STABILITY OF LINEAR DIFFERENTIAL EQUATIONS WITH A DISTRIBUTED DELAY

We present some new stability results for 
the scalar linear equation with a distributed delay 
 
$\dot{x}(t) + \sum_{k=1}^m \int_{h_k(t)}^t x(s) 
d_s R_k(t,s) =0, h_k(t)\leq t,$ su$p_{t\geq 0}(t-h_k(t))<\infty,$ 
 
 
where the functions involved in the equation are not required to be continuous. 
 
The results are applied to integro-differential equations, 
equations with several concentrated delays and equations of a mixed type.

1. Introduction.It is usually believed that equations with a distributed delay provide a more realistic description for models of population dynamics and mathematical biology in general, see, for example, [1].If a maturation delay is incorporated in the equation, then the maturation time is, generally, not constant, but is distributed around its expectancy value.The same is valid for the digestion delay, as well as for the latency period of most infectious diseases.
Historically, equations with a distributed delay were studied even before relevant models with concentrated delays appeared.For example, Volterra considered the logistic equation with a distributed delay in 1926 [2], before the Hutchinson's equation (the logistic equation with a concentrated delay) was introduced in 1948 [3].
To the best of our knowledge the first systematic study of equations with a distributed delay can be found in the monograph of Myshkis [4], the results obtained by 1993 are summarized in the book of Kuang [1].Presently equations with a distributed delay are intensively studied.However, most of the obtained results are not relevant for non-autonomous models and do not involve equations with a concentrated delay as a special case.
Let us notice that in the present paper we study delay equations in the most general framework from the following points of view.
1. Distributed delays allow us, for an appropriate choice of the distribution functions, to consider integro-differential equations, equations with several variable concentrated delays and equations with both delayed and integral terms.All functions involved in the equation are time dependent.2. Coefficients are not necessarily continuous and the equality is understood in the sense "almost everywhere".For integral terms of equations, this corresponds to measurable locally essentially bounded (not necessarily continuous) kernels of integrals.
Equation with a distributed delay in the same setting were studied, for example, in [5,6].

2.
Preliminaries.We consider a linear scalar differential equation with a distributed delay for t > t 0 ≥ 0.Here we assume that for each t the memory is finite and We consider equation (2.1) for a fixed t 0 ≥ 0 with the initial condition under the following assumptions: (a1) R k : [0, ∞) × IR → IR are functions such that R k (t, •) are left continuous functions of bounded variation for any t; R k (•, s) are locally integrable for any s, R k (t, h k (t)) = 0 and R k (t, s) are constant for s > t (and coincide with the right limits R k (t, t + )); the functions is the variation of the function f on the interval [a, b]; in the case of the right limit b + the variation will also involve the jump as for any ε > 0. For example, if m = 1 and R 1 (t, s) = a(t)χ (t,∞) (s), where χ I is the characteristic function of interval I, then we obtain the ordinary differential equation ẋ(t) + a(t)x(t) = 0.
For some particular choices of R k (t, s) we obtain the following equations as special cases of (2.1) x(s where the kernel and the lower bound h(t) are and Here Then the relevant functions R k (t, s) for (2.5)-(2.7)satisfy (a1); we also assume that (a2) holds for h(t), h k (t), k = 1, • • • , m, and g(t).Now let us proceed to the initial function ϕ.This function should satisfy such conditions that the integral in the left hand side of (2.1) exists almost everywhere.In particular, if R k (t, •) is absolutely continuous for any t (which allows us to write (2.1) as an integro-differential equation), then ϕ can be chosen as a Lebesgue measurable essentially bounded function.If R k (t, •) is a combination of step functions (which corresponds to an equation with concentrated delays) then ϕ should be a Borel measurable bounded function.For any choice of R k the integral exists if ϕ is bounded and continuous.Thus, we assume that (a3) ϕ : (−∞, 0] → IR is a bounded continuous function. Everywhere below we will assume that for all equations and initial conditions hypotheses (a1)-(a3) are satisfied.Definition 2.1.An absolutely continuous function x : IR → IR is called a solution of the problem (2.1), (2.3) if it satisfies equation (2.1) for almost all t ∈ [t 0 , ∞) and conditions (2.3) for t ≤ t 0 .
In addition to linear equation (2.1) we will also consider the non-homogeneous equation where f (t) is a Lebesgue measurable locally essentially bounded function.
Definition 2.2.For each s ≥ t 0 and t ≥ s the solution X(t, s) of the problem is called the fundamental function of equation (2.8).Here X(t, s) = 0, 0 ≤ t < s.
Definition 2.4.Equation (2.1) is (uniformly) exponentially stable, if there exist K > 0, λ > 0, such that the fundamental function X(t, s) defined by (2.9) has the estimate Let us introduce some functional spaces on a halfline.Denote by L ∞ [t 0 , ∞) the space of all essentially bounded on [t 0 , ∞) functions with the essential supremum norm Together with (2.8) we consider an auxiliary equation where for parameters of (2.12) conditions (a1)-(a3) hold.
Denote by X 0 (t, s) the fundamental function of (2.12) and consider the following linear equations and linear operators: , where I is the identity operator.Then equation (2.1) is exponentially stable.
We will also apply in future the Mean Value Theorem for equations with a distributed delay [6].We will use this theorem in the following form.Let us introduce the minimal and the maximal delays (2.16) In particular, if (2.18) for any g(t), h(t) ≤ g(t) ≤ H(t) enjoys anyone of the following properties -all solutions of (2.18) are oscillatory; -there exists a nonoscillatory solution of (2.18); -the zero solution of (2.18) is stable (exponentially stable); then (2.1) has the same property.
Proof.Without loss of generality we can assume t 0 = 0. Let x be a solution of (2.1) such that x(t) = 0, t ≤ 0. Then x(s)d s R k (t, s).
After substituting ẋ from (2.1) we have which means that any solution of (2.1) with the zero initial conditions also satisfies In addition to the operator L defined in (3.5) let us define the following linear operators Let us demonstrate that (3.5) is exponentially stable.Obviously by (3.3) the auxiliary equation with operator L 0 as in (3.6) is exponentially stable.
We have 6 implies that (2.1) is exponentially stable, which completes the proof.Equation (2.7) contains as partial case equation (2.5) with concentrated delays, integro-differential equation (2.6) and all kinds of mixed type equations.Theorem 3.1 gives 2 4 − 1 = 15 different stability conditions for this equation.In the following corollary we obtain 8 such conditions.Here we omit all the cases where the nondelay term is not involved in the chosen set I. For the nondelay term the first integral term in the left hand side of (3.4) vanishes, so excluding this term from set I will never lead to a sharper stability condition.Similar results for equation (2.5) were obtained in [11].
Corollary 1. Suppose there exist β > 0, γ ∈ (0, 1) and t 1 such that for t ≥ t 1 at least one of the following conditions holds: Then equation (2.7) is exponentially stable.
Since in (3.4) R k (t, s) in the terms from the second up to m + 1 are as in (2.8),  |K(τ, s)| ds r 0 > 0, K(t, s) is always nonnegative, R(t, •) is nondecreasing, and both integral terms are normalized:

Inequality (3.3) becomes b(t) +
Then the following result can be immediately deduced from Corollary 1.
Corollary 4. Suppose that at least one of the following conditions holds: Example 2. Let us compare our results and sufficient stability conditions obtained in [12] for the equation in the case when f (x) = 2 h 2 (h − x) for 0 < t < h and f (x) = 0, otherwise, β > α > 0. Thus equation (3.11) can be rewritten as The expectation of f is one can also see that the skewness B(f ) > 0, so f is skewed to the left.Thus, applying Theorem 4.0.5 of [12] we obtain that for In particular, for α = 1/36, β = 1 this gives stability for h < 6, while the estimate of [12] gives h < γ where γ ≈ 4.26.Overall, the two estimates are independent; for example, if β = 2α, we have αh < 3π1.5 c √ 3 ≈ 3.59 and αh 3 < 6, respectively, so one of the estimates can be better than the other depending on α.

4.
Mean value theorem and stability conditions.Next, we consider another approach to obtain stability results for equation (2.1).We will formulate an auxiliary result for an equation with one constant delay.Consider the function where U σ (t) is the solution of the following initial value problem for the autonomous delay equation vavava In [13,14,15] properties of ω(σ) were obtained and its values were tabulated.In particular, it was shown that the constant can be approximately computed where U is defined in (4.1).Then equation (4.3) is exponentially stable.
Let us note that the result of Lemma 4.1 was also obtained in [15] for equation (4.3) with a continuous function a(t) and a piecewise continuous function h(t).Let us compare Corollary 5 with known stability tests for equations with a distributed delay.Stability of equations with a distributed delay was extensively studied in [4]; Theorem 58 in [4] claims that under certain conditions (in particular, these assumptions involve the continuity of the maximal delay, as well as coefficients in the case of several concentrated delays) the inequality sup t≥0 m k=1 a k (t) ≥ β, and (3.4) can be rewritten as m k=1 |a k (t)| t h k (t)   |b(τ )| + m j=1 |a j (τ )| + τ h(τ )