Stability of the Second Order Delay Differential Equations with a Damping Term

Differential Equations and Dynamical Systems, Vol. 16, Nos. 3 & 4, July & Octoer 28, pp. 3-24. Staility of the Second Order Delay Differential Equations with a Damping Term Leonid Berezansky, Elena Braverman and Alexander Domoshnitsky Astract For the delay differential equations ẍ(t) + a(t)ẋ(g(t)) + (t)x(h(t)) =, g(t) t, h(t) t, and ẍ(t) + a(t)ẋ(t) + (t)x(t) + a 1 (t)ẋ(g(t)) + 1 (t)x(h(t)) = explicit exponential staility conditions are otained. c 28 Foundation for Scientific Research and Technological Innovation(FSRTI). All rights reserved. MSC: 34K2 Keywords: Exponential staility; Nonoscillation; Positive fundamental function; Second order delay equations Received on August 7, 28 and in final form on Novemer 5, 28. International Journal for Theory, Real World Modelling and Simulations

4 L. Berezansky, E. Braverman and A. Domoshnitsky 1. Introduction This paper deals with the scalar linear delay differential equation of the second order with a damping term ẍ(t) + a(t)ẋ(g(t)) + (t)x(h(t)) = (1.1) and the equation which also involves nondelay terms ẍ(t) + a(t)ẋ(t) + (t)x(t) + a 1 (t)ẋ(g(t)) + 1 (t)x(h(t)) =. (1.2) Such linear and nonlinear equations attract attention of many mathematicians due to their significance in applications. We mention here the monographs of Myshkis [17], Norkin [18], Ladde, Lakshmikantham and Zhang [15], Györi and Ladas [13], Ere, Kong and Zhang [12], Burton [5], Kolmanovsky and Nosov [14] and references therein. In particular, Minorsky [16] in 1962 considered the prolem of stailizing the rolling of a ship y the activated tanks method in which allast water is pumped from one position to another. To solve this prolem he constructed several delay differential equations with damping of the form (1.1) and (1.2). In spite of ovious importance in applications, there are only few papers on delay differential equations with damping. In [1] the authors considered autonomous equation (1.2) and otained staility results using analysis of the roots of the characteristic equation. In [5] staility of the autonomous equation ẍ(t) + aẋ(t) + x(t τ) = (1.3) was studied using Lyapunov functions. It was demonstrated that if a >, > and τ < a, then equation (1.3) is exponentially stale. Other results otained y Lyapunov functions method can e found in [9, 19]. In [8] Burton and Furumochi applied fixed point theorems to equation (1.1) and otained new staility results. In particular, the equation ẍ(t) + 1 3ẋ(t) + 1 x(t 16) = 48 is exponentially stale, where the condition τ < a does not hold. Here τ = a.

Second Order Delay Equations 5 In [6, 7] some other staility conditions were otained y the fixed point method for (1.1) in the case g(t) t, h(t) = t τ. To the est of our knowledge, there is only one paper [2] where staility of the general nonautonomous equation (1.1) was investigated. In [2] the authors applied the W-method [1] which is ased on the application of the Bohl-Perron type theorem (Lemma 2.3. of the present paper). Note also the paper [11] where nonoscillation of systems of delay differential equations was considered and on this asis several results on nonoscillation and exponential staility of second order delay differential equations were otained. Here we will employ the method of [2] and consider equation (1.2), which was not studied in [2]. Furthermore, we will use a new approach (also ased on a Bohl-Perron type theorem) and otain sharper staility results for equation (1.1) than in [2]. In particular, for (1.3) we otain the same staility condition τ < a as in [5], ut our results are applicale to more general nonautonomous equations as well. 2. Preliminaries We consider the scalar second order delay differential equation (1.1) under the following conditions: (a1) a(t), (t), are Leesgue measurale and essentially ounded functions on [, ); (a2) g : [, ) IR, h : [, ) IR are Leesgue measurale functions, g(t) t, h(t) t, t, lim sup(t g(t)) <, lim sup(t h(t)) <. t t Together with (1.1) consider for each t an initial value prolem ẍ(t) + a(t)ẋ(g(t)) + (t)x(h(t)) = f(t), t t, (2.1) x(t) = ϕ(t), ẋ(t) = ψ(t), t < t, x(t ) = x, ẋ(t ) = x. (2.2) We also assume that the following hypothesis holds (a3) f : [t, ) IR is a Leesgue measurale locally essentially ounded function, ϕ : (, t ) IR, ψ : (, t ) IR are Borel measurale ounded functions.

6 L. Berezansky, E. Braverman and A. Domoshnitsky Definiton 2.1. A function x : IR IR with locally asolutely continuous on [t, ) derivative ẋ is called a solution of prolem (2.1), (2.2) if it satisfies equation (2.1) for almost every t [t, ) and equalities (2.2) for t t. Definiton 2.2. For each s, the solution X(t, s) of the prolem ẍ(t) + a(t)ẋ(g(t)) + (t)x(h(t)) =, t s, x(t) =, ẋ(t) =, t < s, x(s) =, ẋ(s) = 1 (2.3) is called the fundamental function of equation (1.1). We assume X(t, s) =, t < s. Let functions x 1 and x 2 e the solutions of the following equation ẍ(t) + a(t)ẋ(g(t)) + (t)x(h(t)) =, t t, x(t) =, ẋ(t) =, t < t, with initial values x(t ) = 1, ẋ(t ) = for x 1 and x(t ) =, ẋ(t ) = 1 for x 2, respectively. By definition x 2 (t) = X(t, t ). Lemma 2.1. [1] Let (a1)-(a3) hold. Then there exists one and only one solution of prolem (2.1), (2.2) that can e presented in the form x(t) = x 1 (t)x + x 2 (t)x + t X(t, s)f(s)ds t X(t, s)[a(s)ψ(g(s)) + (s)ϕ(h(s))]ds, (2.4) where ϕ(h(s)) = if h(s) > t and ψ(g(s))) = if g(s) > t. Definiton 2.3. Eq. (1.1) is (uniformly) exponentially stale, if there exist M >, µ >, such that the solution of prolem (1.1),(2.2) has the estimate [ ] x(t) M e µ(t t ) x(t ) + sup ( ϕ(t) + ψ(t) ), t t, t<t where M and µ do not depend on t.

Second Order Delay Equations 7 Definiton 2.4. The fundamental function X(t, s) of (1.1) has an exponential estimate if there exist positive numers K >, λ >, such that X(t, s) K e λ(t s), t s. (2.5) For the linear equation (1.1) with ounded delays ((a2) holds) the last two definitions are equivalent. Under (a2) the exponential staility does not depend on values of equation parameters on any finite interval. Remark 2.1. All definitions and Lemma 2.1. can also e applied to equation (1.2), for which we will assume that conditions (a2)-(a3) hold and all coefficients are Leesgue measurale essentially ounded on [, ) functions. Consider the equation ẍ(t) + aẋ(t) + x(t) =, (2.6) where a >, > are positive numers. This equation is exponentially stale. Denote y Y (t, s) the fundamental function of (2.6). Lemma 2.2. [4] Let a >, >. 1) If a 2 > 4 then Y (t, s) ds 1, 2) If a 2 < 4 then Y t (t, s) ds 2a a2 4(a a 2 4). Y (t, s) ds 3) If a 2 = 4 then 4 a 4 a 2, Y t (t, s) ds 2(a + 4 a 2 ) a 4 a 2. Y (t, s) ds 1, Y t (t, s) ds 2. Let us introduce some functional spaces on a semi-axis. Denote y L [t, ) the space of all essentially ounded on [t, ) scalar functions and y C[t, ) the space of all continuous ounded on [t, ) scalar functions with the supremum norm.

8 L. Berezansky, E. Braverman and A. Domoshnitsky Lemma 2.3. [1] Suppose there exists t such that for every f L [t, ) oth the solution x of the prolem ẍ(t) + a(t)ẋ(g(t)) + (t)x(h(t)) = f(t), t t, x(t) =, ẋ(t) =, t t, and its derivative ẋ elong to C[t, ). Then equation (1.1) is exponentially stale. Remark 2.2. A similar result is valid for equation (1.2). Lemma 2.4. [3] If a(t) is essentially ounded on [, ), the fundamental function Z(t, s) of the equation ẋ(t) + a(t)x(g(t)) = (2.7) is positive: Z(t, s) >, t s t and t g(t) δ, then t +δ Z(t, s)a(s)ds 1 for all t t + δ. Lemma 2.5. [3] If a(t) α > is essentially ounded in [, ), lim sup t (t g(t)) < and the fundamental function Z(t, s) of equation (2.7) is positive, then (2.7) is exponentially stale and Z(t, s) has an exponential estimate. Lemma 2.6. [13] Suppose a(t) and g(t) a(s)ds 1 e, t t. Then the fundamental function of (2.7) is positive: Z(t, s) >, t s t.

Second Order Delay Equations 9 3. Staility Conditions, I In this section we consider equation (1.2) as a perturation of an exponentially stale ordinary differential equation for which integral estimations of the fundamental function and its derivative are known. We will start with the main result of [2]. Denote y Y (t, s) the fundamental function of the equation ẍ(t) + a(t)ẋ(t) + (t)x(t) =. (3.1) If equation (3.1) is exponentially stale then there exist Y = sup Y (t, s) ds <, t>t t Denote, g(t) t, a = sup t>t a(t), g(t) t, Y = sup t>t Y t (t, s) ds <. t (3.2), h(t) t, = sup t>t (t), h(t) t, { } h = max sup(t g(t)), sup(t h(t)). t>t t>t Theorem A [2] Suppose a(t), (t), equation (3.1) is exponentially stale and for some t h < Then (1.1) is exponentially stale. 1 a 2 Y + ay + Y. Now we will apply the method of [2] to equation (1.2). Theorem 3.1. Suppose a(t), (t), equation (3.1) is exponentially stale and for some t a 1 Y + 1 Y < 1, where is the norm in the space L [t, ). Then (1.2) is exponentially stale.

1 L. Berezansky, E. Braverman and A. Domoshnitsky Proof. Without loss of generality we can assume t =. Consider the following prolem ẍ(t) + a(t)ẋ(t) + (t)x(t) + a 1 (t)ẋ(g(t)) + 1 (t)x(h(t)) = f(t), t, x(t) = ẋ(t) =, t. (3.3) Let us demonstrate that for every f L the solution x of (3.3) and its derivative are ounded. Due to the zero initial conditions, we can assume that h(t) = g(t) =, t <. The solution x of (3.3) is also a solution of the following prolem ẍ(t) + a(t)ẋ(t) + (t)x(t) = z(t), t, x(t) = ẋ(t) =, t, (3.4) with some function z(t). Then x(t) = Y (t, s)z(s)ds, x (t) = Y t (t, s)z(s)ds, (3.5) where Y (t, s) is the fundamental function of equation (3.1). Hence equation (3.3) is equivalent to the equation z(t) + a 1 (t) g(t) Y (g(t), s)z(s)ds + 1 (t) h(t) Y (h(t), s)z(s)ds = f(t). (3.6) Equation (3.6) has the form z + Hz = f, which we consider in the space L [, ). We have H a 1 Y + 1 Y < 1. Then for the solution of (3.6) we have z L. Equalities (3.5) imply x Y z, x Y z. By Lemma 2.3. equation (1.2) is exponentially stale. Consider now equation (1.2) with a(t) a, (t). By Lemma 2.2. we have the following statement. Corollary 3.2. Suppose for some t one of the following conditions holds:

Second Order Delay Equations 11 1) a 2 > 4, 2) 4 > a 2, 3) a 2 = 4, 2a a2 4(a a 2 4) a 1 + 1 1 < 1, 2(a + 4 a 2 ) a 4 a 1 + 4 a 2 a 4 a 1 < 1, 2 2 a 1 + 1 1 < 1, where is the norm in the space L [t, ). Then equation (1.2) is exponentially stale. Example 3.3. Let us illustrate the exponential staility domain in Corollary 3.2. for two cases. If a = 3, = 2 then a 2 > 4 and the condition of 1) ecomes 3 a 1 +.5 1 < 1 for constant a 1, 1 which corresponds to the domain inside the vertically stretched rhomus in Fig. 1. If a = 3, = 2.5 then a 2 < 4 and the condition of 2) ecomes 2 a 1 + 1 <.75 which is inside a smaller rhomus in Fig. 1. 4. Staility Conditions II In this section we consider equations (1.1) and (1.2) as perturations of an exponentially stale ordinary differential equation for which only an integral estimation of its fundamental function is known. Theorem 4.1. Suppose (3.1) is exponentially stale, (2.7) has a positive fundamental function Z(t, s) >, t s t, a(t) α >, t g(t) δ, t h(t) τ. If for some t + δ [ ( ) ] Y δ a a a + + τ a < 1, (4.1)

12 L. Berezansky, E. Braverman and A. Domoshnitsky 1 2 a = 3, = 2 1-1 -1 1 a 1 a = 3, = 2.5-2 Figure 1: The domain inside the rhomus gives the area of parameters a 1, 1 for the exponential staility of (1.2): for a = 3, = 2 and a = 3, = 2.5, respectively. where Y is denoted in (3.2), is the norm in the space L [, ), then equation (1.1) is exponentially stale. Proof. Consider the following prolem ẍ(t) + a(t)ẋ(g(t)) + (t)x(h(t)) = f(t), x(t) = ẋ(t) =, t. (4.2) As in the proof of Theorem 3.1. we can assume that h(t) = g(t) =, t <. Let us demonstrate that for every f L [, ) the solution of prolem (4.2) and its derivative are ounded. The equation in prolem (4.2) can e rewritten as ẍ(t)+a(t)ẋ(t)+(t)x(t) a(t) g(t) ẍ(s)ds (t) Equation (4.3) is equivalent to the following one h(t) ẋ(s)ds = f(t). (4.3) x(t) Y (t, s)a(s) Y (t, s)(s) s g(s) s h(s) ẍ(τ)dτ ds ẋ(τ)dτ ds = f 1 (t), (4.4)

Second Order Delay Equations 13 where f 1 (t) = Y (t, s)f(s)ds and Y (t, s) is the fundamental function of equation (3.1). Hence f 1 L [, ). The equation in prolem (4.2) can e rewritten in a different form ẋ(t) + Z(t, s)(s)x(h(s))ds = r(t), (4.5) where r(t) = Z(t, s)f(s)ds, Z(t, s) is the fundamental function of (2.7). Since a(t) α > and Z(t, s) >, then y Lemma 2.5. fundamental function Z(t, s) has an exponential estimation. Hence r L [, ). From (4.2) and (4.4) we have x(t) + Y (t, s)a(s) Y (t, s)(s) s g(s) s h(s) [a(τ)ẋ(g(τ)) + (τ)x(h(τ))] dτ ds ẋ(τ)dτds = f 2 (t), (4.6) where f 2 (t) = f 1 (t) + Y (t, s)a(s) s g(s) Since f 2 f 1 + Y δ a f, then f 2 L [, ). Sustituting ẋ from (4.5) into (4.6), we otain f(τ)dτ ds. + x(t) [ s g(s) Y (t, s)a(s) ds a(τ) Y (t, s)(s) g(τ) s h(s) Z(g(τ), ξ)(ξ)x(h(ξ))dξ (τ)x(h(τ)) [ τ ] Z(τ, ξ)(ξ)x(h(ξ))dξ dτds = f 3 (t),(4.7) ] dτ where f 3 (t) = f 2 (t) Y (t, s) [ s s ] a(s) r(g(τ))dτ (s) r(τ)dτ ds. g(s) h(s) Since f 3 f 2 + Y r ( a δ + τ) then f 3 L [, ).

14 L. Berezansky, E. Braverman and A. Domoshnitsky Equation (4.7) has the form x T x = f 3. Lemma 2.4. yields that g(t) t> Z(g(t), s)(s)ds sup sup t> Z(t, s)(s)ds Z(t, s)a(s) (s) a(s) ds a. Condition (4.1) implies T < 1. Hence x C[, ). From (4.5) we have ẋ L [, ). By Lemma 2.3. equation (1.1) is exponentially stale. Remark 4.1. Lemma 2.6. gives explicit conditions for positivity of the fundamental function of equation (2.7). Consider the equation with constant coefficients ẍ(t) + aẋ(g(t)) + x(h(t)) =, (4.8) where a >, >, t g(t) δ, t h(t) τ. Corollary 4.2. Suppose aδ 1 and one of the following conditions e holds: 1) a 2 4, 2δa + τ a < 1, 2) a 2 < 4, 2δa + τ2 a < a 4 a 2. 4 Then equation (4.8) is exponentially stale. Example 4.3. Let us illustrate the exponential staility domain in Corollary 4.2. for oth cases. If a = 3, = 2 then a 2 > 4 and the condition of 1) ecomes 6δ + 2 3 τ < 1. If a = 3, = 2.5 then a2 < 4 and the condition of 2) ecomes 15δ + 25 12 τ < 3 25, or 2δ + τ < 1. Here 4 9 the inequality aδ = 3δ < 1/e should also e satisfied (the area under the horizontal line). Corollary 4.4. Suppose g(t) t, a 2 4, τ < a. Then equation (4.8) is exponentially stale.

Second Order Delay Equations 15 δ a = 3, = 2.5 a = 3, = 2 1 6 1 2 9 25 aδ = 1 e 3 2 τ Figure 2: The domain inside the triangle under the horizontal line aσ = 3σ = 1/e gives the domain of parameters τ, σ for the exponential staility of (4.8) : for a = 3, = 2 and a = 3, = 2.5, respectively. The domain of parameters in the former case involves the domain in the latter case. Remark 4.2. Corollary 4.4. gives the same condition τ < a as was otained y Burton in [5] for autonomous equation (1.3); however, equation (4.8) is not autonomous. In Theorem 4.1. it was assumed that the first order equation (2.7) has a positive fundamental function. In the next theorem we will omit this restriction. Theorem 4.5. Suppose (3.1) is exponentially stale, a(t) α >, t g(t) δ, t h(t) τ. If for some t the inequality δ a < 1 holds and [ (δ a 2 + τ )( ] a + δ ) Y + δ < 1, (4.9) 1 δ a where is the norm in the space L [t, ), then equation (1.1) is exponentially stale. Proof. Without loss of generality we can assume t = and h(t) = g(t) =, t <. As in the proof of Theorem 4.1. we will demonstrate that for every f L [, ) the solution of (4.2) and its derivative are ounded. First we will otain an apriori estimation for the derivative of the solution of (4.2). Equation (4.2) can e rewritten in the form ẍ(t) + a(t)ẋ(t) a(t) g(t) ẍ(s)ds + (t)x(h(t)) = f(t). (4.1)

16 L. Berezansky, E. Braverman and A. Domoshnitsky Sustituting ẍ from (4.2) into (4.1) we have ẍ(t)+a(t)ẋ(t)+a(t) [a(s)ẋ(g(s))+(s)x(h(s))]ds+(t)x(h(t)) = r(t), g(t) (4.11) where r(t) = f(t) + a(t) f(s)ds. Evidently r L g(t) [, ). Equation (4.11) is equivalent to the following one ẋ(t) + e s t s (a(s) a(ξ)dξ [a(τ)ẋ(g(τ)) g(s) + (τ)x(h(τ))]dτ + (s)x(h(s)) ) ds = r 1 (t), (4.12) where r 1 (t) = e s a(ξ)dξ r(s)ds. Since a(t) α >, then r 1 L [, ). Denote y T the norm in the space L [, T ]. From (4.12) we have ẋ T δ( a ẋ T + x T ) + a x T + r 1. Hence we otain the following apriori estimation for ẋ T ẋ T δ + a 1 δ a x T + r 1 1 δ a. (4.13) Equation (4.2) is equivalent to (4.4) which can e rewritten as x(t) + s Y (t, s)a(s) Y (t, s)(s) g(s) s h(s) [a(τ)ẋ(g(τ)) + (τ)x(h(τ))]dτ ds ẋ(τ)dτ ds = f 2 (t), (4.14) where f 2 (t) = f 1 (t) + Y (t, s)a(s) s f(τ)dτds, Y (t, s) is the fundamental function of (3.1). Then f 2 L [, ) g(s) and x T Y [δ a ( a ẋ T + x T ) + τ ẋ T ] + f 2 (4.15) = Y [( [ δ a 2 + τ ) ẋ T + δ x T ] + f2.

Second Order Delay Equations 17 Inequalities (4.13) and (4.15) imply [ (δ a 2 + τ )( ] a + δ ) x T Y + δ x T + C, (4.16) 1 δ a where C is a positive numer. Inequality (4.16) has the form x T A x T + C, where A and C do not depend on T and < A < 1. Hence x L [, ). Inequality (4.13) implies ẋ L [, ). By Lemma 2.3. equation (1.1) is exponentially stale. Now let us apply the method of apriori estimation to equation (1.2). Theorem 4.6. Suppose (3.1) is exponentially stale, a(t) α >, t g(t) δ, t h(t) τ. If for some t the inequality a 1 a < 1 holds and [ a1 ( ) ] + 1a Y a 1 a 1 a + 1 < 1, (4.17) where is the norm in the space L [t, ), Y is defined in (3.2), then equation (1.2) is exponentially stale. Proof. Without loss of generality we can assume t =. Let us demonstrate that for every f L [, ) oth the solution x of (3.3) and its derivative are ounded. Equation (3.3) is equivalent to the following one ẋ(t) = e s a(τ)dτ [a 1 (s)ẋ(g(s)) + 1 (s)x(h(s)) + (s)x(s)]ds + r(t), where r(t) = e s a(τ)dτ f(s)ds. Evidently r L [, ). Hence ẋ T a ( ) 1 ẋ T + a a + 1 a x T + r, where T is the norm in L [, T ]. Then a + 1a ẋ T 1 a 1 x T + a (4.18) r 1 a 1. (4.19) a

18 L. Berezansky, E. Braverman and A. Domoshnitsky Equation (3.3) is also equivalent to x(t) = Y (t, s) [a 1 (s)ẋ(g(s)) + 1 (s)x(h(s))] ds + f 1 (t), (4.2) where f 1 (t) = Y (t, s)f(s)ds, Y (t, s) is the fundamental function of (3.1). The first inequality in (3.2) implies f 1 L [, ). Thus x T Y ( a 1 ẋ T + 1 x T ) + f 1, which together with inequality (4.19) yields x T Y [ a1 ( ) ] a + 1a 1 a 1 + 1 x T + C. a Here C is a positive numer which does not depend on x and T. As in the proof of Theorem 4.5., we otain that x and ẋ are ounded functions. Then y Lemma 2.3. equation (1.2) is exponentially stale. Remark 4.3. If a 1 (t) then the statements of Theorems 3.1. and 4.6. coincide. Corollary 4.7. Suppose a(t) a >, (t) >, a 2 4, t g(t) δ, t h(t) τ and for some t we have a > a 1, a 1 ( + 1 ) < (a a 1 )( 1 ), where is the norm in the space L [t, ). Then equation (1.2) is exponentially stale. Example 4.8. Let us illustrate the exponential staility domain in Corollary 4.7.. If a = 3, = 2 then a 2 > 4 and the condition ecomes 4 a 1 + 3 1 < 6 for constant a 1, 1 (the domain inside the rhomus in Fig. 3) and ounded delays. Corollary 4.9. Suppose a 1 (t), a(t) a >, (t) >, a 2 4, t h(t) τ and for some t we have 1 <, where is the norm in the space L [t, ). Then equation (1.2) is exponentially stale.

Second Order Delay Equations 19 2 1 1-1 1 a 1-1 -2 Figure 3: The domain inside a rhomus gives the area of parameters a 1, 1 for the exponential staility of (1.2) for a = 3, = 2. 5. Explicit Staility Conditions In all previous results we have assumed that the ordinary differential equation (3.1) is exponentially stale and an estimation on the integral of its fundamental function is known. All these conditions can hold, as the following lemma demonstrates. Lemma 5.1. [4] Suppose for some t a = inf t t a(t) >, = inf t t (t) >, B = sup t t (t), a 2 4B. (5.1) Then the fundamental function of (3.1) is positive: Y (t, s) >, t > s t. Moreover, (3.1) is exponentially stale and for its fundamental function we have t Y (t, s)(s)ds 1. (5.2) As an application of Lemma 5.1., we will otain new explicit staility conditions for equations (1.1) and (1.2). Theorem 5.1. Suppose there exists t such that condition (5.1) holds, g(t) a(s)ds 1 e, t t, (5.3)

2 L. Berezansky, E. Braverman and A. Domoshnitsky and t g(t) δ, t h(t) τ for t t. If for some t + δ δ a ( ) a a + + τ a < 1, (5.4) where is the norm in the space L [, ), then equation (1.1) is exponentially stale. Proof. Let us demonstrate that for every f L [, ) the solution x of prolem (4.2) and its derivative are ounded. The equation in prolem (4.2) is equivalent to (4.7), which has the form x Hx = f 3. Condition (5.3) and Lemma 2.6. imply that the fundamental function of equation (2.7) satisfies Z(t, s) >, t t. Lemmas 4 and 6 yield that (Hx)(t) + Y (t, s)(s) a(s) (s) ds [ s ] g(ζ) a(ζ) Z(g(ζ), ξ)a(ξ) (ξ) dξ + (ζ) dζ x g(s) a(ξ) [ s ζ Y (t, s)(s) dζ Z(ζ, ξ)a(ξ) (ξ) ] h(s) a(ξ) dξ ds x [ δ a ( ) ] a a + + τ a x. Inequality (5.4) implies H < 1, hence the solution x of (4.7) and therefore of (1.1) is a ounded function. Equality (4.5) yields that ẋ is a ounded function. By Lemma 2.3. equation (1.1) is exponentially stale. Corollary 5.2. Suppose there exists t such that condition (5.1) holds, g(t) t, t h(t) τ, t t. If τ a < 1, where is the norm in the space L [t, ), then equation (1.1) is exponentially stale. Let us proceed to staility conditions for equation (1.2). Theorem 5.3. Suppose there exists t such that condition (5.1) holds, t g(t) δ, t h(t) τ, t t.

Second Order Delay Equations 21 If a 1 a < 1 and a 1 a + 1a 1 a 1 + 1 a < 1, where is the norm in the space L [t, ), then equation (1.2) is exponentially stale. Proof. We follow the proof of Theorem 4.6.. From (4.19) and (4.2) we have ( a ) 1 x T Y (t, s)(s)ds ẋ T + 1 x T + f 1 ( a 1 + 1a a 1 a 1 + a 1 ) x T + C. The end of the proof is similar to the proof of Theorem 4.6.. Corollary 5.4. Suppose there exists t such that condition (5.1) holds, a 1 (t), t h(t) τ, t t. If 1 < 1, where is the norm in the space L [t, ), then equation (1.2) is exponentially stale. Acknowledgement. The first author was partially supported y Israeli Ministry of Asorption; the second author was partially supported y the NSERC Research Grant. The first and the third author were partially supported y the Israel Science Foundation (grant No. 828/7). References [1] N. V. Azelev and P. M. Simonov, Staility of Differential Equations with Aftereffect. Staility and Control: Theory, Methods and Applications, 2. Taylor & Francis, London, 23. [2] D. Bainov and A. Domoshnitsky, Staility of a second-order differential equation with retarded argument, Dynamics and Staility of Systems 9 (1994), no. 2, 145 151.

22 L. Berezansky, E. Braverman and A. Domoshnitsky [3] L. Berezansky and E. Braverman, On exponential staility of linear differential equations with several delays, J. Math. Anal. Appl. 324 (26), no. 2, 1336 1355. [4] L. Berezansky, E. Braverman and A. Domoshnitsky, Positive solutions and staility of a linear ordinary differential equation of the second order with damping term, arxiv:87.2227v1 [math.ds] (July 14, 28); to appear in Funct. Differ. Equ., 29. [5] T. A. Burton, Staility and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Mathematics in Science and Engineering 178, 1985. [6] T. A. Burton, Staility y Fixed Point Theory for Functional Differential Equations. Dover Pulications, Mineola, New York, 26. [7] T. A. Burton, Fixed points, staility, and exact linearization, Nonlinear Anal. 61 (25), 857 87. [8] T. A. Burton and T. Furumochi, Asymptotic ehavior of solutions of functional differential equations y fixed point theorems, Dynam. Systems Appl. 11 (22), no. 4, 499 519. [9] T. A. Burton and L. Hatvani, Asymptotic staility of second order ordinary, functional, and partial differential equations, J. Math. Anal. Appl. 176 (1993), 261 281. [1] B. Cahlon and D. Schmidt, Staility criteria for certain secondorder delay differential equations with mixed coefficients, J. Comput. Appl. Math. 17 (1994), no. 1, 79 12. [11] A. Domoshnitsky, Componentwise applicaility of Chaplygin s theorem to a system of linear differential equations with delay, Differential Equations 26 (1991), no. 1, 1254 1259. [12] L. N. Ere, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, Basel, 1995. [13] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.

Second Order Delay Equations 23 [14] V. B. Kolmanovsky and V. R. Nosov, Staility of Functional Differential Equations. Academic Press, 1986. [15] G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Argument, Marcel Dekker, New York, Basel, 1987. [16] N. Minorski, Nonlinear Oscillations, Van Nostrand, New York, 1962. [17] A. D. Myshkis, Linear Differential Equations with Retarded Argument, Nauka, Moscow, 1972 (in Russian). [18] S. B. Norkin, Differential Equations of the Second Order with Retarded Argument, Translation of Mathematical Monographs, AMS, V. 31, Providence, R.I., 1972. [19] B. Zhang, On the retarded Liénard equation, Proc. Amer. Math. Soc. 115 (1992), no.3, 779 785. Leonid Berezansky: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 8415, Israel, E-mail: rznsky@cs.gu.ac.il. Elena Braverman: Department of Mathematics and Statistics, University of Calgary, 25 University Drive N.W., Calgary, AB T2N1N4, Canada, E-mail: maelena@math.ucalgary.ca. Alexander Domoshnitsky: Department of Mathematics and Computer Science, The College of Judea and Samaria, Ariel 44837, Israel.