Classes of Fuzzy Real-Valued Double Sequences Related to the Space p

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1 Global Journal of Science rontier Reearch Mathematic and Deciion Science Volume 3 Iue 6 Verion 0 Year 03 Tye : Double Blind Peer Reviewed International Reearch Journal Publiher: Global Journal Inc USA Online ISSN: & Print ISSN: Clae of uzzy Real-Valued Double Sequence Related to the Sace By Bijan Nath & Santanu Roy National Intitute of Technology, Silchar, India Abtract - In thi article, correonding to certain general equence { }, we introduce the fuzzy real valued double equence ace ace where i a double equence of bounded trictly oitive number, cloely related to the ace We tudy their different roertie We tudy ome algebraic and toological roertie of the ace Alo we obtained the neceary and ufficient condition for incluion and equality of, and, Keyword : fuzzy real valued double equence, normal, ymmetric etc GJSR- Claification : MSC 00: 03E7, 6E50 Clae of uzzy Real-Valued Double Sequence Related to the Sace l Strictly a er the comliance and regulation of : 03 Bijan Nath & Santanu Roy Thi i a reearch/review aer, ditributed under the term of the Creative Common Attribution-Noncommercial 30 Unorted Licene htt://creativecommonorg/licene/by-nc/30/, ermitting all non commercial ue, ditribution, and reroduction in any medium, rovided the original wor i roerly cited

2 Ref [] L C Barror, R C Baanezi, and P ATonelli, uzzy modelling in oulation dynamic, Ecol Model, 8 000, 7-33 Clae of uzzy Real-Valued Double Sequence Related to the Sace Bijan Nath α & Santanu Roy σ Abtract - In thi article, correonding to certain general equence we introduce the fuzzy real valued double equence ace ace where i a double equence of bounded trictly oitive number, cloely related to the ace We tudy their different roertie We tudy ome algebraic and toological roertie of the ace Alo we obtained the neceary and ufficient condition for incluion and equality of and Keyword : fuzzy real valued double equence, normal, ymmetric etc I Introduction { }, To overcome limitation induced by vaguene and uncertainty of real life data, neoclaical analyi [6] ha been develoed It extend the coe and reult of claical mathematical analyi by alying fuzzy logic to conventional mathematical object, uch a function, equence and erie etc Since the introduction of the concet of fuzzy et by Zadeh [8] in 965, fuzzy et theory ha become an active area of reearch in cience and engineering The idea of fuzzy et theory have been ued widely not only in many engineering alication, uch a, in the comuter rogramming [0], in quantum hyic [5], in oulation dynamic [], in the control of chao [9], in bifurcation of non-linear dynamical ytem [], but alo in variou branche of mathematic, uch a, theory of metric and toological ace [8], in the theory of linear ytem [7], tudie of convergence of equence of function [5,3,7] and in aroximation theory [] 03 Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 37 Author α σ : Deartment of Mathematic, National Intitute of Technology, Silchar , Aam, India bijannath@yahoocom, antanuroy79@yahooin 03 Global Journal Inc US

3 03 Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 38 Uing the notion of fuzzy real number, different tye of fuzzy real-valued equence ace have been introduced and tudied by everal mathematician The initial wor on double equence of real or comlex term are found in Bromwich [4] Hardy [] introduced the notion of regular convergence for double equence of real or comlex term Moričz [6], Baarir and Solancan [3], Choudhary and Triathy [7], Triathy and Dutta [,3], Triathy and Sarma [4,5] are a few to be named thoe who tudied different aect of double equence of fuzzy real number The ace wa introduced by Sargent [9] He tudied ome roertie of the ace Later on it wa tudied from equence ace oint of view and ome matrix clae with one member a and Sen [6] and many other were characterized by Rath and Triathy [8], Triathy Throughout the thei N, R and C denote the et of natural, real and comlex number reectively and w, denote the ace of all and bounded equence of comlex term reectively A fuzzy real number i a fuzzy et on, R ie a maing : R L [0,] aociating each real number t with it grade of memberhi t A fuzzy real number i aid to be convex if t r min [ r], where t r If there exit normal II t 0 R uch that t 0, then the fuzzy real number i aid to be A fuzzy real number i aid to be uer emi-continuou if for each 0, [ 0, a, for all a L Definition and Bacground i oen in the uual toology of R The et of all uer emi continuou, normal, convex fuzzy number i denoted by RL fuzzy real number we mean that the number belong to RL The -level et of a fuzzy real number, 0, [ ] { t R : t } Throughout the thei, by a i denoted and defined a Ref [4] TJI Bromwich, An Introduction to the Theory of Infinite Serie, Macmillan & Co LtdNew Yor, Global Journal Inc US

4 Every real number r can be exreed a a fuzzy real number r a follow : r t 0 if t r otherwie reectively The additive identity and multilicative identity in RL are denoted by 0 and The abolute value of R L, i defined a one may refer to Kaleva and Seila [4]: 03 t max{ t, t}, 0 if if t 0 t 0 Let D be the et of all cloed bounded interval [ L R ] on the real line R If Y [ Y L, Y R ], then Y if and only if L L R R Y and Y L R L R Alo let d, Y max -, Y -Y Then D, d i a comlete metric ace Let d : R L R L R be defined by d, Y u d [ ],[ Y], for, Y R L 0 Then d define a metric on L R and R L, d i a comlete metric ace A fuzzy real-valued double equence i a double infinite array of fuzzy real number for all n, N and i denoted by, where RL A fuzzy real-valued double equence i aid to be convergent in Pringheim ene to the fuzzy real number 0, if for every 0, there exit n n, 0 0 N uch that d 0, for all n n0 and 0 u d n, A fuzzy real-valued double equence E,,0 A fuzzy real valued double equence ace Y whenever d Y, 0 d, 0 for all n N E 0 0 i aid to be bounded if i aid to be olid or normal if, and E Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 39 A fuzzy real valued double equence ace the canonical re-image of all it te ace E i aid to be monotone if E contain 03 Global Journal Inc US

5 Throughout denote a ermutation over N N or a given equence, 03 Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 40 S denote the et of all ermutation of the element of, that i S { } A fuzzy real valued double equence ace S E, for all E A fuzzy real valued double equence ace Y E, whenever Y E A fuzzy real valued double equence ace Y E, whenever E E E and 0 imlie 0 Y Let denote the et of all ubet of N or any N, uch that doe not contain more than element equence uch that n n, n, E i aid to be ymmetric if i aid to be equence algebra if i aid to be convergence free if denote the cla of all for all n, N i a non-decreaing A BK-ace i a Banach ace of comlex double equence x x co-ordinate ma are continuou, that i, i x x 0, whenever x i x 0 a n, i i where x x for all i N and x The ace introduced by Sargent [9] i defined a x x w : x u x, n Triathy and Sen [6] introduce the equence ace a follow : or or 0, x w : x u x, n, n x w : x u x n, n n n in which the 03 Global Journal Inc US

6 Generalizing the above equence ace, we now introduce the ace a follow: w : u u [ d,0] ;,, r, r r n where i a double equence of bounded trictly oitive real number Theorem The cla of equence i a linear ace Proof With tandard technique, we can eaily rove the reult Theorem The cla of equence i comlete i i i Proof Let be a Cauchy equence in where u u, r, r r n [ d,0] Then for a given 0, there exit n 0 N uch that u u i, r, r i j, for all i, j n0 r n j III Main Reult [ d i j ] d, for all i, j n0, for all n, N i n the equence for all n, N, for all n, N Hence for each fixed, N, i a Cauchy equence in RL Since RL i comlete, the equence converge in RL, for each n, N i i i Let lim, for all n, N, where We now how that i and and ii i rom equation, we get for each fixed and r 03 Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 4 n [ d i j ] r, for all i, j n0, 03 Global Journal Inc US

7 Letting j we get 03 Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 4 n [ d i ] r, for all i n0, u u i [ d ], for all i n0, r, r r n i, for all i n0 i i Hence, ince i a linear ace i Alo for all i n i m, Hence i comlete, 0 Theorem 3 The cla of equence i a B K-ace Proof By Theorem, i a Banach ace i Let 0, a i m, Then for a given 0, there exit n 0 N uch that i m,, for all i n0 u u i [ d ], for all i n0, r, r r n i d, for all i n0, for all n, N d i 0, a i Hence i a B K-ace Thi comlete the roof of the theorem 03 Global Journal Inc US

8 Theorem 4 i The cla of equence i a ymmetric ace If and U S, where S denote the et of all ermutation of the element of, then U, ii The cla of equence i a normal ace Proof The roof of the reult follow from definition 03 lim i Theorem 5 if and only if u Proof Let u K, Then K Let u u r Then, r, r n u u K K [ d, r, r r n Hence m,,0] [ d Converely let m,, If oible let u i,,0] we have to how that u u Then there exit a ubequence i of uch that Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 43 Then for, u, r u, r r n [ d we have,0] u, r u, i ri r i ri n [ d,0] 03 Global Journal Inc US

9 03 Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 44,, a contradiction Thi te conclude the roof of the theorem In view of the above theorem, we formulate the following reult if and only if u and u where Corollary m,, Theorem 7,, if and only if,,, u and u Proof Putting, for all n, N, in Corollary, we get the reult eaily Theorem 8 If 0 q uq, then m Proof Uing the roertie of n, q ace, we get the reult eaily Theorem 9 if 0 q uq and u n,, Proof Uing the roertie of ace and Theorem 5, we get the reult Reference Référence Referencia [] G A Anataiou, uzzy aroximation by fuzzy convolution tye oerator, Comut Math Al, , [] L C Barror, R C Baanezi, and P ATonelli, uzzy modelling in oulation dynamic, Ecol Model, 8 000, 7-33 [3] M Baarir and O Solancan, On ome double equence ace, J Indian Acad Math 999, [4] TJI Bromwich, An Introduction to the Theory of Infinite Serie, Macmillan & Co LtdNew Yor, 965 [5] M Burgin, Theory of fuzzy limit, uzzy Set and Sytem, 5 000, [6] M Burgin, Neoclaical analyi, fuzzy continuity and convergence; uzzy Set and Sytem, , 9-99 [7] B Choudhury and B C Triathy, On fuzzy real valued equence ace, Proc International Conference, held at Salt Lae city, USA, during July -5, 005, Global Journal Inc US

10 [8] M A Erceg, Metric ace in fuzzy et theory, J Math Anal Al, , [9] A L radow, and R J Evan, Control of chao, Method and alication in engineering, Chao, Solution & ractal, 9 005, [0] R A Gile, Comuter rogram for fuzzy reaoning, uzzy Set and Sytem, 980, - 34 [] GH Hardy, On the convergence of certain multile erie, Proc Camb Phil Soc, 9 97, [] L Hong, and J Q Sun, Bifurcation of fuzzy nonlinear dynamical ytem, Commun Nonlinear Sci Numer Simul, 006, - [3] G Jager, uzzy uniform convergence and equicontinuity, uzzy Set and Sytem, , [4] H Kizmaz, On certain equence ace, Canad Math Bull 4 98, [5] J Madore, uzzy hyic, Ann Phyic, Ann Phy, 9 99, [6] Moričz, Extenion of ace c and c 0 from ingle to double equence, Acta Math Hung 57-99, 9-36 [7] K Peeva, uzzy linear ytem, uzzy Set and Sytem, 49 99, [8] D Rath and BC Triathy, Characterization of certain matrix oerator; J Oria Math Soc 8 989, -34 [9] W L C Sargent, Some equence ace related to Soc, , 6-7 ace; Jour London Math [0] M Sen and S Roy, Paranormed tye uzzy real valued cla of equence,, Kyungoo Math J, 5 0, [] B C Triathy, On generalized difference aranormed tatitically convergent equence, Indian J Pure Al Math , Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 45 [] B C Triathy and A J Dutta, uzzy real-valued double equence ace; Soochow Jour Math, , Global Journal Inc US

11 03 Global Journal of Science rontier Reearch Volume III Iue VI V erion I Year 46 [3] B C Triathy and A J Dutta, On fuzzy real-valued double equence ace ; Mathematical and Comuter Modelling, , [4] B C Triathy, and B Sarma, Statitically convergent difference double equence Sace, Acta Mathematica Sinica, , [5] B C Triathy, and B Sarma, Some double equence ace of fuzzy number defined by Orlicz function; Acta Mathematica Scientia, 3B 0, [6] B C Triathy, and M Sen, On a new cla of equence related to the ace Tamang Journal of Mathematic, 33 00, 67-7 [7] K Wu, Convergence of fuzzy et baed on decomoition theory and fuzzy olynomial function, uzzy Set and Sytem, , [8] L A Zadeh, uzzy et, Information and Control, 8 965, ; 03 Global Journal Inc US