New Stability Conditions for Linear Differential Equations with Several Delays

Abstract

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation 𝑥̇(𝑡)+∑^𝑚𝑘=1𝑎_𝑘(𝑡)𝑥(ℎ_𝑘(𝑡))=0 with measurable delays and coefficients. These results are compared to known stability tests.

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 coefficients𝑥̇(𝑡)+^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0,𝑡≥𝑡₀,(1.1) 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: 𝑥̇(𝑡)𝑥+̇𝑥(𝑡̇𝛼)(𝑡+)||+s1in||s𝑡i||n−𝑡||s𝑥in(𝑡𝑡−𝑥𝜏()𝑡=−0𝜏,)=0,𝑡𝑥(𝑡)+𝛼𝑡𝑥2𝑡=0.(1.2) 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 𝑥̇(𝑡)+^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥(𝑡)=0(1.3) 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)𝑥(𝑡)=𝜑(𝑡),𝑡<𝑡₀,𝑥𝑡₀=𝑥₀,(2.1) 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, ℎ_𝑘(𝑡)≤𝑡,limsup_𝑡→∞ℎ_𝑘(𝑡)=∞;(2.2)(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 for 𝑡≥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 side 𝑓(𝑡):𝑥̇(𝑡)+^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=𝑓(𝑡),𝑡≥𝑡₀.(2.3)

Definition 2.2. A solution 𝑋(𝑡,𝑠) of the problem 𝑥̇(𝑡)+^𝑚_𝑘=1𝑥(𝑡)𝑎=_𝑘(0t,)𝑥𝑡ℎ<_𝑘𝑠(,𝑡)=𝑥0(,𝑠)=𝑡≥1,𝑠≥0,(2.4) 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 𝑥(𝑡)=𝑋𝑡,𝑡₀𝑥₀−^𝑡𝑡₀𝑋(𝑡,𝑠)^𝑚_𝑘=1𝑎_𝑘(𝑠)𝜑ℎ_𝑘(𝑠)𝑑𝑠+^𝑡𝑡₀𝑋(𝑡,𝑠)𝑓(𝑠)𝑑𝑠,(2.5) where 𝜑(𝑡)=0, 𝑡≥𝑡₀.

Definition 2.4 (see [22]). Equation (1.1) is stable if for any initial point 𝑡₀ and number 𝜀>0 there exists 𝛿>0 such that the inequality sup_𝑡<𝑡0|𝜑(𝑡)|+|𝑥(𝑡₀)|<𝛿 implies |𝑥(𝑡)|<𝜀, 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 ||𝑥(𝑡)||≤𝑀𝑒^{−𝜇(𝑡−𝑡0)}||𝑥𝑡⁰||+sup_𝑡<𝑡0||𝜑(𝑡)||,𝑡≥𝑡₀,(2.6) where 𝑀 and 𝜇 do not depend on 𝑡₀.

Definition 2.6. The fundamental function 𝑋(𝑡,𝑠) of (1.1) has an exponential estimation if there exist 𝐾>0, 𝜆>0 such that ||𝑋(𝑡,𝑠)||≤𝐾𝑒^{−𝜆(𝑡−𝑠)},𝑡≥𝑠≥0.(2.7)

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]). Suppose 𝑎_𝑘(𝑡)≥0. If ^{𝑡maxℎ(𝑡),𝑡}_{0𝑚𝑖=1}𝑎_𝑖(𝑠)𝑑𝑠≤1𝑒,ℎ(𝑡)=min^𝑘ℎ_𝑘(𝑡),𝑡≥𝑡₀,(2.8) or there exists 𝜆>0, such that 𝜆≥^𝑚_𝑘=1𝐴_𝑘𝑒^𝜆𝜎𝑘,(2.9) where 0≤𝑎_𝑘(𝑡)≤𝐴_𝑘,𝑡−ℎ_𝑘(𝑡)≤𝜎_𝑘,𝑡≥𝑡₀,(2.10) then 𝑋(𝑡,𝑠)>0, 𝑡≥𝑠≥𝑡₀, where 𝑋(𝑡,s) is the fundamental function of (1.1).

Lemma 2.8 (see [3]). Suppose 𝑎_𝑘(𝑡)≥0, liminf_{𝑚𝑡→∞}_𝑘=1limsup_𝑡→∞𝑎_𝑘(𝑡)>0,(2.11)𝑡−ℎ_𝑘(𝑡)<∞,𝑘=1,…,𝑚,(2.12) and there exists 𝑟(𝑡)≤𝑡 such that for sufficiently large 𝑡^{𝑡𝑚𝑟(𝑡)}_𝑘=1𝑎_𝑘(𝑠)𝑑𝑠≤1𝑒.(2.13)
If limsup_{𝑚𝑡→∞}_𝑘=1𝑎_𝑘(𝑡)∑^𝑚𝑖=1𝑎_𝑖(𝑡)|||||^{ℎ𝑟(𝑡)}_{𝑘𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠|||||<1,(2.14) 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 𝑎_𝑘(𝑡)≥0, 𝑘∈𝐼, liminf_𝑡→∞_𝑘∈𝐼𝑎_𝑘(𝑡)>0,(2.15) and the fundamental function of the equation 𝑥̇(𝑡)+_𝑘∈𝐼𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0(2.16) is eventually positive. If limsup_𝑡→∞∑_𝑘∉𝐼||𝑎_𝑘(𝑡)||∑_𝑘∈𝐼𝑎_𝑘(𝑡)<1,(2.17) 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 ^∞0𝑛_𝑘=1||𝑏_𝑘(𝑠)||𝑑𝑠<∞,limsup_𝑡→∞𝑡−𝑔_𝑘(𝑡)<∞,𝑔_𝑘(𝑡)≤𝑡,(2.18) then the equation 𝑥̇(𝑡)+^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)+^𝑛_𝑘=1𝑏_𝑘(𝑡)𝑥𝑔_𝑘(𝑡)=0(2.19) is exponentially stable.

The following elementary result will be used in the paper.

Lemma 2.11. The ordinary differential equation 𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡)=0(2.20) is exponentially stable if and only if there exists 𝑅>0 such that liminf_𝑡→∞^{𝑡𝑡+𝑅}𝑎(𝑠)𝑑𝑠>0.(2.21)

The following example illustrates that a stronger than (2.21) sufficient conditionliminf_{𝑡,𝑠→∞}1𝑡−𝑠^𝑡𝑠𝑎(𝜏)𝑑𝜏>0(2.22) is not necessary for the exponential stability of the ordinary differential equation (2.20).

Example 2.12. Consider the equation 𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡)=0,where𝑎(𝑡)=10,,𝑡𝑡∈∈[[22𝑛𝑛,+2𝑛1,+21𝑛)+,2),𝑛=0,1,2,…(2.23) 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. Suppose 𝑎_𝑘(𝑡)≥0, (2.11), (2.12) hold and limsup_{𝑚𝑡→∞}_𝑘=1𝑎_𝑘(𝑡)∑^𝑚𝑖=1𝑎_𝑖(𝑡)^𝑡ℎ_{𝑘𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠<1+1𝑒.(3.1) Then (1.1) is exponentially stable.

Proof. By (2.11) there exists function 𝑟(𝑡)≤𝑡 such that for sufficiently large 𝑡^{𝑡𝑚𝑟(𝑡)}_𝑘=1𝑎_𝑘(𝑠)𝑑𝑠=1𝑒.(3.2) For this function condition (2.14) has the form limsup_{𝑚𝑡→∞}_𝑘=1𝑎_𝑘(𝑡)∑^𝑚𝑖=1𝑎_𝑖(𝑡)|||||^𝑡ℎ_{𝑘𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠−^{𝑡𝑚𝑟(𝑡)}_𝑖=1=limsup_{𝑚𝑡→∞}𝑎_𝑖(𝑠)𝑑𝑠|||||_𝑘=1𝑎_𝑘(𝑡)∑^𝑚𝑖=1𝑎_𝑖(𝑡)|||||^𝑡ℎ_{𝑘𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠−1𝑒|||||<1.(3.3) The latter inequality follows from (3.1). The reference to Lemma 2.8 completes the proof.

Corollary 3.2. Suppose 𝑎_𝑘(𝑡)≥0, (2.11), (2.12) hold and limsup_𝑡→∞^𝑡min_{𝑘{ℎ𝑘𝑚(𝑡)}}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠<1+1𝑒.(3.4) 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 that 𝑏(𝑡)>0 in the space of Lebesgue measurable essentially bounded functions means 𝑏(𝑡)≥0 and 𝑏(𝑡)≠0 almost everywhere.

Theorem 3.3. Suppose 𝑎_𝑘(𝑡)≥0, condition (3.1) holds, ∑^𝑚𝑘=1𝑎_𝑘(𝑡)>0 and ^∞0𝑚_𝑘=1𝑎_𝑘(𝑡)𝑑𝑡=∞,limsup_𝑡→∞^𝑡ℎ_{𝑘𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠<∞.(3.5) Then (1.1) is asymptotically stable.
If in addition there exists 𝑅>0 such that liminf_𝑡→∞^{𝑡𝑚𝑡+𝑅}_𝑘=1𝑎_𝑘(𝜏)𝑑𝜏>0(3.6) then the fundamental function of (1.1) has an exponential estimation.
If condition (2.12) also holds then (1.1) is exponentially stable.

Proof. Let 𝑠=𝑝(𝑡)∶=∫^𝑡0∑^𝑚𝑘=1𝑎_𝑘(𝜏)𝑑𝜏, 𝑦(𝑠)=𝑥(𝑡), where 𝑝(𝑡) is a strictly increasing function. Then 𝑥(ℎ_𝑘(𝑡))=𝑦(𝑙_𝑘(𝑠)), 𝑙_𝑘(𝑠)≤𝑠, 𝑙_𝑘(𝑠)=∫^{ℎ𝑘0(𝑡)}∑^𝑚𝑖=1𝑎_𝑖(𝜏)𝑑𝜏 and (1.1) can be rewritten in the form 𝑦̇(𝑠)+^𝑚_𝑘=1𝑏_𝑘(𝑠)𝑦𝑙_𝑘(𝑠)=0,(3.7) where 𝑏_𝑘(𝑠)=𝑎_𝑘(𝑡)/∑^𝑚𝑖=1𝑎_𝑖(𝑡), 𝑠−𝑙_𝑘(𝑠)=∫^𝑡ℎ_𝑘(𝑡)∑^𝑚𝑖=1𝑎_𝑖(𝜏)𝑑𝜏. Since ∑^𝑚𝑘=1𝑏_𝑘(𝑠)=1 and limsup_𝑠→∞(𝑠−𝑙_𝑘(𝑠))<∞, then Lemma 3.1 can be applied to (3.7). We have limsup_{𝑚𝑠→∞}_𝑘=1𝑏_𝑘(𝑠)∑^𝑚𝑖=1𝑏_𝑖(𝑠)^𝑠𝑙_{𝑘𝑚(𝑠)}_𝑖=1=limsup_{𝑚𝑠→∞}𝑏_𝑖(𝜏)𝑑𝜏_𝑘=1=limsup_{𝑚𝑡→∞}𝑏_𝑘(𝑠)𝑠−𝑙_𝑘(𝑠)_𝑘=1𝑎_𝑘(𝑡)∑^𝑚𝑖=1𝑎_𝑖(𝑡)^𝑡ℎ_{𝑘𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠<1+1𝑒.(3.8) By Lemma 3.1, (3.7) is exponentially stable. Due to the first equality in (3.5) 𝑡→∞ implies 𝑠→∞. Hence lim_𝑡→∞𝑥(𝑡)=lim_𝑠→∞𝑦(𝑠)=0.
Equation (3.7) is exponentially stable, thus the fundamental function 𝑌(𝑢,𝑣) of (3.7) has an exponential estimation||𝑌(𝑢,𝑣)||≤𝐾𝑒^{−𝜆(𝑢−𝑣)},𝑢≥𝑣≥0,(3.9) with 𝐾>0, 𝜆>0. Since 𝑋(𝑡,𝑠)=𝑌(∫^𝑡0∑^𝑚𝑘=1𝑎_𝑘(𝜏)𝑑𝜏,∫^𝑠0∑^𝑚𝑘=1𝑎_𝑘(𝜏)𝑑𝜏), where 𝑋(𝑡,𝑠) is the fundamental function of (1.1), then (3.9) yields ||𝑋(𝑡,𝑠)||≤𝐾exp−𝜆^𝑡𝑠𝑚_𝑘=1𝑎_𝑘(𝜏)𝑑𝜏.(3.10) Hence |𝑋(𝑡,𝑠)|≤𝐾, 𝑡≥𝑠≥0, which together with lim_𝑡→∞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^{𝑡𝑚𝑡+𝑅}_𝑘=1𝑎_𝑘(𝜏)𝑑𝜏≥𝛼>0,𝑡≥𝑠≥0.(3.11) Hence exp−𝜆^𝑡𝑠𝑚_𝑘=1𝑎_𝑘(𝜏)𝑑𝜏≤exp𝜆𝑅sup_𝑚𝑡≥0_𝑘=1𝑎_𝑘(𝑡)𝑒^{−𝜆𝛼(𝑡−𝑠)/𝑅}.(3.12)
Thus, condition (3.6) implies the exponential estimate for 𝑋(𝑡,𝑠).
The last statement of the theorem is evident.

Remark 3.4. The substitution 𝑠=𝑝(𝑡)∶=∫^𝑡0∑^𝑚𝑘=1𝑎_𝑘(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 equation𝑥̇(𝑡)+^𝑡𝑡₀𝑥(𝑠)𝑑_𝑠𝑟(𝑡,𝑠)=0.(3.13) The ideas of [10] allow to generalize the results of the present paper to equations with a distributed delay.

Corollary 3.5. Suppose 𝑎_𝑘(𝑡)≥0, ∑^𝑚𝑘=1𝑎_𝑘(𝑡)≡𝛼>0, condition (2.12) holds and limsup_{𝑚𝑡→∞}_𝑘=1𝑎_𝑘(𝑡)𝑡−ℎ_𝑘(𝑡)<1+1𝑒.(3.14) Then (1.1) is exponentially stable.

Corollary 3.6. Suppose 𝑎_𝑘(𝑡)=𝛼_𝑘𝑝(𝑡), 𝛼_𝑘>0, 𝑝(𝑡)>0, ∫^∞0𝑝(𝑡)𝑑𝑡=∞ and limsup_{𝑚𝑡→∞}_𝑘=1𝛼_𝑘^𝑡ℎ_𝑘(𝑡)𝑝(𝑠)𝑑𝑠<1+1𝑒.(3.15) Then (1.1) is asymptotically stable.
If in addition there exists 𝑅>0 such that liminf_𝑡→∞^{𝑡𝑡+𝑅}𝑝(𝜏)𝑑𝜏>0,(3.16) 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. Suppose 𝑎(𝑡)≥0, 𝑏(𝑡)≥0, 𝑎(𝑡)+𝑏(𝑡)>0, ^∞0(𝑎(𝑡)+𝑏(𝑡))𝑑𝑡=∞,limsup_𝑡→∞^{𝑡ℎ(𝑡)}limsup_𝑡→∞(𝑎(𝑠)+𝑏(𝑠))𝑑𝑠<∞,𝑏(𝑡)𝑎(𝑡)+𝑏(𝑡)^{𝑡ℎ(𝑡)}(𝑎(𝑠)+𝑏(𝑠))𝑑𝑠<1+1𝑒.(3.17) Then the following equation is asymptotically stable 𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡)+𝑏(𝑡)𝑥(ℎ(𝑡))=0.(3.18)
If in addition there exists 𝑅>0 such that liminf_𝑡→∞∫^{𝑡𝑡+𝑅}(𝑎(𝜏)+𝑏(𝜏))𝑑𝜏>0 then the fundamental function of (3.18) has an exponential estimation.
If also limsup_𝑡→∞(𝑡−ℎ(𝑡))<∞ then (3.18) is exponentially stable.

In the following theorem we will omit the condition ∑^𝑚𝑘=1𝑎_𝑘(𝑡)>0 of Theorem 3.3.

Theorem 3.9. Suppose 𝑎_𝑘(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 that 𝑚=2 and consider the equation 𝑥̇(𝑡)+𝑎(𝑡)𝑥(ℎ(𝑡))+𝑏(𝑡)𝑥(𝑔(𝑡))=0,(3.19) where 𝑎(𝑡)≥0, 𝑏(𝑡)≥0, ∫^∞0(𝑎(𝑠)+𝑏(𝑠))𝑑𝑠=∞ and there exist 𝑡₀≥0, 𝜀>0 such that ^{𝑡min{ℎ(𝑡),𝑔(𝑡)}}(𝑎(𝑠)+𝑏(𝑠))𝑑𝑠<1+1𝑒−𝜀,𝑡≥𝑡⁰.(3.20) Let us find 𝑡₁≥𝑡₀ such that 𝑒^−ℎ(𝑡)<𝜀/4, 𝑒^{−𝑔(𝑡)}<𝜀/4, 𝑡≥𝑡₁, such 𝑡₁ exists due to (a2). Then ∫^{𝑡min{ℎ(𝑡),𝑔(𝑡)}}𝑒^−𝑠𝑑𝑠<𝜀/2, 𝑡≥𝑡₁. Rewrite (3.19) in the form 𝑥̇(𝑡)+𝑎(𝑡)+𝑒^−𝑡𝑥(ℎ(𝑡))+𝑏(𝑡)𝑥(𝑔(𝑡))−𝑒^−𝑡𝑥(ℎ(𝑡))=0,(3.21) where 𝑎(𝑡)+𝑏(𝑡)+𝑒^−𝑡>0. After the substitution 𝑠=∫^𝑡𝑡₁(𝑎(𝜏)+𝑏(𝜏)+𝑒^−𝜏)𝑑𝜏, 𝑦(𝑠)=𝑥(𝑡), (3.21) has the form 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,(3.22) where similar to the proof of Theorem 3.3𝑠−𝑙(𝑠)=^{𝑡ℎ(𝑡)}(𝑎(𝜏)+𝑏(𝜏)+𝑒^−𝜏)𝑑𝜏,𝑠−𝑝(𝑠)=^{𝑡𝑔(𝑡)}(𝑎(𝜏)+𝑏(𝜏)+𝑒^−𝜏)𝑑𝜏.(3.23) First we will show that by Corollary 3.2 the equation 𝑦̇(𝑠)+𝑎(𝑡)+𝑒^−𝑡𝑎(𝑡)+𝑏(𝑡)+𝑒^−𝑡𝑦(𝑙(𝑠))+𝑏(𝑡)𝑎(𝑡)+𝑏(𝑡)+𝑒^−𝑡𝑦(𝑝(𝑠))=0(3.24) 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 ^{𝑠min{𝑙(𝑠),𝑝(𝑠)}}1𝑑𝑠==𝑠−min{𝑙(𝑠),𝑝(𝑠)},𝑠−𝑙(𝑠)=^{𝑡ℎ(𝑡)}(𝑎(𝜏)+𝑏(𝜏)+𝑒^−𝜏)𝑑𝜏_{𝑡ℎ(𝑡)}(𝑎(𝜏)+𝑏(𝜏))𝑑𝜏+^{𝑡ℎ(𝑡)}𝑒^−𝜏𝑑𝜏<1+1𝑒−𝜀+2𝜀=1+1𝑒−2𝜀,𝑡≥𝑡¹.(3.25) The same calculations give 𝑠−𝑝(𝑠)<1+(1/𝑒)−𝜀/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^∞𝑡₁𝑒^−𝑡𝑎(𝑡)+𝑏(𝑡)+𝑒^−𝑡𝑑𝑠=^∞𝑡₁𝑒^−𝑡𝑎(𝑡)+𝑏(𝑡)+𝑒^−𝑡𝑎(𝑡)+𝑏(𝑡)+𝑒^−𝑡𝑑𝑡<∞.(3.26) By Lemma 2.10, (3.22) is exponentially stable. Since 𝑡→∞ implies 𝑠→∞ then lim_𝑡→∞𝑥(𝑡)=lim_𝑠→∞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. Suppose 𝑎(𝑡)≥0, ∫^∞0𝑎(𝑡)𝑑𝑡=∞ and limsup_𝑡→∞^{𝑡ℎ(𝑡)}𝑎(𝑠)𝑑𝑠<1+1𝑒.(3.27) Then the equation 𝑥̇(𝑡)+𝑎(𝑡)𝑥(ℎ(𝑡))=0(3.28) is asymptotically stable. If in addition condition (2.21) holds then the fundamental function of (3.28) has an exponential estimation. If also limsup_𝑡→∞(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 indices 𝐼⊂{1,…,𝑚} such that 𝑎_𝑘(𝑡)≥0, 𝑘∈𝐼, ^∞0_𝑘∈𝐼𝑎_𝑘(𝑡)𝑑𝑡=∞,limsup_𝑡→∞^𝑡ℎ_𝑘(𝑡)_𝑖∈𝐼𝑎_𝑖(𝑠)𝑑𝑠<∞,𝑘=1,…,𝑚,(3.29)_𝑘∉𝐼||𝑎_𝑘(𝑡)||=0,𝑡∈𝐸,limsup_{𝑡→∞,𝑡∉𝐸}∑_𝑘∉𝐼||𝑎_𝑘(𝑡)||∑_𝑘∈𝐼𝑎_𝑘(𝑡)<1,where𝐸=𝑡≥0,_𝑘∈𝐼𝑎_𝑘(𝑡)=0.(3.30)
If the fundamental function 𝑋₀(𝑡,𝑠) of (2.16) is eventually positive then all solutions of (1.1) tend to zero as 𝑡→∞.
If in addition there exists 𝑅>0 such that liminf_𝑡→∞^{𝑡𝑡+𝑅}_𝑘∈𝐼𝑎_𝑘(𝜏)𝑑𝜏>0(3.31) 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 assume 𝑋₀(𝑡,𝑠)>0, 𝑡≥𝑠≥0. Rewrite (1.1) in the form 𝑥̇(𝑡)+_𝑘∈𝐼𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)+_𝑘∉𝐼𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0.(3.32) Suppose first that ∑_𝑘∈𝐼𝑎_𝑘(𝑡)≠0. After the substitution 𝑠=𝑝(𝑡)∶=∫^𝑡0∑_𝑘∈𝐼𝑎_𝑘(𝜏)𝑑𝜏, 𝑦(𝑠)=𝑥(𝑡) we have 𝑥(ℎ_𝑘(𝑡))=𝑦(𝑙_𝑘(𝑠)), 𝑙_𝑘(𝑠)≤𝑠, 𝑙_𝑘(𝑠)=∫^{ℎ𝑘0(𝑡)}∑_𝑖∈𝐼𝑎_𝑖(𝜏)𝑑𝜏, 𝑘=1,…,𝑚, and (1.1) can be rewritten in the form 𝑦̇(𝑠)+^𝑚_𝑘=1𝑏_𝑘(𝑠)𝑦𝑙_𝑘(𝑠)=0,(3.33) where 𝑏_𝑘(𝑠)=𝑎_𝑘(𝑡)/∑_𝑖∈𝐼𝑎_𝑖(𝑡). Denote by 𝑌₀(𝑢,𝑣) the fundamental function of the equation 𝑦̇(𝑠)+_𝑘∈𝐼𝑏_𝑘(𝑠)𝑦𝑙_𝑘(𝑠)=0.(3.34) We have 𝑋₀(𝑡,𝑠)=𝑌₀^𝑡0_𝑘∈𝐼𝑎_𝑘(𝜏)𝑑𝜏,^𝑠0_𝑘∈𝐼𝑌𝑎_𝑘(𝜏)𝑑𝜏₀,(𝑢,𝑣)=𝑋₀𝑝⁻¹(𝑢),𝑝⁻¹(𝑣)>0,𝑢≥𝑣≥0.(3.35)
Let us check that other conditions of Lemma 2.9 hold for (3.33). Since ∑_𝑘∈𝐼𝑏_𝑘(𝑠)=1 then condition (2.15) is satisfied. In addition,limsup_{𝑠→∞,𝑝−1(𝑠)∉𝐸}∑_𝑘∉𝐼||𝑏_𝑘(𝑠)||∑_𝑘∈𝐼𝑏_𝑘(𝑠)=limsup_{𝑡→∞,𝑡∉𝐸}∑_𝑘∉𝐼||𝑎_𝑘(𝑡)||∑_𝑘∈𝐼𝑎_𝑘(𝑡)<1.(3.36) By Lemma 2.9, (3.33) is exponentially stable. Hence for any solution 𝑥(𝑡) of (1.1) we have lim_𝑡→∞𝑥(𝑡)=lim_𝑠→∞𝑦(𝑠)=0. The end of the proof is similar to the proof of Theorem 3.9. In particular, to remove the condition ∑_𝑘∈𝐼𝑎_𝑘(𝑡)≠0 we rewrite the equation by adding the term 𝑒^−𝑡 to one of 𝑎_𝑘(𝑡),𝑘∈𝐼.

Remark 3.12. Explicit positiveness conditions for the fundamental function were presented in Lemma 2.7.

Corollary 3.13. Suppose 𝑎(𝑡)≥0,^∞0𝑎(𝑡)𝑑𝑡=∞,limsup_𝑡→∞^𝑡𝑔_𝑘(𝑡)𝑎(𝑠)𝑑𝑠<∞,_𝑛_𝑘=1||𝑏_𝑘(𝑡)||=0,𝑡∈𝐸,limsup_{𝑡→∞,𝑡∉𝐸}∑^𝑛𝑘=1||𝑏_𝑘(𝑡)||𝑎(𝑡)<1,(3.37) where 𝐸={𝑡≥0,𝑎(𝑡)=0}. Then the equation 𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡)+^𝑛_𝑘=1𝑏_𝑘(𝑡)𝑥𝑔_𝑘(𝑡)=0(3.38) is asymptotically stable. If in addition (2.21) holds then the fundamental function of (3.38) has an exponential estimation. If also limsup_𝑡→∞(𝑡−𝑔_𝑘(𝑡))<∞ then (3.38) is exponentially stable.

Theorem 3.14. Suppose ∫^∞0∑^𝑚𝑘=1|𝑎_𝑘(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 ||𝑋(𝑡,𝑠)||≤exp^𝑡𝑠𝑚_𝑘=1||𝑎_𝑘(𝜏)||𝑑𝜏.(3.39) Then by solution representation formula (2.5) for any solution 𝑥(𝑡) of (1.1) we have ||𝑥(𝑡)||≤exp^𝑡𝑡₀^𝑚_𝑘=1||𝑎_𝑘(𝑠)||𝑑𝑠||𝑥𝑡₀||+^𝑡𝑡₀exp^𝑡𝑠𝑚_𝑘=1||𝑎_𝑘(𝜏)||𝑑𝜏^𝑚_𝑘=1≤exp_∞𝑡0𝑚||𝑎_𝑘(𝑠)||||𝜑ℎ_𝑘(𝑠)||𝑑𝑠_𝑘=1||𝑎_𝑘(𝑠)||𝑑𝑠||𝑥𝑡₀||+^∞𝑡₀^𝑚_𝑘=1||𝑎_𝑘(𝑠)||𝑑𝑠‖𝜑‖,(3.40) where ‖𝜑‖=max_𝑡<0|𝜑(𝑡)|. Then 𝑥(𝑡) is a bounded function.
Moreover, |𝑋(𝑡,𝑠)|≤𝐴∶=exp{∫^∞0∑^𝑚𝑘=1|𝑎_𝑘(𝑠)|𝑑𝑠}, 𝑡≥𝑠≥0. Let us choose 𝑡₀≥0 such that ∫^∞𝑡₀∑^𝑚𝑘=1|𝑎_𝑘(𝑠)|𝑑𝑠<1/(2𝐴), then 𝑋^𝑡(𝑡,𝑡₀)+∑^𝑚𝑘=1𝑎_𝑘(𝑡)𝑋(ℎ_𝑘(𝑡),𝑡₀)=0, 𝑋(𝑡₀,𝑡₀)=1 implies 𝑋(𝑡,𝑡₀)≥1−∫^∞𝑡₀∑^𝑚𝑘=1|𝑎_𝑘(𝑠)|𝐴𝑑𝑠>1−𝐴(1/(2𝐴))=1/2, thus 𝑋(𝑡,𝑡₀) does not tend to zero, so (1.1) is not asymptotically stable.

Theorems 3.11 and 3.14 imply the following results.

Corollary 3.15. Suppose 𝑎_𝑘(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|𝑎_𝑘(𝑡)|𝑑𝑡=∞, then all solutions of (1.1) are bounded by Theorem 3.11. Let ∫^∞0∑_𝑘∈𝐼|𝑎_𝑘(𝑡)|𝑑𝑡<∞. By (3.30) we have ∫^∞0∑_𝑘∉𝐼|𝑎_𝑘(𝑡)|𝑑𝑡≤∫^∞0∑_𝑘∈𝐼|𝑎_𝑘(𝑡)|𝑑𝑡<∞. Then ∫^∞0∑^𝑚𝑘=1|𝑎_𝑘(𝑡)|𝑑𝑡<∞. By Theorem 3.14 all solutions of (1.1) are bounded.

Theorem 3.16. Suppose 𝑎_𝑘(𝑡)≥0. If (1.1) is asymptotically stable, then the ordinary differential equation 𝑥̇(𝑡)+^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥(𝑡)=0(3.41) 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{−∫^𝑡𝑡₀∑^𝑚𝑘=1𝑎_𝑘(𝑠)𝑑𝑠}, so (3.41) is asymptotically stable, as far as ^∞0𝑚_𝑘=1𝑎_𝑘(𝑠)𝑑𝑠=∞(3.42) 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 {𝑡_𝑛}, 𝑡_𝑛→∞, such that^{𝑡𝑛𝑡+𝑛}_𝑛^𝑚_𝑘=1𝑎_𝑘(𝜏)𝑑𝜏<1𝑛<1𝑒,𝑛≥3.(3.43) By (2.12) there exists 𝑛₀>3 such that 𝑡−ℎ_𝑘(𝑡)≤𝑛₀, 𝑘=1,…,𝑚. Lemma 2.7 implies 𝑋(𝑡,𝑠)>0, 𝑡_𝑛≤𝑠≤𝑡≤𝑡_𝑛+𝑛, 𝑛≥𝑛₀. Similar to the proof of Theorem 3.14 and using the inequality 1−𝑥≥𝑒^−𝑥, 𝑥>0, we obtain 𝑋𝑡_𝑛,𝑡_𝑛+𝑛≥1−^{𝑡𝑛𝑡+𝑛}_𝑛^𝑚_𝑘=1𝑎_𝑘(𝜏)𝑑𝜏≥exp−^{𝑡𝑛𝑡+𝑛}_𝑛^𝑚_𝑘=1𝑎_𝑘(𝜏)𝑑𝜏>𝑒^−1/𝑛.(3.44) Inequality (2.7) implies |𝑋(𝑡_𝑛+𝑛,𝑡_𝑛)|≤𝐾𝑒^−𝜆𝑛. Hence 𝐾𝑒^−𝜆𝑛≥𝑒^−1/𝑛, 𝑛≥𝑛₀, or 𝐾>𝑒^{𝜆𝑛−1/3} for any 𝑛≥𝑛₀. The contradiction proves the theorem.

Theorems 3.11 and 3.16 imply the following statement.

Corollary 3.17. Suppose 𝑎_𝑘(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]). Suppose 𝑎(𝑡)≥0, ℎ(𝑡)≤𝑡 are continuous functions and limsup_𝑡→∞^{𝑡ℎ(𝑡)}𝑎(𝑠)𝑑𝑠≤32.(4.1) Then all solutions of (3.28) are bounded.
If in additionliminf_𝑡→∞^{𝑡ℎ(𝑡)}𝑎(𝑠)𝑑𝑠>0,(4.2) and the strict inequality in (4.1) holds then (3.28) is exponentially stable.

Statement 2 (see [7]). Suppose 𝑎(𝑡)≥0, ℎ(𝑡)≤𝑡 are continuous functions, the strict inequality (4.1) holds and ∫^∞0𝑎(𝑠)𝑑𝑠=∞. Then all solutions of (3.28) tend to zero as 𝑡→∞.

Statement 3 (see [9, 10]). Suppose 𝑎(𝑡)≥0, ℎ(𝑡)≤𝑡 are measurable functions, ∫^∞0𝑎(𝑠)𝑑𝑠=∞, 𝐴(𝑡)=∫^𝑡0𝑎(𝑠)𝑑𝑠 is a strictly monotone increasing function and limsup_𝑡→∞^{𝑡ℎ(𝑡)}𝑎(𝑠)𝑑𝑠<sup_{0<𝜔<𝜋/2}𝜔+Φ1(𝜔)≈1.45…,(4.3)Φ(𝜔)=∫^∞0𝑢(𝑡,𝜔)𝑑𝑡, where 𝑢(𝑡,𝜔) is a solution of the initial value problem 𝑦̇(𝑡)+𝑦(𝑡−𝜔)=0,𝑦(𝑡)=0,𝑡<0,𝑦(0)=1.(4.4) Then (3.28) is asymptotically stable.

Note that instead of the equation 𝑦(̇𝑡)+𝑦(𝑡−𝜔)=0 with a constant delay, the equation𝑦̇(𝑡)+𝑦(𝑡−𝜏(𝑡))=0(4.5) 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]). Let 𝜏(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 delay 𝜏(𝑡) can be in the neighbourhood of 𝜔 which is close to 1.

Example 4.1. Consider the equation 𝑥̇(𝑡)+𝛼||sin𝑡||−sin𝑡𝑥(ℎ(𝑡))=0,ℎ(𝑡)≤𝑡,(4.6) 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 ∫^∞0𝑎(𝑠)𝑑𝑠=∞. We have limsup_𝑡→∞^{𝑡ℎ(𝑡)}𝑎(𝑠)𝑑𝑠≤limsup_𝑡→∞^{𝑡𝑡−𝜋}𝑎(𝑠)𝑑𝑠≤−𝛼^𝜋2𝜋2sin𝑠𝑑𝑠=4𝛼.(4.7) If 𝛼<0.25(1+1/𝑒), then condition (3.27) holds, hence all solutions of (4.6) tend to zero as 𝑡→∞.

Statements 1–3 fail for this equation. In Statements 1 and 2 the delay should be continuous. In Statement 3 function 𝐴(𝑡)=∫^𝑡0𝑎(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]). Suppose 𝑎_𝑘(𝑡)≥0, ℎ_𝑘(𝑡)≤𝑡 are continuous functions and limsup_𝑡→∞𝑎_𝑘(𝑡)limsup_𝑡→∞𝑡−ℎ_𝑘(𝑡)≤1.(4.8) Then all solutions of (1.1) are bounded and 1 in the right-hand side of (4.8) is the best possible constant.
If ∑^𝑚𝑘=1𝑎_𝑘(𝑡)>0 and the strict inequality in (4.8) is valid then all solutions of (1.1) tend to zero as 𝑡→∞.
If 𝑎_𝑘(𝑡) are constants then in (4.8) the number 1 can be replaced by 3/2.

Statement 7 (see [7]). Suppose 𝑎_𝑘(𝑡)≥0, ℎ_𝑘(𝑡)≤𝑡 are continuous, ℎ₁(𝑡)≤ℎ₂(𝑡)≤⋯≤ℎ_𝑚(𝑡) and limsup_𝑡→∞^𝑡ℎ_1𝑚(𝑡)_𝑘=1𝑎_𝑘(𝑠)𝑑𝑠≤32.(4.9) Then any solution of (1.1) tends to a constant as 𝑡→∞.
If in addition ∫^∞0∑^𝑚𝑘=1𝑎_𝑘(𝑠)𝑑𝑠=∞, then all solutions of (1.1) tend to zero as 𝑡→∞.

Example 4.2. Consider the equation 𝑥̇(𝑡)+𝛼𝑡𝑥2𝑡−sin𝑡+𝛽𝑡𝑥2𝑡=0,𝑡≥𝑡⁰>0,(4.10) where 𝛼>0, 𝛽>0. Denote 𝑝(𝑡)=1/𝑡, ℎ(𝑡)=𝑡/2−sin𝑡, 𝑔(𝑡)=𝑡/2.

We apply Corollary 3.6. Since lim_𝑡→∞[ln(𝑡/2)−ln(𝑡/2−sin𝑡)]=0, then limsup_𝑡→∞𝛼^{𝑡ℎ(𝑡)}𝑝(𝑠)𝑑𝑠+𝛽^{𝑡𝑔(𝑡)}𝑝(𝑠)𝑑𝑠≤(𝛼+𝛽)ln2.(4.11) 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]). Suppose 𝑎(𝑡), 𝑏_𝑘(𝑡), 𝑡−ℎ_𝑘(𝑡) are bounded continuous functions, there exist 𝛿, 𝑘, 𝛿>0, 0<𝑘<1, such that 𝑎(𝑡)≥𝛿 and ∑^𝑚𝑘=1|𝑏_𝑘(𝑡)|<𝑘𝛿. Then the equation 𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡)+^𝑚_𝑘=1𝑏_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0(4.12) is exponentially stable.

In Corollary 3.13 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 𝑥̇(𝑡)+1𝑡𝑥(𝑡)+𝛼𝑡𝑥2𝑡=0,𝑡≥𝑡⁰>0.(4.13) 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 Theorem 3.11 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 constant 1+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 limsup_{𝑚𝑡→∞}_𝑘=1||𝑎_𝑘(𝑡)||∑^𝑚𝑖=1𝑎_𝑖(𝑡)^𝑡ℎ_{𝑘𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠<1+1𝑒.(4.14) 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 𝑎_𝑘(𝑡)≥0, as in Theorem 3.16. The solution of this problem would improve Theorem 3.16.

Acknowledgments

L. Berezansky is partially supported by Israeli Ministry of Absorption. E. Braverman is partially supported by the NSERC Research Grant.