summary:In this paper, we investigate a class of higher order neutral functional differential equations, and obtain some new oscillatory criteria of solutions

Wang, Min

Wang, Peiguang

English

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumPei Guang Wang; Min WangOscillation of a class of higher order neutral differential equationsArchivum Mathematicum, Vol. 40 (2004), No. 2, 201--208Persistent URL: http://dml.cz/dmlcz/107901Terms of use:© Masaryk University, 2004Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 40 (2004), 201 – 208OSCILLATION OF A CLASS OF HIGHER ORDER NEUTRALDIFFERENTIAL EQUATIONSPEIGUANG WANG AND MIN WANGAbstract. In this paper, we investigate a class of higher order neutral func-tional differential equations, and obtain some new oscillatory criteria of solu-tions.1. IntroductionThe problem determining oscillatory criteria for higher order differential equa-tions has attracted a great deal of attention in the last several years, We mentionhere the literatures of Bainov and Mishev [2], Erbe, Kong and Zhang [3], Ladasand Sficas [4], Grace [5, 6], Agarwal, Grace and O’Regan [1] and references citedtherein. In this paper, we consider a class of higher order nonlinear neutral func-tional differential equations of the form[x(t) + p(t)x(t − τ)](n) + q(t)f(x[g1(t)], . . . , x[gm(t)]) = 0 , t ≥ t0(1.1)in which n ≥ 2 is an even number, τ > 0 is a constant. p(t) ∈ C([t0,∞), R),0 ≤ p(t) ≤ 1; q(t) ∈ C([t0,∞), R+) is not identically zero on any ray [t1,∞),t1 > t0; gi(t) ∈ C([t0,∞), R), and limt→∞gi(t) = ∞; f(u1, . . . , um) ∈ C(Rm, R) isnondecreasing on ui, i ∈ Im = {1, 2, . . . , m}.We restrict our attention to proper solutions of equation (1.1), i.e. to noncon-stant solutions existing on [T,∞) for some T ≥ t0 and satisfying supt≥T |x(t)| > 0.A proper solution x(t) of equation (1.1) is called oscillatory if it does not have thelargest zero, otherwise, it is called nonoscillatory. The existence of oscillatorysolutions for functional differential equations of neutral type can be found in [7].We note that a special case of equations (1.1) is the following equation[x(t) + p(t)x(t − τ)](n) + q(t)x[σ(t)] = 0 , t ≥ t0 > 0 .(1.2)Erbe, Kong and Zhang [3] discussed the problem of oscillation of solution, andgave the following theorem2000 Mathematics Subject Classification: 34K11, 34K40.Key words and phrases: higher order, nonlinear, neutral equation, oscillation.Supported by the Natural Science Foundation of Hebei Province of China (A2004000089).Received June 5, 2002.202 P. WANG, M. WANGTheorem A. Assume that the following conditions holdA1) 0 ≤ p(t) ≤ 1, q(t) ≥ 0, and q(t) is not identical zero on any ray [t1,∞),t ≥ t1 ≥ t0;A2) 0 < σ(t) ≤ t, 0 < σ′(t) ≤ 1, and limt→∞σ(t) = ∞.If there exists a function ρ(t) ∈ C′([t0,∞), (0,∞)) such thatlimt→∞1tm−1∫ tt0(t − s)m−3(ρ′(s)(t − s) − (m − 1)ρ(s))2σ′(s)σn−2(s)ρ(s)ds < ∞ ,(1.3)limt→∞1tm−1∫ tt0(t − s)m−1ρ(s)q(s)(1 − p(σ(s))) ds = ∞ ,(1.4)in which m ≥ 3 is a integer, then every solution of equation (1.2) is oscillatory.The aim of this paper is obtain some new oscillatory criteria for equation (1.1)by introducing parameter functions H(t, s) and h(t, s). The results generalize andimprove Theorem A at the same time.The following Lemmas will be found in [8] which are useful for the proof ofmain results.Lemma 1. Let u(t) be a positive and n times differentiable function on R+. Ifu(n)(t) is of constant sign and not identically zero on any ray [t1, +∞) for (t1 > 0),then there exists a tu ≥ t1 and an integer l (0 ≤ l ≤ n), with n + l even foru(t)u(n)(t) ≥ 0 or n + l odd for u(t)u(n)(t) ≤ 0; and for t ≥ tu,u(k)(t) > 0 , 0 ≤ k < l ; (−1)k−lu(k)(t) > 0 , l ≤ k < n .Lemma 2. Suppose that the conditions of Lemma 1 is satisfied, andu(n−1)(t)u(n)(t) ≤ 0 , t ≥ tu ,then for any θ ∈ (0, 1) and sufficiently large t, sufficiently large t, there existsa constant Mθ satisfying|u′(θt)| ≥ Mθtn−2|u(n−1)(t)| .(1.5)2. Main resultsWe say that a function H = H(t, s) belong to a function class X , denoted byH ∈ X , if H ∈ C′(D, R+), where D = {(t, s)|t ≥ s ≥ t0} which satisfiesH(t, t) = 0 , t ≥ t0 , H(t, s) > 0 , t > s ≥ t0 ;moreover, there exists a function h(t, s) ∈ C(D, R) such thatHs′(t, s) ≤ 0 , and − Hs′(t, s) = h(t, s)√H(t, s) , (t, s) ∈ D .Theorem 1. Suppose that the following conditions hold(H1) there exists a function σ(t) ∈ C′([t0,∞), (0,∞)) such that σ(t) ≤ gi(t) ≤ t,and limt→∞σ(t) = ∞, i ∈ Im;OSCILLATION OF HIGHER ORDER EQUATIONS 203(H2) f(u1, . . . , um) have same sign with u1, . . . , um when u1, . . . , um have samesign, i ∈ Im = 1, 2, . . . , m, and there exist constants N > 0 and λ > 0 suchthatlim inf|u1|→∞∣∣∣f(u1, . . . , um)u1∣∣∣≥ λ , |ui| ≥ N , (i 6= 1) .(2.1)If there exists a function ρ(t) ∈ C′([t0,∞), (0,∞)), such that for any H ∈ Xlim supt→∞1H(t, t0)∫ tt0{λH(t, s)ρ(s)q(s){1 − p[g1(s)]}−[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds = ∞ ,(2.2)in which Mθ is a constant, then every solution of equation (1.1) is oscillatory.Proof. Suppose that x(t) is a nonoscillatory solution of equation (1.1), and x(t) bean eventually positive solution. From the assumption of gi(t) and f(u1, . . . , um),there exists a t1 ≥ t0, such thatx(t − τ) > 0 , x[gi(t)] > 0 , f(x[g1(t)], . . . , x[gm(t)]) > 0 , i ∈ Im .Lety(t) = x(t) + p(t)x(t − τ) ,(2.3)then y(t) > 0, t ≥ t1, and from (1.1), y(n)(t) ≤ 0, and y(n)(t) is not identicalzero on any [t1∞). Thus, from Lemma 1, there are exists a t2 ≥ t1 and a oddl(0 < l < n) such thaty(k)(t) > 0 , 0 ≤ k < l , (−1)k−ly(k)(t) > 0 , l ≤ k < n , t ≥ t2 .Choosing k = 1 and k = n − 1, theny′(t) > 0 , y(n−1)(t) > 0 , t ≥ t2 .(2.4)Furthermore, from Lemma 2, and noting that y(n)(t) ≤ 0, there exists Mθ > 0and t3 ≥ t2, such thaty′[σ(t)2] ≥ Mθσn−2(t)y(n−1)[σ(t)] ≥ Mθσn−2(t)y(n−1)(t) , t ≥ t3 .(2.5)Letz(t) =y(n−1)(t)y[σ(t)2 ],(2.6)thenz′(t) = −q(t)f(x[g1(t)], . . . , x[gm(t)])y[σ(t)2 ]−12σ′(t)z(t)y′[σ(t)2 ]y[σ(t)2 ]= I1 + I2 , t ≥ t3 ,(2.7)204 P. WANG, M. WANGfor I1, from (2.3), (H1) − (H2), y(t) ≥ x(t) and y′(t) > 0, we haveI1 ≤ −λq(t)x[g1(t)]y[σ(t)2 ]≤ −λq(t){1 − p[g1(t)]}y[g1(t)]y[σ(t)2 ]≤ −λq(t){1 − p[g1(t)]} ,for I2, from (2.5) and (2.6), we have12σ′(t)z(t)y′[σ(t)2 ]y[σ(t)2 ]≥Mθ2σ′(t)σn−2(t)z2(t) ,thenz′(t) ≤ −λq(t){1 − p[g1(t)]} −Mθ2σ′(t)σn−2(t)z2(t) , t ≥ t3 .(2.8)For any H ∈ X , from (2.8), for any t > T ≥ t3, we have∫ tTλH(t, s)ρ(s)q(s){1 − p[g1(s)]} ds ≤ −∫ tTH(t, s)ρ(s)z′(s) ds−Mθ2∫ tTH(t, s)ρ(s)σ′(s)σn−2(s)z2(s) ds= H(t, T )ρ(T )z(T ) +∫ tT(ρ′(s)H(t, s) − h(t, s)√H(t, s)ρ(s))z(s) ds−Mθ2∫ tTH(t, s)ρ(s)σ′(s)σn−2(s)z2(s) ds= H(t, T )ρ(T )z(T )−Mθ2∫ tT[√H(t, s)ρ(s)σ′(s)σn−2(s)z(s) −ρ′(s)√H(t, s) − h(t, s)ρ(s)Mθ√ρ(s)σ′(s)σn−2(s)]2ds+∫ tT[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)ds≤ H(t, T )ρ(T )z(T ) +∫ tT[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)ds.Thus∫ tT{λH(t, s)ρ(s)q(s){1 − p[g1(s)]} −[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds≤H(t, T )ρ(T )z(T ) .OSCILLATION OF HIGHER ORDER EQUATIONS 205From H ′s(t, s) ≤ 0, for t3 ≥ t0, we have H(t, t3) ≤ H(t, t0), thus∫ tt3{λH(t, s)ρ(s)q(s){1 − p[g1(s)]} −[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds≤H(t, t3)ρ(t3)z(t3) ≤ H(t, t0)ρ(t3)z(t3) .Furthermore, we have1H(t, t0)∫ tt0{λH(t, s)ρ(s)q(s){1 − p[g1(s)]}−[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds=1H(t, t0)(∫ t3t0{λH(t, s)ρ(s)q(s){1 − p[g1(s)]}−[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds+∫ tt3{λH(t, s)ρ(s)q(s){1 − p[g1(s)]}−[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds)≤ ρ(t3)z(t3) + λ∫ t3t0H(t, s)H(t, t0)ρ(s)q(s){1 − p[g1(s)]} ds≤ z(t3) + λ∫ t3t0ρ(s)q(s){1 − p[g1(s)]} ds .(2.9)Let t → ∞, we havelim supt→∞1H(t, t0)∫ tt3{λH(t, s)ρ(s)q(s){1 − p[g1(s)]}−[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds≤ ρ(t3)z(t3) + λ∫ t3t0ρ(s)q(s){1 − p[g1(s)]} ds ,(2.10)which contradicts (2.2).206 P. WANG, M. WANGIf x(t) is an eventually negative solution of equation (1.1), let x∗(t) = −x(t),then equation (1.1) transfer to(1.1∗) [x∗(t) + p(t)x∗(t − τ)](n) + q(t)f∗(x∗[g1(t)], . . . , x∗[gm(t)]) = 0 , t ≥ t0 ,in which f∗(x∗[g1(t)], . . . , x∗[gm(t)]) = −f(−x∗[g1(t)], . . . ,−x∗[gm(t)]), and x∗(t)is an eventually positive solution of equation (1.1∗), f∗ satisfies (H1)− (H2), thensimilar to the case of x(t) > 0, we can also obtain a contradiction. This completesthe proof of Theorem 1.From Theorem 1, we have the following corollary.Corollary 1. Suppose that the conditions (H1) – (H2) hold, and there existsa function ρ(t) ∈ C′([t0,∞), (0,∞)), such that for any H ∈ Xlim supt→∞1H(t, t0)∫ tt0H(t, s)ρ(s)q(s){1 − p[g1(s)]} ds = ∞ ,(2.11)lim supt→∞1H(t, t0)∫ tt0[ρ′(s)√H(t, s) − h(t, s)ρ(s)]2ρ(s)σ′(s)σn−2(s)ds < ∞ ,(2.12)then every solution of equation (1.1) is oscillatory.Remark 1. When f(x[g1(t)], . . . , x[gm(t)]) ≡ x[σ(t)], choosing H(t, s) = (t −s)m−1, then it is clear that Corollary 1 reduce to Theorem A.Example 1. Consider the high-order equation(∗) [x(t) + (1−1t)x(t− τ)](4) +1tx(t)(1 + x2(t−12))(1 + x2(t− 1)) = 0 , t ≥ 1 ,where p(t) = 1 − 1t, q(t) = 1t, g1(t) = x(t), g2(t) = t −12 , g3(t) = t − 1, m = 3,f(x[g1(t)], x[g2(t)], x[g3(t)]) = f(x, y, z) = x(1 + y2)(1 + z2).Taking H(t, s) = (t−s)k−1, h(t, s) = (k−1)(t−s)k−32 , ρ(s) = s, in which (t, s) ∈D and k > 2 is a constant. It is easy known that the condition of Corollary 1 aresatisfied, then all solutions of equation (*) are oscillatory.When f(x[g1(t)], . . . , x[gm(t)]) =m∑i=1fi(x[gi(t)]), in which fi(x) ∈ C(R, R),xfi(x) > 0 for x 6= 0, i ∈ Im. We have the following theorem.Theorem 2. Suppose that the following conditions hold(H3) there exists a function σ(t) ∈ C′([t0,∞), (0,∞)) such thatσ(t) = infs≥t{s, mini∈Im{gi(s)}}, and limt→∞σ(t) = ∞, i ∈ Im;OSCILLATION OF HIGHER ORDER EQUATIONS 207(H4)fi(x)x≥ λi > 0, x 6= 0, where λi > 0 are some constants, i ∈ Im. If thereexists a function ρ(t) ∈ C′([t0,∞), (0,∞)), such that for any H ∈ Xlim supt→∞1H(t, t0)∫ tt0{H(t, s)ρ(s)q(s)m∑i=1λi{1 − p[gi(s)]}−[ρ′(s)√H(t, s) − h(t, s)ρ(s)]22Mθρ(s)σ′(s)σn−2(s)}ds = ∞ ,(2.13)in which Mθ is a constant, then every solution of equation (1.1) is oscillatory.Proof. Suppose that equation (1.1) has a nonoscillatory solution x(t) > 0. Byusing the same arguments as in the proof of Theorem 1, there exists a t1 ≥ t0 suchthat y(t) > 0, y′(t) > 0, y(n−1)(t) > 0 and y(n)(t) ≤ 0 for t ≥ t1, and there existsMθ > 0 and t2 ≥ t1, such that(2.5) y′[σ(t)2] ≥ Mθσn−2(t)y(n−1)[σ(t)] ≥ Mθσn−2(t)y(n−1)(t) , t ≥ t2 .Let(2.6) z(t) =y(n−1)(t)y[σ(t)2 ],then(2.14) z′(t) = −q(t)m∑i=1fi(x[gi(t)])y[σ(t)2 ]−12σ′(t)z(t)y′[σ(t)2 ]y[σ(t)2 ], t ≥ t2 .From (2.3), (H3) – (H4), y(t) ≥ x(t) and y′(t) > 0, we have−q(t)m∑i=1fi(x[gi(t)])y[σ(t)2 ]≤ −q(t)m∑i=1λix[gi(t)]y[σ(t)2 ]= −q(t)m∑i=1λiy[gi(t)] − p[gi(t)]x[gi(t) − τ ]y[σ(t)2 ]≤ −q(t)m∑i=1λi{1 − p[gi(t)]}y[gi(t)]y[σ(t)2 ]≤ −q(t)m∑i=1λi{1 − p[gi(t)]},then(2.15) z′(t) ≤ −q(t)m∑i=1λi{1 − p[g1(t)]} −Mθ2σ′(t)σn−2(t)z2(t) , t ≥ t2 .The remainder proof is as same as proof of Theorem 1, we omit it. This completesthe proof of Theorem 2.208 P. WANG, M. WANGAcknowledgement. The authors are grateful to the referee for his valuablesuggestions.References[1] Agarwal, R.P., Grace, S. R., O’Regan, D., Oscillation criteria for certain nth order differ-ential equations with deviating arguments, J. Math. Anal. Appl. 262 (2001), 601–622.[2] Bainov, D. D, Mishev, D. P., Oscillation Theory for Neutral Differential Equations withDelay, Adam Hilger, Bristol 1991.[3] Erbe, L. H., Kong, Q. K., Zhang, B. Q., Oscillation Theory for Functional DifferentialEquations, Marcel Dekker, New York 1995.[4] Ladas, G., Sficas, Y. D., Oscillations of higher-order equations, J. Austral. Math. Soc., Ser.B 27 (1986), 502–511.[5] Grace, S. R., Oscillation of even order nonlinear functional differential equation with devi-ating arguments, Funkcial. Ekvac. 32 (1989), 265–272.[6] Grace, S. R., Oscillation theorems of comparison type for neutral nonlinear functional dif-ferential equation, Czechoslovak Math. J. 45, 4 (1995), 609–626.[7] Jaroš, J., Kusano, T., Existence of oscillatory solutions for functional differential equationsof neutral type, Acta Math. Univ. Comenian. (N.S.) 60, No. 2 (1991), 185–194.[8] Philos, Ch. G., A new criterion for the oscillatory and asymptotic behavior of delay differ-ential equations, Bull. Acad. Pol. Sci. Ser. Sci. Mat. 29 (1981), 367–370.College of Electronic and Information EngineeringHebei University, Baoding, 071002People’s Republic of ChinaE-mail: pgwang@mail.hbu.edu.cn

Oscillation of a class of higher order neutral differential equations

Abstract. In this paper, we investigate a class of higher order neutral func-tional dierential equations, and obtain some new oscillatory criteria of solu-tions. 1

Peiguang Wang

Min Wang

CiteSeerX

http://siba-sinmemis.unile.it/journals/AM/04-2/wangrev.pdf

Oscillation of a class of higher order neutral differential equations

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

CiteSeerX