On generalized difference Zweier ideal convergent sequences space defined by Musielak-Orlicz functions

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1 Bol. Soc. Paran. Mat. (3s.) v (2017): c SPM ISSN on line ISSN in ress SPM: doi: /bsm.v35i On generalized difference Zweier ideal convergent sequences sace defined by Musiela-Orlicz functions Karan Tamang and Bian Hazaria abstract: Let M = (M ) be a Musiela-Orlicz function. In this article, we introduce a new class of ideal convergent sequence saces defined by Musiela- Orlicz function, using an infinite matrix, and a generalized difference Zweier matrix oerator B. We investigate some toological structures and algebraic roerties of these saces. We obtain some relations related to these sequence saces. Key Words: ideal convergence; difference sace; Musiela-Orlicz function. Contents 1 Introduction 19 2 Ideal convergence in a locally convex sace 24 3 Difference Zweier ideal convergent sequences in a locally convex sace Introduction The notion of the ideal convergence is the dual (equivelant) to the notion of filter convergence introduced by Cartan in 1937 [4]. The notion of the filter convergence is a generalization of the classical notion of convergence of a sequence and it has been an imortant tool in general toology and functional analysis. Nowadays many authors to use an equivalent dual notion of the ideal convergence. Kostyro et al. [24] and Nuray and Rucle [30] indeendently studied in detalis about the notion of ideal convergence which is based on the structure of the admissible ideal I of subsets of natural numbers N. Later on it was further investigated by many authors, e.g. Triathy and Hazaria [36,37], Hazaria [12], Hazaria and Savaş [11] and references therein. Hazaria [14] introduced the concet of generalized difference ideal convergent sequences of fuzzy numbers and studied some interesting roerties. Esi [6] introduced strongly almost summable sequence saces in 2-normed saces defined by ideal convergence and an Orlicz function and roved some interesting results. Before roceeding let us recall a few concets, which we shall use throughout this aer Mathematics Subject Classification: 40A05; 40B50; 46A19; 46A Tyeset by B S P M style. c Soc. Paran. de Mat.

2 20 Karan Tamang and Bian Hazaria Let X be a non-emty set, then a family of sets I 2 X (the class of all subsets of X) is called an ideal if and only if for each A,B I we have A B I and for each A I and each B A we have B I. A non-emty family of sets F 2 X is a filter on X if and only if φ / F, for each A,B F we have A B F and each A F and each B A we have B F. An ideal I is called non-trivial ideal if I φ and X / I. Clearly I 2 X is a non-trivial ideal if and only if F = F(I) = {X A : A I} is a filter on X. A non-trivial ideal I 2 X is called admissible if and only if {{x} : x X} I. A non-trivial ideal I is maximal if there cannot exists any non-trivial ideal J I containing I as a subset. Recall that a sequence x = (x ) of oints in R is said to be I-convergent to a real number l if { N : x l ε} I for everyε > 0 ([24]). In this case we write I limx = l. Throughout the article w,l,c,c 0, denote for the classes of all, bounded, convergent, null sequences of comlex numbers, resectively. The notion of difference sequence sace was introduced by Kizmaz [23], who studied the difference sequence saces l ( ), c( ), c 0 ( ). The notion was further generalized by Et and Cola [8] introducing the sequence saces l ( ), c( ), c 0 ( ). For a non negative integer, the generalized difference sequence saces are defined as follows. For a given sequence sace Z we have Z( ) = {x = (x ) w : ( x ) Z}, where x = 1 x 1 x +1, 0 x = x, for all N, the difference oerator is equivalent to the following binomial reresentation: x = ( ) ( 1) ν x +ν for all N. ν ν=0 Taing = 1, we get the saces l ( ),c( ),c 0 ( ), introduced and studied by Kizmaz [23]. Triathy and Esi [34] introduced and studied the new tye of generalized difference sequence saces Z( i ) = {(x ) w : i x Z}, for Z = l,c,c 0 where i x = ( i x ) = (x x +i ) for all,i N. Triathy, et al [35] further generalized this notion and introduced the following sequence saces. For 1 and i 1, Z( i ) = {(x ) w : i x Z}, for Z = l,c,c 0.This generalized difference has the following binomial reresentation, ( ) i x = ( 1) ν x +iν for all N. ν ν=0

3 On generalized difference Zweier ideal convergent sequences sace 21 Dutta [5] introduced the following difference sequence saces Z( ) = {(x ) w : x Z} for all,i N, for Z = l,c,c 0 where c,c 0 are the sets of statistically convergent and statistically null sequences, resectively, and x = ( x ) = ( 1 x 1 x i) and 0 x = x for all N, which is equivalent to the following binomial reresentation: ( ) x = ( 1) ν x iν. ν ν=0 Basar and Altay [1] introduced the generalized difference matrixb(r,s)=(b (r,s)) which is a generalization of 1 (1)-difference oerator as follows: r, if = ; b (r,s) = s, if = 1; 0, if 0 < 1 or >. for all, N,r,s R {0}. Basarir and Kayici [2] have defined the generalized difference matrixb of order, which reduced the difference oerator (1) in case r = 1,s = 1 and the binomial reresentation of this oerator is B x = ν=0 ( ) r ν s ν x ν, ν where r,s R {0} and N. Recently Basarir et al., [3] introduced the following generalized difference sequence saces Z(B ) = {(x ) w : B x Z} for all,i N, for Z = l,c,c 0 where c,c 0 are the sets of statistically convergent and statistically null sequences, resectively, and B x = (B x ) = (rb 1 x +sb 1 x i ) and B 0 x = x for all N, which is equivalent to the following binomial reresentation: ( ) B x = r ν s ν x iν. ν ν=0 Let X and Y be two nonemty subsets of the sace w of comlex sequences. Let A = ( ),(n, = 1,2,3,...) be an infinite matrix of comlex numbers. We write Ax = (A n (x)) if A n (x) = x converges for each n. If x = (x ) X Ax = (A n (x)) Y we say that A defines a (matrix) transformation from X to Y and we denote it by A : X Y.

4 22 Karan Tamang and Bian Hazaria Şengönül [33] defined the sequence y = (y ) which is frequently used as the Z-transformation of the sequence x = (x ) i.e. y = αx +(1 α)x 1 where x 1 = 0, 0,1 < < and Z denotes the matrix Z = (z n ) defined by α, if n = ; z n = 1 α, if n 1 = ; 0, otherwise. Şengönül [33] introduced the Zweier sequence saces Z and Z 0 as follows Z = {x = (x ) w : Z(x) c} and Z 0 = {x = (x ) w : Z(x) c 0 }. For details on Zweier sequence saces we refer to [7,15,16,18,19,21,22]. A function M : [0, ) [0, ) is called an Orlicz function if it is continuous, non-decreasing and convex with M(0) = 0,M(x) > 0 as x > 0 and M(x) as x (see [25]). An Orlicz function M can always be reresented in the following integral form: M(x) = x 0 (t)dt where is the nown ernel of M, right differentiable for t 0,(0) = 0,(t) > 0 for t > 0 and (t) as t. If convexity of Orlicz function is relaced by M(x+y) M(x) +M(y) then this function is called the modulus function and characterized by Naano [29], followed by Rucle [32]. An Orlicz function M is said to satisfy 2 condition for all values of u, if there exists K > 0 such that M(2u) KM(u),u 0. Two Orlicz functions M 1 and M 2 are said to be equivalent if there exist ositive constants α,β and x 0 such that for all x with 0 x < x 0. M 1 (α) M 2 (x) M 1 (β) Lindenstrauss and Tzafriri [27] studied some Orlicz tye sequence saces defined as follows: { ( ) } x l M = (x ) w : M <, for some > 0.

5 On generalized difference Zweier ideal convergent sequences sace 23 The sace l M with the norm x = inf { > 0 : ( ) } x M 1 becomes a Banach sace which is called an Orlicz sequence sace. The sace l M is closely related to the sace l which is an Orlicz sequence sace with M(t) = t for 1 <. A sequence M = (M ) of Orlicz functions is called a Musiela-Orlicz function (for details see [10,13,17,20]). Also a Musiela-Orlicz function φ = (φ ) is called a comlementary function of a Musiela-Orlicz function M if φ (t) = su{ t s M (s) : s 0}, for = 1,2,3,... For a given Musiela-Orlicz function M, the Musiela-Orlicz sequence sace l M and its subsace h M are defined as follows: l M = {x = (x ) w : I M (cx) <, for some c > 0}; h M = {x = (x ) w : I M (cx) <, for all c > 0}, where I M is a convex modular defined by I M = M (x ),x = (x ) l M. We consider l M equied with the Luxemburg norm { ( x ) x = inf > 0 : I M or equied with the Orlicz norm x 0 = inf } 1 { } 1 (1+I M(x)) : > 0. The following well-nown inequality will be used throughout the article. Let = ( ) be any sequence of ositive real numbers with 0 su = G, D = max{1,2 G 1 } then a +b D( a + b ) for all N and a,b C. Also a max{1, a G } for all a C. Subsequently Orlicz function was used to define sequence saces by Parashar and Choudhary [31] and many others (see [9,26,28,38]). Remar 1.1. It is well nown if M is a convex function and M(0) = 0, then M(λx) λm(x), for all λ with 0 < λ < 1. Throughout the aer X we denote a locally convex Hausdorff toological linear sace whose toology is determined by a set Q of continuous seminorms q. Also we denote I is an non-trivial admissible ideal of N.

6 24 Karan Tamang and Bian Hazaria 2. Ideal convergence in a locally convex sace In this section we define I-convergence in a locally convex sace X and investigate some basic roerties. Definition 2.1. A sequence x = (x ) in X is said to be I q -convergent to l X if for all q Q and all ε > 0, { N : q(x l) ε} I. In this case we can write I q limx = l. We denote I q = {{ N : q(x l) ε} I}. Further, since X is Hausdorff, the limit of ideal convergent sequence is unique. Definition 2.2. A sequence x = (x ) in X is said to be B (I q)-convergent to l X if for all q Q and all ε > 0, { N : q(b x l) ε} I. In this case we can write I q limb (x) = l. We denote where B (I q) = {{ N : q(b x l) ε} I}., B x = ν=0 ( ) α ν (1 α) ν x iν. ν Definition 2.3. Let M be a Musiela-Orlicz function. We say that a sequence x = (x ) in w I (B,M) if and only if there exists l X such that for all q Q and for every ε > 0, { n N : 1 n ( q(b [M x )] } l) ε I for > 0. (2.1) When (2.1) holds we write x l(w I (B,M)). The condition (2.1) rovides a definition of ideal summability for a sequence in a locally convex sace. Theorem 2.1. Let A = ( ) be a non-negative reguler matrix and u = (u ) be a bounded sequence of ositive real numbers. Let M be a Musiela-Orlicz function. Then x l(w(m,a,u)) imlies that x l(b (I q)(a)).

7 On generalized difference Zweier ideal convergent sequences sace 25 Proof: Let q Q. Assume that x l(w(m,a,u)), then for > 0 we have lim n [ ( q(b M x )] u l) = 0 for l C. Let ε > 0 be given. We define { } K(ε) = N : q(b x l) ε and we write [ ( q(b M x )] u l) r = [ ( q(b M x )] u l) + [ ( q(b M x l) r r K(ε) K(ε) Then we have x l(b (I q)(a)). / K(ε) [ M ( ε r)] u. )] u Theorem 2.2. Let A = ( ) be a non-negative reguler matrix and u = (u ) be a bounded sequence of ositive real numbers. Let M be a Musiela-Orlicz function. If x = (x ) l (B ) and x l(b (I q)(a)), then x l(w(m,a,u)). Proof: Suose that x = (x ) l (B ) and x l((b (I q))(a)). Then there is a set K F(B (I q)) such that Now = K(ε) lim K q(b x l) = 0. )] u [ ( q(b M x l) r [ ( q(b M x )] u l) + [ ( q(b M x l) r r / K(ε) )] u ( q(b = χ K () [M x )] u ( l) q(b + χ r K c() [M x )] u l). r If we consider the regularity of A, K c B (I q) and boundedness of (x ) right side tends to zero. Hence x l(w(m,a,u)).

8 26 Karan Tamang and Bian Hazaria 3. Difference Zweier ideal convergent sequences in a locally convex sace In this section we define some new classes of difference ideal convergent sequences by using infinite matrix and investigate their linear toological structures. Also we find out some relations related to these saces. Let I be an admissible ideal of N, u = (u ) be a bounded sequence of ositive real numbers and A = ( ) be an infinite matrix. Let M be a Musiela-Orlicz function. Further w(x) denotes the sace of all X-valued sequences. For each ε > 0, for all q Q and for > 0 we define the following sequence saces. [Z(A,B,M,u,q)]I = x = (x ) w(x) : [Z 0 (A,B, M,u,q)]I = x = (x ) w(x) : [Z (A,B, M,u,q)]I = q B x l u ε I for l X, x = (x ) w(x) : K > 0s.t. [Z (A,B,M,u,q)] = x = (x ) w(x) : su Particular cases: n N q B x u ε I, q B x u K I, q B x u <. If = 1, then above saces are denoted by [Z(A,B,M,u,q)] I, [Z 0 (A,B, M,u,q)] I, [Z (A,B,M,u,q)] I and [Z (A,B,M,u,q)]. (ii) If i = 1 then above saces are denoted by [Z(A,B,M,u,q)] I, [Z 0 (A,B,M, u,q)] I, [Z (A,B,M,u,q)] I and [Z (A,B,M,u,q)]. (iii) If M (x) = x for all x [0, ), N then we obtain the above saces as [Z(A,B,u,q)]I,[Z 0 (A,B,u,q)]I,[Z (A,B,u,q)]I and[z (A,B,u,q)].

9 On generalized difference Zweier ideal convergent sequences sace 27 (iv) If u = (u ) = (1,1,1...), then above saces are denoted by [Z(A,B,M,q)]I, [Z 0 (A,B,M,q)]I, [Z (A,B,M,q)]I and [Z (A,B,M,q)]. (v) If we tae A = (C,1) ), i.e., the Cesàro matrix, then the above classes of sequences are denoted by [Z(B,M,u,q)]I, [Z 0 (B,M,u,q)]I, [Z (B,M, u,q)] I and [Z (B,M,u,q)]. (vi) If we tae A = ( ) is a de la Vallée Poussin mean, i.e., { 1 = λ n, if I n = [n λ n +1,n]; 0, otherwise. where (λ n ) is a non-decreasing sequence of ositive numbers tending to and λ n+1 λ n + 1,λ 1 = 1, then the above classes of sequences are denoted by [Z(λ,B,M,u,q)]I, [Z 0 (λ,b,m,u,q)]i, [Z (λ,b,m,u,q)]i and [Z (λ,b,m,u,q)]. (vii) By a lacunary sequence θ = ( r ), where 0 = 0, we shall mean an increasing sequence of non-negative integers with r r 1 as r. The intervals determined by θ will be denoted by J r = ( r 1, r ] and we let h r = r r 1. As a final illustration let { 1 = h r, if I r = ( r 1, r ]; 0, otherwise. Then the above classes of sequences are denoted by [Z(θ,B,M,u,q)]I, [Z 0 (θ,b,m,u,q)]i, [Z (θ,b,m,u,q)]i and [Z (θ,b,m,u,q)]. Theorem 3.1. [Z(A,B,M,u,q)]I, [Z 0 (A,B,M,u,q)]I and [Z (A,B,M,u, q)] I are linear saces. Proof: We will roved the result for the sace [Z 0 (A,B,M,u,q)]I only and the others can be roved in similar way. Let x = (x ) and y = (y ) be two elements in [Z 0 (A,B,M,u,q)]I. Then there exist 1 > 0 and 2 > 0 such that and Aε 2 = Bε 2 = q B x u 2 I 1 q B y u 2 I. Let α,β be two scalars in R. Since B is linear and the continuity of the Musiela-Orlicz function M, the following inequality holds: q B (αx u +βy ) α 1 + β 2 2

10 28 Karan Tamang and Bian Hazaria DK D +D ( where K = max{1, q B x α M q B x α 1 + β 2 1 β M q B y α 1 + β 2 1 ) α 1 α 1 + β 2 (, From the above relation we get n N : DK +DK β 2 α 1 + β )}. 2 2 u u q B y 2 q B (αx +βy ) u ε ( α 1 + β 2 ) n N : DK q B x u 2 1 u, q B y u 2. (3.1) Since both of the sets on the right hand of (3.1) are belong to I, this comletes the roof of the theorem. 2 Remar 3.1. It is easy to verify that the sace [Z (A,B,M,u,q)] is a linear sace. Theorem 3.2. Let S = (S ) and T = (T ) be Musiela-Orlicz functions. Then the following holds: [Z 0 (A,B,S,u,q)]I [Z 0 (A,B,T,u,q)]I [Z 0 (A,B,S+T,u,q)]I. Proof: Letx = (x ) [Z 0 (A,B,S,u,q)]I [Z 0 (A,B,T,u,q)]I. Then the result follows from the inequality (S +T ) q B x u

11 On generalized difference Zweier ideal convergent sequences sace 29 D S q B x u +D T q B x. Theorem 3.3. Let S = (S ) and T = (T ) be Musiela-Orlicz functions. Then the following holds: rovided h = infu > 0. [Z 0 (A,B,T,u,q)]I [Z 0 (A,B,ST,u,q)]I Proof: For a givenε>0, we first chooseε 0 > 0 such that su n ( n )max{ε h 0, ε H 0 } < ε. Using the continuity of M, choose 0 < δ < 1 such that 0 < δ < t imlies that S (t) < ε 0 for all N. Let x = (x ) [Z 0 (A,B,T,u,q)]I. For some > 0 we denote n A 5 = T q B x u δ H I. If n / A 5, then we have T q B x u < δ H i.e. T q B x u < δ H for all N ( ) i.e.t q B x < δ for all N ( ) i.e.s T q B x < ε 0 for all N. Consequently, we get ( ) S T q B x ( n u < su )max{ε h 0,εH 0 } < ε. n i.e. ( ) S T q B x u < ε.

12 30 Karan Tamang and Bian Hazaria This shows that n ( ) S T q B x u ε A 5 I. This comletes the roof. Theorem 3.4. The inclusions [Y(A,B 1,M,u,q)] I [Y(A,B,M,u,q)]I, are strict for 1. In general [Y(A,B j,m,u,q)]i [Y(A,B,M,u,q)]I, for j = 0,1,2,..., 1 and the inclusions are strict, where Y = Z 0,Z,Z. Proof: We shall give the roof for [Z 0 (A,B 1,M,u,q)] I only. The others can be roved by similar arguments. Let x = (x ) be any element in the sace [Z 0 (A,B 1,M,u,q)] I. Let ε > 0 be given. Then there exists δ > 0 such that the set q B 1 x ε I. Since M is non-decreasing and convex, it follows that q B x ( = q 2 D DH ( 1 2 M q ( q B 1 ) x +1 ) x +1 B 1 +D +DH ) x +1 B 1 x 2 B 1 ( 1 2 M q ( q ) x B 1 B 1 ) x, where H = max{1,( 1 2 )G }. Thus we have n N : DH n N : DH q B x ε 2 ( q ( q ) x +1 2 B 1 ) x 2 B 1 (3.2)

13 On generalized difference Zweier ideal convergent sequences sace 31 Since both the sets in the right side of (3.2) belongs to I, we get q B x ε 2 I. If follow from the following examle that the inclusion is strict. Examle 3.1. Let A = (C,1), M (x) = x, for all x [0, ),u = 1 for all N and r = 1,s = 1. Consider a sequence x = (x ) = ( ). Then x = (x ) belongs to [Z 0 (A,B,M,u,q)]I but does not belong to [Z 0 (A,B 1,M,u,q)] I, because B x = 0 and B 1 x = ( 1) 1 ( 1)!. Theorem 3.5. (a) Let 0 < infu u 1, then [Z(A,B,M,u,q)]I [Z(A, B,M,q)]I and [Z 0 (A,B,M,u,q)]I [Z 0 (A,B,M,q)]I. (b) If 1 < u suu <, then [Z(A,B,M,q)]I [Z(A,B,M,u,q)]I and [Z 0 (A,B,M,q)]I [Z 0 (A,B,M,u,q)]I. Proof: (a) Let x = (x ) [Z(A,B,M,u,q)]I. Since 0 < infu u 1, we have q B x l q B x l and therefore q B x l ε M q B x l ε I. (b) Let 1 < u suu < and let x = (x ) [Z(A,B,M,q)]I. Then for each 0 < ε < 1 there exists a ositive integer N such that q B x l ε < 1 for all n N. This imlies that q B x l q B x l.

14 32 Karan Tamang and Bian Hazaria Thus we have q B x l ε q B x l ε I. This comletes the roof of the theorem. Corollary 3.1. Let A = (C,1) Cesáro matrix and let M = (M ) be a Musiela- Orlicz function. (a) If 0 < infu u 1, then [Z(B,M,u,q)]I [Z(B,M,q)]I ; (ii) [Z 0 (B,M,u,q)]I [Z 0 (B,M,q)]I. (b) If 1 < u suu <, then [Z(B,M,q)]I [Z(B,M,u,q)]I ; (ii) [Z 0 (B,M,q)]I [Z 0 (B,M,u,q)]I. Theorem 3.6. Let 0 < u v for all N and [Z(A,B,M,v,q)]I [Z(A,B,M,u,q)]I. ( ) v u is bounded, then Proof: The roof of the theorem is straightforward, so we should omitted here. Theorem 3.7. If lim u > 0 and x = (x ) x 0 ([Z(A,B,M,u,q))]I, then x 0 is unique. Proof: Let lim u = u 0 > 0. Suose that (x ) x 0 ([Z(A,B,M,u,q)]I ) and (x ) y 0 ([Z(A,B,M,u,q)]I ). Then there exist 1, 2 > 0 such that B 1 = q B x u x 0 2 I (3.3) 1 and B 2 = q B x u y 0 I. (3.4) 2 1

15 On generalized difference Zweier ideal convergent sequences sace 33 Let = max{2 1,2 2 }. Then we have D q B x x 0 1 Thus from (3.3) and (3.4) we have { n N : [ ( )] u q(x0 y 0 ) M n N : D n N : D u +D [ ( )] u q(x0 y 0 ) M ε} q B x y 0 1 q B x u x q B x u y 0 2 B 1 B 2 I. Also we have [ ( )] u [ ( )] u0 q(x0 y 0 ) q(x0 y 0 ) M M as. Therefore we have [ ( )] u [ ( )] u0 q(x0 y 0 ) q(x0 y 0 ) M M = 0. 1 u. Hence x 0 = y 0. Theorem 3.8. Let M = (M ) be a Musiela-Orlicz function. Then the following statements are equivalent: [Z (A,B,u,q)]I [Z (A,B,M,u,q)]I (ii) [Z 0 (A,B,u,q)]I [Z (A,B,M,u,q)]I (iii) su n n [M ( t )] u < (t, > 0). Proof: (ii) is obvious, because [Z 0 (A,B,u,q)]I [Z (A,B,u,q)]I.

16 34 Karan Tamang and Bian Hazaria (ii) (iii). Suose [Z 0 (A,B,u,q)]I [Z (A,B,M,u,q)]I. We assume that (iii) is not satisfied. Then for some t, > 0 su n [ M ( t )] u =, and therefore there exists a sequence (n j ) of ositive integers such that n j a nj ( )] j 1 u [M > j,j = 1,2,3,... (3.5) Define a sequence x = (x ) by { 1 B x = j, if 1 n j,j = 1,2,3,...; 0, if > n j Then x = (x ) [Z 0 (A,B,u,q)]I but by equation (3.5) we have x = (x ) / [Z (A,B,M,u,q)]I which contradicts (ii). Hence (iii) must hold. (iii) Suose (iii) is satisfied and x [Z (A,B,u,q)]I. Suose that x / [Z (A,B,M,u,q)]I. Then su n q B x u =. (3.6) ( ) Put t = q Z(B x ) for all N. Then by the equation (3.6) we have su n [ M ( t )] u = which contradicts (iii). Hence must hold. Theorem 3.9. Let M = (M ) be a Musiela-Orlicz function. Let 1 u su u <. Then the following statements are equivalent: [Z 0 (A,B,M,q)]I [Z 0 (A,B,u,q)]I (ii) [Z 0 (A,B,M,u,q)]I [Z (A,B,u,q)]I (iii) inf n n Proof: (ii) is obvious. [M ( t )] u > 0 (t, > 0).

17 On generalized difference Zweier ideal convergent sequences sace 35 (ii) (iii). Suose [Z 0 (A,B,M,u,q)]I [Z (A,B,u,q)]I. We assume that (iii) does not hold. Then for some t, > 0 inf n [ M ( t )] u = 0. We can choose an index sequence (n j ) of ositive integers such that n j a nj Define a sequence x = (x ) by [ ( )] u i M > 1,j = 1,2,3,... (3.7) j { B j, if 1 x nj,j = 1,2,3,...; = 0, if > n j Then by equation (3.7) we have x = (x ) [Z 0 (A,B,M,u,q)]I but x = (x ) / [Z (A,B,u,q)]I which contradicts (ii). Hence (iii) must hold. (iii) Let (iii) hold and x [Z 0 (A,B,M,u,q)]I. Then for every ε > 0, we have n q B x u ε I. (3.8) Suose that x / [Z 0 (A,B,u,q)]I. Then for some integer ε 0 > 0, we have Therefore we have { n N : } ( )] [q B x u ε0 / I. [ ( )] u ε0 M q B x u and consequently by the relation (3.8) we have inf n [ ( )] u ε0 M = 0 which contradicts (iii). Hence [Z 0 (A,B,M,q)]I [Z 0 (A,B,u,q)]I.

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