# New Stability Conditions for Linear Differential Equations with Several Delays

Authors: Leonid Berezansky; Elena Braverman
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equationx˙(t)+∑k=1mak(t)x(hk(t))=0 with measurable delays and coefficients. These results are compared to known stability tests.


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## Body

## 1. Introduction

In this paper, we continue the study of stability properties for the scalar linear differential equation with several delays and an arbitrary number of positive and negative coefficients(1.1)ẋ(t)+∑k=1mak(t)x(hk(t))=0,t≥t0,
which was begun in [1–3]. Equation (1.1) 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 nonimpulsive 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 [1–3]. 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 equations: (1.2)ẋ(t)+|sint|x(t-τ)=0,ẋ(t)+α(|sint|-sint)x(t-τ)=0,ẋ(t)+1tx(t)+αtx(t2)=0.
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 (1.1) with nonnegative coefficients and bounded delays. In particular, if this equation is exponentially stable then the ordinary differential equation (1.3)ẋ(t)+∑k=1mak(t)x(t)=0
is also exponentially stable.


## 2. Preliminaries

We consider the scalar linear equation with several delays (1.1) for t≥t0 with the initial conditions (for any t0≥0)(2.1)x(t)=φ(t),t<t0,x(t0)=x0,
and under the following assumptions:(a1)
ak(t) are Lebesgue measurable essentially bounded on [0,∞) functions;(a2)
hk(t) are Lebesgue measurable functions,
(2.2)hk(t)≤t,limsup⁡t→∞hk(t)=∞;(a3)
φ:(-∞,t0)→R is a Borel measurable bounded function.We assume conditions (a1)–(a3) hold for all equations throughout the paper.Definition 2.1.
A locally absolutely continuous fort≥t0 function x:R→R is called a solution of problem (1.1), (2.1) if it satisfies (1.1) for almost all t∈[t0,∞) and the equalities (2.1) for t≤t0.Below we present a solution representation formula for the nonhomogeneous equation with locally Lebesgue integrable right-hand sidef(t):(2.3)ẋ(t)+∑k=1mak(t)x(hk(t))=f(t),t≥t0.Definition 2.2.
A solutionX(t,s) of the problem
(2.4)ẋ(t)+∑k=1mak(t)x(hk(t))=0,t≥s≥0,x(t)=0,t<s,x(s)=1,
is called the fundamental function of (1.1).Lemma 2.3 (see [22, 23]).
Suppose conditions (a1)–(a3) hold. Then the solution of (2.3), (2.1) has the following form
(2.5)x(t)=X(t,t0)x0-∫t0tX(t,s)∑k=1mak(s)φ(hk(s))ds+∫t0tX(t,s)f(s)ds,
where φ(t)=0, t≥t0.Definition 2.4 (see [22]).
Equation (1.1) is stable if for any initial point t0 and number ɛ>0 there exists δ>0 such that the inequality supt<t0|φ(t)|+|x(t0)|<δ implies |x(t)|<ɛ, t≥t0, for the solution of problem (1.1), (2.1).Equation (1.1) is asymptotically stable if it is stable and all solutions of (1.1)-(2.1) for any initial point t0 tend to zero as t→∞.In particular, (1.1) 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 2.5.
Equation (1.1) is (uniformly) exponentially stable, if there exist M>0, μ>0 such that the solution of problem (1.1), (2.1) has the estimate
(2.6)|x(t)|≤Me-μ(t-t0)(|x(t0)|+sup⁡t<t0|φ(t)|),t≥t0,
where M and μ do not depend on t0.Definition 2.6.
The fundamental functionX(t,s) of (1.1) has an exponential estimation if there exist K>0, λ>0 such that
(2.7)|X(t,s)|≤Ke-λ(t-s),t≥s≥0.For the linear (1.1) with bounded delays the last two definitions are equivalent. For unbounded delays estimation (2.7) implies asymptotic stability of (1.1).Under our assumptions the exponential stability does not depend on values of equation parameters on any finite interval.Lemma 2.7 (see [24, 25]).
Supposeak(t)≥0. If
(2.8)∫max⁡{h(t),t0}t∑i=1mai(s)ds≤1e,h(t)=min⁡k{hk(t)},t≥t0,
or there exists λ>0, such that
(2.9)λ≥∑k=1mAkeλσk,
where
(2.10)0≤ak(t)≤Ak,t-hk(t)≤σk,t≥t0,
then X(t,s)>0, t≥s≥t0, where X(t,s) is the fundamental function of (1.1).Lemma 2.8 (see [3]).
Supposeak(t)≥0,
(2.11)lim⁡inf⁡t→∞∑k=1mak(t)>0,(2.12)limsup⁡t→∞(t-hk(t))<∞,k=1,…,m,
and there exists r(t)≤t such that for sufficiently large t(2.13)∫r(t)t∑k=1mak(s)ds≤1e.
If(2.14)limsup⁡t→∞∑k=1mak(t)∑i=1mai(t)|∫hk(t)r(t)∑i=1mai(s)ds|<1,
then (1.1) is exponentially stable.Lemma 2.9 (see [3]).
Suppose (2.12) holds and there exists a set of indices I⊂{1,…,m}, such that ak(t)≥0, k∈I,
(2.15)lim⁡inf⁡t→∞∑k∈Iak(t)>0,
and the fundamental function of the equation
(2.16)ẋ(t)+∑k∈Iak(t)x(hk(t))=0
is eventually positive. If
(2.17)limsup⁡t→∞∑k∉I|ak(t)|∑k∈Iak(t)<1,
then (1.1) is exponentially stable.The following lemma was obtained in [26, Corollary 2], see also [27].Lemma 2.10.
Suppose for (1.1) condition (2.12) holds and this equation is exponentially stable. If
(2.18)∫0∞∑k=1n|bk(s)|ds<∞,limsup⁡t→∞(t-gk(t))<∞,gk(t)≤t,
then the equation
(2.19)ẋ(t)+∑k=1mak(t)x(hk(t))+∑k=1nbk(t)x(gk(t))=0
is exponentially stable.The following elementary result will be used in the paper.Lemma 2.11.
The ordinary differential equation(2.20)ẋ(t)+a(t)x(t)=0
is exponentially stable if and only if there exists R>0 such that
(2.21)lim⁡inf⁡t→∞∫tt+Ra(s)ds>0.The following example illustrates that a stronger than (2.21) sufficient condition(2.22)lim⁡inf⁡t,s→∞1t-s∫sta(τ)dτ>0
is not necessary for the exponential stability of the ordinary differential equation (2.20).Example 2.12.
Consider the equation(2.23)ẋ(t)+a(t)x(t)=0,wherea(t)={1,t∈[2n,2n+1),0,t∈[2n+1,2n+2),n=0,1,2,…
Then liminf in (2.22) equals zero, but |X(t,s)|<ee-0.5(t-s), so the equation is exponentially stable. Moreover, if we consider liminf in (2.22) under the condition t-s≥R, then it is still zero for any R≤1.


## 3. Main Results

Lemma 3.1.
Supposeak(t)≥0, (2.11), (2.12) hold and
(3.1)limsup⁡t→∞∑k=1mak(t)∑i=1mai(t)∫hk(t)t∑i=1mai(s)ds<1+1e.
Then (1.1) is exponentially stable.Proof.
By (2.11) there exists function r(t)≤t such that for sufficiently large t(3.2)∫r(t)t∑k=1mak(s)ds=1e.
For this function condition (2.14) has the form
(3.3)limsup⁡t→∞∑k=1mak(t)∑i=1mai(t)|∫hk(t)t∑i=1mai(s)ds-∫r(t)t∑i=1mai(s)ds|=limsup⁡t→∞∑k=1mak(t)∑i=1mai(t)|∫hk(t)t∑i=1mai(s)ds-1e|<1.
The latter inequality follows from (3.1). The reference to Lemma 2.8 completes the proof.Corollary 3.2.
Supposeak(t)≥0, (2.11), (2.12) hold and
(3.4)limsup⁡t→∞∫min⁡k⁡{hk(t)}t∑i=1mai(s)ds<1+1e.
Then (1.1) is exponentially stable.The following theorem contains stability conditions for equations with unbounded delays. We also omit condition (2.11) in Lemma 3.1.We recall thatb(t)>0 in the space of Lebesgue measurable essentially bounded functions means b(t)≥0 and b(t)≠0 almost everywhere.Theorem 3.3.
Supposeak(t)≥0, condition (3.1) holds, ∑k=1mak(t)>0 and
(3.5)∫0∞∑k=1mak(t)dt=∞,limsup⁡t→∞∫hk(t)t∑i=1mai(s)ds<∞.
Then (1.1) is asymptotically stable.
If in addition there existsR>0 such that(3.6)lim⁡inf⁡t→∞∫tt+R∑k=1mak(τ)dτ>0
then the fundamental function of (1.1) has an exponential estimation.
If condition (2.12) also holds then (1.1) is exponentially stable.Proof.
Lets=p(t):=∫0t∑k=1mak(τ)dτ, y(s)=x(t), where p(t) is a strictly increasing function. Then x(hk(t))=y(lk(s)), lk(s)≤s, lk(s)=∫0hk(t)∑i=1mai(τ)dτ and (1.1) can be rewritten in the form
(3.7)ẏ(s)+∑k=1mbk(s)y(lk(s))=0,
where bk(s)=ak(t)/∑i=1mai(t), s-lk(s)=∫hk(t)t∑i=1mai(τ)dτ. Since ∑k=1mbk(s)=1 and limsups→∞(s-lk(s))<∞, then Lemma 3.1 can be applied to (3.7). We have
(3.8)limsup⁡s→∞∑k=1mbk(s)∑i=1mbi(s)∫lk(s)s∑i=1mbi(τ)dτ=limsup⁡s→∞∑k=1mbk(s)(s-lk(s))=limsup⁡t→∞∑k=1mak(t)∑i=1mai(t)∫hk(t)t∑i=1mai(s)ds<1+1e.
By Lemma 3.1, (3.7) is exponentially stable. Due to the first equality in (3.5) t→∞ implies s→∞. Hence limt→∞x(t)=lims→∞y(s)=0.
Equation (3.7) is exponentially stable, thus the fundamental function Y(u,v) of (3.7) has an exponential estimation(3.9)|Y(u,v)|≤Ke-λ(u-v),u≥v≥0,
with K>0, λ>0. Since X(t,s)=Y(∫0t∑k=1mak(τ)dτ,∫0s∑k=1mak(τ)dτ), where X(t,s) is the fundamental function of (1.1), then (3.9) yields
(3.10)|X(t,s)|≤Kexp⁡{-λ∫st∑k=1mak(τ)dτ}.
Hence |X(t,s)|≤K, t≥s≥0, which together with limt→∞x(t)=0 yields that (1.1) is asymptotically stable.
Suppose now that (3.6) holds. Without loss of generality we can assume that for some R>0, α>0 we have(3.11)∫tt+R∑k=1mak(τ)dτ≥α>0,t≥s≥0.
Hence
(3.12)exp⁡{-λ∫st∑k=1mak(τ)dτ}≤exp⁡{λRsup⁡t≥0∑k=1mak(t)}e-λα(t-s)/R.
Thus, condition (3.6) implies the exponential estimate for X(t,s).
The last statement of the theorem is evident.Remark 3.4.
The substitutions=p(t):=∫0t∑k=1mak(τ)dτ, y(s)=x(t) was first used in [28].Note that in [10, Lemma 2] this idea was extended to a more general equation(3.13)ẋ(t)+∫t0tx(s)dsr(t,s)=0.
The ideas of [10] allow to generalize the results of the present paper to equations with a distributed delay.Corollary 3.5.
Supposeak(t)≥0, ∑k=1mak(t)≡α>0, condition (2.12) holds and
(3.14)limsup⁡t→∞∑k=1mak(t)(t-hk(t))<1+1e.
Then (1.1) is exponentially stable.Corollary 3.6.
Supposeak(t)=αkp(t), αk>0, p(t)>0, ∫0∞p(t)dt=∞ and
(3.15)limsup⁡t→∞∑k=1mαk∫hk(t)tp(s)ds<1+1e.
Then (1.1) is asymptotically stable.
If in addition there existsR>0 such that(3.16)lim⁡inf⁡t→∞∫tt+Rp(τ)dτ>0,
then the fundamental function of (1.1) has an exponential estimation.
If also (2.12) holds then (1.1) is exponentially stable.Remark 3.7.
Let us note that similar results for (3.13) were obtained in [10], see Corollary 3.4 and remark after it, Theorem 4 and Corollaries 4.1 and 4.2 in [10], where an analogue of condition (3.16) was applied. This allows to extend the results of the present paper to equations with a distributed delay.Corollary 3.8.
Supposea(t)≥0, b(t)≥0, a(t)+b(t)>0,
(3.17)∫0∞(a(t)+b(t))dt=∞,limsup⁡t→∞∫h(t)t(a(s)+b(s))ds<∞,limsup⁡t→∞b(t)a(t)+b(t)∫h(t)t(a(s)+b(s))ds<1+1e.
Then the following equation is asymptotically stable
(3.18)ẋ(t)+a(t)x(t)+b(t)x(h(t))=0.
If in addition there existsR>0 such that liminft→∞∫tt+R(a(τ)+b(τ))dτ>0 then the fundamental function of (3.18) has an exponential estimation.
If alsolimsupt→∞(t-h(t))<∞ then (3.18) is exponentially stable.In the following theorem we will omit the condition∑k=1mak(t)>0 of Theorem 3.3.Theorem 3.9.
Supposeak(t)≥0, condition (3.4) and the first inequality in (3.5) hold. Then (1.1) is asymptotically stable.
If in addition (3.6) holds then the fundamental function of (1.1) has an exponential estimation.
If also (2.12) holds then (1.1) is exponentially stable.Proof.
For simplicity suppose thatm=2 and consider the equation
(3.19)ẋ(t)+a(t)x(h(t))+b(t)x(g(t))=0,
where a(t)≥0, b(t)≥0, ∫0∞(a(s)+b(s))ds=∞ and there exist t0≥0, ɛ>0 such that
(3.20)∫min⁡⁡{h(t),g(t)}t(a(s)+b(s))ds<1+1e-ɛ,t≥t0.
Let us find t1≥t0 such that e-h(t)<ɛ/4, e-g(t)<ɛ/4, t≥t1, such t1 exists due to (a2). Then ∫min{h(t),g(t)}te-sds<ɛ/2, t≥t1. Rewrite (3.19) in the form
(3.21)ẋ(t)+(a(t)+e-t)x(h(t))+b(t)x(g(t))-e-tx(h(t))=0,
where a(t)+b(t)+e-t>0. After the substitution s=∫t1t(a(τ)+b(τ)+e-τ)dτ, y(s)=x(t), (3.21) has the form
(3.22)ẏ(s)+a(t)+e-ta(t)+b(t)+e-ty(l(s))+b(t)a(t)+b(t)+e-ty(p(s))-e-ta(t)+b(t)+e-ty(l(s))=0,
where similar to the proof of Theorem 3.3(3.23)s-l(s)=∫h(t)t(a(τ)+b(τ)+e-τ)dτ,s-p(s)=∫g(t)t(a(τ)+b(τ)+e-τ)dτ.
First we will show that by Corollary 3.2 the equation
(3.24)ẏ(s)+a(t)+e-ta(t)+b(t)+e-ty(l(s))+b(t)a(t)+b(t)+e-ty(p(s))=0
is exponentially stable. Since  (a(t)+e-t)/(a(t)+b(t)+e-t)+b(t)/(a(t)+b(t)+e-t)=1, then (2.11) holds. Condition (3.20) implies (2.12). So we have to check only condition (3.4) where the sum under the integral is equal to 1. By (3.20), (3.23) we have
(3.25)∫min⁡{l(s),p(s)}s1ds=s-min⁡{l(s),p(s)},s-l(s)=∫h(t)t(a(τ)+b(τ)+e-τ)dτ=∫h(t)t(a(τ)+b(τ))dτ+∫h(t)te-τdτ<1+1e-ɛ+ɛ2=1+1e-ɛ2,t≥t1.
The same calculations give s-p(s)<1+(1/e)-ɛ/2, thus condition (3.4) holds.
Hence (3.24) is exponentially stable.
We return now to (3.22), t≥t1. We have ds=(a(t)+b(t)+e-t)dt, then(3.26)∫t1∞e-ta(t)+b(t)+e-tds=∫t1∞e-ta(t)+b(t)+e-t(a(t)+b(t)+e-t)dt<∞.
By Lemma 2.10, (3.22) is exponentially stable. Since t→∞ implies s→∞ then limt→∞x(t)=lims→∞y(s)=0, which completes the proof of the first part of the theorem. The rest of the proof is similar to the proof of Theorem 3.3.Corollary 3.10.
Supposea(t)≥0, ∫0∞a(t)dt=∞ and
(3.27)limsup⁡t→∞∫h(t)ta(s)ds<1+1e.
Then the equation
(3.28)ẋ(t)+a(t)x(h(t))=0
is asymptotically stable. If in addition condition (2.21) holds then the fundamental function of (3.28) has an exponential estimation. If also limsupt→∞(t-h(t))<∞ then (3.28) is exponentially stable.Now consider (1.1), where only some of coefficients are nonnegative.Theorem 3.11.
Suppose there exists a set of indicesI⊂{1,…,m} such that ak(t)≥0, k∈I,
(3.29)∫0∞∑k∈Iak(t)dt=∞,limsup⁡t→∞∫hk(t)t∑i∈Iai(s)ds<∞,k=1,…,m,(3.30)∑k∉I|ak(t)|=0,t∈E,limsup⁡t→∞,t∉E∑k∉I|ak(t)|∑k∈Iak(t)<1,whereE={t≥0,∑k∈Iak(t)=0}.
If the fundamental functionX0(t,s) of (2.16) is eventually positive then all solutions of (1.1) tend to zero as t→∞.
If in addition there existsR>0 such that(3.31)lim⁡inf⁡t→∞∫tt+R∑k∈Iak(τ)dτ>0
then the fundamental function of (1.1) has an exponential estimation.
If condition (2.12) also holds then (1.1) is exponentially stable.Proof.
Without loss of generality we can assumeX0(t,s)>0, t≥s≥0. Rewrite (1.1) in the form
(3.32)ẋ(t)+∑k∈Iak(t)x(hk(t))+∑k∉Iak(t)x(hk(t))=0.
Suppose first that ∑k∈Iak(t)≠0. After the substitution s=p(t):=∫0t∑k∈Iak(τ)dτ, y(s)=x(t) we have x(hk(t))=y(lk(s)), lk(s)≤s, lk(s)=∫0hk(t)∑i∈Iai(τ)dτ, k=1,…,m, and (1.1) can be rewritten in the form
(3.33)ẏ(s)+∑k=1mbk(s)y(lk(s))=0,
where bk(s)=ak(t)/∑i∈Iai(t). Denote by Y0(u,v) the fundamental function of the equation
(3.34)ẏ(s)+∑k∈Ibk(s)y(lk(s))=0.
We have
(3.35)X0(t,s)=Y0(∫0t∑k∈Iak(τ)dτ,∫0s∑k∈Iak(τ)dτ),Y0(u,v)=X0(p-1(u),p-1(v))>0,u≥v≥0.
Let us check that other conditions of Lemma2.9 hold for (3.33). Since ∑k∈Ibk(s)=1 then condition (2.15) is satisfied. In addition,(3.36)limsup⁡s→∞,p-1(s)∉E∑k∉I|bk(s)|∑k∈Ibk(s)=limsup⁡t→∞,t∉E∑k∉I|ak(t)|∑k∈Iak(t)<1.
By Lemma 2.9, (3.33) is exponentially stable. Hence for any solution x(t) of (1.1) we have limt→∞x(t)=lims→∞y(s)=0. The end of the proof is similar to the proof of Theorem 3.9. In particular, to remove the condition ∑k∈Iak(t)≠0 we rewrite the equation by adding the term e-t to one of ak(t),k∈I.Remark 3.12.
Explicit positiveness conditions for the fundamental function were presented in Lemma2.7.Corollary 3.13.
Suppose(3.37)a(t)≥0,∫0∞a(t)dt=∞,limsup⁡t→∞∫gk(t)ta(s)ds<∞,∑k=1n|bk(t)|=0,t∈E,limsup⁡t→∞,t∉E∑k=1n|bk(t)|a(t)<1,
where E={t≥0,a(t)=0}. Then the equation
(3.38)ẋ(t)+a(t)x(t)+∑k=1nbk(t)x(gk(t))=0
is asymptotically stable. If in addition (2.21) holds then the fundamental function of (3.38) has an exponential estimation. If also limsupt→∞(t-gk(t))<∞ then (3.38) is exponentially stable.Theorem 3.14.
Suppose∫0∞∑k=1m|ak(s)|ds<∞. Then all solutions of (1.1) are bounded and (1.1) is not asymptotically stable.Proof.
For the fundamental function of (1.1) we have the following estimation
(3.39)|X(t,s)|≤exp⁡{∫st∑k=1m|ak(τ)|dτ}.
Then by solution representation formula (2.5) for any solution x(t) of (1.1) we have
(3.40)|x(t)|≤exp⁡{∫t0t∑k=1m|ak(s)|ds}|x(t0)|+∫t0texp⁡{∫st∑k=1m|ak(τ)|dτ}∑k=1m|ak(s)||φ(hk(s))|ds≤exp⁡{∫t0∞∑k=1m|ak(s)|ds}(|x(t0)|+∫t0∞∑k=1m|ak(s)|ds∥φ∥),
where ∥φ∥=maxt<0|φ(t)|. Then x(t) is a bounded function.
Moreover,|X(t,s)|≤A:=exp{∫0∞∑k=1m|ak(s)|ds}, t≥s≥0. Let us choose t0≥0 such that ∫t0∞∑k=1m|ak(s)|ds<1/(2A), then Xt′(t,t0)+∑k=1mak(t)X(hk(t),t0)=0, X(t0,t0)=1 implies X(t,t0)≥1-∫t0∞∑k=1m|ak(s)|Ads>1-A(1/(2A))=1/2, thus X(t,t0) does not tend to zero, so (1.1) is not asymptotically stable.Theorems3.11 and 3.14 imply the following results.Corollary 3.15.
Supposeak(t)≥0, there exists a set of indices I⊂{1,…,m} such that condition (3.30) and the second condition in (3.29) hold. Then all solutions of (1.1) are bounded.Proof.
If∫0∞∑k∈I|ak(t)|dt=∞, then all solutions of (1.1) are bounded by Theorem 3.11. Let ∫0∞∑k∈I|ak(t)|dt<∞. By (3.30) we have ∫0∞∑k∉I|ak(t)|dt≤∫0∞∑k∈I|ak(t)|dt<∞. Then ∫0∞∑k=1m|ak(t)|dt<∞. By Theorem 3.14 all solutions of (1.1) are bounded.Theorem 3.16.
Supposeak(t)≥0. If (1.1) is asymptotically stable, then the ordinary differential equation
(3.41)ẋ(t)+(∑k=1mak(t))x(t)=0
is also asymptotically stable. If in addition (2.12) holds and (1.1) is exponentially stable, then (3.41) is also exponentially stable.Proof.
The solution of (3.41), with the initial condition x(t0)=x0, can be presented as x(t)=x0exp{-∫t0t∑k=1mak(s)ds}, so (3.41) is asymptotically stable, as far as
(3.42)∫0∞∑k=1mak(s)ds=∞
and is exponentially stable if (3.6) holds (see Lemma 2.11).
If (3.42) does not hold, then by Theorem 3.14, (1.1) is not asymptotically stable.
Further, let us demonstrate that exponential stability of (1.1) really implies (3.6).
Suppose for the fundamental function of (1.1) inequality (2.7) holds and condition (3.6) is not satisfied. Then there exists a sequence {tn}, tn→∞, such that(3.43)∫tntn+n∑k=1mak(τ)dτ<1n<1e,n≥3.
By (2.12) there exists n0>3 such that t-hk(t)≤n0, k=1,…,m. Lemma 2.7 implies X(t,s)>0, tn≤s≤t≤tn+n, n≥n0. Similar to the proof of Theorem 3.14 and using the inequality 1-x≥e-x, x>0, we obtain
(3.44)X(tn,tn+n)≥1-∫tntn+n∑k=1mak(τ)dτ≥exp⁡{-∫tntn+n∑k=1mak(τ)dτ}>e-1/n.
Inequality (2.7) implies |X(tn+n,tn)|≤Ke-λn. Hence Ke-λn≥e-1/n, n≥n0, or K>eλn-1/3 for any n≥n0. The contradiction proves the theorem.Theorems3.11 and 3.16 imply the following statement.Corollary 3.17.
Supposeak(t)≥0 and the fundamental function of (1.1) is positive. Then (1.1) is asymptotically stable if and only if the ordinary differential equation (3.41) is asymptotically stable.
If in addition (2.12) holds then (1.1) is exponentially stable if and only if (3.41) is exponentially stable.


## 4. Discussion and Examples

In paper [2] we gave a review of known stability tests for the linear equation (1.1). 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 [5–7, 9, 10, 22]. First of all we replace the constant 3/2 in most of these tests by the constant 1+1/e. Evidently 1+1/e=1.3678⋯<3/2, so we have a worse constant, but it is an open problem to obtain (3/2)-stability results for equations with measurable coefficients and delays.Consider now (3.28) with a single delay. This equation is well studied beginning with the classical stability result by Myshkis [29]. We present here several statements which cover most of known stability tests for this equation.Statement 1 (see [5]).
Supposea(t)≥0, h(t)≤t are continuous functions and
(4.1)limsup⁡t→∞∫h(t)ta(s)ds≤32.
Then all solutions of (3.28) are bounded.
If in addition(4.2)lim⁡inf⁡t→∞∫h(t)ta(s)ds>0,
and the strict inequality in (4.1) holds then (3.28) is exponentially stable.Statement 2 (see [7]).
Supposea(t)≥0, h(t)≤t are continuous functions, the strict inequality (4.1) holds and ∫0∞a(s)ds=∞. Then all solutions of (3.28) tend to zero as t→∞.Statement 3 (see [9, 10]).
Supposea(t)≥0, h(t)≤t are measurable functions, ∫0∞a(s)ds=∞, A(t)=∫0ta(s)ds is a strictly monotone increasing function and
(4.3)limsup⁡t→∞∫h(t)ta(s)ds<sup⁡0<ω<π/2(ω+1Φ(ω))≈1.45…,Φ(ω)=∫0∞u(t,ω)dt, where u(t,ω) is a solution of the initial value problem
(4.4)ẏ(t)+y(t-ω)=0,y(t)=0,t<0,y(0)=1.
Then (3.28) is asymptotically stable.Note that instead of the equationẏ(t)+y(t-ω)=0 with a constant delay, the equation(4.5)ẏ(t)+y(t-τ(t))=0
can be used as the model equation. For example, the following results are valid.Statement 4 (see [10]).
Equation (4.5) is exponentially stable if |τ(t)-ω|≤k/χ(ω), where k∈[0,ω), 0≤ω<π/2 and χ(ω)=∫0∞|u(t,ω)|dt.Obviously in this statement the delay can exceed 2.Statement 5 (see [10]).
Letτ(t)≤k+ω{t/ω}, where k∈(0,1), 0<ω<1, {q} is the fractional part of q. Then (4.5) is exponentially stable.Here the delayτ(t) can be in the neighbourhood of ω which is close to 1.Example 4.1.
Consider the equation(4.6)ẋ(t)+α(|sint|-sint)x(h(t))=0,h(t)≤t,
where h(t) is an arbitrary measurable function such that t-h(t)≤π and α>0.This equation has the form (3.28) where a(t)=α(|sint|-sint). Let us check that the conditions of Corollary 3.10 hold. It is evident that ∫0∞a(s)ds=∞. We have (4.7)limsup⁡t→∞∫h(t)ta(s)ds≤limsup⁡t→∞∫t-πta(s)ds≤-α∫π2π2sinsds=4α.
If α<0.25(1+1/e), then condition (3.27) holds, hence all solutions of (4.6) tend to zero as t→∞.Statements1–3 fail for this equation. In Statements 1 and 2 the delay should be continuous. In Statement 3 function A(t)=∫0ta(s)ds should be strictly increasing.Consider now the general equation (1.1) with several delays. The following two statements are well known for this equation.Statement 6 (see [6]).
Supposeak(t)≥0, hk(t)≤t are continuous functions and
(4.8)limsup⁡t→∞ak(t)limsup⁡t→∞(t-hk(t))≤1.
Then all solutions of (1.1) are bounded and 1 in the right-hand side of (4.8) is the best possible constant.
If∑k=1mak(t)>0 and the strict inequality in (4.8) is valid then all solutions of (1.1) tend to zero as t→∞.
Ifak(t) are constants then in (4.8) the number 1 can be replaced by 3/2.Statement 7 (see [7]).
Supposeak(t)≥0, hk(t)≤t are continuous, h1(t)≤h2(t)≤⋯≤hm(t) and
(4.9)limsup⁡t→∞∫h1(t)t∑k=1mak(s)ds≤32.
Then any solution of (1.1) tends to a constant as t→∞.
If in addition∫0∞∑k=1mak(s)ds=∞, then all solutions of (1.1) tend to zero as t→∞.Example 4.2.
Consider the equation(4.10)ẋ(t)+αtx(t2-sint)+βtx(t2)=0,t≥t0>0,
where α>0, β>0. Denote p(t)=1/t, h(t)=t/2-sint, g(t)=t/2.We apply Corollary3.6. Since limt→∞[ln(t/2)-ln(t/2-sint)]=0, then (4.11)limsup⁡t→∞(α∫h(t)tp(s)ds+β∫g(t)tp(s)ds)≤(α+β)ln⁡2.
Hence if α+β<(1/ln2)(1+1/e) then (4.10) is asymptotically stable. Statement 4 fails for this equation since the delays are unbounded. Statement 5 fails for this equation since neither h(t)≤g(t) nor g(t)≤h(t) holds.Stability results where the nondelay term dominates over the delayed terms are well known beginning with the book of Krasovskii [30]. The following result is cited from the monograph [22].Statement 8 (see [22]).
Supposea(t), bk(t), t-hk(t) are bounded continuous functions, there exist δ, k, δ>0, 0<k<1, such that a(t)≥δ and ∑k=1m|bk(t)|<kδ. Then the equation
(4.12)ẋ(t)+a(t)x(t)+∑k=1mbk(t)x(hk(t))=0
is exponentially stable.In Corollary3.13 we obtained a similar result without the assumption that the parameters of the equation are continuous functions and the delays are bounded.Example 4.3.
Consider the equation(4.13)ẋ(t)+1tx(t)+αtx(t2)=0,t≥t0>0.
If α<1 then by Corollary 3.13 all solutions of (4.13) tend to zero. The delay is unbounded, thus Statement 8 fails for this equation.In [31] 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 Győri [32] and some results of [33].In Theorem3.11 we obtained a similar result for a linear nonautonomous equation without the assumption that coefficients and delays are continuous.In all the results of the paper we assumed that all or several coefficients of equations considered here are nonnegative. Stability results for (3.28) with oscillating coefficient a(t) were obtained in [34, 35].We conclude this paper with some open problems.(1)
Is the constant1+1/e sharp? Prove or disprove that in Corollary 3.10 the constant 1+1/e can be replaced by the constant 3/2. Note that all known proofs with the constant 3/2 apply methods which are not applicable for equations with measurable parameters.(2)
Suppose (2.11), (2.12) hold and
(4.14)limsup⁡t→∞∑k=1m|ak(t)|∑i=1mai(t)∫hk(t)t∑i=1mai(s)ds<1+1e.
Prove or disprove that (1.1) is exponentially stable. The solution of this problem will improve Theorem 3.3.(3)
Suppose (1.1) is exponentially stable. Prove or disprove that the ordinary differential equation (3.41) is also exponentially (asymptotically) stable, without restrictions on the signs of coefficients ak(t)≥0, as in Theorem 3.16. The solution of this problem would improve Theorem 3.16.



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