Instability for a class of second order delay differential equations

There exists a well-developed stability theory for many classes of functionaldifferential equations and only a few results on their instability. One of the aims of this paper is to fill this gap. Explicit tests for instability of linear delay differential equations of the second order with damping terms are obtained.


Introduction
The ordinary second order differential equation with a nonnegative coefficient p(t) is one of the classical objects in a qualitative theory of linear differential equations.In spite of the quite simple shape of this equation it appears to provide a variety of different oscillatory and asymptotic properties to its solutions.Asymptotic properties of the solutions have been studied in the classical monographs by R. Bellman [5], G. Sansone [26], P. Hartman [13] and I. T. Kiguradze and T. A. Chanturia [17].The latter states the current situation in the subject and at the same time encourages further investigation.Some of the most important results are various generalizations of oscillation and asymptotic results to equations with delayed argument.The second order delay differential equation (DDE) has its own history.Oscillation and asymptotic properties of this equation were considered in the well known monographs by A. D. Myshkis [22], S. B. Norkin [24], G. S. Ladde, V. Lakshmikantham and B. G. Zhang [21], I. Győri and G. Ladas [16], L. N. Erbe, Q. Kong and B. G. Zhang [12].
A delay equation is generally known to inherit oscillation properties of the corresponding ordinary equation.For example, it was proven by J. J. A. M. Brands [7] that for each bounded delay τ(t), equation (1.2) is oscillatory if and only if the corresponding ordinary differential equation (1.1) is oscillatory.Thus, the following question arises: are the asymptotic properties of an ordinary differential equation inherited by a delay equation?The answer is negative.A. D. Myshkis [22] proved that there exists an unbounded solution of the equation for each couple of positive constants p and ε.The problem of unboundedness of solutions in the case of non-constant coefficients was formulated in [22] as one that needs to be solved.The first results on this subject were obtained by A. Domoshnitsky in [10]: if there exists a positive constant ε such that p(t) > ε, τ(t) ≥ ε for t ≥ 0, then there exists an unbounded solution to equation (1.2).A necessary and sufficient condition for the boundedness of all solutions to equation (1.2) in the case of bounded and nondecreasing p(t) ≥ 0 is the following [10] ∞ τ(t)dt < ∞.
It was proved in [23] that if 0 ≤ τ(t) ≤ τ and tp(t) is an integrable function on the semi-axis then equation (1.2) is unstable.
In this paper we consider instability problems for DDE of the second order with damping terms.Various stability and instability results for autonomous DDE of the second order with damping terms were obtained in [9].Other stability and instability results for this class of equations can be found in [8,28] and also in monographs [18,19].All of these results were obtained by analysis of the roots of the characteristic equation.Concerning stability of second order equations, the recent papers [2] (equations with damping terms) and [11] (equations without damping term) can be noted.Results on oscillation and asymptotic properties for functional differential equations of the second order can be found in [3,15,20,25,29] (see also references therein).
There are many mathematical models described by delay differential equations of the second order with damping terms [19].In the case of instability, one of the stabilizing methods is used.Before an application of such methods it is necessary to check that this equation is unstable.So stability/instability investigations are an important part of the theory and applications of DDE.
To the best of our knowledge the problem of instability for non-autonomous DDE of the second order with damping terms, which is the aim of this paper, has not been studied before.For such equations, the method of analysis of the characteristic equation does not work and researchers need other methods for analysis.We consider several methods, in particular perturbations by asymptotically small terms, applications of properties of nonoscillatory equations, reducing a differential equation of the second order to a system of two equations of the first order and some other approaches.Some of these methods are new and can be used to study instability for other classes of functional differential equations.
We discuss several open problems on instability in the last section of this paper.

Preliminaries
We consider the following equation Together with (2.1) consider for each t 0 ≥ 0 an initial value problem We also assume that the following hypothesis holds The fundamental solution X(t, s) is the solution of (2.1) for t ≥ s ≥ 0 with the initial conditions ẋ It is evident that x 2 (t) = X(t, t 0 ).

Lemma 2.2 ([4]
).Let (a1)-(a3) hold.Then there exists one and only one solution of problem (2.2), (2.3) that can be presented in the form Definition 2.3.Equation (2.1) is uniformly exponentially stable, if there exist M > 0, µ > 0, such that the solution of problem (2.2), (2.3) with f ≡ 0 has the estimate where M and µ do not depend on t 0 .We say that the fundamental solution has an exponential estimate if there exist positive numbers It is known that for bounded delays (i.e.(a2) holds), the existence of an exponential estimate for the fundamental solution is equivalent to exponential stability and also it is equivalent to uniform asymptotic stability [14].Definition 2.4.We will say, that equation (2.1) is not exponentially stable (exponentially unstable), if the fundamental solution of the equation has no exponential estimate.
In particular equation (2.1) is not exponentially stable if it has a solution which does not tend to zero as t → ∞.
where g(t) = min k g k (t), g k (t) ≤ t, t ≥ t 0 .Then the fundamental solution Z(t, s) of the equation 3 Asymptotically small coefficients Definition 3.1.We will say that a locally measurable scalar function a is asymptotically small if lim t→∞ t+1 t |a(s)|ds = 0.A matrix-function A is asymptotically small if all elements of the matrix are asymptotically small.
In particular, a function a is asymptotically small if at least one of the following conditions hold: Consider the following equation which is a perturbation of equation (2.1).
Proof.For a vector linear delay differential equation it is known [14] that exponential stability is preserved under asymptotically small perturbations.Hence exponential instability of this equation is also preserved under asymptotically small perturbations.Equations (2.1) and (3.1) can be written in a vector form.Hence exponential instability of equation (2.1) implies exponential instability of equation (3.1).

Corollary 3.3. If there exists
is not exponentially stable.

Application of positivity of the fundamental solution
Theorem 4.1.Assume that the fundamental solution Z(t, s) of the equation is positive and b k (t) ≤ 0, t ≥ t 0 , k = 1, . . ., l.Then (2.1) is not exponentially stable.
Proof.Consider the following problem It is easy to see, that the solution y of this problem has the form Hence y(t) ≥ 0, t ≥ t 0 .But y(t) = Y(t, t 0 ) where Y(t, s) is the fundamental solution of (4.2).Hence the fundamental solution of (4.2) is nonnegative.Since constants are solutions of (4.2) then this equation is not exponentially stable.Equation (4.2) one can write in the form is not exponentially stable.
Proof.The fundamental solution of the ordinary differential equation is positive for any function a.

Equation with a negative damping term
In the previous section we obtained instability conditions of equation (2.1) where b k (t) ≤ 0, t ≥ t 0 .The asymptotic behavior of equation ( 2.1) may be very complicated when a k (t) ≤ 0, t ≥ t 0 .In [19], page 241, the following autonomous equation was studied We consider first the following equation where a(t) ≥ 0, t ≥ t 0 .
Theorem 5.2.Let there exist an absolutely continuous function u such that 0 ≤ u(t) ≤ a(t), t ≥ t 0 and for t ≥ t 0 .Then equation (5.1) is not exponentially stable.
Proof.Suppose that equation (5.1) is exponentially stable.By condition (a2) the delay functions t − h k (t) are bounded hence exponential stability is equivalent (for some N > 0, λ > 0) to the following inequality |X(t, s| ≤ Ne −λ(t−s) , t ≥ s ≥ t 0 , ( where X(t, s) is the fundamental function of equation (5.1).Consider the initial value problem where f is a bounded function on [t 0 , ∞) and greater than some positive number .Lemma 2.2 implies that for the solution x of the problem we have Inequality (5.3) and the boundedness of f imply that solution x of problem (5.4) is a bounded function on [t 0 , ∞).Suppose u(t), a(t) ≥ u(t) ≥ 0 is a solution of (5.2) and z(t) = ẋ(t) − u(t)x(t), z(t 0 ) = 0. Hence We have after substitution of x, ẋ, ẍ in (5.4): t 0 e h k (t) s u(τ)dτ z(s)ds + f (t).
It is evident that x is an unbounded function.We have a contradiction with the assumption, hence equation (5.1) is not exponentially stable.
Then equation (5.1) is not exponentially stable.
For the function f (t) we have Hence for the first integral in (5.8) we have We have a contradiction, since the left-hand side of (5.8) is a unbounded positive function, but the right-hand side is non-positive for t sufficiently large.Thus equation (5.5) is exponentially unstable.
6 Reducing to a system of two first order equations Consider the autonomous system where A k , k = 1, . . ., m are constant n × n matrices, τ k ≥ 0, k = 1, . . ., m.
To apply Lemma 6.1 consider the following autonomous equation where Then equation (6.2) can be written in the form form By Lemma 6.1 system (6.3) is unstable.Hence equation (6.2) is unstable.implies the instability of equation (7.1).
In this paper we obtain several instability conditions for linear delay differential equations of the second order assuming that all or part of the coefficients are asymptotically small and/or negative.We suppose that the results of the paper can be improved or extended for a more general class of equations.Bellow we present some of such problems for a future research.
2. Prove or disprove that there exists a 0 < 0 such that the condition ∑ m k=1 a k (t) < a 0 , t ≥ t 0 implies exponential instability of equation (2.1).

7 Discussion and topics for a future research
For instability we know very little even for the simple nonautonomous equationẋ(t) + b(t)x(h(t)) = 0. (7.1)We know that equation (7.1) is not exponentially stable if b is asymptotically small or b(t) ≤ 0, t ≥ t 0 .We do not know if the last condition can be replaced by an integral one:Also it is known that if b(t) ≥ b 0 > 0, t ≥ t 0 and