Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function

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1 Applied Mathematics, 0,, doi:0.436/am Published Olie April 0 ( Statistically Coverget Double Sequece Spaces i -Normed Spaces Defied by Orlic Fuctio Abstract Vakeel A. Kha, Sabiha Tabassum Departmet of Mathematics, Aligarh Muslim Uiversity, Aligarh, Idia vakha@math.com, sabihatabassum@math.com Received November 6, 00; revised Jauary 5, 0; accepted Jauary 8, 0 The cocept of statistical covergece was itroduced by Stihauss [] i 95. I this paper, we study covergece of double sequece spaces i -ormed spaces ad obtaied a criteria for double sequeces i - ormed spaces to be statistically Cauchy sequece i -ormed spaces. Keywords: Double Sequece Spaces, Natural Desity, Statistical Covergece, -Norm, Orlic Fuctio. Itroductio I order to eted the otio of covergece of sequeces, statistical covergece was itroduced by Fast [] ad Schoeberg [3] idepedetly. Later o it was further ivestigated by Fridy ad Orha [4]. The idea depeds o the otio of desity of subset of. The cocept of -ormed spaces was iitially itroduced by G ahler [5-7] i the mid of 960 s. Sice the, may researchers have studied this cocept ad obtaied various results, see for istace [8]. Let X be a real vector space of dimesio d, where d. A -orm o X is a fuctio.,. : X X R which satisfies the followig four coditios: ), = 0 if ad oly if, are liearly depedet; ), =, : 3), =,, for ay R : 4),,, The pair X,.,. is the called a -ormed space (see [9]). Eample.. A stadard eample of a -ormed space is R equipped with the followig -orm y, := the area of the triagle havig vertices 0,, y. Eample.. Let Y be a space of all bouded real-valued fuctios o R. For f, g i Y, defie f, g = 0, if f, g are liearly depedet, 000 Mathematics Subect Classificatio. 46E30, 46E40, 46B0. f, g = sup f t g t, if f, g are liearly idepedet. tr The.,. is a -orm o Y. We recall some facts coectig with statistical covergece. If K is subset of positive itegers, the K deotes the set { k K : k }. The atural desity K of K is give by K = lim, where K deo tes the umber of elemets i K, provided this limit eists. Fiite subsets have atural desity ero ad K c c = K where K = \ K, that is the complemet of K. If K K ad K ad K have atu- K K Moreover, if ral desities the. K = K =, the K K = (see [0]). A real umber sequece = is statistically coverget to L provided that for every >0 the set : L has atural desity ero. The sequece = is statustically Cauchy sequece if for each >0 there is positive iteger N = N such that : N = 0 (see []). If = is a sequece that satisfies some property P for all ecept a set of atural desity ero, the we say that satisfies some property P for almost all. A Orlic Fuctio is a fuctio M :0, 0, which is cotiuous, odecreasig ad cove with M 0 = 0, M >0 for >0 ad M, as. If coveity of M is replaced by M y M M y, the it is called a Modulus futio (see Maddo []). A Orlic fuctio may be bouded or u- Copyright 0 SciRes.

2 399 bouded. For eample, = p 0< M p is u bouded ad M = is bouded. Lidesstrauss ad Tafriri [3] used the idea of Orlic sequece space; k lm := w: M <, for some > 0 k = which is Baach space with the orm k =if >0: M. M k = The space l M is closely related to the space l p, which is a Orlic sequece space with M = p for p <. A Orlic fuctio M satisfies the coditio ( M f or short ) if there eist costat K ad >0 0 u such that M u KM u wheever u u0. Note that a Orlic fuctio satisfies the iequality for all with 0 < <. M M Orlic fuctio has bee studied by V. A. Kha [4-7] ad may others. Throughout a double sequece = kl is a double ifiite array of elemets kl for kl,. Double sequeces have bee studied by V. A. Kha [8-0], Moric ad Rhoades [] ad may others. A double sequece = called statistically coverget to L if lim, : L, mk, = 0 m, m where the vertical bars idicate the umber of elemets i the set. (see [9]) I this case we write st lim = L.. Defiitios ad Prelimiaries Let be a sequece i -ormed space X,.,.. The sequece is said to be statistically coverget to L, if for every >0, the set : L, has atural desity ero for each oero i X, i other words statistically coverges to L i -ormed space X,.,. if lim : L, =0 for each oero i X. It meas that for every X, L, < aa... I this case we write stlim L, := L,. Eample. Let orm by the formula X = R be equiped with the - i -ormed space,.,., y = y y, =,, y = y, y. Defie the X by, if = k, k N, =, otherwise. ad let L =, ad =, for each i so. If =0 the K = : L, = X, : = k, k : L, is a fiite set, = : = k, k fiite set. Therefore, : L, : = k, k 0 for each i X. Hece, L :, = 0 for every >0 ad X. V. A. Kha ad Sabiha Tabassum [0] defied a double sequece i -ormed space X,.,. to be Cauchy with respect to the -orm if lim pq, = 0 for every X ad k, q., p If every Cauchy sequece i X coverges to some L X, the X is said to be complete with respect to the -orm. Ay complete -ormed space is said to be -Baach space. Eample. Defie the i i -ormed space X,.,. by = 0,0 otherwise. 0, if = k, k N, Copyright 0 SciRes.

3 400 ad let L = 0,0 ad =,. If =0 the :,, 4,9,6,, ; L We have that : L, = 0 for every >0 ad X. This implies that st lim, = L., But the sequece is ot coverget to L. A sequece which coverges statistically eed ot be bouded. This fact ca be see from Eample [.] ad Eample [.]. 3. Mai Results I this paper we defie a double sequece i -ormed space X,.,. to be statistically Cauchy with respect to the -orm if for every >0 ad every oero X there eists a umber p = p, ad q = q, such that lim, NN: pq, m, m, mk, =0 I this case we write st lim L, := L,. Theorem 3.. Let be a double sequece i -ormed space X,.,. ad LL, X. If st lim, = L, ad st lim, = L,, the L = L. Proof. Assume L = L,. The L L =0, so there eists a X, such that L L ad are liearly idepedet. Therefore Now So But L L, =, with > 0. = L L, L, L,., :, < = 0 L, : L, <, : L, <.. Cotradictig the fact that Lstat. Theorem 3.. Let the double sequece ad y i -ormed space X,.,.. If y is a coverget sequece such that = y almost all, the is statistically coverget. (, ) N N: = y = 0 ad Proof. Suppose lim, X. y, = L,. The for every >0 ad, NN: L,, NN: = y. Therefore, NN: L,, NN: y L,, NN: = y. Sice lim y, = L, (3.) for every X, the set, NN: y L, cotais fiite umber of itegers. Hece,, NN: y L, =0. Usig iequality [3.], we get, NN : L, = 0 for every >0 ad X. Cosequetly, st lim L, = L,. Theorem 3.3. Let the double sequece ad y i -ormed space X,.,. ad LL, X ad a. If st lim, = L, ad st lim y, = L,, for every oero X, the ) st lim y, = L L,, for each oero X ad ) st lim a, = al,, for each oero X. Proof ) Assume that st lim, = L,, ad st lim y, = L,, for every oero X. The K =0 ad K =0 where K = K :=, knn : L, K = K :=, knn : y L, for every >0 ad X. Let K = K :=, k NN : y ( LL ),. To prove that K =0, it is sufficiet to prove that K K K. Suppose 0, k0 K. The o 0 y LL, (3.) Copyright 0 SciRes.

4 Suppose to the cotrary that 0, k 0 K K. The 0, k0 K ad 0, k 0 K. If 0, k0 K ad 0, k0 K the, < 00 k L ad, <. 00 k L The, we get y, LL o 0,, < = L y o L 0 which cotradicts [3.]. Hece 0, k0 K K, that is, K K K. ) Let st lim, = L,, a ad a =0. The, NN: L, = 0. a The we have, NN: a al, =, NN: a L, =, NN: L,. a Hece, the right hadside of above equality equals 0. Hece, st lim a, = al,, for every oero X. From Theorem of Fridy [] we have Theorem 3.4. Let be statistically Cauchy sequece i a fiite dimesioal -ormed space X,.,.. The there eists a coverget double sequece y i X,.,. such that = y for almost all. Proof. See proof of Theorem.9 [9]. Theorem 3.5. Let be a double sequece i - ormed space X,.,. The double sequece ( ) is statistically coverget if ad oly if ( ) is a statistically Cauchy sequece. Proof. Assume that st lim, = L, for every oero X ad >0. The, for every X, L, < almost all, p = p, q = q, is chose so that ad if ad pq L, <, the, we have pq, L, Lpq, < almost all. = almost all. 40 Hece, is statistically Cauchy sequece. Coversely, assume that is a statistically Cauchy sequece. By Theorem 3.4, we have st lim, = L, for each X. 4. Refereces [] H. Stihaus, Sur la Covergece Ordiarie et la Covergece Asymptotique, Colloqium Mathematicum, Vol., No., 95, pp [] H. Fast, Sur la Covergece Statistique, Colloqium Mathematicum, Vol., No., 95, pp [3] I. J. Schoeberg, The Itegrability of Certai Fuctios ad Related Summability Methods, America Mathematical Mothly, Vol. 66, No. 5, 959, pp doi:0.307/ [4] J. A. Fridy ad C. Orha, Statistical Limit Superior ad Limit Iferior, Proceedigs of the America Mathematical Society, Vol. 5, No., 997, pp doi:0.090/s [5] S. Gähler, -Merische Räme ud Ihre Topological Struktur, Mathematische Nachrichte, Vol. 6, No. -, 963, pp [6] S. Gähler, Liear -Normietre Räme, Mathematische Nachrichte, Vol. 8, No. -, 965, pp [7] S. Gähler, Uber der Uiformisierbarkeit -Merische Räme, Mathematische Nachrichte, Vol. 8, No. 3-4, 964, pp [8] H. Guawa ad Mashadi, O Fiite Dimesioal - Normed Spaces, Soochow Joural of Mathematics, Vol. 7, No. 3, 00, pp [9] M. Gurdal ad S. Pehliva, Statistical Covergece i -Normed Spaces, Southeast Asia Bulleti of Mathematics, Vol. 33, No., 009, pp [0] A. R. Freedma ad I. J. Sember, Desities ad Summability, Pacific Joural of Mathematics, Vol. 95, 98, pp [] J. A. Fridy, O Statistical Covergece, Aalysis, Vol. 5, No. 4, 985, pp [] I. J. Maddo, Sequece Spaces Defied by a Modulus, Mathematical Proceedigs of the Cambridge Philosophical Society, Vol. 00, No., 986, pp doi:0.07/s [3] J. Lidestrauss ad L. Tafiri, O Orlic Sequece Spaces, Israel Joural of Mathematics, Vol. 0, No. 3, 97, pp doi:0.007/bf [4] V. A. Kha ad Q. M. D. Lohai, Statistically Pre-Cauchy Sequece ad Orlic Fuctios, Southeast Asia Bulleti of Mathematics, Vol. 3, No. 6, 007, pp [5] V. A. Kha, O a New Sequece Space Defied by Orlic Fuctios, Commuicatio, Faculty of Sciece, Uiversity of Akara, Series Al, Vol. 57, No., 008, pp [6] V. A. Kha, O a New Sequece Space Related to the Copyright 0 SciRes.

5 40 Orlic Sequece Space, Joural of Mathematics ad Its Applicatios, Vol. 30, 008, pp [7] V. A. Kha, O a New Sequece Spaces Defied by Musielak Orlic Fuctios, Studia Mathematica, Vol. 55 No., 00, pp [8] V. A. Kha, Quasi almost Covergece i a Normed Space for Double Sequeces, Thai Joural of Mathematics, Vol. 8, No., 00, pp [9] V. A. Kha ad S. Tabassum, Statistically Pre-Cauchy Double Sequeces ad Orlic Fuctios, Accepted by Southeast Asia Bulleti of Mathematics. [0] V. A. Kha ad S. Tabassum, Some Vector Valued Multiplier Differece Double Sequece Spaces i -Normed Spaces Defied by Orlic Fuctio, Submitted to Joural of Mathematics ad Applicatios. [] F. Moric ad B. E. Rhoades, Almost Covergece of Double Sequeces ad Strog Regularity of Summability Matrices, Mathematical Proceedigs of the Cambridge Philosophical Society, Vol. 04, No., 988, pp doi:0.07/s Copyright 0 SciRes.