We prove oscillation theorems for the nonlinear delay differential equation
$\left( \left\vert y^{\prime }(t)\right\vert ^{\alpha -2}y^{\prime}(t)\right) ^{\prime }+q(t)\left\vert y(\tau (t))\right\vert ^{\beta-2}y(\tau (t))=0, t\geq t_{\ast }>0,$ 
where $\beta >1,$ $\alpha >1,$ $q(t)\geq 0$ and locally integrable on $[t_{\ast },\infty ),$ $\tau (t)$ is a continuous function satisfiying $ 0<\tau (t)\leq t$ and lim$_{t\rightarrow \infty }\tau (t)=\infty .$ The results obtained essentially improve the known results in the literature and can be applied to linear and half-linear delay type differential equations

Cakmak, Devrim

Tiryaki, A.

Electronic journal of qualitative theory of differential equations

Devrim Cakmak

Aydin Tiryaki

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Oscillation criteria for nonlinear delay differential equations of second order

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Electronic Journal of Qualitative Theory of Differential Equations2011, No. 84, 1-9; http://www.math.u-szeged.hu/ejqtde/Oscillation Criteria for Nonlinear DelayDifferential Equations of Second Order∗Devrim C¸akmak†Gazi UniversityFaculty of EducationDepartment of Mathematics Education06500 Teknikokullar, Ankara, Turkeydcakmak@gazi.edu.trAydın TiryakiIzmir UniversityFaculty of Arts and SciencesDepartment of Mathematics and Computer Sciences35350 Uckuyular, Izmir, Turkeyaydin.tiryaki@izmir.edu.trAbstractWe prove oscillation theorems for the nonlinear delay differential equation“|y′(t)|α−2y′(t)”′+ q(t) |y(τ (t))|β−2 y(τ (t)) = 0, t ≥ t∗ > 0,where β > 1, α > 1, q(t) ≥ 0 and locally integrable on [t∗,∞), τ (t) is acontinuous function satisfiying 0 < τ (t) ≤ t and limt→∞τ (t) = ∞. Theresults obtained essentially improve the known results in the literature andcan be applied to linear and half-linear delay type differential equations.AMS Subject Classification: 34C10, 34C15.Keywords: Oscillation; Second order; Ordinary differential equation; Emden-Fowler equation; Delayed argument; Riccati transformation.1 IntroductionIn the last decades, there has been an increasing interest in obtaining suffi-cient conditions for the oscillation and/or nonoscillation of solutions for dif-ferent classes of second order differential equations with or without deviatingarguments. For interested readers we refer to the papers [7, 8, 12, 13, 15] andthe references quoted therein.∗Dedicated to Professor A. Okay C¸elebi on the occasion of his 70th birthday†Corresponding authorEJQTDE, 2011 No. 84, p. 1Before we continue with the description of the content of this paper, wepresent a short survey of the most basic results in the literature.Let us consider the following linear differential equationy′′ + q(t)y = 0, t ≥ t0 ≥ t∗ > 0, (1)where q(t) ≥ 0 is locally integrable on [t0,∞).In 1948, Hille [6] established the following results:Theorem A. If q ∈ L1[t0,∞) andlim supt→∞t∫ ∞tq(s)ds ≤ 14, (2)then equation (1) is nonoscillatory.Theorem B. If q ∈ L1[t0,∞) andlim inft→∞t∫ ∞tq(s)ds >14, (3)then equation (1) is oscillatory.In 1997, Huang [7] obtained the following interval criteria:Theorem C. If there exists t0 ≥ t∗ such that for each n ∈ N0 = {0, 1, 2, ...} ,q(t) satisfies ∫ 2n+1t02nt0q(s)ds ≤ θ02n+1t0, (4)where θ0 = 3− 2√2, then equation (1) is nonoscillatory.Theorem D. If there exists t0 ≥ t∗ such that for each n ∈ N0 = {0, 1, 2, ...} ,q(t) satisfies ∫ 2n+1t02nt0q(s)ds ≥ θ2nt0, (5)where θ > θ0, then equation (1) is oscillatory.In 2004, by replacing the sequence {2n} in Theorems C and D by {λn} withλ > 1, Wong [15] generalized Theorems C and D as follows:Theorem E. Let λ > 1. If there exists some t0 such that for each n ∈ N0 ={0, 1, 2, ...} , q(t) satisfies∫ λn+1t0λnt0q(s)ds ≤ θ(λ− 1)λn+1t0 , (6)where θ ≤ k0(λ) = (√λ− 1)2, then equation (1) is nonoscillatory.Theorem F. Let λ > 1. If there exists some t0 such that for each n ∈ N0 ={0, 1, 2, ...} , q(t) satisfies∫ λn+1t0λnt0q(s)ds ≥ θ(λ− 1)λnt0 , (7)EJQTDE, 2011 No. 84, p. 2where θ > k0(λ), then equation (1) is oscillatory.Furthermore, Wong [15] extended the oscillation criteria (3) and (7) forequation (1) to the following linear delay differential equationy′′(t) + q(t)y(τ(t)) = 0, t ≥ t0, (8)where q(t) ≥ 0 and locally integrable on [t0,∞), and τ(t) is a continuous functionsatisfying 0 < τ(t) ≤ t and limt→∞τ(t) =∞.In 1987, Yan [16] proved the following result for equation (8), but Wong gavean alternative and simpler proof in [15].Theorem G. Suppose that for all sufficiently large t, q(t) satisfies∫ ∞tq(s)τ(s)sds ≥ θt(9)for some fixed constant θ > 14 , then all solutions of equation (8) are oscillatory.Wong also proved the extension of Theorem F for equation (8).Theorem H. Let λ > 1. If there exists t0 ≥ t∗ and for each n ∈ N0 ={0, 1, 2, ...} , q(t) satisfies∫ λn+1t0λnt0q(s)τ(s)sds ≥ θ(λ − 1)λnt0 , (10)where θ > k0(λ). Then all solutions of equation (8) are oscillatory.Now, let us consider the following half-linear differential equation(|y′(t)|α−2 y′(t))′+ q(t) |y(t)|α−2 y(t) = 0, t ≥ t0, (11)where α > 1, q(t) ≥ 0 is locally integrable on [t0,∞).In 1995, Kusano and Yoshida [9] generalized Theorems A and B as follows:Theorem I. If q ∈ L1[t0,∞), andlim supt→∞tα−1∫ ∞tq(s)ds ≤ (α − 1)α−1αα, (12)then equation (11) is nonoscillatory.Theorem J. If q ∈ L1[t0,∞), andlim inft→∞tα−1∫ ∞tq(s)ds >(α − 1)α−1αα, (13)then equation (11) is oscillatory.In 2004, Yang [17] extended Theorems I and J as follows:Theorem K. If q ∈ L1[t0,∞), and for large t > t0,tα−1∫ ∞tq(s)ds ≤ (α− 1)α−1αα, (14)then equation (11) is nonoscillatory.EJQTDE, 2011 No. 84, p. 3Theorem L. If q ∈ L1[t0,∞), and for large t > t0,tα−1∫ ∞tq(s)ds ≥ α0, (15)where α0 >(α−1)α−1αα, then equation (11) is oscillatory.In 2007, Kong [8] extended the results of Wong [15], namely Theorems E andF, for the linear differential equation (1) to the half-linear differential equation(11) as follows:Theorem M. Let λ > 1 and ξ∗ = ξ∗(α). Assume there exists t0 ∈ (0,∞) suchthat for each n ∈ N0 = {0, 1, 2, ...} , q(t) satisfies(∫ λn+1t0λnt0q(s)ds) 1α−1≤ ξ∗(λ− 1)λn+1t0 , (16)then equation (11) is nonoscillatory.Theorem N. Let λ > 1 and ξ∗ = ξ∗(α). Assume there exists t0 ∈ (0,∞) andξ > ξ∗ such that for each n ∈ N0 = {0, 1, 2, ...} , q(t) satisfies(∫ λn+1t0λnt0q(s)ds) 1α−1≥ ξ(λ− 1)λnt0 , (17)then equation (11) is oscillatory.In this paper, by using the same method in Wong [15], we extend TheoremsG, H and N to the following nonlinear delay differential equation(|y′(t)|α−2 y′(t))′+ q(t) |y(τ(t))|β−2 y(τ(t)) = 0, t ≥ t0, (18)where β > 1, α > 1, q(t) ≥ 0 and locally integrable on [t0,∞), τ(t) is continuousfunction satisfying 0 < τ(t) ≤ t and limt→∞τ(t) =∞.Note that the equation (18) with τ(t) = t is referred to as a super-half-linearequation, a sub-half-linear equation and an Emden-Fowler type equation forβ > α, β < α and β 6= α, respectively. We refer the readers to the introductorybooks by Agarwal et al. [2] and by Dosˇly´ and R˘eha´k [4] for the equation (18)with τ(t) = t.To present our results, we need the following lemma which is given by Erbe[5].Lemma P. Assume that τ ∈ C ([t0,∞),R+) , 0 < τ(t) < t for t ≥ t0 andlimt→∞ τ(t) =∞. Let y ∈ C2 ([t0,∞),R+) be such that y′′(t) ≤ 0 for t ≥ T ≥t0. Then for each constant k ∈ (0, 1) , there is a Tk ≥ T such thaty (τ (t))y (t)≥ k τ (t)tfor t ≥ Tk. (19)EJQTDE, 2011 No. 84, p. 42 Main ResultsFirst, we obtain two theorems which concern the oscillatory behaviour of equa-tion (18) with β = α. Next, motivated by the ideas of Agarwal and Grace [1]and C¸akmak [3], we present two other results for β 6= α.Theorem 1 Suppose that for all sufficiently large t, q(t) satisfies∫ ∞tq(s)(τ(s)s)α−1ds ≥ α1tα−1(20)for some fixed constant α1 >(α−1)α−1αα, then all solutions of equation (18) withβ = α are oscillatory.Proof. Assume on the contrary that equation (18) with β = α has a nontrivialnonoscillatory solution y(t), we can assume without loss of generality that y(t) >0 for t ≥ t0. Since limt→∞ τ(t) =∞, there exists t1 ≥ t0 such that y(τ(t)) > 0for t ≥ t1. By equation (18) with β = α, since q(t) ≥ 0, |y′(t)|α−2 y′(t) isnonincreasing on [t1,∞), so is y′(t). This implies that y′(t) > 0 and y′′(t) ≤ 0for t ≥ t1. Define w(t) = |y′(t)|α−2y′(t)|y(t)|α−2y(t), then w(t) satisfies the equationw′(t) + (α− 1) |w(t)| αα−1 + q(t)(y(τ(t))y(t))α−1= 0 (21)on [t1,∞). Thus, by Lemma P, for each constant k ∈ (0, 1) , there exists t2,depending on k, such that for t ≥ t2 ≥ t1,y (τ (t))y (t)≥ k τ (t)t. (22)Substituting (22) into (21), we findw′(t) + (α− 1) |w(t)| αα−1 +(kτ(t)t)α−1q(t) ≤ 0, (23)since q(t) ≥ 0. It follows from the result of Li and Yeh [10, Theorem 3.2] that(23) implies the half-linear differential equation(|u′(t)|α−2 u′(t))′+(kτ(t)t)α−1q(t) |u(t)|α−2 u(t) = 0 (24)is nonoscillatory for every k, 0 < k < 1. Note that µ = kα−1 ∈ (0, 1) for every0 < k < 1 and α > 1. Choose µ sufficiently close to 1 so that µα1 >(α−1)α−1ααwhich is possible since α1 >(α−1)α−1αα; for example, choose µ = 12+(α−1)α−12ααα1< 1.Condition (20) impliesµ∫ ∞t(τ(s)s)α−1q(s)ds ≥ µα1tα−1and µα1 >(α − 1)α−1αα, (25)EJQTDE, 2011 No. 84, p. 5which in turn implies the oscillation criteria (15) given by Yang [17] that equa-tion (24) is oscillatory. This is a contradiction, hence equation (18) with β = αis oscillatory.Using the same argument as in the proof of Theorem 1, we can also provethe following result.Theorem 2 Let λ > 1 and ξ∗ = ξ∗(α). Assume there exists t0 ∈ (0,∞) andξ > ξ∗ such that for each n ∈ N0 = {0, 1, 2, ...} , q(t) satisfies(∫ λn+1t0λnt0q(s)(τ(s)s)α−1ds) 1α−1≥ ξ(λ− 1)λnt0 . (26)Then all solutions of equation (18) with β = α are oscillatory.Proof. We follow the proof of Theorem 1 and conclude that the existence of anonoscillatory solution of (18) with β = α lead to the conclusion that the half-linear differential equation (24) is nonoscillatory for every k, 0 < k < 1. Forevery 0 < k < 1 and α > 1, we can again choose µ = kα−1 ∈ (0, 1) sufficientlyclose to 1 so that µξ > ξ∗. Now the coefficient function of equation (24) satisfiesµ(∫ λn+1t0λnt0q(s)(τ(s)s)α−1ds) 1α−1≥ ξ1(λ− 1)λnt0 , (27)where ξ1 = µξ > ξ∗, so we can apply Theorem N given by Kong [8] to equation(24) and conclude that it is oscillatory for such µ, 0 < µ < 1, but µξ > ξ∗. Thiscontradicts the fact that equation (24) is nonoscillatory for all k, 0 < k < 1.The proof is complete.Remark 3 When α = 2, Theorems 1 and 2 reduce to Theorems G and H,respectively.Remark 4 If the delayed argument is absent, i.e. τ(t) = t, then Theorems 1and 2 reduce to Theorems L and N, respectively. Furthermore, Theorem 1 is anextension of Theorem J.Remark 5 Let α = 2 and τ(t) = t. In this case, Theorem 1 is an extensionof Theorem B. Moreover, Theorem 2 (or Theorem 2 with λ = 2) reduces toTheorem F (or Theorem D).Theorem 6 Suppose that for all sufficiently large t, q(t) satisfiesc∫ ∞tq(s)(τ(s)s)β−1ds ≥ α1tα−1(28)for some fixed constant α1 >(α−1)α−1ααand any constant c > 0, then the followingassertions are true:(i) all unbounded solutions of equation (18) with β > α are oscillatory.(ii) all bounded solutions of equation (18) with β < α are oscillatory.EJQTDE, 2011 No. 84, p. 6Proof. Assume on the contrary that equation (18) with β 6= α has a nontrivialnonoscillatory solution y(t), we can assume without loss of generality that y(t) >0 for t ≥ t0. Since limt→∞ τ(t) =∞, there exists t1 ≥ t0 such that y(τ(t)) > 0for t ≥ t1. By equation (18) with β 6= α, since q(t) ≥ 0, |y′(t)|α−2 y′(t) isnonincreasing on [t1,∞), so is y′(t). This implies that y′(t) > 0 and y′′(t) ≤ 0for t ≥ t1. Define w(t) = |y′(t)|α−2y′(t)|y(t)|α−2y(t), then w(t) satisfies the equationw′(t) + (α− 1) |w(t)| αα−1 + q(t)(y(τ(t))y(t))β−1(y(t))β−α = 0 (29)on [t1,∞). Next, we consider the following two cases:(i) If y(t) is an unbounded nonoscillatory solution of equation (18) withβ > α for t ≥ t0, then there exist a constant k1 > 0 and t2 ≥ t1 ≥ t0 such thaty(t) ≥ k1 for t ≥ t2. Therefore,(y(t))β−α ≥ k1β−α = c1 for t ≥ t2, (30)where c1 is a constant. Using (30) in the equation (29), and proceeding as inthe proof of Theorem 1, we arrive at the desired contradiction.(ii) If y(t) is a bounded nonoscillatory solution of equation (18) with β < αfor t ≥ t0, then there exist a constant k2 > 0 and t2 ≥ t1 ≥ t0 such thaty(t) ≤ k2 for t ≥ t2. Therefore,(y(t))β−α ≥ k2β−α = c2 for t ≥ t2, (31)where c2 is a constant. The rest of the proof is similar to that of previous caseand, hence omitted.Combining some ingredients of the proofs of Theorems 2 and 6, we give thefollowing result for equation (18) with β 6= α, the proof of which is similar tothat of Theorem 6, and hence omitted.Theorem 7 Let λ > 1 and ξ∗ = ξ∗(α). Assume there exists t0 ∈ (0,∞) andξ > ξ∗ such that for each n ∈ N0 = {0, 1, 2, ...} and any constant c > 0, q(t)satisfies (c∫ λn+1t0λnt0q(s)(τ(s)s)β−1ds) 1α−1≥ ξ(λ − 1)λnt0 . (32)Then the following assertions are true:(i) all unbounded solutions of equation (18) with β > α are oscillatory.(ii) all bounded solutions of equation (18) with β < α are oscillatory.Finally, we generalize above results for a class of more general nonlineardelay differential equations as follows:Let α, β, q(t), and τ(t) be as above, and consider(|y′(t)|α−2 y′(t))′+ f(t, y(τ(t))) = 0, t ≥ t0, (33)EJQTDE, 2011 No. 84, p. 7where the function f satisfiessf(t, s) ≥ q(t) |s|β for t ≥ t0 and s ∈ R. (34)The proof of the following results are exactly as in that of above theorems andhence omitted.Theorem 8 In addition to the conditions of Theorem 1 (or Theorem 2), if (34)with β = α holds, then all solutions of equation (33) are oscillatory.Theorem 9 In addition to the conditions of Theorem 6 (or Theorem 7), if (34)with β > α holds, then all unbounded solutions of equation (33) are oscillatory.Theorem 10 In addition to the conditions of Theorem 6 (or Theorem 7), if(34) with β < α holds, then all bounded solutions of equation (33) are oscilla-tory.Remark 11 For another oscillation criteria contain for equation (18) with β ≥α and (33) with f(t, y(τ(t))) = F (y(τ(t))) under different sufficient conditions,the reader is referred to [13].Remark 12 In case the delay is bounded, i.e., 0 ≤ t − τ(t) ≤ M , then τ(t)tinconditions (20), (26), (28) and (32) can be replaced by 1. In other words, Hille’soscillation criterion (3) is also valid oscillation criteria for equations (18) and(33) with β = α = 2 and bounded delay; see [11, 14, 15].References[1] R.P. Agarwal and S.R. Grace, Second-order nonlinear forced oscillations,Dynam. Systems Appl. 10 (2001), no. 4, 455-464.[2] R.P. Agarwal, S.R. Grace and D. O’Regan, Oscillation Theory for SecondOrder Linear, Half-linear, Superlinear and Sublinear Dynamic Equations,Kluwer, Dordrecht, 2002.[3] D. C¸akmak, Oscillation criteria for nonlinear second-order differential equa-tions with damping, Ukrainian Math. J. 60 (2008), no. 5, 799-809.[4] O. Dosˇly´ and P. Rˇeha´k, Half-linear Differential Equations, MathematicsStudies 202, North-Holland, 2005.[5] L. Erbe, Oscillation criteria for second order nonlinear delay equations,Canad. Math. Bull. 16 (1973), no. 1, 49-56.[6] E. Hille, Nonoscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234-252.EJQTDE, 2011 No. 84, p. 8[7] C. Huang, Oscillation and nonoscillation for second order linear differentialequations, J. Math. Anal. Appl. 210 (1997), 712-723.[8] Q. Kong, Nonoscillation and oscillation of second order half-linear differ-ential equations, J. Math. Anal. Appl. 332 (2007), 512-522.[9] K. Kusano and N. Yoshida, Nonoscillation theorems for a class of quasilin-ear differential equations of second order, J. Math. Anal. Appl. 189 (1995),115-127.[10] H.J. Li and C.C. Yeh, Sturmian comparison theorem for half-linear secondorder differential equations, Proc. Roy. Soc. Edin. 125A (1995), 1193-1204.[11] W.E. Mahfoud, Oscillation theorems for a second order delay differentialequation, J. Math. Anal. Appl. 63 (1978), 339-346.[12] A. Tiryaki and Y. Bas¸cı, Interval oscillation criteria for second-order quasi-linear functional differential equations, Dyn. Contin. Discrete Impuls. Syst.Ser. A Math. Anal. 16 (2009), 233-252.[13] A. Tiryaki, Oscillation criteria for a certain second-order nonlinear differ-ential equations with deviating arguments, E. J. Qualitative Theory of Diff.Equ. 61 (2009), 1-11.[14] J.S.W. Wong, Second order oscillation with retarded arguments, in: L.Weiss (Ed.), Ordinary Differential Equations, 1971 NRL-MRC Conference,Academic Press, New York, 1972, pp. 581-596.[15] J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291(2004), 180-88.[16] J. Yan, Oscillatory property for second order linear differential equations,J. Math. Anal. Appl. 122 (1987), 380-384.[17] X. Yang, Oscillation criterion for a class of quasilinear differential equations,Appl. Math. Comput. 153 (2004), 225-229.(Received April 25, 2011)EJQTDE, 2011 No. 84, p. 9

Oscillation criteria for nonlinear delay differential equations of second order

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