / k x/ N ε <ε} m I Z for all ε>0. Then the set C ε = {k N : x / k [x/ N ε ε, x / N ε + ε]} m I Z for all ε>0. Let J ε =[x / N ε ε, x / N ε + ε].if we fix an >0thenwehaveC m I Z as well as C ε mi 2 Z. Hence C ε C ε mi 2 Z.This implies that J = J ε J ε 6= φ 2 that is {k N : x / k J} mi Z that is diamj diamj ε where the diam of J denotes the length of interval J. In this way, by induction we get the sequence of closed intervals J ε = I 0 I 1... I k... with the property that diami k 1 2 diami k 1 for (k=2,3,4,...) and {k N : x / k I k} m I Z for (k=1,2,3,4,...). Then there exists a ξ I k where k N such that ξ / = I lim x /, that is L = I lim x /. Theorem 2.4.Let I be an admissible ideal.then the following are equivalent. (a) (x k ) Z I ; (b) there exists (y k ) Z such that x k = y k, for a.a.k.r.i;

9 On Zweier I-convergent sequence spaces 267 (c) there exists (y k ) Z and (z k ) Z I 0 such that x k = y k + z k for all k N and {k N : y k L } I ; (d) there exists a subset K = {k 1 <k 2...} of N such that K (I) and lim n x k n L =0. Proof.(a) implies (b). Let (x k ) Z I. Then there exists L C such that {k N : x / k L ε} I.Let (m t ) be an increasing sequence with m t N such that Define a sequence (y k )by {k m t : x / k L 1 t } I. y k = x k, for all k m 1. For m t <k m t+1,t N. ( y k = x k, if x / k L <t 1, L, otherwise. Then (y k ) Z and form the following inclusion {k m t : x k 6= y k } {k m t : x / k L } I. We get x k = y k, for a.a.k.r.i. (b) implies (c).for (x k ) Z I. Then there exists (y k ) Z such that x k = y k, for a.a.k.r.i. Let K = {k N : x k 6= y k }, then K I. Define a sequence (z k )by ( xk y z k = k, if k K, 0, otherwise. Then z k Z I 0 and y k Z.

10 268 Vakeel A. Khan, Khalid Ebadullah and Yasmeen (c) implies (d).let P 1 = {k N : z k ε} I and.thenwehave lim n x k n L =0. K = P c 1 = {k 1 <k 2 <k 3 <...} (I) (d) implies (a). Let K = {k 1 <k 2 <k 3 <...} (I) and lim n x k n L =0. Then for any >0, and by Lemma, we have {k N : x / k L } Kc {k K : x / k L }. Thus (x k ) Z I. Theorem 2.5. The inclusions Z I 0 ZI Z I are proper. Proof: Let(x k ) Z I. Then there exists L C such that I lim x / k L =0 We have x / k 1 2 x/ k L L. Taking the supremum over k on both sides we get (x k ) Z. I The inclusion Z0 I ZI is obvious. Theorem 2.6. The function h : m I Z R is the Lipschitz function,where m I Z = ZI Z, and hence uniformly continuous. Proof:Let x, y m I Z, x 6= y.then the sets A x = {k N : x / k h(x/ ) x / y / } I, A y = {k N : y / k h(y/ ) x / y / } I. Thus the sets, B x = {k N : x / k h(x/ ) < x / y / } m I Z, B y = {k N : y / k h(y/ ) < x / y / } m I Z. Hence also B = B x B y m I Z,sothatB 6= φ.

11 On Zweier I-convergent sequence spaces 269 Now taking k in B, h(x / ) h(y / ) h(x / ) x / k + x/ k y/ k + y/ h(y / ) 3 x / y /. Thus h is a Lipschitz function. For m I Z 0 the result can be proved similarly. Theorem 2.7. If x, y m I Z,then(x.y) mi Z and h(xy) = h(x) h(y). Proof: For >0 B x = {k N : x / h(x / ) <ε} m I Z, B y = {k N : y / h(y / ) <ε} m I Z. Now, x /.y / h(x / ) h(y / ) = x /.y / x / h(y / )+x / h(y / ) h(x / ) h(y / ) (2.4) x / y / h(y / ) + h(y / ) x / h(x / ) As m I Z Z, there exists an M R such that x / < M and h(y / ) <M. Using eqn(5) we get x /.y / h(x / ) h(y / ) Mε+ Mvarε =2Mε For all k B x B y m I Z. Hence (x.y) m I Z and h(xy) = h(x) h(y). For m I Z 0 the result can be proved similarly. Theorem 2.8. The spaces Z I 0 and mi Z 0 are solid and monotone. Proof: We prove the result for the case Z I 0. (2.5) Let (x k ) Z I 0.Then I lim k x/ k =0

12 270 Vakeel A. Khan, Khalid Ebadullah and Yasmeen Let (α k ) be a sequence of scalars with α k 1 for all k N. Then the result follows from (6) and the following inequality α k x / k α k x / k x/ k for all k N. That the space Z I 0 is monotone follows from the Lemma For m I Z 0 the result can be proved similarly. Theorem 2.9.The spaces Z I and m I Z are neither monotone nor solid, if I is neither maximal nor I = I f in general. Proof: Herewegiveacounterexample. Let I = I δ. Consider the K-step space X K of X defined as follows, Let (x k ) X and let (y k ) X K be such that ( (y / k )= (x / k ), if k is odd, 1,otherwise. Consider the sequence (x / k )defined by (x/ k )=k 1 for all k N. Then (x k ) Z I but its K-stepspace preimage does not belong to Z I. Thus Z I is not monotone. Hence Z I is not solid. Theorem The spaces Z I and Z I 0 are sequence algebras. Proof: WeprovethatZ I 0 is a sequence algebra. Let (x k ), (y k ) Z I 0.Then I lim x / k =0 and Then we have I lim y / k =0 I lim (x / k.y/ k ) =0 Thus (x k.y k ) Z I 0

13 On Zweier I-convergent sequence spaces 271 Hence Z I 0 is a sequence algebra. For the space Z I, the result can be proved similarly. Theorem The spaces Z I and Z I 0 are not convergence free in general. Proof: Herewegiveacounterexample. Let I = I f. Consider the sequence (x / k )and(y/ k )defined by x / k = 1 k and y/ k = k for all k N Then (x k ) Z I and Z I 0, but (y k) / Z I and Z I 0. Hence the spaces Z I and Z I 0 are not convergence free. Theorem If I is not maximal and I 6= I f, then the spaces Z I and Z0 I are not symmetric. Proof: LetA I be infinite. If x / k = ( 1, for k A, 0,otherwise. Then by lemma x k Z I 0 ZI.LetK N be such that K/ I and N K/ I.Let φ : K A and ψ : N K N A be bijections, then the map π : N N defined by π(k) = ( φ(k), for k K, ψ(k),otherwise. is a permutation on N, but x π(k) / Z I and x π(k) / Z I 0. Hence Z I and Z I 0 are not symmetric. Theorem The sequence spaces Z I and Z0 I are linearly isomorphic to the spaces c I and c I 0 respectively, i.e ZI = c I and Z0 I = c I 0.

14 272 Vakeel A. Khan, Khalid Ebadullah and Yasmeen Proof.We shall prove the result for the space Z I and c I. The proof for the other spaces will follow similarly. We need to show that there exists a linear bijection between the spaces Z I and c I.Define a map T : Z I c I such that x x / = Tx where x 1 =0,p6= 1,1<p<. Clearly T is linear. T (x k )=px k +(1 p)x k 1 = x / k Further,it is trivial that x =0=(0, 0, 0,...), whenever Tx =0and hence injective. Let x / k ci and define the sequence x = x k by kx x k = M ( 1) k i N k i x / i i=0 (i N), where M = 1 p and N = 1 p p. Then we have lim k px k +(1 p)x k 1 kx = p lim M ( 1) k i N k i x / i +(1 p) lim k i=0 k 1 M X k i=0 ( 1) k i N k i x / i which shows that x Z I. = lim k x/ k Hence T is a linear bijection. Also we have x = Z p x c. Therefore, x =sup px k +(1 p)x k 1, k N

15 On Zweier I-convergent sequence spaces 273 =sup pm k N Hence Z I = c I. kx ( 1) k i N k i x / k 1 i +(1 p)m X ( 1) k i N k i x / i i=0 i=0 =sup x / k = x/ c I. k N Acknowledgments: The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper. References [1] Altay, B.,Başar, F. and Mursaleen. On the Euler sequence space which include the spaces l p and l, Inform.Sci., 176 (10), pp , (2006). [2] Başar, F. and Altay, B. On the spaces of sequences of p-bounded variation and related matrix mappings. Ukrainion Math. J. 55. (2003). [3] Buck, R. C.: Generalized Asymptotic Density, Amer. J. Math. 75, pp , (1953). [4] Connor, J. S.: it The statistical and strong P-Cesaro convergence of sequences,analysis. 8(1988), [5] Connor, J. S. : On strong matrix summability with respect to a modulus and statistical convergence, Cnad. Math. Bull. 32, pp , (1989). [6] Connor, J., Fridy, J. A. and Kline, J. Statistically Pre-Cauchy sequence, Analysis. 14, pp , (1994). [7] Fast, H. : Sur la convergence statistique, Colloq. Math. 2, pp , (1951). [8] Kamthan, P. K. and Gupta, M. : Sequence spaces and series. Marcel Dekker Inc, New York.(1980).

16 274 Vakeel A. Khan, Khalid Ebadullah and Yasmeen [9] Khan, V. A. and Ebadullah, K. On some I-Convergent sequence spaces defined by a modullus function. Theory and Application of Mathematiccs and Computer Science. 1 (2), pp , (2011). [10] Khan, V. A., Ebadullah, K and Ahmad, A. I-Pre-Cauchy Sequences and Orlicz Function. Journal of Mathematical Analysis. 3 (1), pp , (2012). [11] Khan, V.A. and Ebadullah,K.I-Convergent difference sequence spaces defined by a sequence of moduli. J. Math. Comput.Sci. 2 (2), pp , (2012). [12] Kostyrko, P., Šalát, T.,Wilczyński,W.I-convergence. Real Analysis Exchange, 26 (2), pp , (2000). [13] Malkowsky, E. Recent results in the theory of matrix transformation in sequence spaces. Math. Vesnik. (49), pp , (1997). [14] Ng, P., N. and Lee P., Y. Cesaro sequence spaces of non-absolute type. Comment. Math. Pracc. Math. 20 (2), pp , (1978). [15] Šalát,T., Tripathy, B. C., Ziman, M. On some properties of I- convergence. Tatra Mt. Math. Publ., (28), pp , (2004). [16] Šalát,T.,Tripathy,B.C.,Ziman,M.OnI-convergencefield. Ital. J. Pure Appl. Math., (17), pp (2005). [17] Schoenberg, I. J. : The integrability of certain functions and related summability methods, Amer. Math. Monthly. 66 : pp , (1959). [18] Şengönül, M. On The Zweier Sequence Space, DEMONSTRATIO MATHEMATICA, Vol.XL No.(1), pp , (2007). [19] Tripathy, B. C. and Hazarika, B. Paranorm I-Convergent sequence spaces, Math Slovaca. 59 (4), pp (2009). [20] Tripathy, B. C. and Hazarika,B. Some I-Convergent sequence spaces defined by Orlicz function., Acta Mathematicae Applicatae Sinica. 27 (1), pp , (2011). [21] Tripathy, B. C. and Hazarika, B. I -convergent sequence spaces associated with multiplier sequence spaces ; Mathematical Inequalities and Appl;ications ; 11 (3), pp , (2008).

17 On Zweier I-convergent sequence spaces 275 [22] Tripathy, B. C. and Mahanta, S. On Acceleration convergence of sequences ; Journal of the Franklin Institute, 347, pp , (2010). [23] Tripathy,B.C. and Hazarika, B. I-monotonic and I-Convergent sequences, Kyungpook Math. Journal. 51 (2) (2011), pp , (2011). [24] Tripathy, B. C. Sen, M. and Nath, S. I-Convergence in probabilistic n-normed space ; Soft Comput, 16, pp , (2012). [25] Tripathy, B. C. Hazarika,B and Choudhry, B. Lacunary I-Convergent sequences, Kyungpook Math. Journal. 52 (4), pp , (2012). [26] Tripathy,B. C. and Sharma, B. On I-Convergent double sequences of fuzzy real numbers, Kyungpook Math. Journal. 52 (2), pp , (2012). [27] Tripathy, B. C. and Ray, G. C. Mixed fuzzy ideal topological spaces ; Applied mathematics and Computaions; 220, pp , (2013). Vakeel A. Khan Department of Mathematics, Aligarh Muslim University Aligarh , India vakhanmaths@gmail.com Khalid Ebadullah Department of Applied Mathematics, Z. H. College of Engineering and Technology, Aligarh Muslim University, Aligarh , India khalidebadullah@gmail.com and

18 276 Vakeel A. Khan, Khalid Ebadullah and Yasmeen Yasmeen Department of Mathematics, Aligarh Muslim University, Aligarh , India khany9828@gmail.com