LINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM

Transcription

1 Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp ISSN: URL: or ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM FAIZ AHMAD Abstract. W study a linar dlay diffrntial quation with a singl positiv and a singl ngativ trm. W find a ncssary condition for th oscillation of all solutions. W also find sufficint conditions for oscillation, which improv th known conditions. 1. Introduction Ladas and co-workrs [3, 6, 8] studid th dlay diffrntial quation u(t) + pu(t τ) qu(t σ) = 0, (1.1) whr p, q, τ, σ R +. Thy obtaind th following st of sufficint conditions for th oscillation of all solutions to this quation: p > q, τ σ, q(τ σ) 1, (1.2) (p q)τ > 1 [1 q(τ σ)]. Chuanxi and Ladas [4] gnralizd th abov conditions to a non-autonomous quation in which p and q ar rplacd by continuous functions P (t) and Q(t). Th dlay quation with an arbitrary numbr of positiv and ngativ trms has bn studid, for th autonomous cas, by Agwo [2] and Ahmad and Alhrbi [1]. Rcntly Elabbasy t al. [5] hav applid a tchniqu in Li [9] to gnraliz conditions (1.2) for a non-autonomous quation having arbitrary numbr of positiv and ngativ trms. A common fatur among rsults prtaining to non-autonomous gnralizations of (1.1) is that, in th limit of constant cofficints, thy rduc to a strongr vrsion of conditions (1.2) in which th fourth condition is rplacd by (p q)τ > 1. Although this st of conditions undrwnt a gradual improvmnt in [3, 6, 8], aftr 1991 no attmpt has bn succssful in improving ths conditions (as far as th author is awar). Th first and scond conditions of (1.2) ar ncssary for oscillation [6, p41]. Howvr it is obvious that th third condition is unncssarily rstrictiv sinc, no 1991 Mathmatics Subjct Classification. 34K15, 34C10. Ky words and phrass. Dlay quation, oscillation, ncssary condition, sufficint conditions. c 2003 Txas Stat Univrsity-San Marcos. Submittd Jun 21, Publishd Sptmbr 8,

2 2 FAIZ AHMAD EJDE 2003/92 mattr how larg th cofficint of th ngativ trm and th diffrnc btwn th dlays might b, all solutions of th quation must b oscillatory providd th cofficint of th positiv trm is sufficintly larg. Idally th oscillation conditions should involv what Hunt and York [7] hav trmd as th torqu associatd with th dlay diffrntial quation. For (1.1) this paramtr is pτ qσ. In this papr w shall sk sufficint conditions which dpnd ithr on th diffrnc or th ratio of pτ and qσ. W also show that th condition pτ qσ > 1 is ncssary for th oscillation of all solutions of (1.1). W xpct that this work will stimulat furthr rsarch in th qust of ncssary and sufficint conditions for oscillation. 2. Basic rsults and prliminary lmmas Th following proprtis of th xponntial function ar asily stablishd. Lt x 0. Thn ax a x + 1 a, if a 1, (2.1) ax a x + 1 a, if a < 1, (2.2) ax bx + b a [1 ln b ] if a > 0 and b > 0. (2.3) a If q = 0 or τ = σ, (1.1) rducs to a dlay quation with a singl positiv trm. If σ = 0, thn a ncssary and sufficint condition for oscillation is [6, p40] pτ qτ > 1. Lt 0 < q < p and 0 < σ < τ. For x 0, dfin g(x) = q σx p τx, x 1 = 1 τ σ ln(p q ), x 2 = 1 τp ln( τ σ σq ), k = (1 σ pτ )q( τ qσ ) σ τ σ. (2.4) A simpl calculation shows that g is incrasing and concav on th intrval [0, x 2 ]. It vanishs at x 1 and attains its maximum valu, k, at x 2. Th charactristic quation corrsponding to (1.1) is x + p τx q σx = 0, (2.5) or x = g(x). It is wll-known that a dlay diffrntial quation posssss a non-oscillatory solution if and only if its charactristic quation has a ral root. Sinc p > q, it follows that x = 0 dos not satisfy (2.5). W shall invstigat th xistnc of positiv or ngativ roots in th following lmmas. Lmma 2.1. Th charactristic quation dos not hav a positiv root if x 1 k i.. ln( p (τ σ)2 ) q( pτ q τ qσ ) σ τ σ.

3 EJDE 2003/92 LINEAR DELAY DIFFERENTIAL EQUATION 3 Proof. Lt f(x) = g(x) x. Thn f is ngativ on [0, x 1 ] and for c > 0, w hav f(x 1 + c) = g(x 1 + c) x 1 c k x 1 c < 0. Thus f rmains ngativ on [0, ) implying that th charactristic quation dos not hav a positiv root. Lmma 2.2. Th charactristic quation has no positiv root if ln( p (τ σ)2 ) > q( p q τ q ) σ (τ σ) τ σ. τ Proof. Sinc g is concav on [x 1, x 2 ], thr is at most on point, call it c, in [x 1, x 2 ] whr th tangnt to th curv y = g(x) is paralll to th lin y = x. Th intrcpt of th tangnt with th x axis is c g(c). If this numbr is positiv, th lin y = x dos not intrsct th curv y = g(x). This implis no positiv root of th charactristic quation if g(c) < c, or q σc p τc < c. (2.6) Sinc th slop at th point (c, g(c)) is unity, w hav From (2.6) (2.7), w gt σq σc + pτ τc = 1. (2.7) τc + q(τ σ) σc < 1. (2.8) Th xprssion on th lft sid of (2.8), as a function of c, is dcrasing. Hnc it will hold on th ntir intrval [x 1, x 2 ] providd it dos so for c = x 1. This givs τ τ σ ln(p σ ) + q(τ σ) τ σ ln( p q ) < 1, (2.9) q or ln( p (τ σ)2 ) > q( p q τ q ) σ (τ σ) τ σ. τ Lmma 2.3. Th charactristic quation dos not hav a ngativ root if pτ qσ > 1/b or q(τ σ) 1, whr b is th largr root of th quation and 1 < b. x(1 ln x) = q(τ σ) pτ qσ, Proof. A ngativ root of th charactristic quation (2.5) is quivalnt to a positiv root of th quation x + p τx q σx = 0. (2.10) In th abov quation, th chang of variabl y = τx yilds y + pτ y qτ σ τ y = 0. Suppos y > 0 is a root of this quation. On making us of (2.2), w gt 0 y + pτ y qτ( σ τ y + 1 σ τ ) = y + (pτ qσ)y q(τ σ). (2.11) Now w us (2.3) to obtain, for arbitrary b > 0, 0 y q(τ σ) + (pτ qσ)[by + b(1 ln b)]. (2.12)

4 4 FAIZ AHMAD EJDE 2003/92 Choos b such that 1 < b and b(1 ln b) = q(τ σ) pτ qσ. Inquality (2.12) bcoms 0 y[ 1 + b(pτ qσ)]. Sinc y > 0, thr will b a contradiction if pτ qσ > 1/b. Now dfin z = y + q(τ σ). Inquality (2.11) bcoms 0 z + (pτ qσ) q(τ σ) z z + (pτ qσ) q(τ σ) z = z[ 1 + (pτ qσ) 1 q(τ σ) ], which will lad to a contradiction if pτ qσ > q(τ σ) Main Rsults It is wll-known that th first two conditions (1.2) ar ncssary for th oscillation of all solutions of (1.1). In this sction w first prov a ncssary condition which should b usful in stimating how far from th bst possibl position a sufficint condition actually is. Thorm 3.1 (A ncssary condition for oscillation). Lt p, q, τ, σ R +, p > q and τ σ. If all solutions of (1.1) ar oscillatory thn pτ qσ > 1/. Proof. W shall prov that othrwis a ral root of th charactristic quation must xist indicating a positiv solution of th dlay quation. Dnot th lft sid of th charactristic quation, i.. (2.5), by f(x). First assum pτ qσ = 1/. Sinc f(0) = p q > 0 and f( 1/τ) = 1/τ + p q σ σ τ = q[ τ σ τ ] 0, hnc f has a zro in [ 1/τ, 0). Nxt assum pτ qσ < 1/. If qσ = 0, it is asy to s that a zro of f will xist in [ 1/τ, 0). Lt qσ > 0 and dfin a = qσ. Thr xists an ɛ > 0 such that pτ qσ = 1 2ɛ < 1 ɛ. Without loss of gnrality w can choos ɛ < a. Thus pτ < qσ + 1 ɛ W writ f(x) = f 1 (x) + f 2 (x), whr = a + 1 ɛ. (3.1) f 1 (x) = (a + 1 ɛ)x + p τx, f 2 (x) = (a ɛ)x q σx. Sinc an quation of th form x + c dx = 0, has a ral root if and only if cd 1/, it follows from (3.1) that f 1 (x) = 0 has a ral root, say x 0, whil qσ = a/ > a ɛ, shows that f 2 (x) = 0 dos not hav any ral root. Sinc f 2 (0) < 0 it follows that f 2 (x 0 ) < 0. Now f(x 0 ) = f 1 (x 0 ) + f 2 (x 0 ) < 0.

5 EJDE 2003/92 LINEAR DELAY DIFFERENTIAL EQUATION 5 Also f(0) = p q > 0, hnc th charactristic quation posssss a root in (0, x 0 ). This provs th ncssity of th condition pτ qσ > 1/ for th oscillation of all solutions of (1.1). Th proof of Thorm 1 is complt. Putting th rsults of Lmmas togthr, w obtain th following Thorm. Thorm 3.2 (Sufficint conditions for oscillation). All solutions of (1.1) will b oscillatory if p > q > 0, τ > σ 0, ln( p σ)2 ) > min{(τ q( pτ q τ qσ ) σ (τ σ) 2 τ σ, q( p τ q ) σ (τ σ) τ σ }, τ whr b is th largr root of th quation pτ qσ > min{1/b, q(τ σ) 1 }, x(1 ln x) = q(τ σ) pτ qσ, and 1 < b. If q = 0 and/or τ = σ thn b =. A glanc at (2.9) indicats that it can b satisfid for arbitrary q(τ σ) if p is larg nough. For xampl if q = 3, τ = 4, σ = 2, th inquality of Lmma 2.2 will hold for p Also pτ qσ = (p q)τ + q(τ σ), hnc th condition is quivalnt to pτ qσ > q(τ σ) 1, (p q)τ > q(τ σ) 1 q(τ σ). (3.2) Sinc on [0, 1], x x 1 x, it is clar that whn conditions (1.2) hold, th numbr on th right sid of th fourth condition is largr than its countrpart on th right sid of (3.2). Rfrncs [1] Ahmad, F. and Alhrbi, R. A., Oscillation of solutions of a dlay diffrntial quation with positiv and ngativ cofficints, submittd for publication. [2] Agwo H, A., On th oscillation of dlay diffrntial quations with ral cofficints, Intrnat. J. Math. and Math. Sci. 22 (1999), [3] Arino, O., Ladas, G., and Sficas, Y. G.; On oscillation of som rtardd diffrntial quations SIAM J. Math. Anal. 18 (1987), [4] Chuanxi, Q. and Ladas, G. Oscillation in diffrntial quations with positiv and ngativ cofficints, Canad. Math. Bull. 33 (1990), [5] Elabbasy, El M., Hgazi, A. S., and Sakr, S. H.; Oscillation of solutions to dlay diffrntial quations with positiv and ngativ cofficints, Elctronic Journal of Diffrntial Equations, 2000 (2000), No.13, [6] Gyori, I. and Ladas, G. Oscillation Thory of Dlay Diffrntial Equations, Clarndon Prss, Oxford (1991). [7] Hunt, B. R. and York, J. A., Whn all solutions of x = n j=1 q j(t)x(t T j (t)) oscillat, J. Diffrntial Equations 53 (1984), [8] Ladas, G., and Sficas, Y. G., Oscillations of dlay diffrntial quations with positiv and ngativ cofficints, in Procdings of th Intrnational Confrnc on Qualitativ Thory of Diffrntial Equations, Univrsity of Albrta, (1984), [9] Li, B. Oscillation of first ordr dlay diffrntial quations, Proc. Amr. Math. Soc. 124 (1996),

6 6 FAIZ AHMAD EJDE 2003/92 Faiz Ahmad Dpartmnt of Mathmatics, Faculty of Scinc, King Abdulaziz Univrsity, P. O. Box 80203, Jddah 21589, Saudi Arabia addrss: