Some New I-Lacunary Generalized Difference Convergent Sequence Spaces in 2-Normed Spaces

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1 Inf. Sci. Lett. 4, No. 3, ) 35 Information Sciences Letters An International Journal Some New I-Lacunary Generalized Difference Convergent Sequence Spaces in 2-Normed Spaces M. Aiyub Department of Mathematics, University of Bahrain, P. O. Box-32038, Kingdom of Bahrain Received: 25 Jan. 205, Revised: 3 Aug. 205, Accepted: 5 Aug. 205 Published online: Sep. 205 Abstract: In this paper, we introduce examine the properties of some new class of ideal lacunary convergent sequence spaces using, an infinite matrix witespect to a modulus function F = ) in 2-normed linear space. We study these spaces for some topological structures algebraic properties. We also give some relations related to these sequence spaces. Keywords: Difference sequence, lacunary sequence, I-convergent, infinite matrix, 2-normed space. AMS subject classification 2000): 40A05, 46B70,46A45 Introduction Preliminaries The idea of difference sequence spaces was introduced by Kizmaz 9]. Who studied the difference sequence spaces l ),c ) c 0 ). The the idea was further generalized by Et Çolak 4] for l n ),c n ) c 0 n ). Let ω be the space of all complex or real sequence x=x k ) m,s be non-negative integers, then Z =l,c c 0, we have sequence spaces, Z m s x k)= x=x k ) ω : m s x k) Z, ) where s mx) = s mx k) = s m x k s m x k+ s 0 x k = x k ) for all k N, which is equivalent to the following binomial representation s m m ) x k = ) v m x v k+sv v=0 P.Kostyrko et al 0] introduced the concept of I- convergence of sequence in metric space studied some properties of such convergence. Since then many author have been studied these subject obtained various results 29, 30, 3, 32,?] Note that I-convergence is an interesting generalization of statistical convergence. The concept of 2-normed space was initially introduced by Gähler 7] as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors, 8]. Recently a lot of activities have been started to study summability, sequence spaces related topics in these nonlinear space. Sahiner et al., 23] introduce I-convergence in 2-normed space. Give that I 2 N be trivial ideal in N. The sequence x n ) n N in X is said to be I-convergent to x X, if for each ε > 0 then the set, Aε)= n N: x n x I 0,] Let X be a real vector space of dimension d, where 2 d <. A 2-norm on X is a function.,. : X X R. Which satisfies: i) x,y = 0 if only if x y are linearly dependent, ii) x,y = y,x, iii) αx,y = α x,y, α R, iv) x,y+z x,y + x,z. The pair X,.,. ) is called a 2- normed spaces 8]. As an example of a 2-normed space we may take X =R 2 being equipped with stard Euclid 2-norm on R 2 Corresponding author maiyub2002@gmail.com c 205 NSP

2 36 M. Aiyub: Some New I-Lacunary Generalized Difference... are given by, ) x x,x 2 E = abs x 22 x 2 x 22 ) We know that X,.,. is 2-Banach space if every Cauchy sequence in X is convergent to some x X. By an ideal we mean a family I 2 X of subsets a non-empty set X satisfying: i)φ I ii)a,b I implies A B I iii)a I, B A impliy B I While an admissible ideal I of X further satisfies y I for each y X 0]. By Lacunary sequence we mean an increasing sequence θ = k r of positive integers satisfying; k 0 = 0 = k r k r as r. We denote the intervals, which θ determines by I r =k r k r ]. A sequence space X is said to be solid or normal if α k x k ) X whenever x k ) X for all sequences of scalarα k ) with α k. We recall that a modulus f is a function from0, ) 0, ) such that i) fx)=0 if only if x=0, ii) fx+y) fx)+ fy) for all x 0, y 0, iii) f is increasing, iv) f is continuous from right at 0. It follows fromi) iv) that f) must be continuous every where on0, ). For a sequence of moduli F = ), we give the following conditions. v)sup k x)< for all x>0 vi)lim x o x)=0, uniformly in k I. We remark that in case = f for all k, where f is a modulus, the conditions v) vi) are automatically fulfilled. 2 Main Results In this article using the lacunary sequence notion of ideal, we aimed to introduced some new ideal convergent sequence space by combining an infinite matrix with respect to a modulus function F = ) study their linear topological structures. Also we give some relations related to these sequence spaces. Let I be an admissible ) ideal, F = ) be a sequence of moduli, X,., be 2-normed space, p = p k ) be a sequence of strictly positive real numbers A = a nk ) be an infinite matrix of complex numbers. We write Ax = A n x)) n= if A nx) = n= a nkx)) converges for each n N. By ω2 X) we denotes the space of all ) sequences defined over 2-normed space X,.,. Now we define the following sequence spaces: N I θ A, s m,f, p,,., ] 0 = x=x k ) ω2 X) : ε > 0, I each z X r N: A k m s x k ),z N I θ A, m s,f, p,,., ]= x=x k ) ω2 X) : ε > 0, A k s m x k ) L,z I for some L>0 each z X Where A k m s x k)= k= a nk m s x k If Fx)=x, we get for all n N N I θ A, m s p,,., ] 0 = x=x k ) ω2 X) : ε > 0, )] A k s m x pk k),z I each z X, N I θ A, m s, p,,., ]= x=x k ) ω2 X) : ε > 0, A k s m x k ) L,z I for some L>0 each z X If p=p k )= for all k N, N I θ A, m s,f,,., ] 0 = x=x k ) ω2 X) : ε > 0, A k m s x k),z )] ε, I each z X, N I θ A, m s,f,,., ]= x=x k ) ω2 X) : ε > 0, A k s m x k ) L,z )] ε I for some L>0 each z X c 205 NSP

3 Inf. Sci. Lett. 4, No. 3, ) / 37 The following inequality will be used throughout the paper. Let p=p k ) be a positive sequence of real ) numbers with 0< p k sup p k = G, D=max,2 G. Then for all a k,b k C for all k N. We have N I θ A, m s, p,,.,,z] 0 N I θ A, m s,f, p,,.,,z] 0 a k + b k p k D a k p k + b k p k ) ) Theorem 2.. Let F = ) be a sequence of modulus functions, p=p k ) is bounded then, N I θ A, m s,f, p,,., ] N I θ A, m s,f, p,,., ] 0 are linear space over the complex field C. Proof. We shall give the proof only for N I θ A, m s,f, p,,., ] 0 other can be proved by the same technique. Let x,y N I θ A, m s,f, p,,., ] 0 α,β C, there exist M α N β such that α M α β N β. Since,., is 2-norm F = ) is a modulus function for all k from equation ),the following inequality holds: )] A k s m αx pk k+ β y k ),z D.M α ) H )] A k s m x pk k),z +D.N β ) H A k m s β y k),z On the other h from the above inequality we get. Proof. Let x N I θ A, m s, p,,.,,z], then for some L> 0 for each z X )] A k s m x pk Now let ε > 0, we choose 0<δ < such that for every t with 0 t δ, we have t) < ε for all k. Now by using Lemma ), we get )] A k m pk s x k L),z = hr.maxε h,ε H ] ) max 2 fk )δ ) h, 2 fk )δ ) H )] A k s m x pk This completes the proof. The other case can be proved similarly. Theorem 2.3. Let F = ) be a sequence of moduli. If lim t sup t) t = η > 0 for all k then )] A k s m αx pk k+ β y k ),z D.M α ) H )] A k s m x pk k),z D.N β ) H A k m s y k),z. Two sets on the right side belongs to I, so this completes the proof. Lemma. Let f be a modulus function, let 0 < δ <. Then for each x > δ we have fx) 2 f)δ x 6] Theorem 2.2. Let F = ) be sequence of a moduli 0<in p k = h p k sup k p k = H <. N I θ A, m s, p,,.,,z] N I θ A, m s,f, p,,.,,z] N I θ A, m s, p,,.,,z] 0 = N I θ A, m s,f, p,,.,,z] 0 N I θ A, m s, p,,.,,z]=ni θ A, m s,f, p,,.,,z] Proof. In Theorem 2.2, it is shown that, N I θ A, m s,f, p,,.,,z] NI θ A, m s, p,,.,,z] we must show that N I θ A, m s, p,,.,,z] NI θ A, m s,f, p,,.,,z] For any modulus function there exist a positive limit B > 0 x N I θ A, m s,f, p,,.,,z]. Since B > 0 for every t) Bt for all k. Hence )] A k s m x pk B H )] A k m pk s x k L),z c 205 NSP

4 38 M. Aiyub: Some New I-Lacunary Generalized Difference... This completes the proof. Corollary. Let F = ) F 2 = g k ) be sequences of moduli. If implies that lim t sup t) g kt) < N I θ A, m s,f 2, p,,.,,z] 0 = N I θ A, m s,f 2, p,,.,,z] 0 N I θ A, m s,f, p,,.,,z]=n I θ A, m s,f 2, p,,.,,z] ) ) Theorem 2.4. Let X,.,. S X,.,. E be stard Euclid 2- normed spaces respectively then N I θ A, m s,f, p,,., S] N I θ A, m s,f, p,,., E] ) N I θ A, m s,f, p ],., S +,., E Proof. We have the following inclusion. )] A k.,. S +.,. E m pk s x k L),z D r N:D A k m s x k L),z S A k m s x E by using equation ). This complete the proof. Theorem 2.5. Let F = ) F 2 = g k ) be sequences of moduli. Then i) N I θ A, m s,f, p,,.,,z] 0 N I θ A, m s,f of 2, p,,.,,z] 0 ii) N I θ A, m s,f, p,,.,,z] N I θ A, m s,f of 2, p,,.,,z] N I θa, m s,f, p,,.,,z] 0 N I θa, m s,f 2, p,,.,,z] 0 N I θa, m s,f +F 2, p,,.,,z] 0 N I θa, m s,f, p,,.,,z] N I θa, m s,f 2, p,,.,,z] N I θa, m s,f + F 2, p,,.,,z] Proof. i) Let x k N I θ A, m s,f, p,,.,,z]. Let 0 < ε < 0 < δ <, such that f) k t) < ε for 0< t < δ. Let y k = g k A k s mx. Let fk y k ) ] p k = fk y k ) ] p k + fk y k ) ] p k 2 where the first summation is over y k δ the second summation is over y k > δ. Then fk y k ) ] p k ε H for y k > δ, we use the fact that, y k < y k δ < + y k δ ) where z denotes the integer part of z. From the properties of modulus function, we have for y k > δ Hence. 2 y k )< + y k δ )) y k ) 2 δ fk y k ) ] p k 2 ) δ ] H ] pk yk 2 which together with fk y k ) ] p k ε H yields. fk y k ) ] p k ε H +max, 2 ) ] ) H ] pk δ fk y k k Ir This completes the proof. ii) Let x k N I θ A, m s,f, p,,.,,z] N I θ A, m s,f 2, p,,.,,z]. The fact is that )] + g k ) A k s m x pk gives the results D. )] A k s m x pk +D )] g k A k m pk s x k L),z c 205 NSP

5 Inf. Sci. Lett. 4, No. 3, ) / Statistical Convergent The notion of statistical convergence of sequences was introduced by Fast 5]. Later on it was studied from sequence space linked with summability theory by Fridy 6], Salat 28] many others. The notion depends on the density of subsets of the set N of natural numbers. A subset E of N is said to have density δe) if δe) = lim n n n τ E k) exist, where τ E is the characteristics function of E. A complex number sequence x = x k ) is said to be statistically convergent to the number L if for every Kε) ε > 0, lim n n = 0, where Kε) denotes the number of elements in the set Kε)= k N: x k L ε. A complex number sequence x = x k ) is said to be strongly generalized difference S λ A, s m )-statistically convergent to the number L if for every ε > 0, lim n λn KA s m,ε) = 0, where KA s m,ε) denotes the number of elements in the set KA s m,ε)= k I n : A k s m x k ) L ε. The set of all strongly generalized difference statistically convergent sequences is denoted by S λ A, s m ), If m = 0, = 0, then S λ A, s m ) reduces to S λ A), which was defined studied by Bilgin Altin ]. If A is identity matrix, λ n = n, s=0, S λ A, s m) reduces to S λ m ) which was defined by Et Nuray 3]. If m=0, s=0 λ n = n then S λ A, s m ) reduces to S A, which was defined by Esi 2]. If m=0, s=0 A is identity matrix λ n = n, strongly generalized difference S λ A, s m )-statistically convergent sequences reduces to ordinary statistically convergent sequences. Theorem 3. Let F = ) be a sequence of modulus functions. Then N I θ A, m s,f, p,,.,,z] S λ A, m s ) min, A k m s x k L),z >S, A k m s x k L),z >S, A k m s x k L),z >S )] A k s m x pk ] pk ε) min ε) h, ε) H) ε) h, ε) H) KA m s,ε) Hence x S λ A, m s ) Theorem 3.2 Let F = ) be a sequence of modulus functions. Then N I θ A, m s,f, p,,.,,z]=s λ A, m s ) Proof. By Theorem 3., it is sufficient to show that N I θ A, m s,f, p,,.,,z] Sλ A, m s ). Let x k S λ A, s m). Since F = ) is bounded so that there exist an integer ) K > 0, such that A k s mx K. Then for given ε > 0, we have = )] A k m pk s x k L),z, A k m s x k L),z S +, A k m s x k L),z >S max )] A k s m pk x k L),z )] A k s m x pk ε) h, ε) H) + K H KA m s,ε) Taking limit as ε 0 n, it follows that x N I θ A, m s,f, p,,.,,z] Proof. Let x N I θ A, m s,f, p,,.,,z], then References )] A k m pk s x k L),z ] T. Bilgil Y. Altun., Strongly V λ,a, p)-summable sequence spaces defined by a modulus. Math. Model. Anal ), c 205 NSP

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