"Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an Olicz Function Mahmut KARAKUŞ Yüzüncü Yıl Univesity, Science Faculty, Depatment of Mathematics. matfons@gmail.com Tunay BİLGİN Yüzüncü Yıl Univesity, Education Faculty, Depatment of Mathematics. tbilgin@yyu.edu.t Abstact: In this study, we intoduced the concepts asymptotically I p M -lacunay equivalence with ode α and asymptotically I M- lacunay statistical equivalence with ode α by using a non-tivial ideal I, an Olicz function M and a sequence of positive eal numbes p = (p ). In addition to these definitions, we also pesented some inclusion theoems. Keywods: Asymptotically equivalence, Ideal convegence, Lacunay sequence, Olicz function.. Intoduction Let s be the space of eal valued sequences and any subspace of s is also called a sequence space. l and c denote the spaces of all bounded and convegent sequences, espectivel A lacunay sequence is an inceasing sequence θ = ( ) such that 0 = 0, h = as.the intevals detemined by θ will be denoted by I = (, ] and q = /.These notations will be used toughout the pape. The sequence space of lacunay stongly convegent sequences N θ was defined by Feedman et al.5], as following: N θ = x = (x i ) s : lim h x i s = 0 fo some s}. i I Olicz 8] used the idea of Olicz function to constuct the space L M. An Olicz function is a function M : 0, ) 0, ),which is continuous,
336 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal nondeceasing and convex with 0) = 0, x) > 0 and x) as x. An Olicz function M is said to satisfy the 2 -condition fo all values of u, if thee exists constant K > 0, such that 2u) Ku) (u 0). It is also easy to see that K > 2 aways holds. The 2 -condition is equivalent to the satisfaction of inequality Lu) KLu) fo all values of u and L >. Rema. An Olicz function M satisfies the inequality λx) < λx) fo all λ with 0 < λ <. The following well nown inequality will be used toughout the pape; () a i + b i p i T ( a i p i + b i p i ), whee a i and b i ae complex numbes, T = max(, 2 H ), and H = sup p i <. Maouf pesented definitions fo asymptotically equivalent sequences and asymptotic egula matices in 7]. Patteson extended these concepts by pesenting an asymptotically statistical equivalent analog of these definitions and natual egulaity conditions fo nonnegative summability matices in 9]. Subsequently,many authos have shown thei inteest to solve diffeent poblems aising in this aea (see 2], and 0] ). The concept of I convegence was intoduced by Kostyo et al. 5], which is a genealization of statistical convegence. Recently, Das et al. 2,3] unified these two appoaches to intoduce new concepts such as I- statistical convegence and I-lacunay statistical convegence and investigated some of thei consequences. Moe investigations in this diection and moe applications can be found in, 3, 4, ]. In this pape we intoduce the concepts asymptotically I p M -lacunay equivalence with ode α and asymptotically I M -lacunay statistical equivalence with ode α,by using a non-tivial ideal I, an Olicz function M,and a sequence of positive eal numbes p = (p ) and also some inclusion theoems ae poved. 2. Definitions and Notations In this section, we ecall the basic definitions and concepts. Fo simplicity, below and in what follows limits un to, that is, we use lim instead of lim.
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 337 Definition 2.. Two nonnegative sequences x = ( ) and y = ( ) ae x said to be asymptotically equivalent if lim =,(denoted by x y). Definition 2.2. Two nonnegative sequences x = ( ) and y = ( ) ae said to be asymptotically statistical equivalent of multiple L povided that fo evey ε > 0, lim n n } n : ε = 0 (denoted by x S y) and simply asymptotically statistical equivalent, if L =. Definition 2.3.Two nonnegative sequences x = ( ) and y = ( ) ae said to be stong asymptotically equivalent of multiple L povided that, lim n n n = 0 = (denoted by x w y) and simply stong asymptotically equivalent, if L =. Definition 2.4. Let θ be a lacunay sequence; the two nonnegative sequences x = ( ) and y = ( ) ae said to be asymptotically lacunay statistical equivalent of multiple L povided that fo evey ε > 0, } lim h I : ε = 0 (denoted by x S θ y) and simply asymptotically lacunay statistical equivalent, if L =. Definition 2.5. Let θ be a lacunay sequence; the two nonnegative sequences x = ( ) and y = ( ) ae said to be stong asymptotically lacunay equivalent of multiple L povided that, lim h = 0 (denoted by x N θ y) and simply stong asymptotically lacunay equivalent, if L =. Definition 2.6. Let M be any Olicz function; the two nonnegative sequences x = ( ) and y = ( ) ae said to be M-asymptotically equivalent of multiple L povided that, lim = 0
338 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal fo some ρ > 0, (denoted by x M y) and simply M asymptotically equivalent, if L =. Definition 2.7. Let M be any Olicz function; the two nonnegative sequences x = ( ) and y = ( ) ae said to be stong M-asymptotically equivalent of multiple L povided that, lim n n n = 0 = fo some ρ > 0, (denoted by x wm y) and simply stong M asymptotically equivalent, if L =. Definition 2.8. Let M be any Olicz function and θ be a lacunay sequence; the two nonnegative sequences x = ( ) and y = ( ) ae said to be stong M-asymptotically lacunay equivalent of multiple L povided that, lim h = 0 fo some ρ > 0, (denoted by x N θ M y) and simply stong M-asymptotically lacunay equivalent, if L =. Let X be a non-empty set. Then P (X) denote the powe set of X, that is, the space of all subsets of X. Definition 2.9. A family I P (X) is said to be an ideal in X if the following conditions hold: (i) I; (ii) A, B I imply A B I and (iii) A I, B A imply B I. Definition 2.0. A non-empty family F P (X) is said to be a filte in X if the following conditions hold: (i) / F ; (ii) A, B F imply A B F and (iii) A F, B A imply B F. An ideal I is said to be non-tivial if I } and X / I. A non-tivial ideal I is called admissible if it contains all the singleton sets. Moeove, if I is a non-tivial ideal on X, then F = F (I) = X A : A I} is a filte on X and convesel The filtef (I) is called the filte associated with the ideal I.
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 339 Definition 2.. Let I P (N) be a non-tivial ideal in N. A sequence x = ( ) in X is said to be I-convegent to L if fo each ε > 0, the set N : L ε} I. In this case, we wite I lim = L. Let x = ( ) be a sequence in X. Then it is said to be I null if L = 0. In this case we wite I lim = 0. Definition 2.2. A sequence x = ( ) of numbes is said to be I- statistical convegent o S(I)-convegent to L, if fo evey ε > 0 and δ > 0, we have n N; n n : x } L ε} δ I. In this case, we wite L(S(I)) o S(I) - lim = L. Definition 2.3 Let I P (N) be a non-tivial ideal in N. The two non-negative sequences x = ( ) and y = ( ) ae said to be stongly asymptotically equivalent of multiple L with espect to the ideal I povided that fo each ε > 0, n N; } n n ε I, = (denoted by x I(w) y) and simply stongly asymptotically equivalent with espect to the ideal I, if L =. Definition 2.4. Let I P (N) be a non-tivial ideal in N and θ = ( ) be a lacunay sequence. The two nonnegative sequences x = ( ) and y = ( ) ae said to be asymptotically lacunay statistical equivalent of multiple L with espect to the ideal I povided that fo each ε > 0 and γ > 0, } } h I : ε γ I (denoted by x I( S θ ) y) and simply asymptotically lacunay statistical equivalent with espect to the ideal I, if L =. Definition 2.5. Let I P (N) be a non-tivial ideal in N and θ = ( ) be a lacunay sequence. The two non-negative sequences x = ( ) and y = ( ) ae said to be stongly asymptotically lacunay equivalent of multiple L
340 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal with espect to the ideal I povided that fo ε > 0, } h ε I (denoted by x I(N θ) y) and simply asymptotically lacunay equivalent with espect to the ideal I, if L =. Savas] have given the following defini- Quite ecently Bilgin ] and tions. Definition 2.6. Let I P (N) be a non-tivial ideal in N, M be any Olicz function, θ = ( ) be a lacunay sequence and p = (p ) be a sequence of positive eal numbes. Two sequences x = ( ) and y = ( ) ae said to be (M, p)-asymptotically lacunay equivalent of multiple L with espect to the ideal I povided that fo each ε > 0, h ε I I(N (M,p) θ ) fo some ρ > 0, (denoted by x y) and simply (M, p) -asymptotically lacunay equivalent with espect to the ideal I, if L =. Definition 2.7. Let I P (N) be a non-tivial ideal in N,and θ = ( ) be a lacunay sequence. Two sequences x = ( ) and y = ( ) ae said to be asymptotically I-lacunay statistical equivalent of ode α, whee 0 < α, to multiple L povided that fo any ε > 0 and δ > 0, } } I : ε δ I. In this case we wite x SL θ (I)α Definition 2.8. Let I P (N) be a non-tivial ideal in N, θ = ( ) be a lacunay sequence and p = (p ) be a sequence of positive eal numbes. Two sequences x = ( ) and y = ( ) ae said to be stongly asymptotically I -lacunay equivalent of ode α, whee 0 < α, to multiple L fo the sequence p povided that fo any ε > 0,
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 34 } ε I fo some ρ > 0, (denoted by x N Lp θ (I) α y). 3. Main Results In this section we shall give some new definitions and also pove some inclusion elations. We begin with the following definitions. Definition 3.. Let I P (N) be a non-tivial ideal in N, M be any Olicz function,and θ = ( ) be a lacunay sequence. Two numbe sequences x = ( ) and y = ( ) ae said to be asymptotically I M -lacunay statistical equivalent of ode α, whee 0 < α, to multiple L povided that fo any ε > 0 and δ > 0. } } I : ε δ I fo some ρ > 0. In this case we wite x I M (Sθ if we put x) = x fo x 0, we wite x I(Sα θ Hence x I(Sα θ ) y is the same as the x SL θ (I)α y of Savas ]. Definition 3.2. Let I P (N) be a non-tivial ideal in N, M be any Olicz function, θ = ( ) be a lacunay sequence and p = (p ) be a sequence of positive eal numbes. Two sequences x = ( ) and y = ( ) ae said to be asymptotically I p M -lacunay equivalent of ode α, whee 0 < α, to multiple L fo the sequence p povided that fo any ε > 0, ε I fo some ρ > 0, (denoted by x Ip M (N θ y). If we tae α =, we wite x Ip M (N θ y instead of x Ip M (N θ y. Hence x Ip M (N θ ) y is the same as the (M,p) I(N θ ) x y of Bilgin ]. Also if we put x) = x fo x 0, we wite x Ip (Nθ y instead of x Ip M (N θ Hence x Ip (Nθ y is the same as the x N Lp θ (I) α y of Savas ].
342 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal If we tae ( ) instead of ( ),denoted by x Ip M (wα ) y and simply stongly n α Cesao I p M -asymptotically equivalent of ode α, if L =. If we tae p = fo all N, we wite x I M (N θ y instead of x Ip M (N θ We now pove some inclusion theoems. Theoem 3.. Let I P (N) be a non-tivial ideal in N, M be any Olicz function, θ = ( ) be a lacunay sequence, p = (p ) be a sequence of positive eal numbes and 0 < α α 2 then x Ip M (N α θ y implies x Ip M (N α 2 θ Poof. Let 0 < α α 2 and x Ip M (N α θ ) Since h = as,we can actually choose, so that h α h α 2 and h α 2 h α. Hence h α 2 h α. And so, h α 2 ε h α ε. Finally, we have h α 2 ε I Hence x Ip M (N α 2 θ One can have the following esult by setting α 2 = in Theoem 3.. Coolla Let I P (N) be a non-tivial ideal in N, M be any Olicz function, θ = ( ) be a lacunay sequence, p = (p ) be a sequence of positive eal numbes, and 0 < α, then x I M (N α θ y implies x I M (N θ Theoem 3.2. Let I P (N) be a non-tivial ideal in N, M, M 2 be any Olicz functions, θ = ( ) be a lacunay sequence, p = (p ) be a
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 343 sequence of positive eal numbes and 0 < α. Then x I M M 2 ((Nθ implies x I (M +M 2 )(Nθ α Poof. Now suppose that x I (M M 2 )(Nθ α ) y and ε > 0. Let the set A = M ( < ε/2 and B = M 2 ( < ε/2 be given fo some ρ > 0. Then we have, ] (M + M 2 )( = M ( + M 2( ]p with T = max(, 2 H ), and H = sup p i <. So ] (M + M 2 )( ] T M ( + ] M 2 ( }, /ρ) T + ] M ( + ] M 2 ( } < T ε = ε. It follows that fo any ε > 0, ] (M + M 2 )( which yields that x I (M +M 2 )(N α θ ) < ε } F (I) y
344 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal The next theoem shows elationship between the stongly I p M asymptotically equivalence and the I p M asymptotically lacunay equivalence with espect to the ideal I. Theoem 3.3. Let I P (N) be a non-tivial ideal in N, M be any Olicz function, θ = ( ) be a lacunay sequence, p = (p ) be a sequence of positive eal numbes and 0 < α. Then following popositions ae tue. (i) If sup ( α m m ) α = B(say) < then x Ip M (N θ y implies x Ip M (wα ) y, m= (ii) If sup = C(say) < then x Ip M (wα ) y implies x Ip M (N θ Poof. (i): Now suppose that x Ip M (N θ y and ε > 0. Let A = < ε, fo some ρ > 0. Hence,fo all j A and fo some ρ > 0, we have H j = h α j < ε. I j Choose n is any intege with n > whee A. Now wite n α = α = α α h α n = α = +... + I I +... + ( ) α = } α α H + ( 2 ) α H 2 +... + ( ) α H
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 345 = α m= m m h α m = α I m ( m m ) α sup m= H j j A BsupH j j A < Bε = ε. It follows that fo any ε > 0, n ] n N; n α = < ε } F (I) since fo any set A F (I), n : < n <, A} F (I). which yields that x Ip M (wα ) (ii): Let x Ip M (wα ) Let us tae ε > 0 and define the set, A = = < ε, fo some ρ > 0. We have A F (I), which is the filte of the ideal I. Fo each A, we have, fo some ρ > 0, = = = = =
346 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal < Cε = ε It follows that fo any ε > 0, ] < ε } F (I) which yields that x Ip M (N θ Now we give elation between asymptotically I-lacunay statistical equivalence and I p M-asymptotically lacunay equivalence of ode α. Theoem 3.4. Let I P (N) be a non-tivial ideal in N, M be any Olicz function, θ = ( ) be a lacunay sequence, p = (p ) be a sequence of positive eal numbes and 0 < α. If h = inf p sup p = H <, then x Ip M (N θ y implies x I(Sα θ Poof. Tae ε > 0 and let denote the sum ove I, with L ε. Then, h α h α } I : ε ε/ρ) p } I : ε minε/ρ) h, ε/ρ) H } and } I : ε } γ ) γ minε/ρ) h, ε/ρ) H } I. But then, by definition of an ideal, late set belongs to I, and theefoe x I(Sα θ
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 347 Theoem 3.5. Let I P (N) be a non-tivial ideal in N, M be any Olicz function, θ = ( ) be a lacunay sequence, p = (p ) be a sequence of positive eal numbes and 0 < α. If h = inf p sup p = H <, then x Ip M (N θ y implies x I M (Sθ y, Poof. Tae ε > 0 and let denote the sum ove I with L ε. Then, h α h α } I : ε (ε) p } I : ε minε h, ε H } and } I : ε } γ γ minε h, ε H } I. But then, by definition of an ideal, late set belongs to I and theefoe x I M (Sθ Theoem 3.6. Let I P (N) be a non-tivial ideal in N, M be any Olicz function that satisfy the 2 -condition, θ = ( ) be a lacunay sequence, and 0 < α. Then x I(Sα θ y implies x I M (Sθ Poof. Let x I(Sα θ y and ε > 0. We choose 0 < δ < such that u) < ε/2 fo evey u with 0 u δ. Let ( L δ, then L < ε/2. Fo ( L > δ we use the fact that ( L < ( L/δ < + ( L/δ. Since M is non-deceasing and convex, it follows that
348 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal L < +( L/δ) < 2)+ 2( 2 2 L/δ). Since M satisfies the 2 -condition, theefoe L < K(( 2 L/δ) 2)+ K(( 2 L/δ)2) = K( L/δ)2). Hence. K( L /δ)2) + ε/2. Thus } I : ε } γ } I : ερδ/2k2) } γ. Since x I(Sα θ ) y it follows the late set, and hence, the fist set in above expession belongs to I. This poves that x I M (w α ) Let p = p fo all, t = t fo all and 0 < p t. Then we can give following theoem. Theoem 3.7. Let I P (N) be a non-tivial ideal in N, M be an Olicz function, θ = ( ) be a lacunay sequence and 0 < α then x It M (N θ y implies x Ip M (N θ Poof. Let x It M (N θ It follows fom Holde s inequality, p ( t ) p/t and p ε }
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 349 t L ε t/p I. Thus we have x Ip M (N θ We now conside that (p ) and (t ) ae not constant sequences. Theoem 3.8. Let I P (N) be a non-tivial ideal in N, M be an Olicz function, θ = ( ) be a lacunay sequence, 0 < p t fo all, (t /p ) be bounded, and 0 < α, then x It M (N θ y implies x Ip M (N θ Poof. Let x It M (N θ z = tand L] λ = (p / t ), so that fo 0 < λ λ, we define the sequences (u ) and (v ) as follows. Fo z ;let u = z and v = 0 and fo z < ; let v = z and u = 0. Then, we have, z = u + v ; z λ = u λ + vλ. Now it follows that u λ u z and v λ v λ. Theefoe, Now fo each ; z λ v λ = and so = (u λ h α z + + vλ ) v λ. ( v h α ) λ ( ) λ ( ( ] /λ v h α ) λ ) λ ( ( ] / λ ) λ ) λ h α < ( h = v ) λ. z λ = z + ( v h α ) λ z + ( z, z z ) λ, z <
350 Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 2( z, z I z ) λ, z <. If then ε t L] ε, z t ( L ε /λ 2), z <. Hence ε } t ( ε ) } } /λ min ε, I. 2 Thus we have x Ip M (N α θ ) Refeences ] T. Bilgin. Asymptotically Lacunay Equivalent Sequence spaces defined by ideal, VII Intenational Confeence Mathematical Analysis, Diffeential Equations and thei Applications (MADEA-7), Bau 205 2] T. Bilgin, M.Kaauş. On The Space of Asymptotically Lacunay Equivalent Sequences obtained fom an Olicz Function, to appea.. 3] P. Das, E. Savaş, S. Ghosal. On genealizations of cetain summability methods using ideals, Appl. Math. Lett. 24 (20), 509-54 4] P. Das, E. Savaş. On I-statistical and I-lacunay statistical convegence of ode alpha, Bull. Ianian Math. Soc. 40 (2) (204), 459-472.
Asymptotically Lacunay Statistical Equivalent Sequence Spaces defined by ideal 35 5] A.R. Feedman, J. J. Sembe, M. Raphel. Some Cesáo-type summability spaces, Poc. London Math. Soc. 37 (3) (978), 508-520. 6] P.Kostyo, T, Šalát, W.Wilczyńsi. I-convegence, Real Anal. Exchange 26 (2) (2000), 669-686. 7] M. Maouf. Asymptotic equivalence and summability, Int.J. Math. Math. Sci. 6 (4) (993), 755-762. 8] W. Olicz. Ube Raume L M, Bull. Int. Acad. Polon. Sci. Sé. A (936), 93-07. 9] R. F. Patteson. On asymptotically statistically equivalent sequences, Demonstatio Math. 36 () (2003), 49-53. 0] R.F. Patteson, E. Savaş. On asymptotically lacunay statistically equivalent sequences, Thai J. Math. 4 (2) (2006), 267-272. ] E. Savaş. A Genealization on I-Asymptotically Lacunay Statistical Equivalent Sequences, Thai J. Math. 6 () (206), 43-5. Published: Volume 207, Issue / Novembe 25, 207