Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson

Transcription

1 PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode, N Penningon, and A Peerson Universiy of Nebraska-Lincoln Lincoln, NE , USA Absrac The Euler-Cauchy differenial equaion and difference equaion are well known Here we sudy a more general Euler-Cauchy dynamic equaion For his more general equaion when we have complex roos of he corresponding characerisic equaion we for he firs ime wrie soluions of his dynamic equaion in erms of a generalized exponenial funcion and generalized sine and cosine funcions This resul is even new in he difference equaion case We hen spend mos of our ime sudying he oscillaion properies of he Euler-Cauchy dynamic equaion Several oscillaion resuls are given and an open problem is posed Inroducion In his paper we will assume ha he reader is familiar wih he elemenary conceps and noaion used in he calculus on ime scales see, for example, Bohner and Peerson [] We are concerned wih he so-called Euler Cauchy dynamic equaion σx + ax + bx 0, on a ime scale T closed subse of he reals R, where we assume 0 inf T > 0 We will assume hroughou he regressiviy condiion for T κ The equaion σ aµ + bµ 0 λ + aλ + b 0 3 is called he characerisic equaion of he Euler Cauchy dynamic equaion and he roos of 3 are called he characerisic roos of We now give an alernae shorer proof of Theorems 363 and 366 in [] Our proof combines he proofs of hese wo heorems in a novel way Theorem Assume λ, λ are soluions of he chacerisic equaion 3 If λ λ, hen x c e λ, 0 + c e λ, 0 is a general soluion of If λ λ, hen is a general soluion of x c e λ, 0 + c e λ, 0 0 s + λ µs s 99 Mahemaics Subec Classificaion Primary: 39A0; Secondary: 34B0 Key words and phrases Euler Cauchy dynamic equaion, ime scale, oscillaion

2 S HUFF, G OLUMOLODE, N PENNINGTON, AND A PETERSON Proof Noe if we le D denoe he dela derivaive operaor wih respec o, hen he Euler Cauchy dynamic equaion can be wrien in he facored form D λ D λ x 0 4 This facorizaion was noed by Akin Bohner and Bohner [] and [3], Chaper when hey discovered how o define and solve he n-h order Euler Cauchy dynamic equaion Assume ha x is a soluion of he dynamic equaion 4 and le y D λ x, 5 hen by 4, y is a soluion of D λ y 0, which is equivalen o he dynamic equaion y λ y 6 The regressiviy condiion ensures ha λ, λ R, where R is he regressive group [], page 58 Hence he dynamic equaion 6 is regressive and so we ge y c e λ, 0 I follows from 5 ha x saisfies he dynamic equaion or equivalenly D λ x c e λ, 0, D λ x c e λ, 0 7 Using he variaion of consans formula [], page 77, we ge ha x c e λ, 0 + c, σs s e λ s, 0 s e λ 0 c e λ, 0 + c e λ, 0 c e λ, 0 + c e λ, 0 c e λ, 0 + c e λ, 0 c e λ, 0 + c e λ, s e λ 0, σse λ s, 0 s s e λ σs, 0 e λ s, 0 s s + λ µs e λ s, 0 e λ s, 0 s 0 s + λ µs e λ λ Firs noe ha if λ λ, hen we ge he desired resul ha x c e λ, 0 + c e λ, 0 s, 0 s 0 s + λ µs s Nex assume ha λ λ, hen using he formula 0 s + λ µs e [ ] λ λ s, 0 s e λ λ λ λ, 0 we are led o he final resul

3 3 Nex we would like o show ha if our characerisic roos are complex, hen here is a nice form for all real-valued soluions of he Euler Cauchy dynamic equaion in erms of he generalized exponenial and rigonomeric funcions Even in he difference equaions case he complex roos are no considered see Kelley and Peerson [7] Theorem Assume ha he characerisic roos of are complex, ha is λ, α ± i, where > 0, and α R Then x c e α, 0 cos +αµ, 0 + c e α, 0 sin +αµ is a general soluion of he Euler Cauchy dynamic equaion, 0 Proof Assume λ, α ± i, where > 0, are he characerisic roos Then by Theorem, e α+i, 0, e α i, 0 are soluions of We wan o find so ha Solving his equaion we ge Hence if is defined by 9, hen 8 holds Similarly I follows ha is a soluion Similarly x e α+i e α α + i α i 8 + αµ 9 α i α i e α, 0 i, 0 + e, α i 0, 0 + e α ei e α, 0 cos, 0 e α, 0 cos i, 0 + e i +αµ x e α, 0 sin, 0 +αµ, 0, 0, 0 is a soluion Since x, x are linearly independen soluions on T we ge he desired resul Oscillaion Resuls In his secion we will be concerned wih he oscillaion of he Euler Cauchy dynamic equaion We assume hroughou his secion ha T is now unbounded above We now show if he characerisic roos of are complex how a general soluion can be wrien in erms of he classical exponenial funcion and classical rigonomeric funcions A ime scale for which every poin in he ime scale is isolaed is called a discree ime scale

4 4 S HUFF, G OLUMOLODE, N PENNINGTON, AND A PETERSON Lemma If he characerisic roos are complex, ha is λ, α ± i, where > 0, hen where x A c cos B + c sin B, Rξ A e 0 µ α+i > 0, B 0 I α + i ξ µ 0 is a general soluion of he Euler Cauchy dynamic equaion If, in addiion, T is a discree ime scale, hen for T, A ρ + µα + µ, B 0 ρ 0 Arcan µ + αµ Proof Noe ha see page 59, Bohner and Peerson [] he generalized exponenial e α+i, 0 e e ξ 0 µ α+i Rξ 0 µ α+i Ae ib +iiξ µ α+i A cos B + i sin B I follows ha he imaginary par and real par x : A cos B, x : A sin B are soluions of Since hey can be shown o be linearly independen on T he resul follows Now assume ha every poin in T is isolaed, hen α + i ξ µ µ Log + µ α + i µ log + αµ + i µ + i + αµ µ Arg + i µ µ log + αµ + µ + i µ µ Arcan + αµ Hence α + i R ξ µ µ log + αµ + µ and α + i I ξ µ µ µ Arcan + αµ

5 5 I follows from 0 and ha A e 0 µ log +αµ + µ e ρ 0 log +αµ + µ ρ 0 + αµ + µ I follows from 0 and ha µ B 0 µ Arcan which is he desired resul ρ 0 Arcan + αµ µ, + αµ Definiion If he characerisic roos of are complex, hen we say he Euler Cauchy dynamic equaion is oscillaory iff B is unbounded As a well-known example noe ha if T is he real inerval [, and he Euler Cauchy equaion has complex roos, hen he Euler Cauchy equaion is oscillaory This follows from wha we said here because in his case by 0 B d log which is unbounded If T q N0, where q >, hen by Lemma ρ µ B Arcan + αµ ρ q Arcan + αq q k Arcan q + αq which is unbounded and hence he Euler Cauchy dynamic equaion on T q N0 is oscillaory when he characerisuc roos are complex If T N, hen B Arcan, k + α which can be shown o be unbounded and hence he Euler Cauchy dynamic equaion on T N is oscillaory when he characerisuc roos are complex These las wo examples where shown in Bohner and Saker [4], Erbe, Peerson, and Saker [6], and Erbe and Peerson [5], bu here we esablished hese resuls direcly Theorem 3 Comparison Theorem Assume has complex roos α ± i, > 0 Le T : { 0,, } and T : {s 0, s, }, where { n } and {s n } are sricly increasing sequences of posiive numbers wih limi If he Euler Cauchy equaion is oscillaory on T and α < sn equaion is oscillaory on T µs n n µ n,, for n 0, hen he Euler Cauchy

6 6 S HUFF, G OLUMOLODE, N PENNINGTON, AND A PETERSON Proof Since s n µs n n µ n, for n 0, we have ha for n 0, and herefore using α < s n µs n + α n µ n + α sn µs n 0 < s n + µs n α µs n n + µ n α µ n for n 0 Taking reciprocals and muliplying by we obain This implies ha Arcan µs n s n + µs n α µ n n + µ n α µsn µn Arcan s n + αµs n n + αµ n for n 0 This implies ha n µsk B s B s n : Arcan s k + αµs k k0 n µk B B n : Arcan k + αµ k k0 Since we are assuming ha he Euler Cauchy equaion is oscillaory on T we ge ha lim n B s n and herefore from he above inequaliy lim n B n, which implies ha he Euler Cauchy equaion is oscillaory on T µ exiss as a finie Theorem 4 Assume T is discree ime scale and lim number, hen he Euler Cauchy equaion in he complex characerisic roos case is oscillaory on T Proof In his case lim B µ Arcan + µα 0 Arcan 0 µ + α I follows ha B is unbounded and hence he Euler Cauchy equaion in he complex characerisic roos case is oscillaory on T Theorem 4 does no cover he case when T is a discree ime scale where lim The nex heorem considers a discree ime scale where lim µ µ Theorem 5 Le p 0 and le T p : { n : 0, n+ n + p n, n N 0} In he complex characerisic roos case, he Euler Cauchy dynamic equaion is oscillaory on T p

7 7 Proof If p 0, hen T N and he resul was proved earlier in his paper Assume p > 0 Since n+ n + p, 3 n for n N 0, he sequence { n } is sricly increasing Assume lim n n L, where L is a posiive consan Then from 3 we ge L L + L p and his implies ha 0 L P, which is a conradicion, which proves ha he sequence { n } is unbounded above Now for k N pick N k so ha Nk is he smalles elemen in T ha is in he real inerval [k, k + Then for N k N k+, k < k + Noe ha N k+ N k is he number of elemens of T in he real inerval [k, k + Since k + < k and since + + p Therefore Consider we have ha k + p < µ p k p k p N k+ N k k + p n µ B B n Arcan 0 + µ α n Arcan + α 0 p+ To prove ha is oscillaory i suffices o show ha Arcan p+ To show his noe ha 0 Arcan p+ p+ k k N k+ N k 3 3p+ p+ 3 3p+ k p k + p k + p+ 3k 3p+

8 8 S HUFF, G OLUMOLODE, N PENNINGTON, AND A PETERSON One migh hink ha one could use he argumen in he proof of Theorem 5 o show ha if here is an increasing unbounded sequence of poins { } in T wih µ, p hen he Euler Cauchy equaion is oscillaory on T in he complex characerisic roos case The following example shows ha he same ype of argumen does no work Example Assume ha he Euler Cauchy dynamic equaion has complex characerisic roos α ± i, > 0 and T : n[n +, n ] To see if is oscillaory or no in his case we have by 0 + lim B n + I n n n+ n n + α + i I ξ µ α + i ξ µ α + i I ξ µ d 4 Consider he firs erm on he righ hand side of equaion 4 This is he erm we ge by looking a he righ scaered poins in T Noe ha for n 0 sufficienly large n n n n 0 n n 0 n < n + α + i I ξ µ n µn Arcan Arcan n + µn α n + α Arcan n + α Arcan n + α + Arcan nn 0 + nn 0 n + α n + α Hence unlike in he proof of Theorem 5 we do no ge ha his erm corresponding o he righ-scaered poins is infinie Bu now consider he second erm on he

9 9 righ hand side of equaion 4 n+ I n n + n+ α + i ξ µ n d n n+ n + n + d log n n + n + n n + log + log n log + + n n n log + n + n n + Hence we ge ha our equaion is oscillaory α + i I d log + n n + Conecure If he ime scale T is an unbounded subse of he real inerval 0, and he Euler Cauchy equaion has complex characerisic roos, hen is oscillaory on T The auhors would like o hank Seve Haaaa for his help in he proof of Theorem 5 Research suppored by NSF REU Gran and NSF Gran REFERENCES [] E Akin-Bohner and M Bohner, Miscellaneous dynamic equaions, Mehods and Applicaions of Analysis, o appear [] M Bohner and A Peerson, Dynamic Equaions on Time Scales, Birkhauser, Boson, 00 [3] M Bohner and A Peerson, Advances in Dynamic Equaions on Time Scales, Birkhauser, Boson, 003 [4] M Bohner and S H Saker, Oscillaion of second order nonlinear dynamic equaions on ime scales, Rocky Mounain J of Mah, o appear [5] L Erbe and A Peerson, Boundedness and oscillaion for nonlinear dynamic equaions on a ime scale, Proceedings of he American Mahemaics Sociey, [6] L Erbe, A Peerson, and S H Saker, Oscillaion crieria for second order nonlinear dynamic equaions on ime scales, J London Mah Soc, 003 o appear [7] W Kelley and A Peerson, Difference Equaions: An Inroducion wih Applicaions, Second Ediion, Harcour, Academic Press, San Diego, 00