Džurina, Jozef

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

Mathematica SlovacaJozef DžurinaUnbounded oscillation of the second-order neutral differential equationsMathematica Slovaca, Vol. 51 (2001), No. 4, 441--447Persistent URL: http://dml.cz/dmlcz/136813Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2001Institute 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.czMathematica Slovaca ©2001 • . _ • , - • .--• /«.-,«.. \ >• - - - . . - - - Mathematical Insti tute Math . SlOVaCa, 5 1 ( 2 0 0 1 ) , NO. 4 , 4 4 1 - 4 4 7 Slovák Academy of Sciences UNBOUNDED OSCILLATION OF THE SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS JOZEF DZURINA (Communicated by Milan Medved!) ABSTRACT. The aim of this paper is to present sufficient conditions for all unbounded solutions of the second order neutral differential equation (x(t)+px(t - r))" - q(t)x(a(t)) = 0 to be oscillatory. Some new sufficient conditions for unbounded oscillation are derived along with some comparison result. Introduction Let us consider the second order neutral differential equation of the form: (x(t) +px(t - T))" - q(t)x(a(t)) = 0 , t> t0 , (1) under the assumption: (i) p ^ 0 and r > 0 are constants; (ii) < 7 G C 1 ( R f , R + ) , a(t)>t, a'(t)>0; (hi) qeC(R+,R+). In what follows, by a solution of (1) is meant a continuous function x(t) defined on an interval [tx,oo), tx > t0, such that x(t) + px(t — r ) is a twice continuously differentiable and x(t) satisfies (1) for all sufficiently large t and sup{|x(£)| : t > T} > 0 for any T >tx. Such a solution is called oscillatory if it has a sequence of zeros tending to infinity; otherwise it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory. In the past two decades research on the oscillation theory of differential equa-tions of neutral types has attracted the attention of many authors. The prob-lems of asymptotic and oscillatory behavior of solutions of neutral differential 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Primary 34C10. Key words: neutral equation, delayed argument. Research is supported by S.G.A, Grant No. 1/7466/20. 441 JOZEF D2URINA equations is of both theoretical and practical interest. For recent contributions regarding the theoretical part and providing the systematic treatment of os-cillation and nonoscillation theory of neutral differential equations we refer to the books by D. D. B a i n o v and D. P. M i s h e v [1], by I. G y o r i and G. L a d a s [8], and by L. H. E r b e , Q. K o n g and B. G. Z h a n g [6]. The books summarize some important work in this area and the main effort is de-voted to the first and second order differential equations. The results presented in the mentioned books cover in sufficient way the theory of equations involving delayed arguments (a(t) < t). However the known results touch only little upon advanced types of neutral equations (see [6], [7] and [11]). The aim of this paper is to establish unbounded oscillation criteria for the second order neutral differential equations of unstable type. For the corresponding bounded oscillation criteria for the second order delay differential equations of the form (x(t) -px(t - T ) ) " - q(t)x(a(t)) = 0 the reader is referred to [5] and [6]. In this paper all functional inequalities that we write are assumed to hold eventually, that is, for sufficiently large t. Main results In the sequel we shall use the following function r(t) = a-1[a(t)-r], (2) where a~l(t) is the inverse function to a(t). It is easy to see that r(t) < t and r'(t) > 0. We begin with the following unbounded oscillation criterion: THEOREM 1. Let r(t) be defined by (2) and r"(t) > 0. Assume that a(t) limsup / (a(t)-s)q*(s) ds>l+p, (3) t where q*(t) = mm{q(t),q[r(t)\(rr(t))2} . Then every unbounded solution of equa-tion (1) is oscillatory. P r o o f . Assume for the sake of contradiction, that equation (1) has an eventually positive unbounded solution x(t). That is, there exists a t0 ^ 0 such that x(t) > 0 for t ^ t0. Set z(t) =x(t)+px(t-T). 442 UNBOUNDED OSCILLATION OF NEUTRAL DIFFERENTIAL EQUATIONS Then z(t) > 0 for t ^ tx = t0 + r , z(t) is unbounded and z"(t) = q(t)x[a(t)}. Thus z"(t) > 0 and this implies that z'(t) is of constant sign. But if we admit zf(t) < 0, then z(t) would be bounded. Therefore, z'(t) > 0 for t^ t2 ^ tx. We put y(t) = z(t)+pz[r(t)]. Now, using properties of z(t) and r(t), we see that ?/(£) > 0, y'(t) > 0 for all large t and moreover y"(t) = z"(t)+pz"[r(t)} (r'(t)f+pz'[r(t)}r"(t) > q(t)x(a(t)) +pq[r(t)} (r'(t)fx[a(t) - r] >q*(t)[x(a(t))+px(a(t)~T)} = q*(t)z(a(t)) q*(t) >- ^----[z(a(t)) +pz[r(a(t)))] V{*(t)) • 1+P 1+P Hence, y(t) is a positive solution of the differential inequality y"(t)>(^y(a(t)), t>t3. An integration of (4) from t tow yields U y'(ч)-y'(t)ž Jţ^y(a(S))dS. Now integrating in u from t to a(t) we are lead to <r(t) Î/И*)) - У(*) - У\t)(a(t) ~t)> I (<т(t) - s)ţ^y(a(S)) ds. Consequently, using the monotonicity of y(t) one gets <7(t) 0 > y(a(t)) J(a(t)-S)q-^ldS-l (4) We have arrived at a contradiction with (3). The proof is complete now. • Remark 1. For differential equation without neutral term (p = 0) Theorem 1 provides the well-known result of C h a n t u r i a and K o p l a t a d z e [2]. 443 JOZEF D2URINA EXAMPLE 1. Consider the following neutral differential equation of the form (x(t) + px(t - T))N -^x(X2t) = 0 , A > 1 , t ^ l . (5) Now r(t) =-1 —r/A2 and q*(t) = 1/t2. Simple computation shows that condition (3) for equation (5) reduces to A 2 - 2 1 n A - 2 - p > 0 . (6) A natural question arises what happen if (3) is violated. The following com-parison theorem provides the partial answer. THEOREM 2. Let r(t) and q*(t) be the same as in Theorem 1. Assume that there exists a couple of the functions aeC1 ((t0, oo)) , a(t) > 0 , o!(t) < 0 , (7) PeCl{(t0ioo)), p(t)>t, /?'(*)> 0, (8) such that <r(t)>/3(/3(t)), (9) q*(t)>(l+p)a(t)a(l3(t))P'(t). (10) // the first order differential inequality v'(t)-a{p(t))f3'(t)v{f3(t))>0 (11) has no eventually positive solutions, then all unbounded solutions of (1) are os-cillatory. P r o o f . Assume that x(t) > 0 is an unbounded solution of (1). Let z(t) and y(t) be the same functions as in the proof of Theorem 1. Then proceeding exactly as in the proof of Theorem 1 we are led to (4). We put b(t)=y'(t) + a(t)y((3(t)). Then b(t) is positive and b'(t)-a(t)f}'(t)b{(3(t)) = y"(t) + a'(t)y{P(t)) ~ a(t)a{(3(t))P'{t)y{(3{p(t))) • Hence, taking into account (4), (7), (8), (9), (10) one gets for all large t b'(t) - ^b(t) - a(t)/3'(t)b(/3(t)) = y"(t) ~ ^V'it) - a(t)a((3(t))P'(t)y(p((3(t))) >y"(t)-a(t)a(p(t))(3'(t)y{(3((3(t))) >y"(t)-(^-y{cr(t))>o. 1 + P 444 UNBOUNDED OSCILLATION OF NEUTRAL DIFFERENTIAL EQUATIONS Setting b(t) = a(t)v(t) we conclude that v(t) is a positive solution of (11). This is a contradiction and the proof is complete. • COROLLARY 1. Let r(t) and q*(t) be the same as in Theorem 1. Assume that (7)-(10) hold. If liminf fa(P(s))P'(s) ds > - , (12) t->oo J e t then (1) does not allow unbounded nonoscillatory solutions. P r o o f . It is known (see [10; Theorem 2.4.1]) that (12) is sufficient for (11) to have no eventually positive solutions. So, the assertion of this corollary follows from Theorem 2. • R e m a r k 2. Theorem 2 permits to apply any sufficient condition for absence of eventually positive solutions of (11) to get unbounded oscillation criterion for equation (1) provided that (7)-(10) hold. We have established the relationship between unbounded oscillation of the second order neutral equation of unstable type and the absence of an eventu-ally positive solution of the first order differential inequality without neutral term. The following example is intended to show that Theorems 1 and 2 are independent. E X A M P L E 2. We consider once more neutral differential equation (5). We let a ( 0 = t / | , and P(t) = Xt. Then all conditions of Corollary 1 are satisfied and (12) takes the form v^+p A > e • . (13) Thus (13) is sufficient for unbounded oscillation of (5). R e m a r k 3 . For equation (5) we shall compare unbounded oscillation criteria (6) and (13). It is easy to verify that for p = 0.5 conditions (6) and (13) are satisfied for A > 1.962 and A > 1.57, respectively. On the other hand, for p = 24 conditions (6) and (13) are satisfied for A > 5.42 and A > 6.29, respectively. So we conclude that neither Corollary 1 is included in Theorem 1 nor The-orem 1 in Corollary 1. The following results provides unbounded oscillation criteria for special types of equation (1). 445 JOZEF D_?URINA COROLLARY 2. Let r(t) and q*(t) be the same as in Theorem 1. If a > 0 and q*(t) > ' ^ 2 (1 + p) for some e > 0, then all unbounded solutions of (x(t) + px(t - T ) ) " - q(t)x(t + a) = 0 are oscillatory. P r o o f . We set a(t) = ^f and (i(t) = t + f in Corollary 1. • COROLLARY 3 . If a > 0 and q > ^ 2 ( 1 + p ) , ^ e n all unbounded solutions of (x(t) + px(t - T ) ) " - qx(t + a) = 0 (14) are oscillatory. P r o o f . It is easy to see that q*(t) = q. Since q > ^ 2 ( 1 + P ) , there exists e > 0 such that q > ^ff-(l +p). Hence we put a(t) = ^f and (i(t) = t + f in Corollary 1. • Remcirk 4. For differential equation (14) we get the same result as G r a c e in [5]. COROLLARY 4. Let r(t) and q*(t) be the same as in Theorem 1. Suppose that q*(t) > ^72 and A > e« . Then all unbounded solutions of (x(t)+px(t - r ) ) " - q(t)x(\2i) = 0 are oscillatory. P r o o f . Put a(t) = \ and (3(t) = Xt in Corollary 1. • In this paper we have established two independent results for unbounded oscillation of delay-advanced differential equations of neutral type. Recently the advanced equations became a discussed theme in qualitative theory of functional differential equations. This paper complements the recent works in this direction [2], [3], [7]. Moreover the presented results complement and generalize those in the well-known monograph by T. A. C h a n t u r i a and R. G. K o p l a t a d z e [2]. R E F E R E N C E S [1] BAINOV, D. D — MISHEV, D. P . : Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, Bristol, 1991. [2] CANTURIA, T. A.—KOPLATADZE, R. G.: On Oscillatory Properties of Differential Equations with Deviating Arguments, Tbilisi Univ. Press, Tbilisi, 1977. (Russian) 446 UNBOUNDED OSCILLATION OF NEUTRAL DIFFERENTIAL EQUATIONS [3] DOMOSHNITSKY, A. I.: Extension of Sturm's theorem to apply to an equation with time lag, Differentsial'nye Uravneniya 19 (1983), 1475-1482. [4] DOSLA, Z .—KIGURADZE, I. T . : On boundedness and stability of solutions of second order equations with advanced arguments, Adv. Math. Sci. Appl. 9 (1999), 1-24. [5] DZURINA, J .—MIHALIKOVA, B .: A note on unstable neutral differential equations of the second order differential equations, Fasc. Math. 29 (1999), 17-22. [6] ERBE, L. H .—KONG, Q.—ZHANG, B. G.: Oscillation Theory for Functional Differential Equations, Dekker, New York, 1995. [7] GRACE, S. R . : Oscillation criteria for n-th order neutral functional differential equations, J. Math . Anal . Appl . 184 (1994), 44-55. [8] GYORI , I .—LADAS, G.: Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991. [9] LADDE, G. S .—LAKSHMIKANTHAM, V .—ZHANG, B . G.: Oscillation Theory of Dif-ferential Equations with Deviating Arguments, Dekker, New York, 1987. [10] KULCSAR, S.: On the asymptotic behavior of solutions of the second order neutral dif-ferential equations, Publ . Math . Debrecen 57 (2000), 153-161. [11] TANAKA, S.: Oscillation properties of solutions of second order neutral differential equa-tions with deviating arguments, Analysis 17 (1997), 99-111. Received October 28, 1999 Department of Mathematical Analysis Revised February 8, 2000 Faculiy °f Sciences Safdrik University Jesennd 5 SK-041 54 Kosice SLOVAKIA E-mail: dzurina@kosice.upjs.sk 447 

Unbounded oscillation of the second-order neutral differential equations

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Unbounded oscillation of the second-order neutral differential equations

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