Generalized I-convergent DifferenceSequence Spaces defined by a ModuliSequence

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1 Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 9 Version 1.0 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA Online ISSN: & Print ISSN: Generalized I-convergent Difference Sequence Spaces defined by a Moduli Sequence By M. Aiyub University of Bahrain, Kingdom of Bahrain Abstract - In this article we introduce the sequence space c I 0(F, n and l I (F, n for the sequence of F = (f and given some inclusion relations. Keywords : Ideal, filter, sequence of moduli, difference sequence space, I-convergent sequence space. AMS subject classication (2000 : 40C05, 46A45 Generalized I-convergent DifferenceSequence Spaces defined by a ModuliSequence Strictly as per the compliance and regulations of :. M. Aiyub. This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original wor is properly cited.

2 Ref. Generalized I-convergent Difference Sequence Spaces Defined by a Moduli Sequence 10. H.Kizmaz, on certain sequence spaces, Canadian Math.Bull.,24(1981, M. Aiyub Abstract - In this article we introduce the sequence space c I 0(F, n and l I (F, n for the sequence of F = (f and given some inclusion relations. Keywords and phrases : Ideal,filter, sequence of moduli,difference sequence space, I-convergent sequence space I. Introduction and Preliminaries Let ω, l, c 0 be the set of all sequences of complex numbers, the linear spaces of bounded, convergent and null sequences x = (x with complex terms, respectively, normed by x = sup x, wherek N = 1, 2, 3... The idea of difference sequence spaces was introduced by H. Kizmaz [10]. In 1981, Kizmaz defined the sequence spaces as follow; where These are Banach space with the norm l ( = {x = (x ω : ( x l }, c( = {x = (x ω : ( x c}, c 0 ( = {x = (x ω : ( x c 0 }, x = (x x +1 and 0 x = (x, x = x 1 + x. Later Cola and Et [2] defined the sequence spaces: l ( n = {x = (x ω : ( n x l }, 33 c( n = {x = (x ω : ( n x c}, Author : Department of Mathematics, University of Bahrain, P.O. Box-32038, Kingdom of Bahrain. maiyub2002@yahoo.com Global Journals Inc. (US

3 c 0 ( n = {x = (x ω : ( n x c 0 }, where n N, 0 x = (x, x = (x x +1, n x = ( n x = ( n 1 x n 1 x +1 and so that n [ ] n x = ( 1 v n x v +v, v=0 and so that these are Banach space with the norm n x = x i + n x. i=1 Ref. 34 The idea of modulus was defined by Naano [15] in A function f : [0, [0, is called a modulus if (i f(t = 0 if and only if t = 0, (ii f(t + u f(t + f(u, for all t, u 0, (iii f is increasing and (iv f is continuous from the right at 0. Let X be a sequence spaces.then the sequence spaces X(f is defined as for a modulus f. Maddox and Rucle[14,16] X(f = {x = (x : (f( x X} Kola[11,12] gave an extension of X(f by considering a sequence of moduli F = (f,that is X(F = {x = (x : (f ( x X}. After then Gaur and Mursaleen[9] defined the following sequence spaces for a sequence of moduli F = (f. we defined the following sequence spaces: l (F, = {x = (x : ( x l (F }, c 0 (F, = {x = (x : ( x c 0 (F }, l (F, n = {x = (x : ( n x l (F }, c 0 (F, n = {x = (x : ( n x c 0 (F }, for a sequence of moduli F = (f. We will give the necessary and sufficient conditions for the inclusion relations between X( n and Y (F, n, where X, Y = l or c 0. Sequence of moduli have been studied by C.A.Betas and R. Cola[1] and many other authours. The notion of statical convergence was introduced by H.Fast[6]. Later on it was studied by J.A.Fridy [7,8] from the sequence space point view and lined with the summability theory. The notion of I-convergence is a generalization of the statical convergence. It was studied at initial stage by Kostyro, Salat and Wilezynsi [13]. Later on it was studied by Salat [19],Salat, Tripathy and Ziman [20], Demric[3] 15. H.Naano, Concave modulars, J.Math Soc. Japan., 5(1953, Global Journals Inc. (US

4 Let N be a non empty set. Then a family of sets I 2 N (power set of N is said to be an ideal if I is additive i.e (A, B I (A B I and i.e A I, B A B I. A non empty family of sets (I 2 N is said to be filter on N if and only if Φ / (I for A, B (I we have (A B (I and for each A (I and A B implies B (I. Ref. An ideal I 2 N is called non trivial if I 2 N. A non trivial ideal I 2 N is called admissible if {(x : x N} I. A non trivial ideal is maximal if there cannot exist any non-trivial ideal J I containing I as a subset. For each ideal I, there exist a filter (I corresponding to I, i.e (I = {K N : K c I},where K c = N K. Definition 1.1. A sequence (x ω is said to be I-convergent to a number L if for every ɛ > 0. { N : x L ɛ} I.In this case we write I lim x = L. Definition 1.2. A sequence (x ω is said to be I-null if L=0. In this case we write I lim x = C.A.Betas, R.Cola, Generalized difference sequence spaces defined by a sequence of moduli, Soochow. J.Math., 29(2(2003, Definition 1.3. A sequence (x ω is said to be I-cauchy if for every ɛ > 0, there exist a number m = m(ɛ such that { N : x x m ɛ} I. Definition 1.4. A sequence (x ω is said to be I-bounded if there exist M > 0 such that {K N : x M}. We need the following Lemmas. Lemma 1.5. The condition sup f (t <, t > 0 hold if and only if there is a point t 0 > 0 such that sup f (t 0 < (see [1,9]. Lemma 1.6. The condition inf f (t > 0 hold if and only if there exist is a point t 0 > 0 such that inf f (t 0 > 0 (see [1,9]. Lemma 1.7. Let K (I and M N. If M I then M K I (see [20]. Lemma 1.8. If I 2 N and M N. If M I then M K I (see [13]. II. Main Results In this article we introduce the following classes of sequence spaces. c I 0(F, n = {(x ω : I lim f ( n x = 0} I, l I (F, n = {(x ω : I sup f ( n x < } I Theorem 2.1. For a sequence F = f of moduli, the following statements are equivalent: (a l I ( n l I (F, n, (b c I 0( n c I 0(F, n, (c sup f (t <, (t > 0. Proof. (a implies (b is obvious. (b implies (c. Let c I 0( n c I 0(F, n. Suppose that (c is not true. Then by Lemma (1.5 sup f (t =, for all t > 0, 35 Global Journals Inc. (US

5 and therefore there is a sequence ( i of positive integers such that f i ( 1 > i, for each i = 1, 2, 3... (1 i Define x = (x as follow { 1 x = i if = i, i = 1, 2, 3...; 0 otherwise. Then x c I 0( n but by (1, x / l I (F, n which contradicts (b. Hence (c must hold. (c implies (a. Let (c be satisfied and x l I (F, n. If we suppose that x / l I (F, n then sup f ( n x = for n x l I Notes 36 If we tae t = n x then sup f (t = which contradicts (c. Hence l I ( n l I (F, n. Theorem 2.2. For a sequence F = f is a sequence of moduli, the following statements are equivalent: (a c I 0(F, n c I 0( n, (b c I 0(F, n l I ( n, (c inf f (t > 0, (t > 0. Proof. (a implies (b is obvious. (b implies (c. Let c I 0(F, n l I ( n. Suppose that (c is not true. Then by Lemma (1.6 inf f (t = 0, (t > 0 and therefore there is a sequence ( i of positive integers such that Define x = (x as follow f i (i 2 < 1 i for each i = 1, 2, 3... { i x = 2, if = i i = 1, 2, 3...; 0 otherwise. By (2 x c I 0(F, n but x / l I ( n which contradicts (b. Hence (c must hold. (c implies (a. Let (c be satisfied and x c I 0(F, n that is I lim f ( n x = 0. Suppose that x / c I 0( n. Then for some number ɛ 0 > 0 and positive integer 0 we have n x ɛ 0 for > 0. Therefore f (ɛ 0 f ( n x for > 0 and hence lim f (ɛ 0 > 0, which contradicts our assumption that x / c I 0( n. Thus c I 0(F, n c I 0( n. Theorem 2.3.The inclusion l I (F, n c I 0( n holds if and only if lim f (t = for t > 0. (3 (2 Global Journals Inc. (US

6 Proof. Let l I (F, n c I 0( n such that lim f (t = for t > 0 doesn t hold. Then there is a number t 0 > 0 and a sequence ( i of positive integer such that f i (t 0 M <. (4 define the sequence x = (x by Notes { t0, if = x = i i = 1, 2, 3...; 0 otherwise. Thus x l I (F, n by (4. But x / c I 0( n, so that (3 must hold.if l I (F, n c I 0( n. Conversely, let (3 hold. If x l I (F, n,then f ( n x M <, for = 1, 2, 3...Suppose that x / c I 0( n. Then for some number ɛ 0 > 0 and positive integer 0 we have n x < ɛ 0 for 0. Therefore f (ɛ 0 f ( n x M for 0, which contradicts(3. Hence x c I 0( n. Theorem 2.4. The inclusion l I ( n c I 0(F, n holds if and only if 37 lim f (t = 0, for t > 0. (5 Proof. Suppose that l I ( n c I 0(F, n but (5 doesn t hold, Then Define the sequence x = (x by lim f (t 0 = l 0, for some t 0 > 0 (6. n [ ] x = t 0 ( 1 n n + v 1 v v=0 for = 1, 2, 3,... Then x / c I 0(F, n by (6. Hence (5 must hold. conversely, let x l I ( n and suppose that (5 holds. Then n x M < for K = 1, 2, 3... There for f ( n x f (M for = 1, 2, 3... and lim f ( n x lim f (M = 0 by (5. Hence x c I 0(F, n References Références Referencias 1. C.A.Betas, R.Cola, Generalized difference sequence spaces defined by a sequence of moduli, Soochow. J.Math., 29(2(2003, R.Cola and M.ve Et, on some generalized difference sequence spaces and related matrix transformations, Hoaido Math.J.,26(3(1997, K.Demerici, I-limit superior and limit inferior, Math.Commun., 6(2001, K.Dems, On I-Cauchy sequences, Real Analysis Exchange., 30(2005, A.Esi and M.Isi, Some generalized difference sequence spaces, Thai J.Math.,3(2(2005, H.Fast, Sur Ia convergence statistique, Colloq.Math., 2(1951, J.A.Fridy, On statical convergence, Analysis., 5(1985, J.A.Fridy, Statistical limit points, Proc. Amer.Math.Soc., 11(1993, A.K. Gaur and M.Mursaleen, Di_erence sequence spaces defined by a sequence of moduli, Demonstratio Math., 31(1998, Global Journals Inc. (US

7 H.Kizmaz, on certain sequence spaces, Canadian Math.Bull.,24(1981, E.Kola, On strong boundedness and summability with respect to a sequence of moduli, Acta Comment.Univ.Tartu.,960(1993, E.Kola, Inclusion theorems for some sequence spaces defined by a sequence of modulii, Acta Comment.Univ. Tartu., 970(1994, P.ostyro.T.Salat and W.Wilczynsi, I-Convergence, Real Analysis Exchange., 26 (2(2000, I.J.Maddox, Sequence spaces defined by a modulus, Math.Camb.Phil.Soc., 100(1986, H.Naano, Concave modulars, J.Math Soc. Japan., 5(1953, W.H.Rucle, On perfect Symmetric BK-spaces,Math.Ann.,175(1968, W.H.Rucle, Symmetric coordinate space and symmetric bases,canad,j.math., 19(1967, W.H.Rucle, FK-spaces in which the sequence of coordinate vectors is bounded. Canad.J.Math.,25(5(1973, T.salat, On statitical convergent sequences of real numbers, Math. Solvaca., 30(1980, T.salat, B.C.Tripathy and M.Ziman, On some properties of I-convergence, Tatra Mt. Math.Publ., 28(2004, Notes Global Journals Inc. (US