Tauberian Conditions in Terms of General Control Modulo of Oscillatory Behavior of Integer Order of Sequences

Transcription

1 Iteratioal Mathematical Forum, 2, 2007, o. 20, Tauberia Coditios i Terms of Geeral Cotrol Modulo of Oscillatory Behavior of Iteger Order of Sequeces İ. Çaak Departmet of Math., Ada Mederes Uiversity, 09010, Aydi, Turkey ibrahimcaak@yahoo.com Abstract Let (u ) be a sequece of real umbers ad let L be ay (C, 1) regular ad additive limitable method. I this paper we prove that if the classical cotrol modulo of the oscillatory behavior of (u ) belogig to some space of sequeces is a Tauberia coditio for L, the covergece or (C, 1) covergece of (u ) out of L-limitability of (u )is recovered depedig o the coditios o the geeral cotrol modulo of the oscillatory behavior of iteger order m of (u ). Mathematics Subject Classificatio: 40E05 Keywords: Geeral cotrol modulo, geeral limitable methods, Tauberia coditios. 1 Itroductio Tauber s first theorem [5] asserted that a Abel limitable sequece u =(u ) is coverget if ω (0) (u) =Δu. (1) To describe this, we say that the coditio (1) is a Tauberia coditio for the Abel limitable method. Tauber [5] proved further that the weaker coditio σ (1) (ω (0) (u)) = 1 +1 kδu k (2) k=0 is also a Tauberia coditio for the Abel limitable method. Meyer-Koig ad Tietz [3] showed that if (1) is a Tauberia coditio for ay regular ad additive method L, the (2) is a Tauberia coditio for L. Recetly, Çaak,

2 958 İ. Çaak Dik ad Dik [1] have used the cocept of the geeral cotrol modulo defied by Staojević [4] ad obtaied several results which state that for specially choose spaces A ad B of sequeces, if (ω (0) (u)) Ais Tauberia coditio for ay (C, 1) regular ad additive method L, the (ω (1) (u)) Bis a Tauberia coditio for L. Çaak ad Totur [2] exteded the results i [1] ad proved that if (ω (0) (u)) Ais Tauberia coditio for ay (C, 1) regular ad additive method L, the for ay iteger m 1,(ω (m) (u)) Bis a Tauberia coditio for L. I this paper we prove that if (ω (0) (u)) belogig to some space of sequeces is a Tauberia coditio for L, the covergece or (C, 1) covergece of (u ) out of L-limitability of (u ) is recovered depedig o the coditios o (ω (m) (u)) for ay iteger m 1. 2 Defiitios ad Notatios For a sequece u =(u ) ad for each iteger m 1 we defie (Δ) m u = Δ((Δ) m 1 u ) = (Δ) m 1 (Δu ) where (Δ) 0 u = u ad (Δ) 1 u = Δu := (u u 1 ). The symbol u = o(1) meas that u 0as. The classical cotrol modulo of the oscillatory behavior of (u ) is deoted by ω (0) (u) =Δu. The geeral cotrol modulo of the oscillatory behavior of order m of (u ) is defied iductively by ω (m) (u) =ω (m 1) (u) σ (1) (ω (m 1) (u)) where σ (1)(u) = 1 k=0 u +1 k. The Kroecker idetity u σ (1) (0) (u) =V (Δu) where V (0) (Δu) = 1 k=0 kδu +1 k is well kow ad used i various steps of proofs of theorems. For each iteger m 1 ad for all oegative itegers, we iductively defie sequeces related to (u ) such as V (m) (Δu) = σ (1)(V (m 1) (Δu)) ad σ (m) (u) =σ(1) (σ(m 1) (u)) where σ (0)(u )=u. A sequece (u ) is Abel limitable to s if lim x 1 (1 x) =0 u x = s ad (C, 1) limitable to s if lim σ (1) (u) =s. A sequece (u ) is slowly oscillatig if lim λ 1 + lim max kj=+1 +1 k [λ] Δu j = 0 where [λ] deotes the iteger part of λ. A sequece (u ) is moderately oscillatory if for λ>1, lim max kj=+1 +1 k [λ] Δu j < +. Let L be ay limitable method. If u =(u )isl limitable to s, we write L lim u = s. A limitatio method L is called additive if L lim u = s ad L lim v = t imply L lim (u + v )=s + t. The limitable method L is said to be regular if lim u = s implies L lim u = s. The limitable method L is said to be (C, 1) regular if L lim u = s implies L lim σ (1) (u) = s. That every regular limitable method is (C, 1) regular is clear. A coditio o (u )is(c, 1) Tauberia coditio for L if we recover oly covergece of (C, 1) meas of (u ) out of L-limitability of (u ) ad the coditio o (u ).

3 Tauberia Coditios i Terms of Geeral Cotrol Modulo Mai Results Throughout this paper we require L to be (C, 1) regular ad additive. Let B, N, S ad M deote the space of sequeces bouded, covergig to 0, slowly oscillatig ad moderately oscillatig, respectively. Deote by A the space of sequeces such that N A. We prove the followig theorems. Theorem 3.1 If (ω (0) (u)) A is a Tauberia coditio for L, the (ω (0) (u))) Ais a Tauberia coditio for L. (σ (1) Theorem 3.2 If (ω (0) (u)) Ais a Tauberia coditio for L, the for ay iteger m 1, (ω (m) (u)) Ais also a Tauberia coditio for L. Theorem 3.3 Let U ad V be spaces of sequeces such that (u σ (1)(u)) U for every (u ) V ad N U.If(ω (0) (u)) U is a Tauberia coditio for L, the for ay iteger m 1, (ω (m) (u)) V is a Tauberia coditio for L. Theorem 3.4 Let U ad V be spaces of sequeces such that (σ (1) (u)) U for every (u ) Vad N U.If(ω (0) (u)) Uis a (C, 1) Tauberia coditio for L, the for ay iteger m 1, (ω (m) (u)) V is a Tauberia coditio for L. 4 Proof of Theorems To prove this theorems, we eed the followig observatio. Observatio 4.1 For each iteger m 1, ω (m) (u) =(Δ) mv (m 1) (Δu). Proof We do the proof by iductio. By defiitio, for m = 1 we have ω (1) (u) =ω (0) (u) σ (1) (ω (0) (u)) = Δu V (0) (Δu) =ΔV (0) (Δu). Assume the observatio is true for m = k. That is, assume that ω (k) (u) =(Δ) k V (k 1) (Δu). (3) We must show that the observatio is true for m = k + 1. That is, we must show that ω (k+1) (u) = (Δ) k+1 V (k) (Δu). Agai by defiitio, ω (k+1) (u) =ω (k) (u) σ(1) (ω(k) (u)). By (3), ω (k+1) (u) =(Δ) k V (k 1) (Δu) (Δ) k V (k) (Δu) = (Δ) k (V (k 1) (Δu) V (k) (Δu)) = (Δ) k (ΔV (k) = (Δ) k+1 V (k) (Δu).

4 960 İ. Çaak Thus, we coclude that observatio is true for each iteger m 1. Proof of Theorem 3.1 Assume that (ω (0) (u)) Ais a Tauberia coditio for L. Let (σ (1)(ω(0) (u))) Aad L lim u = s. Sice L is (C, 1) regular, we have L lim V (0) (Δu) =s. From the idetity σ(1) (ω(0) (u)) = ΔV (0)(Δu), we deduce that V (0) (Δu) = 0 by the assumptio. Sice N A, we coclude that (V (0) (Δu)) = (Δσ(1) (u)) A. It the follows that lim σ (1) (u) =s by (C, 1) regularity of L. By Kroecker idetity, we obtai lim u = s. I the case A = N, we have Meyer-Koig ad Tietz s theorem [3]. I Theorem 3.1, A ca be also take as B, S or M. Proof of Theorem 3.2 Assume that (ω (0) (u)) Ais a Tauberia coditio for L. Let (ω (m)(u)) A. Sice L lim (Δ) m 1 V (m 1) (Δu) =0, (Δ) m 1 V (m 1) (Δu) =o(1) (4) by assumptio. Sice L lim (Δ) m 2 V (m 1) (Δu) = 0, by (4) we obtai From the idetity (Δ) m 1 V (m 1) (Δu), (4) ad (5), we coclude that V (m 1) (Δ) m 2 V (m 1) (Δu) =o(1). (5) (Δu) = (Δ) m 2 V (m 2) (Δu) (Δ) m 2 (Δ) m 2 V (m 2) (Δu) =o(1). (6) Similarly, we obtai (Δ) m 3 V (m 3) (Δu) =o(1). (7) Cotiuig i this vei, we deduce that V (0) (Δu) =o(1). (8) Sice V (0) (Δu) =Δσ (1) (u) =o(1) ad L lim σ (1) (u) =L lim u,we have lim σ (1) (u) =L lim u (9) by assumptio. This completes the proof. Proof of Theorem 3.3 Assume that (ω (0) (u)) U is a Tauberia coditio for L. Let (ω (m) (u)) V. The Sice L lim (Δ) m V (m) (Δu) =0, (Δ((Δ) m V (m) (Δu))) U. (10) (Δ) m V (m) (Δu) =o(1) (11)

5 Tauberia Coditios i Terms of Geeral Cotrol Modulo by assumptio. Sice L lim (Δ) m 1 V (m) (Δu) =0, by (11) we obtai (Δ) m 1 V (m) (Δu) =o(1). (12) From the idetity (Δ) m V (m) (Δu) =(Δ) m 1 V (m 1) (Δu) (Δ) m 1 V (m) (Δu), (11) ad (12), we have (Δ) m 1 V (m 1) (Δu) =o(1). (13) Cotiuig i this vei, we deduce that V (0) (Δu) =o(1). (14) Sice V (0) (Δu) =Δσ(1) (u) =o(1) ad L lim σ (1)(u) =L lim u,we coclude lim σ (1) (u) =L lim u (15) by assumptio. This completes the proof. As special cases to sequece spaces i Theorem 3.3, spaces U ad V ca be take as follows. i) U = B, V = S ii) U = B, V = M iii) U = V = B iv) U = V = N. Proof of Theorem 3.4 Assume that (ω (0) (u)) U is a Tauberia coditio for L. Let (ω (m) (u)) V. The (σ (1) (ω(m) (u))) = ((Δ) m V (m) (Δu)) U. (16) Additivity ad (C,1) regularity of L ad L lim u = s imply L lim (Δ) m 1 (Δu) =0. Sice (Δ((Δ) m 1 V (m) (Δu))) U, we obtai V (m) (Δ) m 1 V (m) (Δu) =o(1) (17) by assumptio. From the idetity (Δ) m V (m) (Δu) =(Δ) m 1 V (m 1) (Δu) (Δ) m 1 V (m) (Δu), (16) ad (17), it follows that ((Δ) m 1 V (m 1) (Δu)) U. Additivity ad (C,1) regularity of L ad L lim u = s imply L lim (Δ) m 2 V (m 1) (Δu) =0. Sice (Δ((Δ) m 2 V (m 1) (Δu))) U, we coclude by assumptio that (Δ) m 2 V (m 1) (Δu) =o(1). (18) Additivity ad (C,1) regularity of L ad L lim u = s imply L lim (Δ) m 3 V (m 1) (Δu) =0. Sice (Δ((Δ) m 3 V (m 1) (Δu))) U, we deduce by assumptio that (Δ) m 3 V (m 1) (Δu) =o(1). (19)

6 962 İ. Çaak From the idetity (Δ) m 2 V (m 1) (Δu), (18) ad (19), it follows that V (m 1) (Δu) = (Δ) m 3 V (m 2) (Δu) (Δ) m 3 (Δ) m 3 V (m 2) (Δu) =o(1). (20) Cotiuig i this vei, we have ΔV (2) (Δu) =o(1). (21) From L lim V (2) (Δu) = 0, we have by assumptio V (2) (Δu) =o(1). (22) From the idetity ΔV (2) (Δu) =V (1) (Δu) V (2) (Δu), (21) ad (22), we have V (1) (Δu) =o(1). (23) Sice V (1) (Δu) =Δσ(2) (u) =o(1) ad L lim σ (1)(u) =L lim σ (2)(u), we have lim σ (2) (u) =L lim σ(1) (u). (24) From the idetity σ (1) (1) (u) σ(2) (u) =V (Δu), (23) ad (24), we have lim σ (1) (u) =L lim σ (1) (u). (25) As special cases to sequece spaces i Theorem 3.4, spaces U ad V ca be take as follows. i) U = S, V = S ii) U = S, V = M iii) U = V = B iv) U = V = N. Refereces [1] [2] İ. Çaak, M. Dik ad F. Dik, O a Theorem of W. Meyer-Koig ad H. Tietz, It. J. Math. Math. Sci. 15 (2005), İ. Çaak ad Ü. Totur, Tauberia coditios for a geeral limitable method, Submitted (2006). [3] W. Meyer-Koig ad H. Tietz, O Tauberia coditios of type o, Bull. Amer. Math. Soc. 73 (1967), [4] Č. V. Staojević, Aalysis of Divergece: Applicatios to the Tauberia Theory, Graduate Research Semiar, Uiversity of Missouri - Rolla (1999). [5] A. Tauber, Ei Satz aus der Theorie der uedliche Reihe, Moatsh. f. Math. 8 (1897), Received: August 17, 2006