ASYMPTOTIC STABILITY IN DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

In this paper we consider a functional differential equation of the form (1.1) x = F (t, x, t 0 C(at − s)x(s)ds) where a is a constant satisfying 0 < a < ∞. Thus, the integral represents the memory of past positions of the solution x. We make the assumption that ∞ 0 |C(t)|dt < ∞ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of a. Very different properties of solutions emerge as we vary a and we are interested in developing an approach which handles them in a unified way. Our study is based on a Liapunov functional V (t, x t) and wedges W i satisfying (1.2) W 1 (|x(t)|) ≤ V (t, x t) ≤ W 2 (|x(t)|) + t 0 D(t, s)W 3 (|x(s)|)ds and (1.3) V (t, x t) ≤ −b(t)W 4 (|x(t)|). The goal is to formulate a set of conditions on these relations which will imply asymp-totic stability of the zero solution of (1.1). In this context there are four main challenges that a theorem needs to meet: (i) x may be unbounded for x bounded. (ii) The necessarily fading memory must be utilized. (iii) W 4 may be unrelated to W 3. (iv) b(t) may be near zero some of the time. Volumes have been written about (i) through (iv) and we will give only a brief summary , together with references, so that the interested reader may trace them down. It was recognized early in the theory of Liapunov's direct method that if the zero solution of an ordinary differential equation was stable, but not uniformly stable, then the existence of a positive definite Liapunov function with a negative definite derivative was insufficient to conclude that bounded solutions converge to zero. The difficulty is that there can be an annular ring around the point x = 0 through which a solution can pass infinitely often, moving so quickly that integration of V does not send V to minus infinity effecting a desired contradiction. The mechanics are discussed in Burton [1, p. 161], for example, while a recent paper of Hatvani [7] details many of the advances. But the first significant contribution was that of Marachkov [9] who showed that it was sufficient to ask that x be bounded for x bounded to ensure that solutions …

In this paper we consider a functional differential equation of the form (1.1) x = F (t, x, where a is a constant satisfying 0 < a < ∞.Thus, the integral represents the memory of past positions of the solution x.We make the assumption that ∞ 0 |C(t)|dt < ∞ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of a. Very different properties of solutions emerge as we vary a and we are interested in developing an approach which handles them in a unified way.
Our study is based on a Liapunov functional V (t, x t ) and wedges W i satisfying The goal is to formulate a set of conditions on these relations which will imply asymptotic stability of the zero solution of (1.1).In this context there are four main challenges that a theorem needs to meet: ( i) x may be unbounded for x bounded.
( ii) The necessarily fading memory must be utilized.
( iii) W 4 may be unrelated to W 3 .
( iv) b(t) may be near zero some of the time.
Volumes have been written about (i) through (iv) and we will give only a brief summary, together with references, so that the interested reader may trace them down.
It was recognized early in the theory of Liapunov's direct method that if the zero solution of an ordinary differential equation was stable, but not uniformly stable, then the existence of a positive definite Liapunov function with a negative definite derivative was insufficient to conclude that bounded solutions converge to zero.The difficulty is that there can be an annular ring around the point x = 0 through which a solution can pass infinitely often, moving so quickly that integration of V does not send V to minus infinity effecting a desired contradiction.The mechanics are discussed in Burton [1, p. 161], for example, while a recent paper of Hatvani [7] details many of the advances.
But the first significant contribution was that of Marachkov [9] who showed that it was sufficient to ask that x be bounded for x bounded to ensure that solutions will tend to zero.That assumption became the basic one both for ordinary and functional differential equations.Much work has been done to relax that condition and in two recent papers Burton and Makay [4,5] have shown that x could be unbounded of order t ln t and we could still conclude asymptotic stability.
It is easy to understand that the boundedness requirement on x is highly objectionable because it is clear that in many systems the unboundedness of x actually drives the solution to zero faster than would be possible with a bounded x .Our work here on asymptotic stability completely avoids mention of boundedness of x .
G. Seifert [11] seems to have been the first to offer an example which shows clearly the necessity of a fading memory for asymptotic stability.And most of our work here EJQTDE, 1999 No. 13, p. 2 will focus on the fading memory.The reader may also consult a recent paper by Huang and Zhang [8] for further results on asymptotic stability and fading memory.
To show the ideas most clearly we will focus on the scalar equation under the assumption that so that there results the relation Here, h and b can be unbounded.
When C ∈ L 1 [0, ∞), a is a positive constant whose magnitude will dictate the manner in which the memory fades.The integral in (1.5) can be thought of as a weighted average of x.The future position of x depends on this memory of past x.
For a = 1, we have the well-studied convolution case, and if |C| decreases monotonically to zero as t → ∞, then: a ) at s = t we have C(0)x(t) so that x(t) is weighted maximally.This seems prudent since the future position should be most influenced by the present position.b ) at s = 0 we have C(t)x(0) so that x(0) is weighted mimimally; again, this seems right since things that happened long ago should have the least effect.
c ) It is crucial to notice that, in contrast to the case of a > 1, the memory, that is the integral, t 0 C(at − s)ds, never fades away entirely; it tends to a constant as t → ∞ for C ∈ L 1 [0, ∞).This means that, on one hand, we could have asymptotic stability even when h and b are zero, if the stability comes from the integal.On the other EJQTDE, 1999 No. 13, p. 3 hand, if the stability does not come from the integral, then the h must be as large as a positive constant for uniform asymptotic stability.
Very different behavior emerges if a > 1.In this case, we have strongly fading memory; the integral t 0 C(at − s)ds= t (a−1)t C(v)dv tends to zero as t → ∞ for C ∈ L 1 [0, ∞).This will allow us to prove asymptotic stability when h(t) and b(t) are unbounded, and at the same time, when they are close to zero for extended periods of time.
The fact that W 4 may be unrelated to W 3 , Item (iii) above, has played a major role in the study of stability and boundedness.Most motivating examples by investigators deal with problems similar to (1.4), but with b(t) = 0.In this case, asymptotic stability comes from the term h(t) when (1.6) is strengthened by adding a positive constant α to the right hand side, resulting in V (t, x t ) ≤ −αx 2 .This makes the derivative of V closely related to the memory in V and has resulted in simple proofs of asymptotic stability without any mention of boundedness of x .Details of this type of work and references may be found in Burton [2, p. 311] for systems with unbounded delay, or Burton and Hatvani [3], Busenberg and Cooke [6], and Hatvani [7] for systems with bounded delay.
But when (1.6) is not strengthened and we have (1.7) as stated, then different ideas are needed.We use the fading memory, together with integral inequalities, to avoid asking that close relationship between V and the memory in order to conclude asymptotic stability without reference to boundedness of x .
In the next section we present a sample of representative theorems for the particular example (1.4).Much of what is done here can also be obtained when b(t)x 3 is replaced by b(t)x n where n is the quotient of odd positive integers.In that case the analysis of asymptotic stability and uniform asymptotic stability would be modified in the inequalities using (1/p) + (1/q) = 1, while with our choice we always have p = q = 2.
The case of 0 < a < 1 is also very interesting from the point of view of fading memory.
It will require that h(t) grow in the relation (1.6) in order to establish stability.But we then get more than simple stability; even when V is zero, we find that we almost have asymptotic stability.
The kernel in the integral term, C(at − s), determines the kind of memory of the equation.
For example, let a > 1; if C is positive and decreases monotonically to zero, then: a ) at s = 0, C(at) is the weight on x(0), a small amount for large t; b ) at s = t, the weight on x(t) is C((a − 1)t) which is larger than at s = 0; (Compare with the convolution discussion above.)c ) but for bounded x, the memory will gradually fade to zero.
The constant a is the index of fading memory.
Our first result contrasts (2.1) with the convolution case in that the integral is essentially a dissipative term.Even with h = 0, that integral is eventually dominated by b(t)x n for b 0 > 0 and n ≥ 1. THEOREM for some α.Thus, for t ≥ T we have V ≤ −β < 0 for some β whenever |x(t)| ≥ A − δ for some δ > 0. We can not have |x(t)| ≥ A − δ for all large t or V → −∞.Thus, there is a t 1 > T with |x(t 1 )| < A − δ.But there is a t 2 > t 1 with |x(t 2 )| = A − δ and |x(t)| < A − δ on [t 1 , t 2 ).This will give a contradiction just as before.Hence, A does not exist and the solution tends to zero.This proves Theorem 1.
In the convolution case, if C is taken as a general perturbation, it will require that h(t) be greater than a positive constant in order for −h(t)x to dominate the integral term.Here, we see that even if x = −h(t)x is only stable, that can be sufficient to make (2.1) stable.THEOREM 2. Let (2.2) hold and let Then the zero solution of (2.1) is stable.
The next result illustrates how the fading memory is used to show asymptotic stability and relates the derivative of V to the kernel of the Liapunov functional.It is here that the conditions on C change if the exponent in b(t)x n changes, as mentioned in the introduction.
THEOREM 3. Let (2.2) hold with b(t) > 0, ∞ 0 b(s)ds = ∞, and let (2.3) hold.Suppose also that there is a B such that for each T > 0 if t > T , then Then the zero solution of (2.1) is asymptotically stable.
Proof.We have already shown that the solution is stable and we have defined a Liapunov functional in (2.4) with negative derivative in (2.5).

Let x(t) be a solution with |x
Now we want to prove that V (t) → 0. If it does not, then as above there is a µ> 0 If we can find a t f such that for t > t f we have Let us find a t f such that for t > t f condition (2.7) is satisfied.
Without loss of generality, let µ < 1.Let B be that of condition (2.6) Since |x(t)| < 1 , the first integral,(2.9),satisfies Also, since C(t) ∈ L 1 [0, ∞), we can pick a t f (which depends on the fixed T ) such that In the second integral, (2.10), we have for .
EJQTDE, 1999 No. 13, p. 9 Thus, by (2.6) and (2.8) Putting together the two integrals we see that for As a > 1 the proof is complete.

QED.
Remark.Note that when b(t) ≥ b 0 > 0 and when (2.3) holds, then the growth condition on C(t), (2.6), is all we need for asymptotic stability.
Proof.For any t 0 we have Let > 0 be given and take a δ < such that Assume that we have |φ| < δ on [0, t 0 ] and let x(t) be the solution x(t, t 0 , φ). Then, Thus for any |φ| < δ, and any t 0 the solution x(t, t 0 , φ) satisfies |x(t)| < .QED REMARK: We can use the same Liapunov functional for all values of a and the proof of uniform stability always proceeds the same way.Here, for a > 1 the function EJQTDE, 1999 No. 13, p. 10 ∞ t |C(u)|du is not quite required to be integrable to infinity.Our condition holds, for example, if this function is 1/(t + 1).But for a = 1 integrability to infinity is the exact requirement of this method.When 0 < a < 1 we can not find a condition at all for uniform stability.Uniform stability means that the behavior of solutions with similar initial functions, but different starting times, is much the same.And this is consistent with the rapidly fading memory with a > 1.But for a < 1 more and more of the memory is retained.In fact, upon change of variable, the limits on the memory become (a − 1)t to at.If C is even and if |C(t)| decreases monotonically as t increases, then that interval always includes the largest part of C. Thus, there is increasingly more weight on the initial function so that different behavior must be expected as t 0 increases.
Proof.The zero solution is uniformly stable by the previous theorem.
Let us prove uniform asympototic stability.For = 1, find the δ of uniform stability.
Let γ > 0 be given.We will find In the following, x(t) will denote any solution x(t, t 0 , φ) described above and V (t) will denote V (t, x t ).Since V (t) ≤ 0, if we find a t f such that (2.12) for all t ≥ t f .We will now find a T so that for any such solution, there will be a Also, since C(t) ∈ L 1 [0, ∞), we can pick a T 3 (which depends on the fixed T 2 ) such that for t > T 3 then (2.16) By (2.16), the first integral, (2.14), satisfies In the second integral, (2.15), because of (2.13), we have for Notice that we have It is clear that x 4 (t) ∈ L 1 .Furthermore, there is a T 4 > 0 such that in all intervals of length T 4 there is at least one point t i such that |x(t i )| 2 < γ 2 /3.( T 4 depends on the γ but does not depend on the particular x(t).)Indeed, if for all t > t 1 ≥ t 0 , |x(t)| 2 ≥ γ 2 /3 then V (t 1 ) ≤ V (t 0 ) and Let us define T = T 3 + T 4 .Then for t > t 0 + T 3 the two integrals are small, and there

QED.
Example of a function C(u) that satisfies conditions (2.6) and (2.11) In the following we will neglect constants of integration.
We have which is bounded for all t > 0 as long as λ > 3/2, and so condition (2.6) in Theorem 3 is satisfied.EJQTDE, 1999 No. 13, p. 13 Also we have For λ > 2, by making t large, the expression can be made as small as we want.
Therefore the integral conditions in Theorems 4 and 5 are satisfied.
Let us take b(t) = 1 1 + t > 0. Then The integral is bounded for λ > 2. Indeed, and the primitive of the first term is, for 0 ≤ s ≤ t,