In this article we introduce the sequence spaces cI0( f ), cI ( f ) and lI1 ( f ) for a modulus function f and study some of theproperties of these spaces

Ebadullah, Khalid

Khan, Vakeel A.

English

Aurel Vlaicu University of Arad Editing House: Journals

Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30On Some I-Convergent Sequence Spaces Defined by a Modulus FunctionVakeel A. Khana,∗, Khalid EbadullahaaDepartment of Mathematics, A. M. U. Aligarh-202002, India.AbstractIn this article we introduce the sequence spaces cI0( f ), cI( f ) and lI∞( f ) for a modulus function f and study some of theproperties of these spaces.Keywords: Ideal, filter,modulus function, Lipschitz function, I-convergence field, I-convergent, monotone and solid spaces.2000 MSC: 40A05, 40A35, 40C05, 46A45.1. IntroductionThroughout the article N,R,C and ω denotes the set of natural,real,complex numbers and the class ofall sequences respectively. The notion of the statistical convergence was introduced by (Fast, 1951). Lateron it was studied by (Fridy, 1985) and (Fridy, 1993) from the sequence space point of view and linked itwith the summability theory. The notion of I-convergence is a generalization of the statistical convergence.At the initial stage it was studied by Kostyrko, Salat and Wilezynski in (Kostyrko et al., 2000). Later on itwas studied by Salat, Tripathy and Ziman in (Šalát et al., 2004) and Demirci in (Demirci, 2001).Here we give some preliminaries about the notion of I-convergence. Let X be a non empty set. A setI ⊆ 2X (2X denoting the power set of X) is said to be an ideal if I is additive i.e A, B ∈ I ⇒ A ∪ B ∈ I andhereditary i.e A ∈ I, B ⊆ A⇒ B ∈ I.A non-empty family of sets £(I) ⊆ 2X is said to be filter on X if and only if Φ < £(I),for A, B ∈ £(I) wehave A ∩ B ∈ £(I) and for each A ∈ £(I) and A ⊆ B implies B ∈ £(I).An Ideal I⊆ 2X is called non-trivial if I , 2X .A non-trivial ideal I ⊆ 2X is called admissible if {{x} : x ∈ X} ⊆ I.A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J , I containing I as a subset.For each ideal I, there is a filter £(I) corresponding to I. i.e £(I) = {K ⊆ N : Kc ∈ I},where Kc = N − K.The idea of modulus was structured in 1953 by Nakano. (See (Nakano, 1953)).A function f : [0,∞) −→ [0,∞) is called a modulus if:(1) f (t) = 0 if and only if t = 0,∗Corresponding authorEmail addresses: vakhan@math.com (Vakeel A. Khan), khalidebadullah@gmail.com (Khalid Ebadullah)Vakeel A. Khan et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30 23(2) f (t + u) ≤ f (t) + f (u) for all t, u ≥ 0,(3) f is nondecreasing, and,(4) f is continuous from the right at zero.Ruckle in (Ruckle, 1968) used the idea of a modulus function f to construct the sequence spaceX( f ) = {x = (xk) : ( f (|xk|)) ∈ X} (1.1)This space is an FK space, and (Ruckle, 1967) proved that the intersecton of all such X( f ) spaces is φ,the space of all finite sequences.The space X( f ) is closely related to the space l1 which is an X( f ) space with f (x) = x for all real x ≥ 0.Thus (Ruckle, 1973) proved that, for any modulus f .X( f ) ⊂ l1 and X( f )α = l∞ (1.2)WhereX( f )α = {y = (yk) ∈ ω :∞∑k=1f (|ykxk|) < ∞} (1.3)The space X( f ) is a Banach space with respect to the norm||x|| =∞∑k=1f (|xk|) < ∞.(See (Ruckle, 1973)). (1.4)Spaces of the type X( f ) are a special case of the spaces structured by B.Gramsch in (Gramsch, n.d.).From the point of view of local convexity, spaces of the type X( f ) are quite pathological. Therefore symmet-ric sequence spaces, which are locally convex have been frequently studied by D. J. H Garling in (Garling,1966) and (Garling, 1968), G. Kothe in (Köthe, 1970)and W. H. Ruckle in (Ruckle, 1968) and (Ruckle,1967).Definition 1.1. A sequence space E is said to be solid or normal if (xk) ∈ E implies (αkxk) ∈ E for allsequence of scalars (αk) with |αk| < 1 for all k ∈ N.Definition 1.2. A sequence space E is said to be monotone if it contains the cannonical preimages of all itsstepspaces.Definition 1.3. A sequence space E is said to be covergence free if (yk) ∈ E whenever (xk) ∈ E and xk = 0implies yk = 0.Definition 1.4. A sequence space E is said to be a sequence algebra if (xkyk) ∈ E whenever (xk) ∈ E,(yk) ∈ E.Definition 1.5. A sequence space E is said to be symmetric if (xπ(k)) ∈ E whenever (xk) ∈ E where π(k) isa permutation onN.Definition 1.6. A sequence (xk) ∈ ω is said to be I-convergent to a number L if for every ε > 0. {k ∈ N :|xk − L| ≥ ε} ∈ I. In this case we write I − lim xk = L.The space cI of all I-convergent sequences to L is given bycI = {(xk) ∈ ω : {k ∈ : |xk − L| ≥ ε} ∈ I, for some L ∈ C}. (1.5)24 Vakeel A. Khan et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30Definition 1.7. A sequence (xk) ∈ ω is said to be I-null if L = 0. In this case we write I − lim xk = 0.Definition 1.8. A sequence (xk) ∈ ω is said to be I-cauchy if for every ε > 0 there exists a number m = m(ε)such that {k ∈ N : |xk − xm| ≥ ε} ∈ I.Definition 1.9. A sequence(xk) ∈ ω is said to be I-bounded if there exists M >0 such that {k ∈ N : |xk| >M} ∈ IDefinition 1.10. A modulus function f is said to satisfy 42 condition if for all values of u there exists aconstant K > 0 such that f (Lu) ≤ KL f (u) for all values of L > 1.Definition 1.11. Take for I the class I f of all finite subsets of N. Then I f is a non-trivial admissible idealand I f convergence coincides with the usual convergence with respect to the metric in X.(see (Kostyrko etal., 2000)).Definition 1.12. For I = Iδ and A ⊂ N with δ(A) = 0 respectively. Iδ is a non-trivial admissible ideal,Iδ-convergence is said to be logarithmic statistical covergence.(see (Kostyrko et al., 2000)).Definition 1.13. A map ~ defined on a domain D ⊂ X i.e ~ : D ⊂ X → R is said to satisfy Lipschitzcondition if |~(x) − ~(y)| ≤ K|x − y| where K is known as the Lipschitz constant. The class of K-Lipschitzfunctions defined on D is denoted by ~ ∈ (D,K) (see (Šalát et al., 2004)).Definition 1.14. A convergence field of I-covergence is a setF(I) = {x = (xk) ∈ l∞ : there exists I − lim x ∈ R}. (1.6)The convergence field F(I) is a closed linear subspace of l∞ with respect to the supremum norm, F(I) =l∞ ∩ cI (See (Šalát et al., 2004)).Define a function ~ : F(I) → R such that ~(x) = I − lim x, for all x ∈ F(I), then the function~ : F(I)→ R is a Lipschitz function (see (Šalát et al., 2004)).(c.f (Connor & Kline, 1996), (Dems, 2005), (Gurdal, 2004), (Jones & Retherford, 1967), (Kamthan &Gupta, 1980), (Maddox, 1970), (Maddox, 1986), (Maddox, 1969), (Šalát, 1980), (Singer, 1970), (Wilansky,1964))Throughout the article l∞, cI , cI0,mI and mI0 represent the bounded , I-convergent, I-null, bounded I-convergent and bounded I-null sequence spaces respectively.In this article we introduce the following classes of sequence spaces:cI( f ) = {(xk) ∈ ω : I − lim f (|xk|) = L for some L} ∈ I (1.7)cI0( f ) = {(xk) ∈ ω : I − lim f (|xk|) = 0} ∈ I (1.8)lI∞( f ) = {(xk) ∈ ω : supkf (|xk|) < ∞} ∈ I (1.9)We also denote bymI( f ) = cI( f ) ∩ l∞( f ) (1.10)andmI0( f ) = cI0( f ) ∩ l∞( f ) (1.11)The following Lemmas will be used for establishing some results of this article:Vakeel A. Khan et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30 25Lemma 1.1. Let E be a sequence space. If E is solid then E is monotone.Lemma 1.2. Let K ∈ £(I) and M ⊆ N. If M < I, then M ∩ N < I.Lemma 1.3. If I ⊂ 2N and M ⊆ N. If M < I, then M ∩ N < I.2. Main resultsTheorem 2.1. For any modulus function f , the classes of sequences cI( f ), cI0( f ),mI( f ) and mI0( f ) are linearspaces.Proof. We shall prove the result for the space cI( f ). The proof for the other spaces will follow similarly.Let (xk), (yk) ∈ cI( f ) and let α, β be scalars. ThenI − lim f (|xk − L1|) = 0, for some L1 ∈ c; (2.1)I − lim f (|yk − L2|) = 0, for some L2 ∈ c; (2.2)That is for a given ε > 0, we haveA1 = {k ∈ N : f (|xk − L1|) >ε2} ∈ I, (2.3)A2 = {k ∈ N : f (|yk − L2|) >ε2} ∈ I. (2.4)Since f is a modulus function, we havef (|(αxk + βyk) − (αL1 + βL2)|) ≤ f (|α||xk − L1|) + f (|β||yk − L2|) ≤ f (|xk − L1|) + f (|yk − L2|) (2.5)Now, by (2.3) and (2.4),{k ∈ N : f (|(αxk + βyk) − (αL1 + βL2)|) > ε} ⊂ A1 ∪ A2. (2.6)Therefore(αxk + βyk) ∈ cI( f ). (2.7)Hence cI( f ) is a linear space.Theorem 2.2. A sequence x = (xk) ∈ mI( f ) I-converges if and only if for every ε > 0 there exists Nε ∈ Nsuch that{k ∈ N : f (|xk − xNε |) < ε} ∈ mI( f ) (2.8)Proof. Suppose thatL = I − lim x. ThenBε = {k ∈ N : |xk − L| <ε2} ∈ mI( f ). For allε > 0. (2.9)Fix an Nε ∈ Bε .Then we have|xNε − xk| ≤ |xNε − L| + |L − xk| <ε2+ε2= ε (2.10)26 Vakeel A. Khan et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30which holds for all k ∈ Bε . Hence{k ∈ N : f (|xk − xNε |) < ε} ∈ mI( f ). (2.11)Conversely, suppose that {k ∈ N : f (|xk − xNε |) < ε} ∈ mI( f ).That is {k ∈ N : (|xk − xNε |) < ε} ∈ mI( f ) for all ε > 0. Then the setCε = {k ∈ N : xk ∈ [xNε − ε, xNε + ε]} ∈ mI( f )for allε > 0. (2.12)Let Jε = [xNε − ε, xNε + ε]. If we fix an ε > 0 then we have Cε ∈ mI( f ) as well as C ε2∈ mI( f ). HenceCε ∩C ε2 ∈ mI( f ). This implies thatJε ∩ J ε2 , φ (2.13)that is{k ∈ N : xk ∈ J} ∈ mI( f ) (2.14)that isdiamJ ≤ diamJε (2.15)where the diam of J denotes the length of interval J.In this way, by induction we get the sequence of closed intervalsJε = I0 ⊇ I1 ⊇ ..... ⊇ Ik ⊇ ........... (2.16)with the property that diamIk ≤ 12 diamIk−1 for (k=2,3,4,.....) and {k ∈ N : xk ∈ Ik} ∈ mI( f ) for (k =1, 2, 3, 4, ......).Then there exists a ξ ∈ ∩Ik where k ∈ N such that ξ = I − lim x. So that f (ξ) = I − lim f (x), that isL = I − lim f (x).Theorem 2.3. Let f and g be modulus functions that satisfy the 42-condition.If X is any of the spacescI , cI0,mI and mI0 etc, then the following assertions hold.(1) X(g) ⊆ X( f .g),(4) X( f ) ∩ X(g) ⊆ X( f + g).Proof. (1) Let (xk) ∈ cI0(g).ThenI − limkg(|xk|) = 0 (2.17)Let ε > 0 and choose δ with 0 < δ < 1 such that f (t) < ε for 0 < t < δ. Write yk = g(|xk|) and considerlimkf (yk) = limkf (yk)yk<δ + limkf (yk)yk>δ. We havelimkf (yk) ≤ f (2) limk(yk). (2.18)For yk > δ, we have yk <ykδ < 1 +ykδ . Since f is non-decreasing, it follows thatf (yk) < f (1 +ykδ) <12f (2) +12f (2ykδ). (2.19)Since f satisfies the 42-condition, we havef (yk) <12Kykδf (2) +12Kykδf (2) = Kykδf (2). (2.20)Vakeel A. Khan et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30 27Hencelimkf (yk) ≤ max(1,K)δ−1 f (2) limk(yk). (2.21)From (2.17), (2.18) and (2.21), we have (xk) ∈ cI0( f .g).Thus cI0(g) ⊆ cI0( f .g). The other cases can be proved similarly.(2) Let (xk) ∈ cI0( f ) ∩ cI0(g). ThenI − limkf (|xk|) = 0 (2.22)andI − limkg(|xk|) = 0 (2.23)The rest of the proof follows from the following equalitylimk( f + g)(|xk|) = limkf (|xk|) + limkg(|xk|). (2.24)Corollary 2.4. X ⊆ X( f ) for X = cI , cI0,mI and mI0.Theorem 2.5. The spaces cI0( f ) and mI0( f ) are solid and monotone .Proof. We shall prove the result for cI0( f ). Let xk ∈ cI0( f ).ThenI − limkf (|xk|) = 0 (2.25)Let (αk) be a sequence of scalars with |αk| ≤ 1 for all k ∈ N. Then the result follows from (2.25) and thefollowing inequalityf (|αkxk|) ≤ |αk| f (|xk|) ≤ f (|xk|) for all k ∈ N. (2.26)That the space cI0(f) is monotone follows from the Lemma 1.1.For mI0( f ) the result can be proved similarly.Theorem 2.6. The spaces cI( f ) and mI( f ) are neither solid nor monotone in general.Proof. Here we give a counter example.Let I = Iδ and f (x) = x2 for all x ∈ [0,∞). Consider the K-step space XK( f ) of X defined as follows.Let (xk) ∈ X and let (yk) ∈ XK be such that(yk) ={(xk), if k is even,0, otherwise.(2.27)Consider the sequence (xk) defined by (xk) = 1 for all k ∈ N. Then (xk) ∈ cI( f ) but its K-stepspace preimagedoes not belong to cI( f ). Thus cI( f ) is not monotone. Hence cI( f ) is not solid.Theorem 2.7. The spaces cI( f ) and cI0( f ) are sequence algebras.28 Vakeel A. Khan et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30Proof. We prove that cI0( f ) is a sequence algebra. Let (xk), (yk) ∈ cI0( f ). ThenI − lim f (|xk|) = 0andI − lim f (|yk|) = 0. (2.28)Then we haveI − lim f (|(xk.yk)|) = 0. (2.29)Thus (xk.yk) ∈ cI0( f ) is a sequence algebra. For the space cI( f ), the result can be proved similarly.Theorem 2.8. The spaces cI( f ) and cI0( f ) are not convergence free in general.Proof. Here we give a counter example.Let I = I f and f (x) = x3 for all x ∈ [0,∞). Consider the sequence (xk) and (yk) defined byxk =1kand yk = k for all k ∈ NThen (xk) ∈ cI( f ) and cI0( f ), but (yk) < cI( f ) and cI0( f ).Hence the spaces cI( f ) and cI0( f ) are not convergence free.Theorem 2.9. If I is not maximal and I , I f , then the spaces cI( f ) and cI0( f ) are not symmetric.Proof. Let A ∈ I be infinite and f (x) = x for all x∈ [0,∞).Ifxk ={1, for k ∈ A,0, otherwise.Then by lemma 1.3 xk ∈ cI0( f ) ⊂ cI( f )let K ⊂ N be such that K < I and N − K < I. Let φ : K → A and ψ : N − K → N − A be bijections, thenthe map π : N→ N defined byπ(k) ={φ(k), for k ∈ K,ψ(k), otherwise.is a permutation on N, but xπ(k) < cI( f ) and xπ(k) < cI0( f ).Hence cI0( f ) and cI( f )are not symmetric.Theorem 2.10. Let f be a modulus function. Then cI0( f ) ⊂ cI( f ) ⊂ lI∞( f ) and the inclusions are proper.Proof. Let xk ∈ cI( f ). Then there exists L ∈ C such thatI − lim f (|xk − L|) = 0. (2.30)We have f (|xk|) ≤ 12 f (|xk − L|) + f12 (|L|).Taking the supremum over k on both sides we get xk ∈ l∞( f ).The inclusion cI0( f ) ⊂ cI( f ) is obvious.Theorem 2.11. The function ~ : mI( f ) → R is the Lipschitz function,where mI( f ) = cI( f ) ∩ l∞( f ), andhence uniformly cotinuous.Vakeel A. Khan et al. / Theory and Applications of Mathematics & Computer Science 1 (2) (2011) 22–30 29Proof. Let x, y ∈ mI( f ), x , y. Then the setsAx = {k ∈ N : |xk − ~(x)| ≥ ||x − y||} ∈ I, (2.31)Ay = {k ∈ N : |yk − ~(y)| ≥ ||x − y||} ∈ I. (2.32)Thus the sets,Bx = {k ∈ N : |xk − ~(x)| < ||x − y||} ∈ mI( f ), (2.33)By = {k ∈ N : |yk − ~(y)| < ||x − y||} ∈ mI( f ). (2.34)Hence also B = Bx ∩ By ∈ mI( f ), so that B , φ. Now taking k in B,|~(x) − ~(y)| ≤ |~(x) − xk| + |xk − yk| + |yk − ~(y)| ≤ 3||x − y||. (2.35)Thus ~ is a Lipschitz function. For mI0( f ) the result can be proved similarly.Theorem 2.12. If x, y ∈ mI( f ), then (x.y) ∈ mI( f ) and ~(xy) = ~(x)~(y).Proof. For ε > 0Bx = {k ∈ N : |xk − ~(x)| < ε} ∈ mI( f ), (2.36)By = {k ∈ N : |yk − ~(y)| < ε} ∈ mI( f ). (2.37)Now,|xkyk − ~(x)~(y)| = |xkyk − xk~(y) + xk~(y) − ~(x)~(y)|≤ |xk||yk − ~(y)| + |~(y)||xk − ~(x)| (2.38)As mI( f ) ⊆ l∞( f ), there exists an M ∈ R such that |xk| < M and |~(y)| < M. Using (2.38) we get|xkyk − ~(x)~(y)| ≤ Mε + Mε = 2Mε.For all k ∈ Bx ∩ By ∈ mI( f ). Hence (x.y) ∈ mI( f ) and ~(xy) = ~(x)~(y). 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On Some I-Convergent Sequence Spaces Defined by a Modulus Function

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On Some I-Convergent Sequence Spaces Defined by a Modulus Function

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