Key words: Fractional difference equation, oscillatory solutions,

Transcription

1 OSCILLATION PROPERTIES OF SOLUTIONS OF FRACTIONAL DIFFERENCE EQUATIONS Musafa BAYRAM * ad Ayd SECER * Deparme of Compuer Egeerg, Isabul Gelsm Uversy Deparme of Mahemacal Egeerg, Yldz Techcal Uversy * Correspodg auhor; E-mal: mbayram@gelsmedur I hs arcle, suded he properes of he oscllao of fracoal dfferece equaos, ad we oba some resuls The resuls we obaed are a expaso ad furher developme of hghly kow resuls The we showed hem wh examples Key words: Fracoal dfferece equao, oscllaory soluos, oscllao heory Iroduco ad Prelmares I he vesgaos of qualave properes for dffereal equaos, research o me scales of he dyamc equaos, oscllao of dffereal (or dfferece) equaos ad fracoal dffereal equaos have bee a very mpora ssue he scece ad egeerg We refer o [-5] ad he refereces here We frs vesgaed followg fracoal dfferece equaos, a x q x s We ca rewre equao () as s s () a x q G ( ) N,,, ( s) where Gs xs posve egers, a ad () ad q are posve coeffce sequeces ad are he dvso of wo odd demosrae ha he Rema-Louvlle fracoal dfferece operaor of order where Therefore, our resuls we use he followg codos: / / C s / s ad s / a s / / C s / s ad s / a s, () (4) By a soluo of equao (), we mea a real-valued sequece x sasfyg equao () for

2 A soluo x of equao () s called oscllaory f s eher eveually posve or eveually egave, oherwse s called o-oscllaory Equao () s called oscllaory f all s soluos are oscllaory Defo [6] We defe vh fracoal sum f as (5) where we defe f for v v v f v s f s, v s amod, sa v f for a vmod ad v v The fracoal sum v f maps fucos defed o o fucos defed o a a v,,,, where Defo [6] Le m m ad v, where m deoes a posve eger, m Se vm The we defe ha -h fracoal dfferece as mv m v f f f (6) Oscllao Properes of () I hs seco, we work he oscllao properes of () Lemma [] Suppose ha he x be a soluo of equao () ad le G s xs (7) s G x (8) Theorem Assume C holds ad furhermore, for all sufcely large, ad / q s s s a (9) The every soluo of () s eher oscllaory or / s q s / s () lm G Proof Assume ha he corary ha x () s o-oscllaory soluo of () The whou loss of geeraly, we may assume ha here s a soluo x of () such ha x o,, where s suffcely large, so ha G o, Ad all of q 's are o decally zero o for,,, From (), we have,

3 I ha case a x a x q G () we udersad ha x ad x s a eveually o-creasg sequece o So, are ulmaely of oe sg For s bg eough, x ad x have a fxed sg o followg codos: Case x ad x ; Case x ad x ; Case x ad x ; Case 4 x ad x For he Case, we have,, We he cosder he / s xs / / s s / x s s G G G The, by C, we oba lm G For he Case, we have from (9), x x whch coradcs wh G a s s x s / s a s x a x / s a s / / The, by C, we oba lm x whch coradcs wh x For he Case, we have lm Gk ad x k suppose ha G k, he lm If we k for Therefore, If we sum boh sdes of () from o, we oba a x q s G s k q s s s ha s k a x q s s If we sum boh sdes of he () from o, we have / ()

4 whch meas for k k x k q s a s s / / / ( ) x k a s q s s If we sum boh sdes of he () from o, we oba Therefore, by (9), we oba lm For he Case 4, we have () G G k q s s s a / G wh coradcs wh / G / / G G s x s x s Tha s, s s / / s s / x G / s The from (), / q x a / x s s (4) If we sum boh sdes of he (4) from o, we oba / x q s a x s / s If we ake, we ge a coradco wh equao () Therefore, he proof of he Theorem s complee Theorem Suppose ha C, (9) ad () hold Furhermore, for all suffcely large, ad s q / s s a (5) 4 / s q / / s as a (6) 4 Therefore, each soluo of () s eher lm G or oscllaory Proof Le's he corary ha x () s o-oscllaory soluo of () The whou loss of geeraly, / / /

5 , x of () such ha x we assume ha here s a soluo,, where s suffcely large, so ha G o I appears ha all of q 's are o decally zero o s a eveually o- for,,, creasg sequece o For he Case, we have o From (), we obaed ha a x, / / /, G s x s s x s s s / / / / s s x s K s The from he las equaly ad (), we oba / (7) a x q K s s If we sum boh sdes of he (7) from o, ha s a x K q s, / s s / K / a (8) s s x q s If we sum boh sdes of he (8) from o, he we ge K s s as / x q G s a s / K s q (9) / If we sum boh sdes of he he (9) from 4 o, we have s K GG4 q / s s 4 a By (4), we oba lm G due o For he Case, ad / / K, whch coradcs wh G / / s xs / G x s / s s / / s

6 Thereore, we have x Thus, from (), we oba s a s s x s / a s / / s a s / s a s a x a x K / s a G s / / / / / s s K a s s / / () / a x q K s a s s s If we sum wo sdes of he () from o, we have / s a x q s K / / s s a The / s 4 x4 q K / / 4 leg, we oba s a s a / s q / / s K a s a 4 whch coradcs wh (6) The res of he proof s made smlar o he proof of he Theorem Thus he proof of he heorem s compleed / / Applcao Le's cosder he followg fracoal dfferece equao as a example 7, () / x s x s s Ths correspods o () wh, 7, /,,, a /, ( ), q /,, ad However,

7 ad The C holds So, we have ad s / s s / s a s s s / / /7 q / s s s a s s s s q s / s s s s s s lm G Therefore, (9) ad () holds, ad he we say ha () s Theorem Coucluso or oscllaory by I hs work, we suded he qualave behavor of soluos of olear fracoal dfferece equaos (FDE) wh fracoal Rema Louvlle dfferece operaor Because here was a gap for he oscllaory soluos of FDE uder he codo (C) he leraure, we cosdered he equao wh he codos (C) ad (C) By usg some echques, we obaed some oscllao resuls The obaed resuls mproved he may crera he leraüre Refereces [] Hassa, T S, Oscllao of hrd order olear delay dyamc equaos o me scales Mahemacal ad Compuer Modellg, 49, (9),7-8, pp [] Öğrekç, S, Ierval Oscllao Crera For Secod-Order Fucoal Dffereal Equaos, Sgma, 6,(8),, pp 5-59 [] Msr, A, & Öğrekç, S, Oscllao Crera for a Class of Secod Order Nolear Dffereal Equaos, Gaz Uversy Joural of Scece, 9, (6),4, pp 9-97 [4] Erbe, L, e al, Oscllao of hrd order olear fucoal dyamc equaos o me scales, Dffereal Equaos ad Dyamcal Sysems, 8,(),-, pp 99-7 [5] G E Chazaraks, e al, Oscllaory soluos of olear fracoal dfferece equaos, I J Dff Equ, (8) ( press) [6] Bayram, M, e al, Oscllaory behavor of soluos of dffereal equaos wh fracoal order, Applead Mahemacs & Iformao Sceces,, (7),, pp [7] Bayram, M, e al, O he oscllao of fracoal order olear dffereal equaos, Sakarya Uversy Joural of Scece,, (7), 6, pp - [8] Erbe, L, e al, Oscllao of hrd order fucoal dyamc equaos wh mxed argumes o me scales, Joural of Appled Mahemacs ad Compug, 4 (),-, pp 5-7 [9] Secer, A, & Adguzel, H,, Oscllao of soluos for a class of olear fracoal

8 dfferece equaos, The Joural of Nolear Scece ad Applcaos, 9, (6),, pp [] Ba, Z, & Xu, R, The asympoc behavor of soluos for a class of olear fracoal dfferece equaos wh dampg erm, Dscree Dyamcs Naure ad Socey, 8, (8) [] Baculíková, B, & Džura, J, Oscllao of hrd-order eural dffereal equaos, Mahemacal ad Compuer Modellg, 5, (),-, pp 5-6 [] Che, D-X, Oscllao crera of fracoal dffereal equaos, Advaces Dfferece Equaos,, (), arcle, 8 pages [] Lu, T, e al, Oscllao o a class of dffereal equaos of fracoal order, Mahemacal Problems Egeerg,, () [4] Feg, Q, Oscllaory Crera For Two Fracoal Dffereal Equaos, WSEAS Trasacos o Mahemacs,, (4), pp 8-8 [5] Q, H, Zheg, B, Oscllao of a class of fracoal dffereal equaos wh dampg erm, Sc World J, (), Arcle ID 6856 [6] Ogrekc, S, Ierval oscllao crera for fucoal dffereal equaos of fracoal order, Advaces Dfferece Equaos, 5, (5), [7] Bayram, M, e al, Oscllao of fracoal order fucoal dffereal equaos wh olear dampg, Ope Physcs,, (5), [8] Bayram, M, e al, Oscllao crera for olear fracoal dffereal equao wh dampg erm, Ope Physcs,4,(6),, pp 9-8 [9] Zheg, B, Oscllao for a class of olear fracoal dffereal equaos wh dampg erm, Joural of Advaced Mahemacal Sudes, 6, (),, pp 7--5 [] XU, R Oscllao Crera for Nolear Fracoal Dffereal Equaos, Joural of Appled Mahemacs,, Arcle ID 9757, 7 pages [] L, W N, Forced oscllao crera for a class of fracoal paral dffereal equaos wh dampg erm, Mahemacal Problems Egeerg, 5, (5) [] Sagayaraj, M R, e al, O he oscllao of olear fracoal dfferece equaos, Mah Aeera, 4, (4), pp 9-99 [] Selvam, A G M, e al, Oscllaory behavor of a class of fracoal dfferece equaos wh dampg, Ieraoal Joural of Appled Mahemacal Research,, (4),() pp -4 [4] L, W N, Oscllao resuls for cera forced fracoal dfferece equaos wh dampg erm, Advaces Dfferece Equaos,, (6), pp -9 [5] Sagayaraj, M R, e al, Oscllao Crera for a Class of Dscree Nolear Fracoal Equaos, Bulle of Socey for Mahemacal Servces ad Sadards ISSN,, (4),, pp 7-5 [6] Ac, F M, Eloe, P W, Ial value problems dscree fracoal calculus, Proceedgs of he Amerca Mahemacal Socey, 7, (8),, pp