A Linearized Oscillation Result for Neutral Delay Differential Equations

Math. Nachr. 163 (1993) 101-107 A Linearized Oscillation Result for Neutral Delay Differential Equations ') By JIANSHE Yu and ZHICHENG WANG of Changsha (Received October 6, 1992) Abstract. We consider the first order nonlinear neutral delay differential equation (1) d dt -(x(t)--(t)g(x(t-z)))+ Q(t)h(x(t-g))=O, and establish a linearized oscillation result of Eq. (1) when P ( t ) 2 1, which answers partially an open problem proposed by GYORI and LADAS. 1. Introduction Consider the first order nonlinear neutral delay differential equation where (2) P , Q ~ c ( [ t ~ , m ) , R ) , g , h € C ( R , R ) and z>O, a20. Recently, a linearized oscillation result of Eq. (1) has been established by LADAS, etc, [3, 41. For some further research work, we refer to [l, 2, 51. As we have seen in [3, 41, it seems that the following assumption lim sup P(t) = p o ~(0, l), liminfP(t) = p ~ ( 0 , l ) t'rn t'a, is always assumed to hold. But, the case P(t) < 0 or P(t) 2 1 has not yet been handled. Therefore, GYORI and LADAS put forth the following question in [3, Problem 6. 12. 71: Obtain linearized oscillation results for Eq. (1) when the coefficient P(t) < 0 for t 2 to or P ( t ) 2 1 for t 2 t,. For the case P(t) < 0, a linearized oscillation result of Eq. (1) has been obtained by 1) Project supported by the National Natural Science Foundation of China. 102 Math. Nachr. 163 (1993) Yu and CHEN [6]. Our aim in this paper is to answer the above problem when P(t) 2 1 for t 2 to. Our main result is the following theorem. Theorem 1. Assume that (2) holds and that P(t) 2 1 for t 2 to and lim sup P(t) = p o E (1, co), lim Q(t) = 4 E (0, co), t+ m (3) (4) f +m (6) uh(u) > 0 for u # 0, Ih(u)l 2 h, > 0 for Iu( sufliciently large and (7) lim h(u)/u = 1 . I4'W I f every solution of the linear equation with constant coefficients oscillates, then every solution of Eq. (1 ) also oscillates. The proof of Theorem 1 will be given in section 2. Let p = max(z, a). By a solution of Eq. (1) we mean a function x E C([ t , - p, co), R), for some t , 2 to, such that x ( t ) - P(t )g(x( t - z)) is continuously differentiable on [ t l , 00) and such that Eq. (1) is satisfied for t 2 t,. Let t , 2 to and let Q)E C([ t , - p, t,], R) be a given initial function. Then we can easily find by the method of steps that Eq. (1) has a unique solution x E C([t , - p, co), R ) such that x ( t ) = @(t) for t , - p I t I t , . As usual, a solution of Eq. (1) is called oscillatory if it has arbitrarily large zeros, In the sequel, for convenience, when we write a functional inequality without specify- and nonosciElatory if it is eventually positive or eventually negative. ing its domain of validity, we assume that it holds for all sufficiently large t. !' 2. Proof of Theorem 1 The following lemma will be useful in the proof of Theorem 1. Lemma 1. Assume that (9) p 0 ~ ( 1 , w),q,zE(O, and o€[O, co) and that every solution of the equation d -(x(t) - pox(t - z)) + qx(t - a) = 0 dt Yu/Wang, Neutral Delay Differential Equations 103 oscillates. Then there exists an E E (0, q) such that every solution of the equation d -(x(t)-((p, + E ) x ( t - T ) ) + ( q - E ) X ( t - f Y ) = O dt (10) also oscillates. Proof . It is well known that every solution of Eq. (8) oscillates if and only if its characteristic equation (1 1) has no real roots. As p o > 1, it follows that (1 1) has no nonpositive real roots. Therefore, every solution of Eq. (8) oscillates if and only if its characteristic equation (11) has no positive roots. Also since f(0) = q > 0 and {(a) = co, we have - I r f (A) = 1(l- poe ) + qe-Au = 0 m:= inf f ( A ) > 0. A10 Thus (12) f ( 1 ) 2 m for all 120. Set E* = 43, g(1) = e*(Ae-" + e-Au). Then f(1) - g ( 1 ) = A(1- (Po + E * ) e-A7) + (q - E * ) e - lU -, co and so there exists 1, > 0 such that f(1) - g(A) 2 m/2 for 1 2 A,. (13) has no positive roots. in fact, for A2 1, we have as A + co Let E = min ( E * , m/2(A0 + l)}. We claim that the characteristic equation of Eq. (10) f , ( A ) = A( 1 - (Po + E ) e - ") + (q - E ) e - f , ( A ) =f (A) - ~ ( l e - ~ ' + e-Au) rf(A) -&*(Ae-ar + e-Au) = 0 m 2 =f(1) - g(1) 2 - > 0 and for 0 < 1 < 1, we have and so (13) has no positive roots. This shows that every solution of Eq. (10) oscillates. The proof of Lemma 1 is completed. Now we are ready to proof Theorem 1 by using the Banach contraction principle. Proof of Theorem 1. Assume, for the sake of contradiction, that Eq. (1) has a nonoscillatory solution x( t ) . We will assume that x ( t ) is eventually positive. The case where x ( t ) is eventually negative is similar and will be omitted. Choose t , 2 to to be such that x ( t - ~ ) > O , x ( t - o ) > O for t k t , . y*(t) = x ( t ) - P ( t ) g ( x ( t - 7)). y*'(t) = - Q(t ) h(x(t - 0)) I O( f 0) for t 2 t , . Set (14) Then (15) 104 Math. Nachr. 163 (1993) So, y*(t) is decreasing and so either (16) Y*(t) < 0 (17) y*(t) > 0. or We claim that (16) holds. Otherwise (17) holds which implies that x( t ) > P(t )g(x( t - 7)) for t 2 t , which, together with (3) and (5), yields x ( t ) > x( t - z) for t 2 t , and so x( t ) 2 M : = min x(s) f l - r s s < t 1 for t 2 t l . Substituting this into (15), we get y*’(t) I - Q ( t ) ,?pin Ih(u)l, for t 2 t , + a which, combining (4) and (6), yields which contradicts (17) and so (16) holds. Since y*(t) is decreasing, it follows that there exist t , 2 t , and a > 0 such that That is, We claim that y*(t)-+ - 00 as t -+ co y * ( t ) I - a for t 2 t,. x ( t ) - P(t)g(x(t - z)) I - a for t 2 t,. p:= inf x( t ) > 0. tzt2 Otherwise, p = 0 and hence there exists a sequence {sn} such that s, -+ co as n + co and x(s,) -+ 0 as n -+ co. Noting g(x(s,)) + 0 as n --+ 00, we have O ~ l i m i n f x ( s , + z ) ~ lim (P(s,+z)g(x(s,))-a)= - a < O n+m n+ m which is a contradiction and so B > 0. Noting that x ( t ) 2 p for t 2 t,, we have y*’(t) I - Q ( t ) min Ih(u)l, for t 2 t , + a which, together (4) and (6), implies y*(t) --* - co as t + 00 which also yields that (19) x( t )+ 03 as t -+ 00 . comes (20) IU lZB Set P*(t) = P(t)g(x(t - z))/x(t - z), Q*(t) = Q(t)h(x(t - a))/x(t - a). Then Eq. (19) be- d dt - (x(t) - P*(t) ~ ( t - 7)) + Q*(t) ~ ( t - C) = 0. Yu/Wang, Neutral Delay Differential Equations 105 From (3)-(5), (7) and (19), we have lim sup P*(t) = p o , lim Q*(t) = q . Integrating (20) from T 2 t , to t 2 IT; we have t-00 t+ m (21) t ~ ( t ) - P*(t) ~ ( t - t) - y*(T) + S Q*(s) X(S - 0) ds = 0 T or equivalently Q*(s) X(S - 0) ds - y*(T) , t 2 T - Z, 1 1 x( t ) = P*(t + t) (22) where y*(T) c 0. every solution of Eq. (10) also oscillates. Hence (23) A ( l - ( P o + ~ ) e - A r ) + ( q - ~ ) e - A " > O for all A E R . For this E > 0, let a > 1 be such that (a - 1) po < E or p o < (po + &)/a. Then, by (21) there exists t , > t , such that Since every solution of Eq. (8) oscillates, by Lemma 1 there is an E E (0,q) such that 1 P*(t) c - Ipo + E ) , Q*(t) > q - E , for t 2 t , - 0 . U Substituting this into the right side of (22), we get 1 t + r ~ ( t ) > ~ [ x ( t + z ) + ( q - ~ ) x(s-~)ds-y*(t,) , trt,. Let X be the Banach space of all bounded and continuous functions defined on Po + E (24) [t, - 0 - 2, co) with the sup-norm. Then is a bounded, closed and convex subset of X . Define a mapping S : A -+ X as follows: A = { y € X : 01y(t)11 for t 2 t , - 0 - ~ } (SY) ( t ) = [ ( S y ) ( t , ) , t , - 0 - t I t I t , . Since for any y E A and t 2 t,, we have by (24) I t + r ~ ( t + t) + (4 - E ) X ( S - 0) ds - y*(t3) f3 s 1 0 I ( S y ) ( t ) I ( P o + 4 x@) 1 I - < l , M 106 Math. Nachr. 163 (1993) it follows that 0 I (Sy)(t) I 1 for all t 2 t , - r~ - z and so S maps A into itself. Next we claim that S is a contraction on A. In fact, for any y,, y2 E A and t 2 t , we have I(SY 1) ( t ) - ( S Y J (01 1 S ; l l Y , -Y,ll Hence 1 5 ; llYl - Y2II . Since a > 1, it follows that S is a contraction on A. Therefore, by the Banach contraction principle S has a fixed point Y E A, i.e., Y (0 = Clearly, y(t) > 0 for all t 2 t , - (r - z. Set z ( t ) = x ( t ) y(t) . Then z(t) is a positive and con- tinuous function on [t , - r~ - z, 00) and satisfies ~ ( t ) = ~ [ z(t+z)+(q-&) z(s-o)ds-y*(t,) , t r t , I t + o P o + E or equivalently t (PO + E ) ~ ( t - z ) = ~ ( t ) +(4 - E ) j Z(S -0)ds -y*(t3), t 2 t 3 + T . t3 Yu/Wang, Neutral Delay Differential Equations 107 Differentiating it, we have d dt -(z(t) - (Po + E ) z(t - z)) + (4 - E ) z(t - 0) = 0, t 2 t , + z which contradicts (23) and the proof is completed. References [l] Q. CHUANXI and G. LADAS, Linearized Oscillation for Equations with Positive and Negative Co- [2] Q. CHUANXI and G. LADAS, Linearized Oscillations for Even Order Neutral Differential Equations, [3] I. GYORI and G. LADAS, Oscillation Theory of Delay Differential Equations with Applications, [4] G. LADAS, Linearized Oscillations for Neutral Equations, Differential Equation, Proceedings of [5] G. LADAS and C. QIAN, Linearized Oscillations for Odd-order Neutral Delay Differential Equations, [6] J. S. Yu and M. P. CHEN, Linearized Oscillations for First Order Neutral Delay Differential Equa- efficients, Hiroshima Math. J., 20 (1990), 331-340 J. Math. Anal. Appl., 159 (1991), 237-250 Clarendon Press, Oxford, 1991 the 1987 Equadiff Conference, Dekker, New York, pp. 379-387 J. Differential Equations, 88 (1990), 238-247 tions, PanAmerican Mathematical Journal, to appear. Department of Applied Mathematics Hunan University Changsha, Hunan 410082 People’s Republic of China