(1) [x(t)-px(t-r)í+ YJQi(t)x(t-ir) = 0, '' = -^, i=i

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 110, Number 4, December 1990 OSCILLATION OF NEUTRAL DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS QINGGUANG HUANG AND SHAOZHU CHEN (Commuicated by Keeth R. Meyer) Abstract. We establish a ecessary ad sufficiet coditio for the oscillatio of all solutios to the eutral differetial equatio [x(t) - p(x)(t - r)]' + 2 q.(t)x(t - ir) = 0, where 0 < p < 1, r > 0 are costats ad q (t) > 0, i = 1,...,«, cotiuous r-periodic fuctios. are 1. Itroductio We are cocered with the eutral differetial equatio (1) [x(t)-px(t-r)í+ YJQi(t)x(t-ir) = 0, '' = -^, where 0 < p < 1, r > 0 are costats ad q (t) > 0, i = 1,...,«, are cotiuous, r-periodic fuctios o R. By a solutio of equatio ( 1 ) o [T r, oo), T G R, we mea a cotiuous fuctio x(t) o [7" r, oo) for which x(t) - px(t - r) is cotiuously differetiable ad (1) is satisfied for t > T. A solutio x(t) of equatio (1) is called oscillatory if it has arbitrarily large zeros. Otherwise, it is called ooscillatory. If every solutio of ( 1 ) is oscillatory, the we will call equatio (1) oscillatory, ad otherwise ooscillatory. Let Qi = f^q^ds, i = 1,...,, ad assume Q > 0. We will call the followig equatio (2) ;r(i_^-ar) + Q e-ixr = 0 the characteristic equatio of ( 1 ). This termiology comes from the autoomous case where each fuctio qxt) is idetically equal to a costat, i = 1,...,. Received by the editors August 14, Mathematics Subject Classificatio (1985 Revisio). Primary 34K15; Secodary 34K25. Key words ad phrases. Oscillatio, eutral equatios, periodic coefficiets, characteristic equatios. 997 j 1990 America Mathematical Society /90 $1.00+ $.25 per page

2 998 QINGGUANG HUANG AND SHAOZHU CHEN There has recetly bee extesive literature about oscillatios of both autoomous ad oautoomous eutral differetial equatios (see [1-4], [6], [8] ad the refereces cited therei). For oscillatios i autoomous cases, may authors have i particular obtaied ecessary ad sufficiet coditios ivolvig the oexistece of real characteristic roots [1], [4], [6], [8]. It is atural to ask whether or, more precisely, to what extet it is possible to do the same for oautoomous eutral equatios. I this paper we will aswer the questio partially, that is, we will prove that equatio ( 1 ) is oscillatory if ad oly if equatio (2) has o real roots. To the best of the authors' kowledge, this is the first paper that characterizes the oscillatio of eutral equatios with periodic coefficiets. The referee has iformed the authors that similar results were foud idepedetly by G. Ladas, Ch. G. Philos, ad Y. G. Sficas. For delay differetial equatios with periodic coefficiets with or without sig restrictios o the coefficiets we refer to [7] or [5], respectively. 2. The mai result The followig two lemmas will be eeded. For the proof of the first oe we refer to [2]. Lemma 1. If x(t) is a solutio of equatio (1), the v(t) = x(t) - px(t - r) is also a solutio of (1). Moreover, if x(t) is evetually positive, the v(t) ad -v'(t) are also evetually positive. Lemma 2. If x(t) is a evetually positive solutio of equatio (1), the there exists a costat a such that (3) lim sup x(t)eal > 0. t»oo Proof. By Lemma 1, there is a T gr such that x(t) > v(t), v'(t) < 0 for t > T. From (1), we have «(4) 0 = v'(t) + ^2qi(t)x(t-ir)>v'(t) + q(t)v(t-r), t > T + r. /=i Sice q(t) is r-periodic ad Q> 0, there must be a b G (0, r) such that (5) f q(s)ds>k = ^Q for all?. Jt-b Itegratig (4) from / to t + b yields /t+b q(s)ds /t+b q(s)v(s-r)ds <0, t > T + r. From (5) ad (6) we have v(t)>kv(t-(r-b)), t>t + r

3 OSCILLATION OF NEUTRAL EQUATIONS 999 ad ca the prove without difficulty that there is a M > 0 such that v(t) > Mexp I-j- l/c Thus (3) holds for a = (-lk)f(r-b) sice x(t) > v(t). The proof of Lemma 2 is complete. The followig theorem is our mai result. Theorem. Equatio (1) is oscillatory if ad oly if its characteristic equatio (2) has o real roots. Proof. The ecessity of the theorem is obvious, for if (2) has a real root X, the the fuctio x(t) = exp \-pe-xr^xjo~! Jo " r, rt _ Xr As)e ' ' ds is a ooscillatory solutio of (1). To prove the sufficiecy, we assume (1) is ooscillatory. Without loss of geerality, let x(t) he a evetually positive solutio of (1). Let v(t) x(t) - px(t-r). I view of Lemma 1, v(t) is a solutio of (1), v(t) > 0, ad v'(t) < 0 for all large t, say, t > T >0. Let u G C([T, oo), R+) be such that v(t) = v(t)exp(- f^u(s)ds). The u(t) satisfies (7) u(t) = pu(t - r) exp u(s)ds Jt-r.i + ^2qi(t)exp u(s)ds, t>tx = T + r,=i J'-'r Set z0(t) = 0 ad for k = 0, 1, 2,... set (8) zk+x(t)=pzk(t-r)exp I Jt-r zk(s)ds ci + Y^ Q (t) exp / zk(s)ds, i=\ J'-'r *^+i(í ). The, it follows by iductio that T<t<tx. (9) zk(t)<zk+x(t), t>t, ad (10) zk(t)<u(t), t>tk = T + kr, k = 0, 1,2,.... For /' = 1,..., «, set Aq ' = 0 ad set (11) 4'i i = P^ exp I Q$ I + exp (/ QjXf, k = 0, 1, 2,... t>t.,

4 1000 QINGGUANG HUANG AND SHAOZHU CHEN From (8) ad (11), it the follows by iductio computatio) that (here we omit the tedious (12) zk(t) = }~2qi(t)Xk, t>t, k = 0,I,2,... Iductively, we also obtai from (11) that (13) A?é4+i. í=l,...,", k = 0, I,... ad (14) 4 < X[M), k = 0, 1,..., i=\,...,-\. We claim that Xk) is bouded above as k oo. If this is ot true, the for ay M > 0, there is a k > 1 such that Xk)Q > Mr. (12) we have for t > tk + r, Hece, from (10) ad v(t) = v(t)exp(- u(s)ds) =v(t)exp - u(s)ds- u(s)ds\ <Bexpl-j! zk(s)ds) <Bexpi-X[) J' q(s)ds) < Bexp(-X[)Q[(t - tk)/r]) < B exp(-m(t -tk- r)), where B > 0 is a costat ad [(t - tk)/r] is the greatest iteger i (t - tk)/r. This cotradicts Lemma 2. Thus, Xk) is bouded above as k oo. The (13) ad (14) allow us to assure that Xk teds to a fiite limit q( as k oo for each i 1,...,. Lettig k > oo i ( 11 ) produces (15) at= pa^+e'ß, i=i,...,, where p = Yl"=\ Q,a - Multiplyig (15) by ß( ad the summig up over i 1,...,, we get (16) p=ppe" + J2Qieifi. The equality (16) implies that -p/r completes the proof. Refereces is a real root of (2). The cotradictio 1. S. Che ad Q. Huag, Necessary ad sufficiet coditios for the oscillatio of solutios to systems of eutral fuctioal differetial equatios, Fukcial. Ekvac. (to appear). 2. M. K. Grammatikopoulos, G. Ladas, ad Y. G. Sficas, Oscillatio ad asymptotic behavior of eutral equatios with variable coefficiets, Rad. Mat. 2 (1986), E. A. Grove, M.R.S. Kuleovic, ad G. Ladas, Sufficiet coditios for oscillatio ad ooscillatio of eutral equatios, J. Differetial Equatios 68 (1987), E. A. Grove, G. Ladas, ad A. Meimaridou, A ecessary ad sufficiet coditio for the oscillatio of eutral equatios, J. Math. Aal. Appl. 126 (1987),

5 OSCILLATION OF NEUTRAL EQUATIONS Q. Huag, Necessary ad sufficiet coditios for the oscillatio of a class of differetial differece equatios with sig-variable periodic coefficiets, Kexue Togbao 33 (1988), M. R. S. Kuleovic, G. Ladas, ad A. Meimaridou, Necessary ad sufficiet coditios for oscillatios of eutral differetial equatios, J. Austral. Math. Soc. Ser. B 28 (1987), G. Ladas, Y. G. Sficas, ad I. P. Stavroulakis, Nooscillatory fuctioal differetial equatios, Pacific J. Math. 115 (1984), Y. G. Sficas ad I. D. Stavroulakis, Necessary ad sufficiet coditios for oscillatios of eutral differetial equatios, J. Math. Aal. Appl. 123 (1987), departmet of mathematics, shadog uiversity, jla, shadog, , people's Republic of Chia