Triple sets of χ 3 -summable sequences of fuzzy numbers defined by an Orlicz function

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1 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ PURE MATHEMATICS RESEARCH ARTICLE Trile sets o χ 3 -summable sequences o uzzy numbers deined by an Orlicz unction Vandana, Deemala * and N Subramanian 3 Received: 03 Setember 06 Acceted: 03 November 06 First Published: 9 November 06 *Corresonding author: Deemala, SQC and OR Unit, Indian Statistical Institute, 03 B T Road, Kolkata , West Bengal, India dmrai3@gmailcom Reviewing editor: Hari M Srivastava, University o Victoria, Canada Additional inormation is available at the end o the article Abstract: In this aer we introduce the χ 3 uzzy numbers deined by an Orlicz unction and study some o their roerties and inclusion results Subjects: Science; Mathematics & Statistics; Foundations & Theorems Keywords: gai sequence; analytic sequence; Orlicz unction; trile sequences; comleteness; solid sace; symmetric sace; Orthogonal olynomials and secial unctions MSC 00 No: 40A05; 40C05; 40D05 Introduction A trile sequence real or comlex can be deined as a unction x:n N N RC, where N, R and C denote the set o natural numbers, real numbers and comlex numbers, resectively Some initial work on double series is ound in Aostol 978, Alzer, Karayannakis, and Srivastava 006, Bor, Srivastava, and Sulaiman 0, Choi and Srivastava 99, Liu and Srivastava 006 and double sequence saces are ound in Hardy 97, Deemala Subramanian, and Mishra in ress, Deemala, Mishra, and Subramanian 06 and many others Later on some initial work ABOUT THE AUTHORS Vandana is a Research Scholar at School o Studies in Mathematics, Pt Ravishankar Shukla University, Raiur-4900, CG India Her research interests are in the areas o alied mathematics including Otimization, Mathematical Programming, Inventory control, Suly Chain Management, Oeration Research, etc She is member o several scientiic committees, advisory boards as well as member o editorial board o a number o scientiic journals Deemala is Visiting Scientist at SQC & OR Unit at Indian Statistical Institute, Kolkata, India Her research interests are in the areas o Otimization, Mathematical Programming, Fixed Point Theory and Alications, Oerator theory, Aroximation Theory etc She is member o several scientiic committees and also member o editorial board o a number o scientiic journals N Subramanian received PhD degree in Mathematics rom Alagaa University at Karaikudi, Tamil Nadu, India and also getting Doctor o Science DSc degree in Mathematics rom Berhamur University, Berhamur, India His research interests are in the areas o summability through unctional analysis o alied mathematics and ure mathematics PUBLIC INTEREST STATEMENT In this aer, we introduced the χ 3 uzzy numbers deined by an Orlicz unction and study some o their roerties with inclusion results Furthermore we rovided an examle o trile sequence o gai which is not symmetric, not solid, not monotone and not convergent ree Our result uniies the results o several author s in the case o classical Orlicz saces One can extend our results or more general saces 06 The Authors This oen access article is distributed under a Creative Commons Attribution CC-BY 40 license Page o 8

2 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ on trile sequence saces is ound in Sahiner, Gurdal, and Duden 007, Esi 04, Esi and Necdet Catalbas 04, Esi and Savas 05, Subramanian and Esi 05 and many others A sequence x =x is said to be trile analytic i su m,n,k x m+n+k < The vector sace o all trile analytic sequences are usually denoted by Λ 3 A sequence x =x is called trile entire sequence i x m+n+k 0 as m, n, k A sequence x =x is called trile chi sequence i m + n + k! x m+n+k 0 as m, n, k The trile gai sequences will be denoted by χ 3 This aer deals with introducing the χ 3 -uzzy number deined by an Orlicz unction and study some toological roerties, inclusion relations and give some examles Some interesting results may be seen in Alzer et al 006, Bor et al 0, Choi and Srivastava 99, Liu and Srivastava 006 Deinitions and reliminaries Deinition An Orlicz unction see Kamthan & Guta, 98 is a unction M: [ 0, [ 0, which is continuous, non-decreasing and convex with M 0 = 0, Mx > 0, or x > 0 and Mx as x I convexity o Orlicz unction M is relaced by Mx + y Mx + My, then this unction is called modulus unction Lindenstrauss and Tzariri 97 used the idea o Orlicz unction to construct Orlicz sequence sace Throughout a trile sequence is denoted by, a trile ininite array o uzzy real numbers Let D denote the set o all closed and bounded intervals X = [ a, a, a 3 ] on the real line R For X = [ a, a, a 3 ] D and Y = [ b, b, b 3 ] D, deine dx, Y = max a b, a b, a 3 b 3 It is known that D, d is a comlete metric sace A uzzy real number X is a uzzy set on R, that is, a maing X:R R R I I I = [ 0, ] associating each real number t with its grade o membershi Xt The α-level set [ X ] α, o the uzzy real number X, or 0 <α ; is deined by [ ] α X = t R:Xt α} The 0-level set is th closure o the strong 0-cut that is, cl t R:Xt > 0 } A uzzy real number X is called convex i Xt Xs Xr Xv = minxs, Xr, Xv}, where s < t < r < v I there exists t 0 R such that X t 0 = then, the uzzy real number X is called normal A uzzy real number X is said to be uer-semi continuous i, or each ε <0, X [ 0, a + ε is oen in the usual toology o R or all a I The set o all uer-semi continuous, normal, convex uzzy real numbers is denoted by LR The absolute value, X o X LR is deined by max Xt, X t}, i t 0; X t = 0, i t < 0 Page o 8

3 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ Let d:lr LR LR R R R be deined by dx, Y = su d [ X ] α [ ] α, Y 0 α Then, d deines a metric on LR and it is well-known that LR, d is a comlete metric sace A sequence LR is said to be null i d X, 0 = 0 A trile sequence o uzzy real numbers is said to be gai in Pringsheim s sense to a uzzy number 0 i lim m,n,k m + n + k!x m+n+k = 0 A trile sequence is said to χ regularly i it converges in the Prinsheim s sense and the ollowing limts zero: m+n+k lim m + n + k!x = 0 or each m, n, k N m,n,k A uzzy real-valued double sequence sace E F is said to be solid i Y E F whenever E F and Y or all m, n, k N Let K = m i, n i, k i :i N; m < m < m 3 and n < n < n 3 and k < k < k 3 < } N N N and E F be a trile sequence sace A K-ste sace o E F is a sequence sace λ E = K X mi n i k w : } X i E F A canonical re-image o a sequence X miniki E F is a sequence Y deined as ollows: Y = X, i m, n, k K, 0, otherwise A canonical re-image o a ste sace λ E K is a set o canonical re-images o all elements in λe K A sequence set E F is said to be monotone i E F contains the canonical re-images o all its ste saces A sequence set E F is said to be symmetric i X πm,πn,πk E F whenever E F, where π is a ermutation o N N N A uzzy real-valued sequence set E F is said to be convergent ree i Y E F whenever E F and = 0 imlies Y = 0 Λ We deine the ollowing classes o sequences: = X : su d Also, we deine the classes o sequences χ R as ollows : A sequence χ R X m+n+k χ = X : lim d m + n + k!x, 0 } <, LR } 0 = 0 i x χ and the ollowing limits hold lim d m + n + k!x 0 = 0 or each m N m lim n d m + n + k!x lim k d m + n + k!x 0 = 0 or each n N 0 = 0 or each k N Page 3 o 8

4 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ Main results m Theorem 3 Let N = min n 0 : su n0 d + n + k! X Y P } 0 } < N = min n 0 : su n0 < and N = max N, N, N 3 i χ R is not a aranormed sace with gx = lim su d m + n + k! X Y P M N 0 N i and only i μ >0, where μ = lim N in N and M = max, su N 3 ii χ R is comlete with the aranorm 3 Proo i Necessity: Let χ R be a aranormed sace with 3 and suose that μ = 0 Then α = in N = 0 or all N N and g λx = lim N su N λ M = or all λ 0, ], where X = α χ R whence λ 0 does not imly λx θ, when X is ixed But this contradicts to 3 to be a aranorm Suiciency: Let μ>0 It is trivial that gθ = 0, g X = gx and g X + Y + Z, 0 g X, 0 + g Y, 0 + g Z, 0 Since μ >0 there exists a ositive number β such that >β or suiciently large ositive integer m, n, k Hence or any λ C, we may write λ max λ M, λ β or suciently large ositive integers m, n, k N Thereore, we obtain g λx, 0 max λ, λ β M g X Using this, one can rove that λx θ, whenever X is ixed and λ 0 or λ 0 and X θ, or λ is ixed and X θ Because a aranormed sace is a vector sace χ R is a set o sequences o uzzy numbers But the set w F = :X LR } o all sequences o uzzy numbers is not a vector sace That is why, in order to say that χ R is a vector subsace that is a sequence sace it is not suicient to show that is closed under addition and scalar multilication Consequently since w F is not a vector sace, χ R is not a vector subsace so that it is not a sequence sace Thereore it cannot be a ara- then χ R normed sace Proo ii Let X be kl a Cauchy sequence in χ R, where X kl = there exists a ositive integer s 0 such that g X kl X rt or all k, l, r, t > s 0 X kl Then or every ε >0 0 <ε< m,n,k N m = lim su d + n + k! X kl N Xrt 0 < ε N 3 By 3 there exists a ositive integer n 0 such that m su d + n + k! X kl Xrst 0 < ε N or all k, l, r, t > s 0 and or N > n 0 Hence we obtain m d + n + k! X kl Xrt 0 < ε < so that m d + n + k! X kl Xrt 0 m < d + n + k! X kl Xrt 0 < ε 35 Page 4 o 8

5 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ or all k, l, r, t > s 0 This imlies that Hence the sequence X kl kl N X kl is a Cauchy sequence in C or each ixed m, n, k n 0 kl N is convergent to say, lim kl Xkl = or each ixedm, n, k > n 0 36 Getting, we deine X = From 3 we obtain g X kl X as r, t, or all k, l, r, t > s 0 By 36 This imlies that lim kl X kl = X Now we show that X = R Since X kl χ R or each k, N N N or every ε >0 0 <ε< there exists a ositive integer n N such that By 36 and 37 we obtain < ε max + ε = ε or k, l > s 0, s and m, n, k > max n0, n This imlies that X χ R Proosition 3 The class o sequences Λ is symmetric but the classes o sequences χ and χ R are not symmetric Proo Obviously the class o sequences Λ is symmetric For the other classes o sequences, consider the ollowing examle Examle Consider the class o sequences χ Let X = X and consider the sequence be deined by and or m >, m = lim su d + n + k! X kl X N 0 < ε N m d ε + n + k!x 0 < or every m, n, k > n d m P M m + n + k! X 0 d + n + k! X kl 0 X nk t+ +n+k, +n+k! t +n+k +n+k! or t =,, or t =, 0, otherwise t+ m+n+k, t m+n+k, or t =, or t =, 0, otherwise Let Y be a rearrangement o X deined by Y nnn t+ 3n 3n! t 3n 3n!, or t =,, or t =, 0, otherwise m + d + n + k! X kl 0 Page 5 o 8

6 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ and or m n k, t+ m+n+k, or t =, Y t = t m+n+k, or t =, 0, otherwise Then, but Y χ Hence, χ is not symmetric Similarly other sequences are also not symmetric Proosition 33 The classes o sequences Λ, χ and χ R are solid Proo Consider the class o sequences χ Let and Y χ be such that m d + n + k!y 0 d m + n + k!x 0 As is non-decreasing, we have lim d m + n + k!y 0 lim d m + n + k!x 0 Hence, the class o sequence χ is solid Simlarly it can be shown that the other classes o sequences are also solid Proosition 34 The classes o sequences χ and χ R are not monotone and hence not solid Proo The result ollows rom the ollowing examle Examle Consider the class o sequences χ and X = X Let J = m, n, k :m n k } N N N Let be deined by t+3 m+n+k, or 3 < t, mt m+n+k 3m + m+n+k 3m m+n+k 3m m+n+k, or t + m, 0, otherwise or all m, n, k N Then Let Y be the canonical re-image o X or the subsequence J o J N N N Then Y = X, or m, n, k J, 0, otherwise Then, Y χ Hence χ is not monotone Similarly, it can be shown that the other classes o sequences are also not monotone Hence, the classes o sequences χ and χ R are not solid Proosition 35 i χ χ χ χ 3 + +, ii χ R 3 χ R χ R 3 χ R Proo It is easy, so omitted Proosition 36 Let, and be three Orlicz unctions, then, i χ iiiλ Λ χ, ii χ R χ R, Proo We rove the result or the case χ χ, the other cases are similar Let ε >0 be given As is continuous and non-decreasing, so there exists η >0, such that η = ε Let Then, there exist m 0, n 0, k 0 N, such that Page 6 o 8

7 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ d m m+n+k, + n + k!x 0 <η, or all m m 0, n n 0, k k 0 d m + n + k!x 0 Hence, Thus, χ χ or all m m 0, n n 0, k k 0 <ε, Proosition 37 i χ Λ, ii χ R Λ, the inclusions are strict Proo The inclusion i χ Λ ii χ R Λ is obvious For establishing that the inclusions are roer, consider the ollowing examle Examle We rove the result or the case χ sequence be deined by or m > n > k, mt m m+n+k m m+n+k 3 t m+n+k and or m < n < k Then, Λ but χ, or + m t,, or < t 3, 0, otherwise mt m+n+k m m+n+k t+ m+n+k, or m t,, or t, 0, otherwise Λ, the other case similar Let X = X Let the Proosition 38 The classes o sequences Λ, χ and χ R are not convergent ree Proo The result ollows rom the ollowing examle Examle Consider the classes o sequences χ Let X = X and consider the sequence deined by + n + k +n+k!x nk = 0, and or other values, m+n+k, or 0 t, mt m+n+k m+ m+n+k +m+ m+n+k +m m+n+k, or < t + m, 0, otherwise Let the sequence Y be deined by + n + k!ynk +n+k = 0, and or other values, Y m+n+k, or 0 t, m t m+n+k m m+n+k, or < t m, 0, otherwise Then, but Y χ Hence, the classes o sequences χ are not convergent ree Similarly, the other saces are also not convergent ree 4 Conclusion The χ 3 uzzy numbers deined by an Orlicz unction and discuss inclusion relation Furthermore, the given examle o trile sequence o gai is not symmetric, not solid, not monotone and not convergent ree Page 7 o 8

8 Vandana et al, Cogent Mathematics 06, 3: 659 htt://dxdoiorg/0080/ Acknowledgements The authors are extremely grateul to the anonymous learned reerees or their keen reading, valuable suggestion and constructive comments or the imrovement o the manuscrit The authors are thankul to the editors and reviewers o Cogent Mathematics The third author NS wish to thank the Deartment o Science and Technology, Government o India or the inancial sanction towards this work under FIST rogram SR/ FST/MSI-07/05 The research o the second author Deemala is suorted by the Science and Engineering Research Board SERB, Deartment o Science and Technology DST, Government o India under SERB National Post-Doctoral ellowshi scheme File Number: PDF/05/ Funding The authors received no direct unding or this research Author details Vandana vdrai988@gmailcom Deemala dmrai3@gmailcom N Subramanian 3 nsmaths@yahoocom School o Studies in Mathematics, Pt Ravishankar Shukla University, Raiur 4900, Chhattisgarh, India SQC and OR Unit, Indian Statistical Institute, 03 B T Road, Kolkata , West Bengal, India 3 Deartment o Mathematics, SASTRA University, Thanjavur 63 40, India Citation inormation Cite this article as: Trile sets o χ 3 -summable sequences o uzzy numbers deined by an Orlicz unction, Vandana, Deemala & N Subramanian, Cogent Mathematics 06, 3: 659 Reerences Alzer, H, Karayannakis, D, & Srivastava, H M 006 Series reresentations or some mathematical constants Journal o Mathematical Analysis and Alications, 30, 45 6 Aostol, T 978 Mathematical analysis London: Addison-Wesley Bor, H, Srivastava, H M, & Sulaiman, W T 0 A new alication o certain generalized ower increasing sequences Filomat, 6, doi:098/fil0487b Choi, J, & Srivastava, H M 99 Certain classes o series involving the Zeta unction Journal o Mathematical Analysis and Alications, 3, 9 7 Deemala, Mishra, L N, & Subramanian, N 06 Characterization o some Lacunary X Convergence Auv o order α with -metric deined by sequence o moduli Musielak Alied Mathematics & Inormation Sciences Letters, 43 Deemala, Subramanian, N, & Mishra, V N in ress Double almost λ m µ n in X -Riesz sace Southeast Asian Bulletin o Mathematics Esi, A 04 On some trile almost lacunary sequence saces deined by Orlicz unctions Research and Reviews: Discrete Mathematical Structures,, 6 5 Esi, A, & Necdet Catalbas, M 04 Almost convergence o trile sequences Global Journal o Mathematical, Analysis,, 6 0 Esi, A, & Savas, E 05 On lacunary statistically convergent trile sequences in robabilistic normed sace Alied Mathematics & Inormation Sciences, 9, Hardy, G H 97 On the convergence o certain multile series Proceedings o the Cambridge Philosohical Society, 9, Kamthan, P K, & Guta, M 98 Sequence saces and series, Lecture notes, Pure and Alied Mathematics New York, NY: 65 Marcel Dekker Inc Lindenstrauss, J, & Tzariri, L 97 On Orlicz sequence saces Israel Journal o Mathematics, 0, Liu, G D, & Srivastava, H M 006 Exlicit ormulas or the Nörlund olynomials B x and n bx Comuters and n Mathematics with Alications, 5, Sahiner, A, Gurdal, M, & Duden, F K 007 Trile sequences and their statistical convergence Selcuk Journal o Alied Mathematics, 8, Subramanian, N, & Esi, A 05 Some new semi-normed trile sequence saces deined by a sequence o moduli Journal o Analysis & Number Theory, 3, The Authors This oen access article is distributed under a Creative Commons Attribution CC-BY 40 license You are ree to: Share coy and redistribute the material in any medium or ormat Adat remix, transorm, and build uon the material or any urose, even commercially The licensor cannot revoke these reedoms as long as you ollow the license terms Under the ollowing terms: Attribution You must give aroriate credit, rovide a link to the license, and indicate i changes were made You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use No additional restrictions You may not aly legal terms or technological measures that legally restrict others rom doing anything the license ermits Page 8 o 8