On some spaces of Lacunary I-convergent sequences of interval numbers defined by sequence of moduli

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1 Poyecciones Jounal of Mathematics Vol. 36, N o 2, pp , June Univesidad Católica del Note Antofagasta - Chile On some spaces of Lacunay I-convegent sequences of inteval numbes defined by sequence of moduli Mohd Shafiq Govt. Degee College, India and Ayhan Esi Adiyaman Univesity, Tuey Received : Octobe Accepted : Januay 2017 Abstact In this aticle we intoduce and study some spaces of I-convegent sequences of inteval numbes with the help of a sequence F =f ) of moduli, a bounded sequence p =p ) of positive eal numbes and a lacunay sequence θ = { } of inceasing integes. We study some topological and algebaic popeties and some inclusion elations on these spaces. Keywods and Phases : Inteval numbes, Ideal, filte, I-convegent sequence, solid and monotone space, Banach space, modulus function.

2 326 Mohq Shafiq and Ayhan Esi 1. Intoduction and Peliminaies Let N, R and C be the sets of all natual, eal and complex numbes, espectively. Let, c and c 0 be denote the Banach spaces of bounded, convegent and null sequences, espectively with nom x =sup x. We denote ω = {x =x ):x RoC}, the space of all eal o complex sequences. It is an admitted fact that the eal and complex numbes ae playing a vital ole in the wold of mathematics. Many mathematical stuctues have been constucted with the help of these numbes. In ecent yeas, since 1965 fuzzy numbes and inteval numbes also managed thei place in the wold of mathematics and cedited into account some alie stuctues. Inteval aithmetic was fist suggested by P.S.Dwye [12] in Futhe development of inteval aithmetic as a fomal system and evidence of its value as a computational device was povided by R.E.Mooe [22] in 1959 and Mooe and Yang [23] and othes and have developed applications to diffeential equations. Recently, Chiao [11] intoduced sequences of inteval numbes and defined usual convegence of sequences of inteval numbes. Şengönül and Eyılmaz [34] intoduced and studied bounded and convegent sequence spaces of inteval numbes and showed that these spaces ae complete. A set closed inteval) of eal numbes x such that a x b is called an inteval numbe [11]. A eal inteval can also be consideed as a set. Thus, we can investigate some popeties of inteval numbes fo instance, aithmetic popeties o analysis popeties. Let us denote the set of all eal valued closed intevals by IR. Any element of IR is called a closed inteval and it is denoted by Ā =[x l,x ]. IR is a quasilinea space unde the algebaic opeations and a patial ode elation fo IR found in [34] and any subspace of IR is called quasilinea subspace see [34]). Futhe eseach on inteval numbes was caied out by Esi and Baha [6], Esi and Hazaia [8], Esi [1-9] and the efeences theein. The set of all inteval numbes IR is a complete metic space defined by

3 On some spaces of Lacunay I-convegent sequences of ) dā1, Ā2) =max{ x 1l x 2l, x 1 x 2 }, see[23, 34]). whee x l andx be fist and last points of Ā, espectively. In a special case, Ā 1 =[a, a], Ā 2 =[b, b], we obtain the usual metic of R with dā1, Ā2) = a b. Let us define tansfomation f fom N to IR by f) = A, ¹ A ¹ = Ā). The function f is called sequence of inteval numbes, whee Ā is the th tem of the sequence Ā). Let us denote the set of sequences of inteval numbes with eal tems by 1.2) ω A)={ ¹ A ¹ =Ā) :Ā IR}. The algebaic popeties of ω ¹ A) can be found in [11, 34]. The following definitions wee given by Şengönül and Eyılmaz in [34]. A sequence ¹ A =Ā) =[x l,x ]) of inteval numbes is said to be convegent to an inteval numbe Ā0 =[x 0l,x 0 ]iffoeach >0, thee exists a positive intege n 0 such that dā, Ā0) <,foall n 0 and we denote it as lim Ā = Ā0. Thus, lim Ā = Ā0 lim x l = x 0l and lim x = x 0 and it is said to be Cauchy sequence of inteval numbes if fo each >0, thee exists a positive intege 0 such that dā, Am ) <, wheneve, m 0. Let us denote the space of all convegent, null and bounded sequences of inteval numbes by C ¹ A), C ¹ A)and ¹ A), espectively. The sets C ¹ A), C ¹ A)and ¹ A) ae complete metic spaces with the metic bdā, B )=supmax{ x l y l, x y }see[34]).

4 328 Mohq Shafiq and Ayhan Esi If we tae B = 0 in 3) then, the metic b d educes to 1.4) bdā, 0) = sup max{ x l, x }. In this pape, we assume that a nom Ā of the sequence of inteval numbes Ā) is the distance fom Ā) to0andsatisfies the following popeties: Ā, B λ A)and α ¹ R N 1 ) Ā λ A) ¹ {0}, Ā λ A) ¹ > 0, N 2 ) Ā λ A) ¹ =0 Ā = 0, N 3 ) Ā + B λ A) ¹ Ā λ A) ¹ + B λ A) ¹, N 4 ) αā λ A) ¹ = α Ā λ A) ¹, whee λ A) ¹ is a subset of ω A). ¹ Let A ¹ =Ā) =[x l,x ]) be the element of C A), ¹ C A)o ¹ A). ¹ Then, in the light of above discussion, the classes C A), ¹ C A)and ¹ A) ¹ of sequences of inteval numbes ae nomed inteval spaces nomed by 1.5) A=sup ¹ max{ x l, x }see[34]). Thoughout, 0=[0, 0] and Ī =[1, 1] epesent zeo and identity inteval numbes accoding to addition and multiplication, espectively. As a genealisation of usual convegence fo the sequences of eal o complex numbes, the concept of statistical convegent was fist intoduced by Fast [13] and also independently by Buc [10] and Schoenbeg [33]. Late on, it was futhe investigated fom a sequence space point of view and lined with the Summability Theoy by Fidy [15], Šalát [30], Tipathy [35] and many othes. The statistical convegence has been extended to inteval numbes by Esi and othes as follows in [1-9]. Let us suppose that ¹ A =Ā ) ¹ A). If, fo evey >0, 1 1.6) lim n n { N : Ā Ā0, n} =0. Then, the sequence A ¹ =Ā) is said to be statistically convegent to an inteval numbe Ā0, whee vetical lines denote the cadinality of the enclosed set. That is, if δa )) = 0, whee A ) ={ N : Ā Ā0 }. The notion of ideal convegence I-convegence) was intoduced and studied by Kostyo, Mačaj, Salǎt and Wilczyńsi [20, 21]. Late on, it

5 On some spaces of Lacunay I-convegent sequences of was studied by Šalát, Tipathy and Ziman [31, 32], Esi and Hazaia [8], Tipathy and Hazaia [36], Khan etal [16, 17], Musaleen and Sunil [25] and many othes. Definition 1.1. Let N be the set of natual numbes. Then, a family of sets I 2 N powe set of N) issaidtobeanideal,if i) I is additive. That is, A, B I A B I, ii) I is heeditay. That is A IandB A B I. A non-empty family of sets I) 2 N is said to be filte on N, if and only if i) Φ / I), ii) A, B I) we have A B I), iii) A I) and A B B I). An Ideal I 2 N is called non-tivial if I 6= 2 N. A non-tivial ideal I 2 N is called admissible if {{x} : x N} I. Let us suppose that I be an ideal. Then, a sequence ¹ A =Ā) ¹ A) ω ¹ A) i)issaidtobei-convegent to an inteval numbe Ā0 if fo evey >0, { N : Ā Ā0 } I. In this case, we wite I lim Ā = Ā0. IfĀ 0 = Ō. Then, the sequence A ¹ =Ā) A) ¹ is said to be I-null. In this case, we wite I lim Ā = Ō. ii) is said to be I-Cauchy, if fo evey >0, thee exists a numbe m = m ) such that { N : Ā A m } I. iii) is said to be I-bounded, if thee exists some M>0 such that { N : Ā M} I. We now that fo each ideal I, theeisafilte I) coesponding to I. That is, I) ={K N : K c I}, wheek c = N \ K. Definition 1.2. Asequencespaceλ ¹ A) of inteval numbes is iv) said to be solidnomal), if α Ā ) λ ¹ A), wheneve Ā) λ ¹ A) and fo any sequence α ) of scalas with α 1, foall N, v)saidtobesymmetic,ifāπ)) λ ¹ A), wheneve Ā λ ¹ A), whee π

6 330 Mohq Shafiq and Ayhan Esi is a pemutation on N, vi) said to be sequence algeba, if Ā) B )=Ā. B ) λ A) ¹ wheneve Ā), B ) λ A), ¹ vii)saidtobeconvegencefee,if B ) λ A) ¹ wheneve Ā) λ A) ¹ and Ā = Ō implies B = Ō, fo all. Definition 1.3. Let K = { 1 < 2 < 3...} N. TheK-step space of the λ A) ¹ is a sequence space µ λ A) ¹ K = {Ā n ) ω A):Ā) ¹ λ A)}. ¹ Definition 1.4. A canonical pe-image of a sequence Ā n ) µ λ A) ¹ K is a sequence B ) ω A) ¹ defined by B = Ā, if K, Ō, othewise. A canonical peimage of a step space µ λ A) ¹ K is a set of canonical peimages of all elements in µ λ A) ¹ K. That is, B ¹ is in the canonical peimage of µ λ A) ¹ K iff B ¹ is the canonical peimage of some A ¹ µ λ A) ¹ K. Definition 1.5. Asequencespaceλ A) ¹ is said to be monotone, if it contains the canonical peimages of its step space. Definition 1.6. Afunctionf : [0, ) [0, ) is called a modulus function if 1) ft) =0if and only if t =0, ft + u) ft)+fu) fo all t, u 0, 3) f is inceasing, and 4) f is continuous fom the ight at zeo. A modulus function f is said to satisfy 2 Condition fo all values of u if thee exists a constant K>0such that flu) KLfu) fo all values of L>1. The idea of modulus function was intoduced by Naano in 1953, See [26], Naano, 1953). Rucle [27-29] used the idea of a modulus function f to constuct the sequence space 1.7) Xf) = n x =x ): X =1 o n ³ o f x ) < = x = x : f x ) X.

7 On some spaces of Lacunay I-convegent sequences of Afte then, E.Kol [18-19] gave an extension of Xf) by consideing a sequence of modulii F =f ) and defined the sequence space n ³ o 1.8) XF) = x =x ): f x ) X. Musaleen and Noman [24] intoduced the notion of λ-convegent and λ- bounded sequences. We extended this concept to the sequence of inteval numbesasfollows. Let λ =λ ) =1 be a stictly inceasing sequence of positive eal numbes tending to infinity. That is, 1.9) 0 <λ 0 <λ 1 <λ 2 <λ 3..., λ as. The sequence ¹ A =Ā ) ¹ A) is λ-convegent to an inteval numbe Ā 0, called the λ-limit of ¹ A,ifΛ m ¹ A) Ā0, asm,whee Λ m ¹ A)= 1 λ m mx λ λ 1 )Ā, N). =1 Hee and in the sequel, we shall use the convention that any tem with a negative subscipt is equal to naught. Fo example, λ 1 =0. In paticula, ¹ A = Ā ) ¹ A) is said to be λ-null, if Λ m ¹ A) 0, as m. The sequence A ¹ =Ā) A) ¹ is λ-bounded, if sup Λ m A) ¹ <. m It can be seen that if lim Ā m = Ā in the odinay sense of convegence of m inteval numbes, then 1.10) Ã Ã 1 m!! X lim λ λ 1 ) m λ Ā Ā =0. m =1 This implies that 1.11) lim m Λ m ¹ A) Ā = lim m 1 λ m mx λ λ 1 )Ā Ā) = 0. =1 which yields that lim Λ m A) ¹ = Ā and hence A ¹ = Ā) A) ¹ is λ- m convegent to Ā.

8 332 Mohq Shafiq and Ayhan Esi Let us denote the classes of I-convegent, I-null, bounded I-convegent and bounded I-null sequences of inteval numbes with C I A), ¹ C I A), ¹ M I C A) ¹ and M I C A), ¹ espectively. By a lacunay sequence we mean an inceasing intege sequence θ = { } such that 0 =0and h := 1 as. Thoughout this pape the intevals detemined by θ will be denoted by I := 1, ) and 1 the atio will be abbeviated by q The space of lacunay stongly convegent sequence N θ was defined by Feedman etal[14] as: N θ = x = x :lim x fo some } h We need the following popula inequalities thoughout the pape. Let p =p ) be the bounded sequence of positive eals numbes. Fo any complex λ, wheneve H =supp ) <, we have λ p max1, λ H ) Also, wheneve H =supp ), we have whee D =max1, 2 H 1 ). a + b p C a p + b p ). Fo any modulus function f, wehavetheinequalities fx) fy) fx y) and fnx) nfx), fo allx, y [0, ]. Now, we give some impotant Lemmas. Lemma.1.7. Evey solid space is monotone. Lemma.1.8. Let K I) andm N. If M / I, thenm K / I, whee I) 2 N filte on N. Lemma.1.9. If I 2 N and M N. IfM/ I, thenm N / I. Let us give a most impotant definitions fo this pape.

9 On some spaces of Lacunay I-convegent sequences of Definition 1.10 see [37]). Let X be a quasilinea space of inteval numbes. A function g : X R is called paanom on X, if fo all A, B X, P 1 )ga) =0ifA = 0, P 2 )ga) 0, P 3 )g A) =ga), P 4 )ga + B) ga)+gb), P 5 )ifλ n ) is a sequence of scalas with λ n λ n )anda n ),A 0 X with ga n ) ga 0 )n ), then gλ n A n ) gλa 0 )n ), P 6 )IfA B, thenga) gb). Definition 1.11 see [37]). Suppose that X is a quasilinea space and Y X. Y is called a subspace of X wheneve Y is a quasilinea space with the same patial odeing and same opeations on X. = Theoem 1.12 see [37]). Y is a subspace of a quasilinea space X if and only if fo evey x, y Y andα, β R,αx+ βy Y. In this aticle, we intoduce and study the following classes of sequences; C I ¹ A,θ,Λ, F,p) ¹A =Ā) ¹ A):{ N : 1.12) C I 0 ¹ A,θ,Λ, F,p) = 1.13) ¹A =Ā) ¹ A):{ N : ) f Λ ¹ p A) Ā h } I,fo someā ) f Λ ¹ p A) } I ; h = I ¹ A,θ,Λ, F,p) ¹A =Ā) ¹ A): K >0s.t.{ N : 1.14) h ³ f Λ ¹ A) p K} I ) ;

10 334 Mohq Shafiq and Ayhan Esi A,θ,Λ, ¹ F,p) ) = ¹A =Ā) A):sup ¹ f Λ ¹ p 1.15) A) <. h We also denote M I C ¹ A,θ,Λ, F,p)= ¹ A,θ,Λ, F,p) C I ¹ A,θ,Λ, F,p) and M I C ¹ A,θ,Λ, F,p)= ¹ A,θ,Λ, F,p) C I 0 ¹ A,θ.Λ, F,p), whee p =p ) is a bounded sequence of positive eal numbes, F =f ) is a sequence of modulus functions, θ = { } is a lacunay sequence and ¹ A =Ā) ¹ A) ω ¹ A) is a bounded sequence of inteval numbes. 2. Main Results Theoem 2.1. Let F =f ) be a sequence of modulus functions and p =p ) be the bounded sequence of positive eal numbes.then, the sets C I A,θ,Λ, ¹ F,p), C0 I A,θ,Λ, ¹ F,p), M I C A,θ,Λ, ¹ F,p)andM I C A,θ,Λ, ¹ F,p) ae quasilinea spaces ove the field of eal numbes. Poof. We shall pove the esult fo the space C I ¹ A,θ,Λ, F,p). Rests will follow similaly. Fo, let ¹ A =Ā), ¹ B = B ) C I ¹ A,θ,Λ, F,p)andα, β be scalas. Then, thee exist positive integes M α and N β such that α M α and β N β. Now, since ¹ A = Ā), ¹ B = B ) C I ¹ A,θ,Λ, F,p), then, thee exists Ā, B IR such that the sets 2.1) and A 1 = { N : f Λ ¹ p A) h Ā } I A 2 = { N : f Λ B) ¹ h B p 2.2) } I. Since, each f, N is modulus function, we have, f Λ α A ¹ + β ¹ p B) αā h + βā)

11 On some spaces of Lacunay I-convegent sequences of DM α ) H f Λ ¹ p A) h Ā + DN β ) H f Λ B) ¹ h B p 2.3) The above inequality futhe implies that ) N : f Λ α A ¹ + β ¹ p B) αā h + βā) N : N : ) f Λ ¹ p A) h Ā ) f Λ B) ¹ h B p = A 2 A 2 I The esult follows fom the above inclusion and the definition of quasilinea space. Theoem 2.2. Let F =f ) be a sequence of modulus functions and p = p ) be the bounded sequence of positive eal numbes. Then, the classes of sequences M I C ¹ A,θ,Λ, F,p)andM I C ¹ A,θ,Λ, F,p) ae paanomed spaces, paanomed by g ¹ A)=gĀ)) = sup f h Λ p A ) M, wheem =max{1, sup p }. Poof. Let A ¹ =Ā), B ¹ = B ) M I C A,θ,Λ, ¹ F,p). P 1 )ItisCleathatg A)=0, ¹ if A ¹ = θ. P 2 )Itisalsoobviousthatg A) ¹ 0. P 3 ) g A)=g ¹ A)isobvious. ¹ P 4 )Since p M 1andM>1, using Minowsi s inequality, we have g ¹ A + ¹ B)=gĀ + B )

12 336 Mohq Shafiq and Ayhan Esi sup =sup =sup f Λ Ā + h B p )) M f Λ Ā)+Λ h B p )) M f Λ Ā) p M h sup +sup + Λ B )) p f Λ Ā) M h f h ³ = g ¹ A)+g ¹ B). Λ B ) p M Theefoe, g ¹ A + ¹ B) g ¹ A)+g ¹ B), fo all ¹ A, ¹ B M I C ¹ A,θ,Λ, F,p). P 5 )Letλ ) be a sequence of scalas with λ ) λ )and Ā), Ā0 M I C ¹ A,θ,Λ, F,p) such that p M in the sense that Ā Ā0 ), gā Ā0) 0 ). Note that gλ ¹ A) max{1, λ }g ¹ A). Then, since the inequality gā) gā Ā0)+gĀ0) holds by subadditivity of g, the sequence {gā)} is bounded. Theefoe, gλ Ā ) gλā0) = gλ Ā ) gλā)+gλā) gλā0) λ λ p M g ³Ā ) + λ p M g Ā ) gā0) 0 as ). That is to say that scala multiplication is continuous. P 6 ). Since each f, N is an inceasing function, it is clea that g ¹ A) g ¹ B), if ¹ A ¹ B.

13 On some spaces of Lacunay I-convegent sequences of Hence M I C ¹ A,θ,Λ, F,p) is a paanomed space. Fo M I C ¹ A,θ,Λ, F,p), the esult is simila. Theoem 2.3. The set M I C ¹ A,θ,Λ, F,p) is closed subspace of ¹ A,θ,Λ, F,p). Poof. Let A ¹ that n) =Ān) )beacauchysequenceinmi C ¹ A,θ,Λ, F,p)such Ā n) Ā. We show that Ā MI C A,θ,Λ, ¹ F,p). Since A ¹ n) =Ān) ) MI C A,θ,Λ, ¹ F,p). Then, thee exists A n such that { N : f Λ A ¹ n) ) h p 2.4) A n } I We need to show that 1) A n ) conveges to Ā0. 1 P ³ 2) If U = { N : h f Λ ¹ p A) Ā0 < }, thenu c I. 1) Since A ¹ n) =Ā n) ) is Cauchy sequence in MI C A,θ,Λ, ¹ F,p) fo a given >0, thee exists 0 N such that sup f Λ A ¹ n) ) Λ ¹ p A q) ) M < h 3, fo alln, q 0 whee M =max{1, sup p }. Fo >0, we have B nq = { N : B q = { N : f Λ A ¹ n) ) Λ A ¹ q) p ) < h 3 )M }, f Λ A ¹ q) p ) h Āq < 3 )M }, B n = N : f Λ A ¹ n) ) h p A n < ) 3 )M. Then, Bnq, c Bq,B c n c I

14 338 Mohq Shafiq and Ayhan Esi Let B c = B c nq B c q B c n,whee B = { N : f h Āq p A n < }. Then, B c I. We choose 0 B c. Then, fo each n, q 0,wehave 1 P ³ n N : h Āq A p o n < f " { N : f h Āq Λ A ¹ q) p ) < 3 )M } { N : f Λ A ¹ n) ) Λ A ¹ q) p ) < h 3 )M } { N : f Λ A ¹ n) ) h p A n < # 3 )M }. Then, A n ) is a Cauchy sequence of inteval numbes, so thee exists some inteval numbe Ā0 such that A n Ā0 as n. 2) Let 0 <δ<1 be given. Then, we show that, if then, U c I. Since ¹ A U = { N : n) =Ān) f Λ ¹ p A) h Ā0 <δ}, ) Ā then, thee exists q 0 N such that P = { N : f Λ A ¹ q0) ) Λ ¹ p 2.5) A) < δ h 3D )M } implies P c I, whee D =max{1, 2 H 1 },H =supp 0. The numbe q 0 can be chosen that togethe with 20), we have Q = { N : 1 h X f ³ Λ Āq 0 ) Ā0 p < δ 3D )M }

15 On some spaces of Lacunay I-convegent sequences of such that Q c I. 1 P ³ Since { N : h f Λ A ¹ q0) ) Λ Āq 0 ) have a subset S of N such that S c I, whee S = { N : p δ} I. Then, we f Λ A ¹ q0) p ) Λ Āq h 0 ) < δ 3D )M }. Let U c = P c Q c S c,whee 1 P ³ U = { N : h f Λ ¹ p A) Ā0 <δ}. Theefoe, fo each U c,wehave 1 P ³ { N : h Λ A) ¹ p Ā 0 <δ} f [{ N : { N : f Λ A ¹ q0) ) Λ ¹ p A) < δ h 3D )M } f Λ A ¹ q0) p ) Λ Āq h 0 ) < δ 3D )M } p 2.6) { N : f Λ Āq h 0 ) Ā0 < δ 3D )M }]. Then, the esult follows fom 21). Since the inclusions M I C A,θ,Λ, ¹ F,p) A,θ,Λ, ¹ F,p)and M I C A,θ,Λ, ¹ F,p) A,θ,Λ, ¹ F,p) ae stict so in view of Theoem 2.3) we have the following esult. Theoem 2.4. The spaces M I C A,θ,Λ, ¹ F,p)andM I C A,θ,Λ, ¹ F,p)ae nowhee dense subsets of A,θ,Λ, ¹ F,p). Theoem 2.5. The spaces C0 I A,θ,Λ, ¹ F,p)andM I C A,θ,Λ, ¹ F,p)ae both solid and monotone. Poof. We shall pove the esult fo C I 0 ¹ A,θ,Λ, F,p). Fo M I C ¹ A,θ,Λ, F,p), the esult follows similaly. Fo, let ¹ A =Ā) C I 0 ¹ A,θ,Λ, F,p)andα ) be a sequence of scalas with α 1, fo all N. Since α p max{1, α G } 1, fo all N, wehave

16 340 Mohq Shafiq and Ayhan Esi p f α Λ Ā) p f Λ Ā), fo all N. h h which futhe implies that p { N : f Λ Ā) } h p { N : f α Λ Ā) }. h Thus, α Ā) C0 I A,θ,Λ, ¹ F,p). Theefoe, the space C0 I A,θ,Λ, ¹ F,p) is solid and hence by lemma 1.7), it is monotone. Theoem 2.6. Let F =f )andg =g ) be two sequences of modulus functions and fo each N, f )andg ) satisfying 2 Condition and p =p ) be a bounded sequence of positive eal numbes. Then, a) X ¹ A,θ,Λ, G,p) X ¹ A,θ,Λ, F G,p), b) X ¹ A,θ,Λ, F,p) ¹ A,θ,Λ, G,p) X ¹ A,θ,Λ, F + G,p), fo X = C I, C I, M I C and MI C Poof. a). Let A ¹ =Ā) C I A,θ,Λ, ¹ G,p) be any element. Then, the set µ ) p 2.7) N : g Λ Ā) I. h Let >0andchooseδ with 0 <δ<1 such that f t) <,0 t δ. Let us denote B = µ p 2.8) Λ Ā) h g and conside ³ ³ ³ lim f B = lim f B + lim f B. B δ, N B >δ, N Now, since f fo each N is an modulus function, we have ³ ³ 2.9) lim f B f 2 lim B ). B δ, N B δ, N

17 On some spaces of Lacunay I-convegent sequences of Fo B >δ,wehave B < B δ < 1+ B δ. Now, since each f is non-deceasing and modulus, it follows that f B ) <f 1 + B δ ) < 1 2 f 2) f 2 B δ ). Again, since each f, N satisfies 2 Condition, we have ³ f B < 1 2 K B ) f ³ K B ) f 2). ³ Thus, f B Hence, <K B ) δ f ³2. δ ³ ³ lim f B max{1, Kδ 1 f 2 ) H } lim B ),H =max{1, sup p }. B >δ, N B >δ, N 2.10) Theefoe, fom 23), 24) and 25), we have ¹A =Ā) C I A,θ,Λ, ¹ F G,p). Thus, C I A,θ,Λ, ¹ G,p) C I A,θ,Λ, ¹ F G,p). Hence, X A,θ,Λ, ¹ G,p) X A,θ,Λ, ¹ F G,p)foX = C. I Fo X = C I, M I C and MI C the inclusions can be established similaly. b). Let A ¹ =Ā ) C I A,θ,Λ, ¹ F,p) C I A,θ,Λ, ¹ G,p). Let >0begiven. Then, the sets µ ) p 2.11) N : f Λ Ā) I h and µ ) p 2.12) N : g Λ Ā) I. h Theefoe, fom 26) and 27), we have N : Thus, ¹ A =Ā) C I ¹ A,θ,Λ, F + G,p) µ ) p f + g ) Λ Ā) I. h δ

18 342 Mohq Shafiq and Ayhan Esi Hence, C I ¹ A,θ,Λ, F,p) C I ¹ A,θ,Λ, G,p) C I ¹ A,θ,Λ, F + G,p). Fo X = C I, M I C and MI C, the inclusions ae simila. Fo g x) = x and f x) = fx), x [0, ), we have the following coollay. Coollay 2.7. X ¹ A,θ,Λ,p) X ¹ A,θ,Λ, F,p), fo X = C I, C I, M I C and M I C. Theoem 2.8. Let F =f ) be a sequence of modulus functions. Then, the inclusions C I 0 ¹ A,θ,Λ, F,p) C I ¹ A,θ,Λ, F,p) I ¹ A,θ,Λ, F,p)hold. Poof. Let A ¹ =Ā) C I A,θ,Λ, ¹ G,p) be any element. Then, thee exists some inteval numbe Ā such that the set n N : o f Λ Ā) h Ā )p I. Since, each f is modulus function fo all N, wehave = 1 h X f Λ Ā) ) p h f Λ Ā) Ā + Ā )p f Λ Ā) h Ā )p + f h Ā )p. Taing supemum ove on both sides, we get A ¹ =Ā) I A,θ,Λ, ¹ F,p). The inclusion C0 I A,θ,Λ, ¹ F,p) C I A,θ,Λ, ¹ F,p)isobvious. Hence C I 0 ¹ A,θ,Λ, F,p) C I ¹ A,θ,Λ, F,p) I ¹ A,θ,Λ, F,p). Theoem 2.9. The spaces C I 0 ¹ A,θ,Λ, F,p)andC I ¹ A,θ,Λ, F,p)aesequence algeba. Poof. Let ¹ A =Ā), ¹ B = B ) C I 0 ¹ A,θ,Λ, F,p). Then, the sets 2.13) N : µ ) p f Λ Ā) I h

19 On some spaces of Lacunay I-convegent sequences of and µ ) N : f Λ h B p ) I. Theefoe, fom 28) and 29), we have µ ) N : f Λ Ā. h B p ) I. Thus, ¹ A. ¹ B C I 0 ¹ A,θ,Λ, F,p). Hence, C I 0 ¹ A,θ,Λ, F,p) is a sequence algeba. Similaly, we can pove that C I ¹ A,θ,Λ, F,p) is a sequence algeba. Refeences [1] A. Esi, A new class of inteval numbes, Jounal of Qafqaz Univesity, Mathematics and Compute Science, pp , 2012). [2] A. Esi, Lacunay sequence spaces of inteval numbes, Thai Jounal of Mathemaatics, 10 2), pp , 2012). [3] A. Esi, Double lacunay sequence spaces of double sequence of inteval numbes, Poyecciones Jounal of Mathematics, 31 1), pp , 2012). [4] A.Esi, Stongly almost -convegence and statistically almost - convegence of inteval numbes, Scientia Magna, 7 2), pp , 2011). [5] A.Esi, Statistical and lacunay statistical convegence of inteval numbes in topological goups, Acta Scientaium Technology, 2014)Bas?laca). [6] A.Esi and N.Baha, On asymptotically -statistical equivalent sequences of inteval numbes, Acta Scientaium Technology, 35 3), pp , 2013). [7] A.Esi and A.Esi, Asymptotically lacunay statistically equivalent sequences of inteval numbes, Intenational Jounal of Mathematics and Its Applications, 1 1), pp , 2013).

20 344 Mohq Shafiq and Ayhan Esi [8] Ayhan, Esi and Hazaia, B., Some I-convegent of double Λ-inteval sequences defined by Olicz function., Global, J. of Mathematical Analysis, 1 3), pp , 2013). [9] Ayhan, Esi., λ-sequence Spaces of Inteval Numbes, Appl. Math. Inf. Sci. 8, No. 3, pp , 2014). [10] Buc, R. C., Genealized asymptotic density,ame., J. Math. 75, pp ). [11] Chiao, K. P., Fundamental popeties of inteval vecto max-nom, Tamsui Oxfod Jounal of Mathematical Sciences, 182), pp , 2002). [12] Dwye, P. S., Linea Computation, New Yo, Wiley, 1951). [13] Fast, H., Su la convegence statistique, Colloq. Math. 2, pp , 1951). [14] Feedman, A. R.,Sembe,J.J.and Raphael,M.,Some Cesao type summability spaces, Poc. London Math. Soc., 37, pp , 1978). [15] Fidy, J. A., On statistical convegence, Analysis, 5, pp , 1985). [16] Khan, V. A., Mohd Shafiq and Ebadullah, K., On paanom I- convegent sequence spaces of inteval numbes, J. of Nonlinea Analysis and Optimisation Theoy and Application), Vol. 5. No. 1, pp , 2014). [17] Khan, V. A., Ahyan Esi and Mohd Shafiq, On paanom BV σ I- convegent sequence spaces defined by an Olicz function, Global Jounal of mathematical Analysis, 2 2), pp , 2014). [18] Kol, E., On stong boundedness and summability with espect to a sequence of moduli, Acta Comment.Univ.Tatu., 960, pp , 1993). [19] Kol, E., Inclusion theoems fo some sequence spaces defined by a sequence of modulii, Acta Comment.Univ. Tatu., 970, pp , 1994). [20] Kostyo, P., Mačaj, M.,and Šalát, T., Statistical convegence and I- convegence.real Analysis Exchange. [21] Kostyo, P., Šalát,T.andWilczyńsi, W., I-convegence,Raal Analysis Analysis Exchange. 262), pp ).

21 On some spaces of Lacunay I-convegent sequences of [22] Mooe, R. E., Automatic Eo Analysis in Digital Computation, LSMD-48421, Locheed Missiles and Space Company, 1959). [23] Mooe, R. E. and Yang,C. T., Inteval Analysis I, LMSD , Locheed Missiles and Space Company, Palo Alto, Calif., 1959). [24] Musaleen, M. and Abdullah K. Noman, On the Spaces of λ- Convegent and Bounded Sequences, Thai Jounal of Mathematics Volume 8, Numbe 2, pp , 2010). [25] Musaleen, M. and Sunil K. Shama, Spaces of Ideal Convegent sequences,volume 2014, Aticle ID , 6 pages og/ /2014/ [26] Naano,H., Concave modulas. J. Math Soc. Japan, 5, pp , 1953). [27] Rucle, W. H., On pefect Symmetic BK-spaces., Math. Ann. 175, pp , 1968). [28] Rucle, W. H., Symmetic coodinate spaces and symmetic bases, Canad. J. Math. 19, pp , 1967). [29] Rucle, W. H., FK-spaces in which the sequence of coodinate vectos is bounded, Canad. J. Math. 25 5), pp , 1973). [30] [31] [32] Šalát, T., On statistical convegent sequences of eal numbes,math, Slovaca 30, 1980). Šalát, T., Tipathy, B. C. and Ziman, M., On some popeties of I- convegence Tata Mt. Math. Publ. 28, pp , 2004). Šalát, T., Tipathy, B. C.and Ziman, M., On I-convegence field., Ital. J. Pue Appl. Math. 17, pp , 2005). [33] Schoenbeg, I. J., The integability of cetain functions and elated summability methods, Ame. Math. Monthly, 66, pp , 1959). [34] Sengönül, M. and Eyilmaz, A., On the Sequence Spaces of Inteval Numbes, Thai J. of Mathematics, Volume 8, No. 3, pp , 2010). [35] Tipathy, B. C., On statistical convegence, Poc.Estonian Acad. Sci. Phy. Math. Analysis, pp , 1998).

22 346 Mohq Shafiq and Ayhan Esi [36] Tipathy, B. C. and Hazaia, B., Paanom I-convegent sequence spaces, Math. Slovaca 59 4), pp , 2009). Mohd Shafiq Depatment of Mathematics Govt. Degee College,Poonch Jammu and Kashmi INDIA shafiqmaths7@gmail.com and Ayhan Esi Adiyaman Univesity Depatmet of Mathematics Adiyaman TURKEY aesi237@hotmail.com