2000 Mathematics Subject Classification: 34K15.This paper is concerned with the oscillatory behavior of first-order delay differential equation of the form x'(t) + p(t)x (τ(t)) = 0

Elabbasy, E.

Hassan, T.

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 30 (2004), 71–86OSCILLATION CRITERIA FOR FIRST ORDER DELAYDIFFERENTIAL EQUATIONSE. M. Elabbasy, T. S. HassanCommunicated by I. D. IlievAbstract. This paper is concerned with the oscillatory behavior of first-order delay differential equation of the form.x (t) + p (t)x (τ (t)) = 0,where p, τ ∈ C [[t0,∞) , R+], R+ = [0,∞), τ (t) is nondecreasing, τ (t) < tfor t ≥ t0 and limt→∞τ (t) =∞. Let the numbers k and l be defined byk = limt→∞inf∫ tτ(t)p(s)ds and l = limt→∞sup∫ tτ(t)p(s)ds.It is proved here that when l < 1 and 0 < k ≤ 1eall solutions of this equationoscillate in several cases in which the conditionl >e− 1e− 2(k +1λ1)− 1e− 2 ,holds, where λ1 is the smaller root of the equation λ = ekλ.2000 Mathematics Subject Classification: 34K15.Key words: Oscillation, Delay differential equations.72 E. M. Elabbasy, T. S. Hassan1. Introduction. The problem of establishing sufficient conditions forthe oscillation of all solutions of the differential equationx˙ (t) + p (t)x (τ (t)) = 0,(1)where p, τ ∈ C [[t0,∞) , R+], R+ = [0,∞), τ (t) is nondecreasing, τ (t) < t fort ≥ t0 and limt→∞τ (t) = ∞, has been the subject of many investigations. See, forexample, [1]–[26] and the references cited therein.By a solution of Eq.(1) we understand a continuously differentiable func-tion defined on [τ (T0) ,∞) for some T0 ≥ t0 and such that (1) is satisfied fort ≥ T0. Such a solution is called oscillatory if it has arbitrarily large zeros, andotherwise it is called nonoscillatory.The first systematic study for the oscillation of all solutions of Eq. (1)was made by Myshkis [24]. In the 1950 paper [24] he proved that every solutionof (1) oscillates if(C1) limt→∞sup(t− τ(t)) <∞ and limt→∞inf(t− τ(t)). limt→∞inf p(t) >1e.In 1972 Ladas et al [19] proved that the same conclusion holds if(C2) limt→∞sup∫ tτ(t)p(s)ds > 1.In 1979 Ladas [18] and in 1982 Koplatadze et al [13] improved (C1) to(C3) limt→∞inf∫ tτ(t)p(s)ds >1e.Concerning the constant1ein (C3), it should be pointed out that if theinequality ∫ tτ(t)p(s)ds ≤ 1e,eventually holds, then, according to a result in [13] , Eq. (1) has a nonoscillatorysolution.In 1984 Ladas et al [20] and in 1984 Fukagai et al [10] established theoscillation criteria (of the type of conditions (C2) and (C3)) for Eq. (1) withoscillating coefficient p(t).Oscillation criteria for first order delay differential equations 73It is obvious that there is a gap between the conditions (C2) and (C3)when the limitlimt→∞∫ tτ(t)p(s)ds,does not exist.How to fill this gap is an interesting problem which has beenrecently investigating by several authors.In 1988 Erbe et al [9] developed new oscillation criteria by employing theupper bound of the ratiox(τ(t))x(t)for possible nonoscillatory solutions x(t) of Eq.(1). Their result, when formulated in terms of the numbers k and l defined by(C) k = limt→∞inf∫ tτ(t)p(s)ds and l = limt→∞sup∫ tτ(t)p(s)ds,says that all the solutions of Eq. (1) are oscillatory, if 0 < k ≤ 1eand(C4) l > 1− k24.Since then several authors tried to obtain better results by improving theupper bound forx(τ(t))x(t). In 1991 Jian Chao [2] derived the condition(C5) l > 1− k22(1− k) ,while in 1992 Yu et al [27] and [28] obtained the condition(C6) l > 1− 1− k −√1− 2k − k22.In 1990 Elbert et al [7] and in 1991 Kwong [16], using different techniques,improved (C4), in the case where 0 < k ≤ 1e, to the conditions(C7) l > 1−[1− 1√λ1]2,and(C8) l >lnλ1 + 1λ1,74 E. M. Elabbasy, T. S. Hassanrespectively, where λ1 is the smaller root of the equationλ = ekλ.(2)In 1994 Koplatadze et al [14] improved (C6), while in 1999 Philos et al[25], in 1998 Jaros et al [11], in 2000 Kon et al [17] and in 2003 Sficas et al [26]derived new conditions.(C9) l > 1− k22(1 − k) −k22λ1,(C10) l >1 + lnλ1λ1− 1− k −√1− 2k − k22,(C11) l > 2k +2λ1− 1,and(C12) l >lnλ1 − 1 +√5− 2λ1 + 2kλ1λ1,respectively.Following this historical (and chronological) review we also mention thatin the case where∫ tτ(t)p(s)ds ≥ 1eand limt→∞∫ tτ(t)p(s)ds =1e,this problem has been studied in 1995 by Elbert et al [8] and in 1995 by Koza-kiewicz [15], Li [22, 23] and by Domshlak et al [5].The purpose of this paper is to improve the methods previously used toshow that the conditions (C2) and (C4)− (C12) may be weakened to(C13) l >e− 1e− 2(k +1λ1)− 1e− 2 .One has to notice that as k → 0, then all conditions (C4)− (C11) and alsoour condition (C13) reduce to the condition (C2). However the improvement isOscillation criteria for first order delay differential equations 75clear as k → 1e. For illustrative purpose, we give the values of the lower boundon l under these conditions, when k =1e, are(C2) : 1.000000000(C4) : 0.966166179(C5) : 0.892951367(C6) : 0.863457014(C7) : 0.845181878(C8) : 0.735758882(C9) : 0.709011646(C10) : 0.599215896(C11) : 0.471517764(C12) : 0.459987065(C13) : 0.367879441We see that our condition (C13) essentially improves all the known resultsin the literature.2. Main results. In what follows we will denote by k and l the lowerand upper limits of the average∫ tτ(t)p(s)ds as t→∞, respectively, see (C).Setω (t) =x (τ (t))x (t).We begin with the preliminary analysis of asymptotic behavior of the functionω (t) for a possible nonoscillatory solution x (t) of Eq.(1) in the case when k ≤ 1e.For this purpose, assume that (1) has a solution x (t) which is positive for alllarge t. Dividing first Eq. (1) by x (t) and then integrating it from τ (t) to t, weget the integral equalityω (t) = exp∫ tτ(t)p(s)ω (s) ds,(3)which holds for all sufficiently large t.Also setF (t) =p(t)µ(t),76 E. M. Elabbasy, T. S. Hassanwhere the function µ (t) satisfies the following conditions:(i) µ (t) is nonincreasing,(ii)1 ≤ N := limt→∞inf µ(t) ≤ 1e− 2 .For the next lemma see [11].Lemma 1 [11]. Suppose that k > 0 and Eq. (1) has an eventually positivesolution x(t). Thenk ≤ 1eand λ1 ≤ limt→∞inf ω(t) ≤ λ2where λ1 and λ2 are the roots of the equation λ = ekλ.Lemma 2. Let 0 < k ≤ 1eand x(t) be an eventually positive solution ofEq. (1). Assume that there exists θ > 0 such that∫ τ(t)τ(u)F (s)ds ≥ θ∫ tuF (s)ds for all τ (t) ≤ u ≤ t,(4)thenlimt→∞supω(t) ≤ 2N(1− k −√(1− k)2 − 4B) ,(5)where B is given byB =eλ1θk − λ1θk − 1(λ1θ)2 .(6)and λ1 is the smaller root of the equation λ = ekλ.P r o o f. Let t > t0 ≥ 1 be large enough so that τ(t) > t0 and t1 ≡ t1(t) > tsuch thatτ(t1) = t, δ =∫ t1tF (s)ds ≤∫ t1tp(s)ds,where δ : 0 < δ < k is arbitrary close to k.Oscillation criteria for first order delay differential equations 77Integrating (1) from t to t1, we obtainx(t) = x(t1) +∫ t1tp(s)x (τ(s)) ds,and F (s) =p(s)µ(s),x(t) = x(t1) +∫ t1tF (s)µ(s)x (τ(s)) ds,Integrating (1) from τ (s) to t for s < t1, we havex(τ(s)) = x(t) +∫ tτ(s)p(u)x (τ(u)) du= x(t) +∫ tτ(s)F (u)µ(u)x (τ(u)) du.Combining the last two equalities, we obtainx(t) = x(t1) +∫ t1tF (s)µ(s)(x(t) +∫ tτ(s)F (u)µ(u)x (τ(u)) du)ds.(7)Let 0 < λ < λ1, then the functionφ(t) = x(t)eλtRt0F (s)ds,(8)is decreasing for large t ≥ t0 since x(t) is also decreasing. Indeed, by Lemma 1,x(τ (t))x(t)> λ, since µ(t) ≥ 1 for t ≥ t0 ≥ 1,thenµ(t)x (τ(t))x(t)> λ,for all sufficiently large t, and consequently0 = x′(t) + F (t)µ(t)x (τ (t)) ≥ x′ (t) + λF (t)x (t) ,which implies φ′(t) ≤ 0 for sufficiently large t.78 E. M. Elabbasy, T. S. HassanSubstituting into (7), we get for sufficiently large t the inequalityx (t) ≥ x (t1) + δx (t)+∫ t1tF (s)∫ tτ(s)µ (u)F (u)φ (τ (u)) e−λτ(u)Rt0F (ξ)dξdu ds≥ x (t1) + δx (t)+µ (t)φ (τ (t))∫ t1tF (s)∫ tτ(s)F (u) e−λτ(u)Rt0F (ξ)dξdu ds= x (t1) + δx (t)+µ (t)φ (τ (t)) e−λτ(t)Rt0F (s)ds ∫ t1tF (s)∫ tτ(s)F (u) eλτ(t)Rτ(u)F (ξ)dξdu ds.From (8), we havex(t) ≥ x(t1)+δx(t)+µ(t)x(τ(t))∫ t1tF (s)∫ tτ(s)F (u) eλτ(t)Rτ(u)F (ξ)dξdu ds.(9)In view of (4) we obtain∫ tτ(s)F (u) eλτ(t)Rτ(u)F (ξ)dξdu ≥∫ tτ(s)F (u) eλθtRuF (ξ)dξdu=1λθeλθtRτ(s)F (ξ)dξ− 1 .Oscillation criteria for first order delay differential equations 79Thus∫ t1tF (s)∫ tτ(s)F (u)eλτ(t)Rτ(u)F (ξ)dξdu ds ≥ −δλθ+1λθ∫ t1tF (s) eλθtRτ(s)F (ξ)dξds=−δλθ+1λθ∫ tt1F (s)eλθsRτ(s)F (ξ)dξ−λθsRtF (ξ)dξds≥ −δλθ+1λθeλθδ∫ t1tF (s) e−λθsRtF (ξ)dξds=−δλθ+eλθδ(λθ)21− e−λθt1RtF (ξ)dξ=−δλθ+eλθδ(λθ)2(1− e−λθδ)=−δλθ+1(λθ)2(eλθδ − 1).and (9) yieldsx(t) ≥ x (t1) + δx (t) +B∗µ (t) x (τ (t)) ,(10)whereB∗ =eλθδ − λθδ − 1(λθ)2.From (10), we havex(t) ≥ d1µ (t)x (τ(t)) ,whered1 =B∗1− δ .Observe thatx(t1) ≥ d1µ (t1)x (τ(t1)) ≥ d1x(t),80 E. M. Elabbasy, T. S. Hassansince µ(t) ≥ 1 for t ≥ t0 ≥ 1 and therefore (10) yieldsx(t) ≥ d2µ (t)x (τ(t))whered2 =B∗1− d1 − δ ,Following this iterative procedure (cf. [26, 27]), we obtainx(t) ≥ dn+1µ (t) x (τ(t)) ,wheredn+1 =B∗1− dn − δ , n = 1, 2, 3, . . . .It is easy to see that the sequence {dn} is strictly increasing and bounded.Thereforelimt→∞dn = d,exists and satisfiesd2 − (1− δ) d+B∗ = 0,since {dn} is strictly increasing it follows thatd =1− δ −√(1− δ)2 − 4B∗2.Observe that for large t one hasx(t)µ(t)x (τ(t))≥1− δ −√(1− δ)2 − 4B∗2,and since 0 < δ < k is arbitrarily close to k, by letting λ→ λ1 the last inequalityleads to (5).The proof is complete. Remark 1. Assume that τ (t) is continuously differentiable and thatthere exists θ > 0 such thatF (τ (t)) τ ′ (t) ≥ θF (t)(11)Oscillation criteria for first order delay differential equations 81eventually for all t. Then it is easy to see that (11) implies (4). Indeed, thefunctionυ (u) =∫ τ(t)τ(u)F (s)ds− θ∫ tuF (s)ds, τ (t) ≤ u ≤ t,satisfies the conditionsυ (t) = 0,andυ′ (u) = −F (τ (u)) τ ′ (u) + θF (u) ≤ 0.If F (t) > 0 eventually for all t andlimt→∞infF (τ (t)) τ′(t)F (t)= θ0 > 0,then θ can be any number satisfying 0 < θ < θ0.Our main result can now be stated as follows:Theorem 1. Consider the differential equation (1) , Assume that l < 1,0 < k ≤ 1eand there exist θ > 0 such that (4) is satisfied. Assume thatl >lnλ1 + 1λ1−N(1− k −√(1− k)2 − 4B)2(12)where λ1 the smaller root of the equation λ = ekλ and B is given by (6). Thenevery solution of (1) oscillates.P r o o f. Assume, for the sake of contradiction, that x(t) is an eventuallypositive solution of equation (1) .Let σ be any number in(1λ1, 1). From Lemma 1, there is a T1 > t0 suchthatx (τ (t))x (t)> σλ1, t ≥ T1,(13)andx (t)x (τ (t))> σM, t ≥ T1,(14)82 E. M. Elabbasy, T. S. HassanwhereM = limt→∞infx (t)x (τ (t)).Now let t ≥ T1. Since the function g (s) = x (τ (t))x (s)is continuous, g (τ (t)) = 1 <σλ1 and g (t) > σλ1, there is a t∗ (t) ∈ (τ (t) , t) such thatx (τ (t))x (t∗ (t))= σλ1.Dividing (1) by x (t) , integrating from τ (t) to t∗ (t) , and taking into account(13) yields∫ t∗(t)τ(t)p (s) ds ≤ − 1σλ1∫ t∗(t)τ(t)x′ (s)x (s)ds =ln (σλ1)σλ1.(15)Integrating (1) over [t∗ (t) , t] and using (14) and the fact that x (τ (s)) ≥ x (τ (t))if s ≤ t yields ∫ tt∗(t)p (s) ds ≤ x (t∗ (t))x (τ (t))− x (t)x (τ (t))=1σλ1− x (t)x (τ (t))≤ 1σλ1− σM.(16)Adding (15) and (16) yields∫ tτ(t)p (s) ds ≤ ln (σλ1) + 1σλ1− σM.Letting t→∞ yieldsl ≤ ln (σλ1) + 1σλ1− σM.Letting σ → 1 we obtainl ≤ lnλ1 + 1λ1−M.Oscillation criteria for first order delay differential equations 83The last inequality, in view of Lemma 2 contradicts (12). Corollary 1. Assume that the hypotheses of Theorem 1 hold with θ = 1and N =1e− 2 . Then its conclusion holds with (C13).P r o o f. When θ = 1 and N =1e− 2 , we get from (6) and (12) thatB =λ1 − λ1k − 1λ21,andl >kλ1 + 1λ1−1e− 2(1− k −√(1− k)2 − 4λ21(λ1 − λ1k − 1))2=e− 1e− 2(k +1λ1)− 1e− 2 ,respectively. Remark 2. Provided that k =1eone has λ1 = e and(C13) leads to l > 0.367879441 =1e.Example 1. Consider the delay differential equationx˙(t) + px(t− a sin2√t− bp) = 0,(17)where p > 0, a > 0, b =1e, pa =25− b and by taking µ (t) = 1e− 2 , HenceN =1e− 2 . Thenk = limt→∞inf∫ tτ(t)p(s)ds = limt→∞inf∫ tτ(t)pds = limt→∞inf p(a sin2√t+bp)=1e,andl = limt→∞sup∫ tτ(t)p(s)ds = limt→∞sup∫ tτ(t)pds = limt→∞sup p(a sin2√t+bp)= pa+ b =25< 1.84 E. M. Elabbasy, T. S. HassanIn the case that k =1e, then λ1 = e and we finde− 1e− 2(k +1λ1)− 1e− 2 =1e<25= l.Thenl >e− 1e− 2(k +1λ1)− 1e− 2 .Hence, the conditions of Theorem 1 and using the Corollary 1 are satisfied andtherefore every solution of equation (17) oscillates. Observe that none of theresults mentioned in the introduction can be applied to this equation.REFERENCES[1] O. Arino, G. Ladas, Y. G. Sficas. On oscillation of some retarded dif-ferential equations. SIAM J. Math. Anal. 18 (1987), 64–73.[2] J. Chao. On the oscillation of linear differential equations with deviatingarguments. Math. Practice Theory 1 (1991), 32–40.[3] Q. Chuanxi, G. Ladas. Oscillations of neutral differential equations withvariable coefficients. Appl. Anal. 32 (1989), 215–228.[4] Y. Domshlak. Sturmian Comparison method in investigation of the behav-ior of solutions for differential operator equations. (Elm), Baku, USSR, 1986(in Russian).[5] Y. Domshlak, I. P. Stavroulakis. Oscillations of first order delay dif-ferential equations in a critical state. Appl. Anal. 61 (1996), 359–371.[6] J. Dzurina. Oscillation of second order differential equations with mixedargument. J. Math. Anal. Appl. 190 (1995), 821–828.[7] A. Elbert, I. P. Stavroulakis.Oscillations of first order differential equa-tion with deviating arguments. University of Ioannina, T.R.No. 172 (1990),Recent trends in differential equations, 163–178, World Sci. Publishing Co.(1992).Oscillation criteria for first order delay differential equations 85[8] A. Elbert, I. P. Stavroulakis.Oscillation and non-oscillation criteria fordelay differential equations. Proc. Amer. Math. Soc. 123 (1995), 1503–1510.[9] L. H. Erbe, B. G. Zhang. Oscillation for first order linear differentialequations with deviating arguments.Differential Integral Equations 1 (1988),305–314.[10] N. Fukagai, T. Kusano. Oscillation theory of first order functional dif-ferential equations with deviating arguments. Ann. Math. Pura Appl. 136(1984), 95–117.[11] J. Jaros, I. P. Stavroulakis. Oscillation tests for delay equations. RockyMountain J. Math. 28 (1999), 197–207.[12] R. G. Koplatadze. On zeros of solutions of first order delay differentialequations. Proceedings of I. N. Vekua Institute of Applied Mathematics 14(1983), 128–135 (in Russian).[13] R. G. Kaplatadze, T. A. Chantiria. On the oscillatory and monotonesolutions of first order differential equations with deviating arguments. Jour-nal Differential Equations 18 (1982), 1463–1465 (in Russian).[14] R. G. Koplatadze, G. Kvinikadze. On the oscillation of solutions of firstorder delay differential equations. Georgian Math. J. 1 (1994), 675–685.[15] E. Kozakiewicz. Conditions for the absence of positive solutions of a firstorder differential inequality with a single delay. Arch. Math. Brno 31, 4(1995), 291–297.[16] M. K. Kwong. Oscillation of first order delay equations. J. Math. Anal.Appl. 156 (1991), 274–286.[17] M. Kon, Y. G. Sficas, I. P. Stavroulakis. Oscillation criteria for delayequations. Proc. Amer. Math. Soc. 128 (2000), 2989–2997.[18] G. Ladas. Sharp condition for oscillations caused by delays. Appl. Anal. 9(1979), 43–48.[19] G. Ladas, V. Lakshmikantham, L. S. Papadakis. Oscillations of higerorder retarded differential equations generated by the retarded arguments,Delay and functional Differential Equations and their Applications. Acad-emic Press, New York, (1972), 219–231.86 E. M. Elabbasy, T. S. Hassan[20] G. Ladas, Y. G. Sficas, I. P. Stavroulakis. Functional differentialinequalities and equationas with oscillatory coefficients. In: Trends in Theoryand practice of non-linear deferential equations. Lecture Notes in Pure Appl.Math. vol. 90 (1984), 277–284.[21] G. Ladas, I. P. Stavroulakis. On delay differential inequalities of firstorder. Funkc. Ekvacioj, Ser. Int. 25 (1982), 105–113.[22] B. Li. Oscillations of delay differential equations with variable coefficients.J. Math. Anal. Appl. 192 (1995), 312–321.[23] B. Li. Oscillation of first order delay differential equations. Proc. Amer.Math. Soc. 124 (1996), 3729–3737.[24] A. D. Myshkis. Linear homogeneous differential equations of first or-der with deviating arguments. Uspekhi Math. Nauk 5 (1950), 160–162 (inRussian).[25] Ch. G. Philos, Y. G. Sficas. An oscillation criterion for first order lineardelay differential equations. Canad. Math. Bull. 41 (1998), 207–213.[26] Y. G. Sficas, I. P. Stavroulakis. Oscillation criteria for first order delayequations. Bull. London Math. Soc. 35 (2003), 239–246.[27] J. S. Yu, Z. C. Wang. Some further results on oscillation of neutral dif-ferential equations. Bull. Austral. Math. Soc. 46 (1992), 149–157.[28] J. S. Yu, Z. C. Wang, B. G. Zhang, X. Z. Qian. Oscillation of dif-ferential equations with deviating arguments. Panamer. Math. J. 2 (1992),59–78.Department of MathematicsFaculty of ScienceMansoura UniversityMansoura, 35516,Egypte-mail: emelabbasy@mans.edu.ege-mail: tshassan@mans.edu.egReceived October 1, 2003Revised February 12, 2004

Oscillation Criteria for First Order Delay Differential Equations

http://sci-gems.math.bas.bg/jspui/bitstream/10525/1727/1/sjm-vol30-num1-2004-p071-p086.pdf

Oscillation Criteria for First Order Delay Differential Equations

Abstract

Similar works

Full text

Available Versions

Bulgarian Digital Mathematics Library at IMI-BAS