A Tauberian Theorem for (C, 1) Summability Method

Transcription

1 Alied Mathematical Scieces, Vol., 2007, o. 45, A Tauberia Theorem for (C, ) Summability Method İbrahim Çaak Ada Mederes Uiversity Deartmet of Mathematics 0900, Aydi, Turkey ibrahimcaak@yahoo.com Mehmet Dik Rockford College 5050 E.State Street Rockford, IL, 608, USA mdik@rockford.edu Abstract I this aer, we retrieve slow oscillatio of a real sequece u =(u ) out of (C, ) summability of the geerator sequece (V (0) (Δu)) of (u ) uder some additioal coditio. Cosequetly, we recover covergece or subsequetial covergece of (u ) out of (C, ) summability of (u ) uder certai additioal coditios that cotrol oscillatory behavior of the sequece (u ). Mathematics Subect Classificatio: 40E05 Keywords: Slow oscillatio, geeral cotrol modulo, (C,) summability, Tauberia theorem, subsequetial covergece. Itroductio Let u =(u ) be a sequece of real umbers. The classical cotrol modulo of the oscillatory behavior of (u ) is deoted by ω (0) (u) =Δu = (u u ). The geeral cotrol modulo of the oscillatory behavior of order of (u )is defied by ω () (u) =ω (0) (u) σ () (ω (0) (u))

2 2248 İ. Çaak ad M. Dik where σ ()(u) = k=0 u + k. The idetity u σ () (u) =V (0) (Δu) where V (0)(Δu) = k=0 kδu + k is kow as Kroecker idetity. It is ow clear that ω () (u) =ΔV (0) (Δu). Sice σ () (u) = V (0) k (Δu) k= +u k 0, Kroecker idetity ca be rewritte as u = V (0) (Δu)+ k= V (0) k (Δu) + u 0 () k i terms of (V (0)(Δu), the geerator sequece of (u ). A sequece (u )is said to be subsequetially coverget [] if there exists a fiite iterval I(u) such that all accumulatio oits of (u ) are i I(u) ad every oit of I(u) is a accumulatio oit of (u ). A sequece (u )is(c, ) summable to s if lim σ () (u) =s. A sequece (u ) is said to be slowly oscillatig [5] if lim max λ + Δu =0, +k[λ] =+ where [λ] deotes the iteger art of λ. A sequece (u ) is said to be C, summable [3] if for > = Δσ () (u) <. A sequece (u ) is said to be slowly varyig [4] if lim u([λ]) u() = for λ>. We ow establish the mai result ad its cosequeces. As a corollary to the mai result, we recover classical covergece or subsequetial covergece of the sequece (u ). Theorem. Let (V (0) (Δu)) be (C, ) summable to s. If for some > (λ ) the (u ) is slowly oscillatig. [λ] =+ = o(), λ +, (2)

3 A Tauberia theorem for (C, ) summability method 2249 Theorem.2 Let (u ) be (C, ) summable to s. If for some > (λ ) the (u ) is coverget. [λ] =+ The roofs are based o the followig Lemma. Lemma.3 [5] For λ> u σ () (u) = [λ]+ [λ] (σ() [λ] (u) σ() (u)) where [λ] deotes the iteger art of λ. 2 Proofs of Theorems Proof of Theorem. = o(), λ +, (3) [λ] =+ [λ] k=+ =+ Δu, (4) Alyig Lemma.3 to (V (0) (Δu)) we have V (0) () (Δu) V (Δu) [λ]+ V () () [λ](δu) V [λ] (Δu) + max ΔV (0) (Δu). (5) +k[λ] Sice (V (0) (Δu)) is (C, ) summable, the first term o the right-had side of (5) is o() as ad (5) becomes V (0) (Δu) V () (Δu) max +k[λ] =+ ΔV (0) For the secod term o the right-had side of (5) we have max ΔV (0) [λ] (Δu) ΔV (0) (Δu) +k[λ] =+ =+ [λ] ([λ] ) q ([λ] ) q =+ [λ] =+ (Δu). (6), where + q =

4 2250 İ. Çaak ad M. Dik ([λ] ) q [λ] =+ ([λ] ) [λ] q q (λ ) q [λ] =+ =+. (7) From (6) ad (7) we have [λ] V (0) (Δu) V () (Δu) (λ ) q =+. (8) Lettig λ + i (8) ad takig (2) ito accout, we deduce that V (0) () (Δu) V (Δu) 0. (9) From (9) we have lim V (0) (Δu) =s. Sice σ() (u) = V (0) k (Δu) k= + u k 0,it follows by () that (u ) is slowly oscillatig. Furthermore, for some slowly varyig sequece (B ), we have u = O(B ),. Sice u = O(B ),, it follows that there exists a fiite iterval I such that for every r I, there is a subsequece ( ) u (r) such that lim (r) u (r) B((r)) = r. (See [, 2]). B((r)) Notice that sice (u ) is slowly oscillatig, for all oegative itegers m, the sequece (V (m) (Δu)) is subsequetially coverget [2]. As a corollary we have the followig. Corollary 2. Let (V (0) (Δu)) be (C, ) summable to s. If (V (0) (Δu)) is C, summable, the (u ) is slowly oscillatig. Proof of Theorem.2 Alyig Lemma.3 to (u ) we have u σ () (u) [λ]+ [λ] σ () [λ](u) σ() (u) + max +k[λ] =+ Δu. (0)

5 A Tauberia theorem for (C, ) summability method 225 Sice (u )is(c, ) summable to s, the first term o the right-had side of (0) is o() as ad (0) becomes u σ () (u) max Δu. () +k[λ] =+ For the secod term o the right-had side of (0) we have max +k[λ] =+ Δu From () ad (2) we have [λ] =+ Δu ([λ] ) q ([λ] ) q ([λ] ) q ([λ] ) q q (λ ) q [λ] =+ [λ] =+ [λ] =+ [λ] =+ [λ] =+ [λ] u σ () (u) (λ ) q =+, where + q =. (2). (3) Lettig λ + i (3) ad takig (3) ito accout, we deduce that u σ () (u) 0. (4) From (4) we have lim u = s. As a corollary we have the followig. Corollary 2.2 Let (u ) be (C, ) summable. If (v ), where u = σ () (v), is C, summable, the (u ) is coverget.

6 2252 İ. Çaak ad M. Dik Refereces [] F. Dik, Tauberia theorems for covergece ad subsequetial covergece of sequeces with cotrolled oscillatory behavior, Mathematica Moravica, 5 (200), [2] F. Dik, M. Dik, ad Y. Caak, Alicatios of subsequetial Tauberia theory to classical Tauberia theory, Alied Mathematics Letters, 20 (2007), [3] T. M. Flett, O a extesio of absolute summability ad some theorems of Littlewood ad Paley, Proceedigs of the Lodo Mathematical Society, 7 (957), 3-4. [4] J. Karamata, Sur u mode de croissace régulière des foctios, Mathematica (Clu), 4 (930), [5] Č. V. Staoević, Aalysis of Divergece: Cotrol ad Maagemet of Diverget Process, Graduate Research Semiar Lecture Notes, edited by Y. Caak, Uiversity of Missouri - Rolla, 998. Received: Aril 6, 2007