Lacunary I-Convergent Sequences

KYUNGPOOK Math. J. 52(2012), 473-482 http://dx.doi.og/10.5666/kmj.2012.52.4.473 Lacunay I-Convegent Sequences Binod Chanda Tipathy Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boagaon, Gachuk, Guwahati-781 035, Assam, India e-mail : tipathybc@yahoo.com; tipathybc@ediffmail.com Bipan Hazaika Depatment of Mathematics, Rajiv Gandhi Univesity, Itanaga-791 112, Aunachal Padesh, India e-mail : bh_gu@yahoo.co.in; bipanhazaika_gu@ediffmail.com Bisweshwa Choudhay Depatment of Mathematics, Univesity of Botswana, Pivate Bag 0022, Gaboone, Botswana e-mail : choudha2000@yahoo.com Abstact. In this aticle we intoduce the concepts of lacunay I- convegent sequences. We investigate its diffeent popeties like solid, symmetic, convegence fee etc. 1. Intoduction Thoughout this aticle w, c, c 0 l denote the spaces of all, convegent, null and bounded sequences espectively. The notion of I-convegence was studied at initial stage by Kostyko, Salat and Wilczynski [6]. Late on it was studied by Demici [1], Salat, Tipathy and Ziman ([8],[9]), Tipathy and Hazaika ([14], [15], [16], [17]), Tipathy and Mahanta [19], Tipathy and Sama [21] and many othes. A lacunay sequence is an inceasing intege sequence θ = (k ) such that k 0 = 0 and h = k k 1, as. The intevals detemined by θ will be defined by J = (k 1, k ] and the atio k k 1 will be defined by ϕ. Feedman, Sembe and Raphael [3] defined the space N θ in the following way: * Coesponding Autho. Received Febuay 8, 2011; accepted May 14, 2011. 2010 Mathematics Subject Classification: 40A05, 40A35, 46D05, 46A45. Key wods and phases: Ideal, I-convegence, lacunay sequence, solid, symmetic, convegence fee. 473

474 B. C. Tipathy, B. Hazaika and B. Choudhay Fo any lacunay sequence θ = (k ), N θ = (x k ) : lim h 1 k I x k L = 0, fo some L The space N θ is a BK space with the nom (x k ) θ = sup h 1 k I x k. Nθ 0 denotes the subset of those sequences in N θ fo which L = 0 and we have (Nθ 0,. θ) is also a BK-space. The notion of Lacunay convegence has been investigated by Colak [2], Fidy and Ohan ([4], [5]), Tipathy and Bauah [12], Tipathy and Et [13], Tipathy and Mahanta [18] and many othes in the ecent past. 2. Definition and notations Definition 2.1. Let X be a non-empty set. Then a family of sets I 2 X (powe sets of X) is said to be an ideal if I is additive i.e. A, B I A B I and heeditay i.e. A I, B A B I. Definition 2.2. A non-empty family of sets I 2 X is called a filte on X if and only if ϕ / I, fo each A, B I A B I and fo each A I and B A B I. Definition 2.3. An ideal I 2 X is called non-tivial if I 2 X. Definition 2.4. A non-tivial ideal I 2 X is called admissible if and only if I x : x X. Definition 2.5. A non-tivial ideal I is a maximal if thee cannot exists any nontivial ideal J I containing I as a subset. Fo each ideal I thee is a filte I(I) coesponding to I i.e. I(I) = K N : K c I, whee K c = N K. Definition 2.6. A sequence (x k ) of complex numbes is said to be I-convegent to the numbe L if fo evey ε > 0, k N : x k L ε I. We wite I lim x k = L.. Definition 2.7. A sequence (x k ) of complex tems is said to be I-null fo which L = 0, we wite I lim x k = 0. Definition 2.8. A sequence (x k ) of complex numbes is said to be I-Cauchy if fo evey ε > 0 thee exists a numbe m = m(ε) such that k N : x k x m ε I.

Lacunay I- Convegent Sequences 475 Definition 2.9. A sequence (x k ) of complex numbes is said to be I-bounded if thee exists M > 0 such that k N : x k > M I. The usual convegence is a paticula case of I-convegence. In this case I = I f (the ideal of all finite subsets of N). Definition 2.10. A subset A of N is said to have asymptotic density o density δ(a) if δ(a) = lim 1 n n n k=1 χ A(k) exists, whee χ A is the chaacteistic function of E. Definition 2.11 A complex sequence (x k ) is said to be statistically convegent to L if fo evey ε > 0, δ(k N : x k L ε) = 0. We wite stat lim x k = L. The statistical convegence is a paticula case of I-convegence. In this case I = I δ (the ideal of all subsets of N of zeo asymptotic density). The notion of statistical convegence was investigated fom sequence space point of view and linked with summability theoy by Rath and Tipathy [7], Tipathy ([10], [11]), Tipathya and Bauah [12], Tipathy and Sama [20], Tipathy and Sen ([22], [23]) and many othes. Definition 2.12. Let A N and d n (A) = 1 n χ A (k) s n k=1 k fo n N whee s n = n k=1 1 k. If lim n d n(a) exists, then it is called as the logaithmic density of A. I d = A N : d(a) = 0 is an ideal of N. Definition 2.13. Let T = (t nk ) be a egula non-negative matix. Fo A N, define d (n) T (A) = lim t nk χ A (k), fo all n N. If T (A) = d T (A) exists, then k=1 n d(n) d T (A) is called as T density of A. Clealy I dt = E N : d T (A) = 0 is an ideal of N. Note 1: I δ and I d ae paticula cases of I dt. (i) Asymptotic density, fo (ii) Logaithmic density, fo t nk = t nk = 1 n if n k; 0, othewise k 1 s n if n k; 0, othewise Definition 2.14. Let A N be defined as A(t+1, t+s) = cadn A : t+1 n lim inf t + s fo t 0 and s 1. Put β s = t A(t + 1, t + s), βs lim sup = t A(t + 1, t + s). If both u(a) = lim β s s s and u(a) = lim β s s s exist and if u(a) = u(a)(= u(a)), then u(a) is called the unifom density of the subset of A. Clealy I u = A N : u(a) = 0 is a non-tivial ideal and I u convegence is said to be unifom statistical

476 B. C. Tipathy, B. Hazaika and B. Choudhay convegence. Definition 2.15. A sequence space E is said to be solid (o nomal) if (α k x k ) E, wheneve (x k ) E and fo all sequence (α k ) of scalas with α k 1,fo all k N. Definition 2.16. A sequence space E is said to be symmetic if (x k ) E implies (x π(k) ) E, whee π is a pemutation of N. Definition 2.17. A sequence space E is said to be sequence algeba if (x k ) (y k ) = (x k y k ) E wheneve (x k ), (y k ) E. Definition 2.18. A sequence space E is said to be convegence fee if (y k ) E wheneve (x k ) E and x k = 0 implies y k = 0. Definition 2.19. Let K = k 1 < k 2 <... N and E be a sequence space. A K-step space of E is a sequence space λ E K = (x k n ) w : (k n ) E. A canonical peimage of a sequence (x kn ) λ E K is a sequence y n w defined by xn, if n K; y n = 0, othewise Definition 2.20. A canonical peimage of a step space λ E K is a set of canonical peimages of all elements in λ E K, i.e. y is in canonical peimage of λe K if and only if y is canonical peimage of some x λ E K. Definition 2.21. A sequence space E is said to be monotone if it contains the canonical peimages of its step spaces. Definition 2.22. Let θ = (k ) be lacunay sequence. Then a sequence (x k ) is said to be lacunay I-convegent if fo evey ε > 0 such that N : h 1 x k L ε I. k I We wite I θ lim x k = L. Definition 2.23. Let θ = (k ) be lacunay sequence. Then a sequence (x k ) is said to be lacunay I-null if fo evey ε > 0 such that N : h 1 x k ε I. k I We wite I θ lim x k = 0. Definition 2.24. Let θ = (k ) be lacunay sequence. Then a sequence (x k ) is said to be lacunay I-Cauchy if thee exists a Subsequence (x k ()) of (x k ) such that k () J fo each N, () = L and fo evey ε > 0 lim x k

N : h 1 Lacunay I- Convegent Sequences 477 k J x k x k () ε I. Definition 2.25. A lacunay sequence θ = (k ()) is said to be a lacunay efinement of the lacunay sequence θ = (k ) if (k ) (k ()). Thoughout the aticle l I, c I, c I 0, c I θ and (ci 0) θ epesents I-bounded, I- convegent, I-null, lacunay I-convegent and lacunay I-null sequence spaces espectively. The following esult is well known. Lemma 2.1. Evey solid space is monotone. 3. Main esults Theoem 3.1. A sequence (x k ) is I θ -convegent if and only if it is an I θ -Cauchy sequence. Poof. Let (x k ) c I θ be a sequence. Then thee exists L C such that I θ lim x k = L. Wite H (i) = i N : x k L < 1 i, fo each i N. Hence fo each i N, H (i) H (i+1) and We choose k 1 such that k 1, implies Next we choose k 2 > k 1 such that k 2 implies N : H (2) J h < 1 / I. N : H () J h I. / I. 1 N : H (i) J h < 1 Poceeding in this way inductively we can choose k p+1 > k p such that > k p+1 implies J H ( p + 1). Futhe fo all satisfying k p < k p+1, choose k () J H (p) such that This implies lim x k = L. x k L < 1 p. Theefoe, fo evey ε > 0, we have N : h 1 x k x k () ε N : h 1 x k L ε 2 k, k () J k J

478 B. C. Tipathy, B. Hazaika and B. Choudhay i.e. N : h 1 N : h 1 k J x k L ε 2 k, k () J x k x k ε Then (x k ) is a I θ -Cauchy sequence. Convesely suppose (x k ) is a I θ -Cauchy sequence. Then fo evey ε > 0, we have N : h 1 x k L ε N : h 1 x k x k () ε 2 k J k, k () J N : h 1 x k () L ε 2. k () J It follows that (x k ) is a I θ -convegent sequence. Theoem 3.2. If θ is a lacunay efinement of a lacunay sequence θ and (x k ) c I θ, then (x k) c I θ. Poof. Suppose that fo each J of θ contains the points (k, t) η() t=1 of θ such that. I. k 1 < k, 1 < k,2 <... < k, η() = k, whee J, t = (k, t 1, k, t]. Since k (k ()), so, η() 1. Let (Jj ) j=1 be the sequence of intevals (J, t) odeed by inceasing ight end points. Since (x k ) c I θ, then fo each ε > 0, j N : (h j ) 1 x k L ε I. J j J Also since h = k k 1, so h, t = k, t k, t 1. Fo each ε > 0, we have N : h 1 x k L ε k J N : h 1 j N : (h j k J ) 1 x k L ε. J j J k J j Theefoe N : h 1 x k L ε I. k J Hence (x k ) c I θ.

Lacunay I- Convegent Sequences 479 Note 3.1. If θ 1 and θ 2 ae lacunay sequences then the union and intesection of θ 1 and θ 2 need not be a lacunay sequence. Fo example conside θ 1 = (3 1) and θ 2 = (3 ). Theoem 3.3. Let ψ be a set of lacunay sequences (a) If ψ is closed unde abitay union, then c I µ = c I θ, whee µ = θ; (b) If ψ is closed unde abitay intesection, then c I ν = c I θ, whee ν = θ; (c) If ψ is closed unde union and intesection, then c I µ c I θ ci ν. Poof. (a) By hypothesis we have µ ψ which is a efinement of each θ ψ. Then fom theoem 3.2, we have if (x k ) c I µ implies that (x k ) c I θ. Thus fo each θ ψ, we get c I µ c I θ. The evese inclusion is obvious. Hence c I µ = c I θ. (b) By pat (a) and theoem 3.2, we have c I ν = (c) By pat (a) and (b) we get c I µ c I θ ci ν. Result 3.1. Let θ = (k ) be a lacunay sequence. Then the spaces c I θ and (ci 0) θ ae solid and monotone. Poof. Let (α k ) be a sequence of scalas with α k 1, fo all k N. Then we have the space (c I 0) θ is solid by the following elation N : h 1 α k x k ε N : h 1 x k ε. k J k J The space (c I 0) θ is monotone by Lemma 2.1. The othe pat of the esult follows in a simila way. Result 3.2. Let θ = (k ) be lacunay sequence. Then the spaces c I θ and (ci 0) θ ae not symmetic in geneal. Poof of the esult follows fom the following example. Example 3.1. Fo I = I c. Let T = t : t = i 2 o j 3, i, j N, then Let θ = (3 ), and let (x k ) defined as follows: x k = 2, fo k = j 2 ; j N. x k = k, othewise. Then (x k ) c I θ. Let (y k ) be a eaangement of (x k ), defined by (y k ) = (x 1, x 2, x 4, x 3, x 9, x 5, x 16, x 6,...). Then (y k ) / c I θ. Hence c I θ is not symmetic. c I θ. t T t 1 <.

480 B. C. Tipathy, B. Hazaika and B. Choudhay Similaly the othe esult follows. Result 3.3. Let θ be a lacunay sequence. Then the spaces c I θ and (ci 0) θ ae not convegence fee in geneal. Poof of the esult follows fom the following example. Example 3.2. Let θ = (3 ) and let (x k ) and (y k ) be two sequences defined by x k = 1 k, fo all k N and y k = k 2, fo all k N. Then (x k ), (y k ) belongs to c I θ, but x k = 0 does not imply y k = 0, fo all k N. Hence c I θ is not convegence fee. The esult gets in same way. Result 3.4. Let θ be a lacunay sequence. Then the spaces c I θ and (ci 0) θ ae not sequence algeba in geneal. Poof of the esult follows fom the following example. Example 3.3. Fo I = I d. Let θ = (3 ) and let (x k ) and (y k ) be two sequences as x k = 1 + 1 k 2, if k is even; 1, othewise and y k = 1 k 2, if k is odd; 0, othewise Then (x k ) (y k ) belongs to c I θ, but (x k).(y k ) does not belongs to c I θ. Hence c I θ is not a sequence algeba. The othe esult gets in same way. Conclusions. In this aticle we have investigated the notion of lacunay convegence fom I-convegence of sequences point of view. Still thee ae a lot to be investigated on sequence spaces applying the notion of I-convegence. The wokes will apply the techniques used in this aticle fo futhe investigations on I-convegence. Refeences [1] R. Colak, Lacunay stong convegence of diffeence sequences with espect to a modulus function, Filomat, 17(2003), 9-14. [2] K. Demici, On lacunay statistical limit points, Demonstato Mathematica, 35(1)(2002), 93-101.

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482 B. C. Tipathy, B. Hazaika and B. Choudhay [23] B. C. Tipathy, M. Sen, Chaacteization of some matix classes involving paanomed sequence spaces, Tamkang Jou. Math., 37(2)(2006), 155-162.