New Stability Conditions for Linear Differential Equations with Several Delays

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m a_k(t)x(h_k(t))=0 $$ with measurable delays and coefficients. These results are compared to known stability tests.


Introduction
In this paper we continue the study of stability properties for the linear differential equation with several delays and an arbitrary number of positive and negative coefficientṡ which was begun in [1]- [3]. Equation (1) and its special cases were intensively studied, for example, in [4]- [21]. In [2] we gave a review of stability tests obtained in these papers.
In almost all papers on stability of delay differential equations coefficients and delays are assumed to be continuous, which is essentially used in the proofs of main results. In real world problems, for example, in biological and ecological models with seasonal fluctuations of parameters and in economical models with investments, parameters of differential equations are not necessarily continuous.
There are also some mathematical reasons to consider differential equations without the assumption that parameters are continuous functions. One of the main methods to investigate impulsive differential equations is their reduction to a non-impulsive differential equation with discontinuous coefficients. Similarly, difference equations can sometimes be reduced to the similar problems for delay differential equations with discontinuous piecewise constant delays.
In paper [1] some problems for differential equations with several delays were reduced to similar problems for equations with one delay which generally is not continuous.
One of the purposes of this paper is to extend and partially improve most popular stability results for linear delay equations with continuous coefficients and delays to equations with measurable parameters.
Another purpose is to generalize some results of [1,2,3]. In these papers, the sum of coefficients was supposed to be separated from zero and delays were assumed to be bounded. So the results of these papers are not applicable, for example, to the following equationṡ In most results of the present paper these restrictions are omitted, so we can consider all the equations mentioned above. Besides, necessary stability conditions (probably for the first time) are obtained for equation (1) with nonnegative coefficients and bounded delays. In particular, if this equation is exponentially stable then the ordinary differential equatioṅ is also exponentially stable.

Preliminaries
We consider the scalar linear equation with several delays (1) for t ≥ t 0 with the initial conditions (for any t 0 ≥ 0) and under the following assumptions: (a1) a k (t) are Lebesgue measurable essentially bounded on [0, ∞) functions; (a2) h k (t) are Lebesgue measurable functions, We assume conditions (a1)-(a3) hold for all equations throughout the paper. Definition. A locally absolutely continuous function x : R → R is called a solution of the problem (1), (2) if it satisfies the equation (1) for almost all t ∈ [t 0 , ∞) and the equalities (2) for t ≤ t 0 .
Below we present a solution representation formula for the nonhomogeneous equation with locally Lebesgue integrable right-hand side f (t): (3)

Definition.
A solution X(t, s) of the probleṁ is called the fundamental function of (1).
In particular, Eq. (1) is asymptotically stable if the fundamental function is uniformly bounded: |X(t, s)| ≤ K, t ≥ s ≥ 0 and all solutions tend to zero as t → ∞.
We apply in this paper only these two conditions of asymptotic stability.
Definition. Eq. (1) is (uniformly) exponentially stable, if there exist M > 0, µ > 0 such that the solution of problem (1)-(2) has the estimate where M and µ do not depend on t 0 .
Definition. The fundamental function X(t, s) of (1) has an exponential estimation if there exist K > 0, λ > 0 such that For the linear equation (1) with bounded delays the last two definitions are equivalent. For unbounded delays estimation (5) implies asymptotic stability of (1).
Under our assumptions the exponential stability does not depend on values of equation parameters on any finite interval.
or there exists λ > 0, such that where is the fundamental function of equation (1).
lim sup and there exists r(t) ≤ t such that for sufficiently large t then equation (1) is exponentially stable.
The following lemma is a consequence of Corollary 2 [26] obtained for impulsive delay differential equations.

Lemma 5 Suppose for equation (1) condition (9) holds and this equation is exponentially
then the equationẋ is exponentially stable.
The following elementary result will be used in the paper.

Lemma 6
The ordinary differential equatioṅ is exponentially stable if and only if there exists R > 0 such that lim inf t→∞ t+R t a(s)ds > 0.
The following example illustrates that a stronger than (15) sufficient condition is not necessary for the exponential stability of the ordinary differential equation (14).
Then lim inf in (16) equals zero, but |X(t, s)| < e e −0.5(t−s) , so the equation is exponentially stable. Moreover, if we consider lim inf in (16) under the condition t − s ≥ R, then it is still zero for any R ≤ 1.
For this function condition (10) has the form The latter inequality follows from (17). The reference to Lemma 3 completes the proof. ⊓ ⊔ Then equation (1) is exponentially stable.
The following theorem contains stability conditions for equations with unbounded delays. We also omit condition (8) in Lemma 7.
Then equation (1) is asymptotically stable. If in addition there exists R > 0 such that then the fundamental function of equation (1) has an exponential estimation. If condition (9) also holds then (1) is exponentially stable. (1) can be rewritten in the formẏ b k (s) = 1 and lim sup s→∞ (s − l k (s)) < ∞, then Lemma 7 can be applied to equation (21). We have By Lemma 7 equation (21) is exponentially stable. Due to the first equality in (19) (21) is exponentially stable, thus the fundamental function Y (u, v) of equation (21) has an exponential estimation with where X(t, s) is the fundamental function of (1), then (22) yields Hence |X(t, s)| ≤ K, t ≥ s ≥ 0, which together with lim t→∞ x(t) = 0 yields that equation (1) is asymptotically stable.
Suppose now that (20) holds. Without loss of generality we can assume that for some Thus, condition (20) implies the exponential estimate for X(t, s). The last statement of the theorem is evident.
then the fundamental function of equation (1) has an exponential estimation. If also (9) holds then equation (1) is exponentially stable.
Then the following equation is asymptotically stablė If in addition there exists R > 0 such that lim inf t→∞ t+R t (a(τ ) + b(τ ))dτ > 0 then the fundamental function of (27) has an exponential estimation. If also lim sup t→∞ (t − h(t)) < ∞ then equation (27) is exponentially stable.
In the following theorem we will omit condition m k=1 a k (t) = 0 a.e. of Theorem 1.
Theorem 2 Suppose a k (t) ≥ 0, condition (18) and the first inequality in (19) hold. Then equation (1) is asymptotically stable. If in addition (20) holds then the fundamental function of equation (1) has an exponential estimation.

Theorem 3 Suppose there exists a set of indices
If the fundamental function X 0 (t, s) of equation (12) is eventually positive then all solutions of equation (1) (1) has an exponential estimation. If condition (9) also holds then (1) is exponentially stable.
Proof. Without loss of generality we can assume X 0 (t, s) > 0, t ≥ s ≥ 0. Rewrite equation Suppose first that . Denote by Y 0 (u, v) the fundamental function of the equatioṅ y(s) + k∈I b k (s)y(l k (s)) = 0.
We have Let us check that other conditions of Lemma 4 hold for equation (40). Since k∈I b k (s) = 1 then condition (11) is satisfied. In addition, By Lemma 4 equation (40)   where E = {t ≥ 0, a(t) = 0}. Then the equatioṅ is asymptotically stable. If in addition (15) holds then the fundamental function of (41) has an exponential estimation. If also lim sup t→∞ (t − g k (t)) < ∞ then (41) is exponentially stable.  (1) is not asymptotically stable.

Theorem 4 Suppose
Proof. For the fundamental function of (1) we have the following estimation Then by solution representation formula (4) for any solution x(t) of (1) we have where ||ϕ|| = max t<0 |ϕ(t)|. Then x(t) is a bounded function.
, thus X(t, t 0 ) does not tend to zero, so (1) is not asymptotically stable. ⊓ ⊔ Theorems 3 and 4 imply the following results.  (1) is asymptotically stable, then the ordinary differential equationẋ is also asymptotically stable. If in addition (9) holds and (1) is exponentially stable, then (42) is also exponentially stable.
Proof. The solution of (42), with the initial condition x(t 0 ) = x 0 , can be presented as a k (s) ds , so (42) is asymptotically stable, as far as and is exponentially stable if (20) holds (see Lemma 6). If (43) does not hold, then by Theorem 4 equation (1) is not asymptotically stable.
Corollary 8 Suppose a k (t) ≥ 0 and the fundamental function of equation (1)

Discussion and Examples
In paper [2] we gave a review of known stability tests for the linear equation (1). In this part we will compare the new results obtained in this paper with known stability conditions. First let us compare the results of the present paper with our papers [1]- [3]. In all these three papers we apply the same method based on Bohl-Perron type theorems and comparison with known exponentially stable equations.
In [1]- [3] we considered exponential stability only. Here we also give explicit conditions for asymptotic stability. For this type of stability, we omit the requirement that the delays are bounded and the sum of the coefficients is separated from zero. We also present some new stability tests, based on the results obtained in [3].
Compare now the results of the paper with some other known results [5,6,7,9,10,22]. First of all we replace the constant 3 [28]. We present here 3 statements which cover most of known stability tests for this equation.

Then equation (35) is asymptotically stable.
Example 2. Consider the equatioṅ where h(t) is an arbitrary measurable function such that t − h(t) ≤ π and α > 0. This equation has the form (35) where a(t) = α(| sin t| − sin t  (1) with several delays. The following two statements are well known for this equation.
Statement 4 [6]. Suppose a k (t) ≥ 0, h k (t) ≤ t are continuous functions and Then all solutions of (1) are bounded and 1 in the right hand side of (47) is the best possible constant.
If m k=1 a k (t) > 0 and the strict inequality in (47) is valid then all solutions of (1) tend to zero as t → ∞. If a k (t) are constants then in (47) the number 1 can be replaced by 3/2.
Then any solution of (1) tends to a constant as t → ∞.

If in addition
∞ 0 m k=1 a k (s)ds = ∞, then all solutions of (1) tend to zero as t → ∞.
In Corollary 6 we obtained a similar result without the assumption that the parameters of the equation are continuous functions and the delays are bounded.
Example 4. Consider the equatioṅ If α < 1 then by Corollary 6 all solutions of equation (51) tend to zero. The delay is unbounded, thus Statement 6 fails for this equation.
In [30] the authors considered a delay autonomous equation with linear and nonlinear parts, where the differential equation with the linear part only has a positive fundamental function and the linear part dominates over the nonlinear one. They generalized the early result of Györi [31] and some results of [32].
In Theorem 4 we obtained a similar result for a linear nonautonomous equation without the assumption that coefficients and delays are continuous.
We conclude this paper with some open problems. Note that all known proofs with the constant 3 2 apply methods which are not applicable for equations with measurable parameters.
Prove or disprove that for any b k (t), 0 ≤ b k (t) ≤ a k (t), where lim inf   (1) is also exponentially stable.
Obtain similar result for the asymptotic stability.
The solution of the following problems would improve Theorems 1 and 5, respectively.
Open Problem 4. Suppose (1) is exponentially stable. Prove or disprove that the ordinary differential equation (42) is also exponentially (asymptotically) stable, without restrictions on the signs of coefficients a k (t) ≥ 0, as in Theorem 5.