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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 178568, 19 pages
http://dx.doi.org/10.1155/2011/178568
Research Article

New Stability Conditions for Linear Differential Equations with Several Delays

1Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
2Department of Mathematics and Statistics, University of Calgary, 2500 University Drive Northwest, Calgary, AB, Canada T2N 1N4

Received 28 January 2011; Accepted 4 April 2011

Academic Editor: JosefΒ DiblΓ­k

Copyright Β© 2011 Leonid Berezansky and Elena Braverman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation βˆ‘Μ‡π‘₯(𝑑)+π‘šπ‘˜=1π‘Žπ‘˜(𝑑)π‘₯(β„Žπ‘˜(𝑑))=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 coefficientsΜ‡π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=0,𝑑β‰₯𝑑0,(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: ||||ξ€·||||ξ€Έ1Μ‡π‘₯(𝑑)+sin𝑑π‘₯(π‘‘βˆ’πœ)=0,Μ‡π‘₯(𝑑)+𝛼sinπ‘‘βˆ’sin𝑑π‘₯(π‘‘βˆ’πœ)=0,Μ‡π‘₯(𝑑)+𝑑𝛼π‘₯(𝑑)+𝑑π‘₯𝑑2=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 Μ‡π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜(𝑑)π‘₯(𝑑)=0(1.3) is also exponentially stable.

2. Preliminaries

We consider the scalar linear equation with several delays (1.1) for 𝑑β‰₯𝑑0 with the initial conditions (for any 𝑑0β‰₯0)π‘₯(𝑑)=πœ‘(𝑑),𝑑<𝑑0𝑑,π‘₯0ξ€Έ=π‘₯0,(2.1) and under the following assumptions:(a1)π‘Žπ‘˜(𝑑) are Lebesgue measurable essentially bounded on [0,∞) functions;(a2)β„Žπ‘˜(𝑑) are Lebesgue measurable functions, β„Žπ‘˜(𝑑)≀𝑑,limsupπ‘‘β†’βˆžβ„Žπ‘˜(𝑑)=∞;(2.2)(a3)πœ‘βˆΆ(βˆ’βˆž,𝑑0)→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 𝑑β‰₯𝑑0 function π‘₯βˆΆπ‘…β†’π‘… is called a solution of problem (1.1), (2.1) if it satisfies (1.1) for almost all π‘‘βˆˆ[𝑑0,∞) and the equalities (2.1) for 𝑑≀𝑑0.

Below we present a solution representation formula for the nonhomogeneous equation with locally Lebesgue integrable right-hand side 𝑓(𝑑):Μ‡π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=𝑓(𝑑),𝑑β‰₯𝑑0.(2.3)

Definition 2.2. A solution 𝑋(𝑑,𝑠) of the problem Μ‡π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(t)π‘₯π‘˜ξ€Έ(𝑑)=0,𝑑β‰₯𝑠β‰₯0,π‘₯(𝑑)=0,𝑑<𝑠,π‘₯(𝑠)=1,(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 ξ€·π‘₯(𝑑)=𝑋𝑑,𝑑0ξ€Έπ‘₯0βˆ’ξ€œπ‘‘π‘‘0𝑋(𝑑,𝑠)π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(𝑠)πœ‘π‘˜ξ€Έξ€œ(𝑠)𝑑𝑠+𝑑𝑑0𝑋(𝑑,𝑠)𝑓(𝑠)𝑑𝑠,(2.5) where πœ‘(𝑑)=0, 𝑑β‰₯𝑑0.

Definition 2.4 (see [22]). Equation (1.1) is stable if for any initial point 𝑑0 and number πœ€>0 there exists 𝛿>0 such that the inequality sup𝑑<𝑑0|πœ‘(𝑑)|+|π‘₯(𝑑0)|<𝛿 implies |π‘₯(𝑑)|<πœ€, 𝑑β‰₯𝑑0, 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 𝑑0 tend to zero as π‘‘β†’βˆž.

In particular, (1.1) is asymptotically stable if the fundamental function is uniformly bounded: |𝑋(𝑑,𝑠)|≀𝐾, 𝑑β‰₯𝑠β‰₯0 and all solutions tend to zero as π‘‘β†’βˆž.

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, πœ‡>0 such that the solution of problem (1.1), (2.1) has the estimate ||||π‘₯(𝑑)β‰€π‘€π‘’βˆ’πœ‡(π‘‘βˆ’π‘‘0)||π‘₯𝑑0ξ€Έ||+sup𝑑<𝑑0||||ξƒͺπœ‘(𝑑),𝑑β‰₯𝑑0,(2.6) where 𝑀 and πœ‡ do not depend on 𝑑0.

Definition 2.6. The fundamental function 𝑋(𝑑,𝑠) of (1.1) has an exponential estimation if there exist 𝐾>0, πœ†>0 such that ||||𝑋(𝑑,𝑠)β‰€πΎπ‘’βˆ’πœ†(π‘‘βˆ’π‘ ),𝑑β‰₯𝑠β‰₯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 π‘Žπ‘˜(𝑑)β‰₯0. If ξ€œπ‘‘ξ€½maxβ„Ž(𝑑),𝑑0ξ€Ύπ‘šξ“π‘–=1π‘Žπ‘–1(𝑠)𝑑𝑠≀𝑒,β„Ž(𝑑)=minπ‘˜ξ€½β„Žπ‘˜ξ€Ύ(𝑑),𝑑β‰₯𝑑0,(2.8) or there exists πœ†>0, such that πœ†β‰₯π‘šξ“π‘˜=1π΄π‘˜π‘’πœ†πœŽπ‘˜,(2.9) where 0β‰€π‘Žπ‘˜(𝑑)β‰€π΄π‘˜,π‘‘βˆ’β„Žπ‘˜(𝑑)β‰€πœŽπ‘˜,𝑑β‰₯𝑑0,(2.10) then 𝑋(𝑑,𝑠)>0, 𝑑β‰₯𝑠β‰₯𝑑0, where 𝑋(𝑑,s) is the fundamental function of (1.1).

Lemma 2.8 (see [3]). Suppose π‘Žπ‘˜(𝑑)β‰₯0, liminfπ‘šπ‘‘β†’βˆžξ“π‘˜=1π‘Žπ‘˜(𝑑)>0,(2.11)limsupπ‘‘β†’βˆžξ€·π‘‘βˆ’β„Žπ‘˜(𝑑)<∞,π‘˜=1,…,π‘š,(2.12) and there exists π‘Ÿ(𝑑)≀𝑑 such that for sufficiently large π‘‘ξ€œπ‘‘π‘šπ‘Ÿ(𝑑)ξ“π‘˜=1π‘Žπ‘˜1(𝑠)𝑑𝑠≀𝑒.(2.13)
If limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1π‘Žπ‘˜(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–|||||ξ€œ(𝑑)β„Žπ‘Ÿ(𝑑)π‘˜π‘š(𝑑)𝑖=1π‘Žπ‘–|||||(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 πΌβŠ‚{1,…,π‘š}, such that π‘Žπ‘˜(𝑑)β‰₯0, π‘˜βˆˆπΌ, liminfπ‘‘β†’βˆžξ“π‘˜βˆˆπΌπ‘Žπ‘˜(𝑑)>0,(2.15) and the fundamental function of the equation ̇π‘₯(𝑑)+π‘˜βˆˆπΌπ‘Žπ‘˜(ξ€·β„Žπ‘‘)π‘₯π‘˜(𝑑)=0(2.16) is eventually positive. If limsupπ‘‘β†’βˆžβˆ‘π‘˜βˆ‰πΌ||π‘Žπ‘˜||(𝑑)βˆ‘π‘˜βˆˆπΌπ‘Žπ‘˜(𝑑)<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 ξ€œβˆž0π‘›ξ“π‘˜=1||π‘π‘˜||(𝑠)𝑑𝑠<∞,limsupπ‘‘β†’βˆžξ€·π‘‘βˆ’π‘”π‘˜ξ€Έ(𝑑)<∞,π‘”π‘˜(𝑑)≀𝑑,(2.18) then the equation Μ‡π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ+(𝑑)π‘›ξ“π‘˜=1π‘π‘˜ξ€·π‘”(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=0(2.19) is exponentially stable.

The following elementary result will be used in the paper.

Lemma 2.11. The ordinary differential equation Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(𝑑)=0(2.20) is exponentially stable if and only if there exists 𝑅>0 such that liminfπ‘‘β†’βˆžξ€œπ‘‘π‘‘+π‘…π‘Ž(𝑠)𝑑𝑠>0.(2.21)

The following example illustrates that a stronger than (2.21) sufficient conditionliminf𝑑,π‘ β†’βˆž1ξ€œπ‘‘βˆ’π‘ π‘‘π‘ π‘Ž(𝜏)π‘‘πœ>0(2.22) is not necessary for the exponential stability of the ordinary differential equation (2.20).

Example 2.12. Consider the equation ξƒ―[[Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(𝑑)=0,whereπ‘Ž(𝑑)=1,π‘‘βˆˆ2𝑛,2𝑛+1),0,π‘‘βˆˆ2𝑛+1,2𝑛+2),𝑛=0,1,2,…(2.23) Then liminf in (2.22) equals zero, but |𝑋(𝑑,𝑠)|<π‘’π‘’βˆ’0.5(π‘‘βˆ’π‘ ), so the equation is exponentially stable. Moreover, if we consider liminf in (2.22) under the condition π‘‘βˆ’π‘ β‰₯𝑅, then it is still zero for any 𝑅≀1.

3. Main Results

Lemma 3.1. Suppose π‘Žπ‘˜(𝑑)β‰₯0, (2.11), (2.12) hold and limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1π‘Žπ‘˜(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–ξ€œ(𝑑)π‘‘β„Žπ‘˜π‘š(𝑑)𝑖=1π‘Žπ‘–1(𝑠)𝑑𝑠<1+𝑒.(3.1) Then (1.1) is exponentially stable.

Proof. By (2.11) there exists function π‘Ÿ(𝑑)≀𝑑 such that for sufficiently large π‘‘ξ€œπ‘‘π‘šπ‘Ÿ(𝑑)ξ“π‘˜=1π‘Žπ‘˜1(𝑠)𝑑𝑠=𝑒.(3.2) For this function condition (2.14) has the form limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1π‘Žπ‘˜(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–|||||ξ€œ(𝑑)π‘‘β„Žπ‘˜π‘š(𝑑)𝑖=1π‘Žπ‘–ξ€œ(𝑠)π‘‘π‘ βˆ’π‘‘π‘šπ‘Ÿ(𝑑)𝑖=1π‘Žπ‘–|||||(𝑠)𝑑𝑠=limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1π‘Žπ‘˜(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–|||||ξ€œ(𝑑)π‘‘β„Žπ‘˜π‘š(𝑑)𝑖=1π‘Žπ‘–1(𝑠)π‘‘π‘ βˆ’π‘’|||||<1.(3.3) The latter inequality follows from (3.1). The reference to Lemma 2.8 completes the proof.

Corollary 3.2. Suppose π‘Žπ‘˜(𝑑)β‰₯0, (2.11), (2.12) hold and limsupπ‘‘β†’βˆžξ€œπ‘‘minπ‘˜{β„Žπ‘˜π‘š(𝑑)}𝑖=1π‘Žπ‘–1(𝑠)𝑑𝑠<1+𝑒.(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(𝑑)>0 in the space of Lebesgue measurable essentially bounded functions means 𝑏(𝑑)β‰₯0 and 𝑏(𝑑)β‰ 0 almost everywhere.

Theorem 3.3. Suppose π‘Žπ‘˜(𝑑)β‰₯0, condition (3.1) holds, βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)>0 and ξ€œβˆž0π‘šξ“π‘˜=1π‘Žπ‘˜(𝑑)𝑑𝑑=∞,limsupπ‘‘β†’βˆžξ€œπ‘‘β„Žπ‘˜π‘š(𝑑)𝑖=1π‘Žπ‘–(𝑠)𝑑𝑠<∞.(3.5) Then (1.1) is asymptotically stable.
If in addition there exists 𝑅>0 such that liminfπ‘‘β†’βˆžξ€œπ‘‘π‘šπ‘‘+π‘…ξ“π‘˜=1π‘Žπ‘˜(𝜏)π‘‘πœ>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 βˆ«π‘ =𝑝(𝑑)∢=𝑑0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝜏)π‘‘πœ, 𝑦(𝑠)=π‘₯(𝑑), where 𝑝(𝑑) is a strictly increasing function. Then π‘₯(β„Žπ‘˜(𝑑))=𝑦(π‘™π‘˜(𝑠)), π‘™π‘˜(𝑠)≀𝑠, π‘™π‘˜βˆ«(𝑠)=β„Žπ‘˜0(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–(𝜏)π‘‘πœ and (1.1) can be rewritten in the form ̇𝑦(𝑠)+π‘šξ“π‘˜=1π‘π‘˜ξ€·π‘™(𝑠)π‘¦π‘˜ξ€Έ(𝑠)=0,(3.7) where π‘π‘˜(𝑠)=π‘Žπ‘˜βˆ‘(𝑑)/π‘šπ‘–=1π‘Žπ‘–(𝑑), π‘ βˆ’π‘™π‘˜βˆ«(𝑠)=π‘‘β„Žπ‘˜(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–(𝜏)π‘‘πœ. Since βˆ‘π‘šπ‘˜=1π‘π‘˜(𝑠)=1 and limsupπ‘ β†’βˆž(π‘ βˆ’π‘™π‘˜(𝑠))<∞, then Lemma 3.1 can be applied to (3.7). We have limsupπ‘šπ‘ β†’βˆžξ“π‘˜=1π‘π‘˜(𝑠)βˆ‘π‘šπ‘–=1π‘π‘–ξ€œ(𝑠)π‘ π‘™π‘˜π‘š(𝑠)𝑖=1𝑏𝑖(𝜏)π‘‘πœ=limsupπ‘šπ‘ β†’βˆžξ“π‘˜=1π‘π‘˜ξ€·(𝑠)π‘ βˆ’π‘™π‘˜ξ€Έ(𝑠)=limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1π‘Žπ‘˜(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–ξ€œ(𝑑)π‘‘β„Žπ‘˜π‘š(𝑑)𝑖=1π‘Žπ‘–1(𝑠)𝑑𝑠<1+𝑒.(3.8) By Lemma 3.1, (3.7) is exponentially stable. Due to the first equality in (3.5) π‘‘β†’βˆž implies π‘ β†’βˆž. Hence limπ‘‘β†’βˆžπ‘₯(𝑑)=limπ‘ β†’βˆžπ‘¦(𝑠)=0.
Equation (3.7) is exponentially stable, thus the fundamental function π‘Œ(𝑒,𝑣) of (3.7) has an exponential estimation||||π‘Œ(𝑒,𝑣)β‰€πΎπ‘’βˆ’πœ†(π‘’βˆ’π‘£),𝑒β‰₯𝑣β‰₯0,(3.9) with 𝐾>0, πœ†>0. Since βˆ«π‘‹(𝑑,𝑠)=π‘Œ(𝑑0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜βˆ«(𝜏)π‘‘πœ,𝑠0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝜏)π‘‘πœ), where 𝑋(𝑑,𝑠) is the fundamental function of (1.1), then (3.9) yields ||||ξƒ―ξ€œπ‘‹(𝑑,𝑠)≀𝐾expβˆ’πœ†π‘‘π‘ π‘šξ“π‘˜=1π‘Žπ‘˜ξƒ°.(𝜏)π‘‘πœ(3.10) Hence |𝑋(𝑑,𝑠)|≀𝐾, 𝑑β‰₯𝑠β‰₯0, which together with limπ‘‘β†’βˆžπ‘₯(𝑑)=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 haveξ€œπ‘‘π‘šπ‘‘+π‘…ξ“π‘˜=1π‘Žπ‘˜(𝜏)π‘‘πœβ‰₯𝛼>0,𝑑β‰₯𝑠β‰₯0.(3.11) Hence ξƒ―ξ€œexpβˆ’πœ†π‘‘π‘ π‘šξ“π‘˜=1π‘Žπ‘˜ξƒ°ξƒ―(𝜏)π‘‘πœβ‰€expπœ†π‘…supπ‘šπ‘‘β‰₯0ξ“π‘˜=1π‘Žπ‘˜ξƒ°π‘’(𝑑)βˆ’πœ†π›Ό(π‘‘βˆ’π‘ )/𝑅.(3.12)
Thus, condition (3.6) implies the exponential estimate for 𝑋(𝑑,𝑠).
The last statement of the theorem is evident.

Remark 3.4. The substitution βˆ«π‘ =𝑝(𝑑)∢=𝑑0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝜏)π‘‘πœ, 𝑦(𝑠)=π‘₯(𝑑) was first used in [28].

Note that in [10, Lemma 2] this idea was extended to a more general equationξ€œΜ‡π‘₯(𝑑)+𝑑𝑑0π‘₯(𝑠)π‘‘π‘ π‘Ÿ(𝑑,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 π‘Žπ‘˜(𝑑)β‰₯0, βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)≑𝛼>0, condition (2.12) holds and limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1π‘Žπ‘˜ξ€·(𝑑)π‘‘βˆ’β„Žπ‘˜ξ€Έ1(𝑑)<1+𝑒.(3.14) Then (1.1) is exponentially stable.

Corollary 3.6. Suppose π‘Žπ‘˜(𝑑)=π›Όπ‘˜π‘(𝑑), π›Όπ‘˜>0, 𝑝(𝑑)>0, ∫∞0𝑝(𝑑)𝑑𝑑=∞ and limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1π›Όπ‘˜ξ€œπ‘‘β„Žπ‘˜(𝑑)1𝑝(𝑠)𝑑𝑠<1+𝑒.(3.15) Then (1.1) is asymptotically stable.
If in addition there exists 𝑅>0 such that liminfπ‘‘β†’βˆžξ€œπ‘‘π‘‘+𝑅𝑝(𝜏)π‘‘πœ>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 π‘Ž(𝑑)β‰₯0, 𝑏(𝑑)β‰₯0, π‘Ž(𝑑)+𝑏(𝑑)>0, ξ€œβˆž0(π‘Ž(𝑑)+𝑏(𝑑))𝑑𝑑=∞,limsupπ‘‘β†’βˆžξ€œπ‘‘β„Ž(𝑑)(π‘Ž(𝑠)+𝑏(𝑠))𝑑𝑠<∞,limsupπ‘‘β†’βˆžπ‘(𝑑)ξ€œπ‘Ž(𝑑)+𝑏(𝑑)π‘‘β„Ž(𝑑)1(π‘Ž(𝑠)+𝑏(𝑠))𝑑𝑠<1+𝑒.(3.17) Then the following equation is asymptotically stable Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(𝑑)+𝑏(𝑑)π‘₯(β„Ž(𝑑))=0.(3.18)
If in addition there exists 𝑅>0 such that liminfπ‘‘β†’βˆžβˆ«π‘‘π‘‘+𝑅(π‘Ž(𝜏)+𝑏(𝜏))π‘‘πœ>0 then the fundamental function of (3.18) has an exponential estimation.
If also limsupπ‘‘β†’βˆž(π‘‘βˆ’β„Ž(𝑑))<∞ then (3.18) is exponentially stable.

In the following theorem we will omit the condition βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)>0 of Theorem 3.3.

Theorem 3.9. Suppose π‘Žπ‘˜(𝑑)β‰₯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 π‘š=2 and consider the equation Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(β„Ž(𝑑))+𝑏(𝑑)π‘₯(𝑔(𝑑))=0,(3.19) where π‘Ž(𝑑)β‰₯0, 𝑏(𝑑)β‰₯0, ∫∞0(π‘Ž(𝑠)+𝑏(𝑠))𝑑𝑠=∞ and there exist 𝑑0β‰₯0, πœ€>0 such that ξ€œπ‘‘min{β„Ž(𝑑),𝑔(𝑑)}(1π‘Ž(𝑠)+𝑏(𝑠))𝑑𝑠<1+π‘’βˆ’πœ€,𝑑β‰₯𝑑0.(3.20) Let us find 𝑑1β‰₯𝑑0 such that π‘’βˆ’β„Ž(𝑑)<πœ€/4, π‘’βˆ’π‘”(𝑑)<πœ€/4, 𝑑β‰₯𝑑1, such 𝑑1 exists due to (a2). Then βˆ«π‘‘min{β„Ž(𝑑),𝑔(𝑑)}π‘’βˆ’π‘ π‘‘π‘ <πœ€/2, 𝑑β‰₯𝑑1. Rewrite (3.19) in the form ξ€·π‘ŽΜ‡π‘₯(𝑑)+(𝑑)+π‘’βˆ’π‘‘ξ€Έπ‘₯(β„Ž(𝑑))+𝑏(𝑑)π‘₯(𝑔(𝑑))βˆ’π‘’βˆ’π‘‘π‘₯(β„Ž(𝑑))=0,(3.21) where π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘>0. After the substitution βˆ«π‘ =𝑑𝑑1(π‘Ž(𝜏)+𝑏(𝜏)+π‘’βˆ’πœ)π‘‘πœ, 𝑦(𝑠)=π‘₯(𝑑), (3.21) has the form ̇𝑦(𝑠)+π‘Ž(𝑑)+π‘’βˆ’π‘‘π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘π‘¦(𝑙(𝑠))+𝑏(𝑑)π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘π‘’π‘¦(𝑝(𝑠))βˆ’βˆ’π‘‘π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘π‘¦(𝑙(𝑠))=0,(3.22) where similar to the proof of Theorem 3.3ξ€œπ‘ βˆ’π‘™(𝑠)=π‘‘β„Ž(𝑑)(π‘Ž(𝜏)+𝑏(𝜏)+π‘’βˆ’πœ)ξ€œπ‘‘πœ,π‘ βˆ’π‘(𝑠)=𝑑𝑔(𝑑)(π‘Ž(𝜏)+𝑏(𝜏)+π‘’βˆ’πœ)π‘‘πœ.(3.23) First we will show that by Corollary 3.2 the equation ̇𝑦(𝑠)+π‘Ž(𝑑)+π‘’βˆ’π‘‘π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘π‘¦(𝑙(𝑠))+𝑏(𝑑)π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘π‘¦(𝑝(𝑠))=0(3.24) is exponentially stable. Since (π‘Ž(𝑑)+π‘’βˆ’π‘‘)/(π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘)+𝑏(𝑑)/(π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘)=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 ξ€œπ‘ min{𝑙(𝑠),𝑝(𝑠)}ξ€œ1𝑑𝑠=π‘ βˆ’min{𝑙(𝑠),𝑝(𝑠)},π‘ βˆ’π‘™(𝑠)=π‘‘β„Ž(𝑑)(π‘Ž(𝜏)+𝑏(𝜏)+π‘’βˆ’πœ)=ξ€œπ‘‘πœπ‘‘β„Ž(𝑑)ξ€œ(π‘Ž(𝜏)+𝑏(𝜏))π‘‘πœ+π‘‘β„Ž(𝑑)π‘’βˆ’πœ1π‘‘πœ<1+π‘’πœ€βˆ’πœ€+21=1+π‘’βˆ’πœ€2,𝑑β‰₯𝑑1.(3.25) The same calculations give π‘ βˆ’π‘(𝑠)<1+(1/𝑒)βˆ’πœ€/2, thus condition (3.4) holds.
Hence (3.24) is exponentially stable.
We return now to (3.22), 𝑑β‰₯𝑑1. We have 𝑑𝑠=(π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘)𝑑𝑑, thenξ€œβˆžπ‘‘1π‘’βˆ’π‘‘π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘ξ€œπ‘‘π‘ =βˆžπ‘‘1π‘’βˆ’π‘‘π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘ξ€·π‘Ž(𝑑)+𝑏(𝑑)+π‘’βˆ’π‘‘ξ€Έπ‘‘π‘‘<∞.(3.26) By Lemma 2.10, (3.22) is exponentially stable. Since π‘‘β†’βˆž implies π‘ β†’βˆž then limπ‘‘β†’βˆžπ‘₯(𝑑)=limπ‘ β†’βˆžπ‘¦(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 π‘Ž(𝑑)β‰₯0, ∫∞0π‘Ž(𝑑)𝑑𝑑=∞ and limsupπ‘‘β†’βˆžξ€œπ‘‘β„Ž(𝑑)1π‘Ž(𝑠)𝑑𝑠<1+𝑒.(3.27) Then the equation Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(β„Ž(𝑑))=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 limsupπ‘‘β†’βˆž(π‘‘βˆ’β„Ž(𝑑))<∞ 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 πΌβŠ‚{1,…,π‘š} such that π‘Žπ‘˜(𝑑)β‰₯0, π‘˜βˆˆπΌ, ξ€œβˆž0ξ“π‘˜βˆˆπΌπ‘Žπ‘˜(𝑑)𝑑𝑑=∞,limsupπ‘‘β†’βˆžξ€œπ‘‘β„Žπ‘˜(𝑑)ξ“π‘–βˆˆπΌπ‘Žπ‘–(𝑠)𝑑𝑠<∞,π‘˜=1,…,π‘š,(3.29)π‘˜βˆ‰πΌ||π‘Žπ‘˜||(𝑑)=0,π‘‘βˆˆπΈ,limsupπ‘‘β†’βˆž,π‘‘βˆ‰πΈβˆ‘π‘˜βˆ‰πΌ||π‘Žπ‘˜(||𝑑)βˆ‘π‘˜βˆˆπΌπ‘Žπ‘˜ξƒ―ξ“(𝑑)<1,where𝐸=𝑑β‰₯0,π‘˜βˆˆπΌπ‘Žπ‘˜ξƒ°.(𝑑)=0(3.30)
If the fundamental function 𝑋0(𝑑,𝑠) of (2.16) is eventually positive then all solutions of (1.1) tend to zero as π‘‘β†’βˆž.
If in addition there exists 𝑅>0 such that liminfπ‘‘β†’βˆžξ€œπ‘‘π‘‘+π‘…ξ“π‘˜βˆˆπΌπ‘Žπ‘˜(𝜏)π‘‘πœ>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(𝑑,𝑠)>0, 𝑑β‰₯𝑠β‰₯0. Rewrite (1.1) in the form ̇π‘₯(𝑑)+π‘˜βˆˆπΌπ‘Žπ‘˜(ξ€·β„Žπ‘‘)π‘₯π‘˜(ξ€Έ+𝑑)π‘˜βˆ‰πΌπ‘Žπ‘˜(ξ€·β„Žπ‘‘)π‘₯π‘˜(𝑑)=0.(3.32) Suppose first that βˆ‘π‘˜βˆˆπΌπ‘Žπ‘˜(𝑑)β‰ 0. After the substitution βˆ«π‘ =𝑝(𝑑)∢=𝑑0βˆ‘π‘˜βˆˆπΌπ‘Žπ‘˜(𝜏)π‘‘πœ, 𝑦(𝑠)=π‘₯(𝑑) we have π‘₯(β„Žπ‘˜(𝑑))=𝑦(π‘™π‘˜(𝑠)), π‘™π‘˜(𝑠)≀𝑠, π‘™π‘˜βˆ«(𝑠)=β„Žπ‘˜0(𝑑)βˆ‘π‘–βˆˆπΌπ‘Žπ‘–(𝜏)π‘‘πœ, π‘˜=1,…,π‘š, and (1.1) can be rewritten in the form ̇𝑦(𝑠)+π‘šξ“π‘˜=1π‘π‘˜ξ€·π‘™(𝑠)π‘¦π‘˜ξ€Έ(𝑠)=0,(3.33) where π‘π‘˜(𝑠)=π‘Žπ‘˜βˆ‘(𝑑)/π‘–βˆˆπΌπ‘Žπ‘–(𝑑). Denote by π‘Œ0(𝑒,𝑣) the fundamental function of the equation ̇𝑦(𝑠)+π‘˜βˆˆπΌπ‘π‘˜(𝑙𝑠)π‘¦π‘˜(𝑠)=0.(3.34) We have 𝑋0(𝑑,𝑠)=π‘Œ0ξƒ©ξ€œπ‘‘0ξ“π‘˜βˆˆπΌπ‘Žπ‘˜ξ€œ(𝜏)π‘‘πœ,𝑠0ξ“π‘˜βˆˆπΌπ‘Žπ‘˜ξƒͺ,π‘Œ(𝜏)π‘‘πœ0(𝑒,𝑣)=𝑋0ξ€·π‘βˆ’1(𝑒),π‘βˆ’1ξ€Έ(𝑣)>0,𝑒β‰₯𝑣β‰₯0.(3.35)
Let us check that other conditions of Lemma 2.9 hold for (3.33). Since βˆ‘π‘˜βˆˆπΌπ‘π‘˜(𝑠)=1 then condition (2.15) is satisfied. In addition,limsupπ‘ β†’βˆž,π‘βˆ’1(𝑠)βˆ‰πΈβˆ‘π‘˜βˆ‰πΌ||π‘π‘˜||(𝑠)βˆ‘π‘˜βˆˆπΌπ‘π‘˜(𝑠)=limsupπ‘‘β†’βˆž,π‘‘βˆ‰πΈβˆ‘π‘˜βˆ‰πΌ||π‘Žπ‘˜||(𝑑)βˆ‘π‘˜βˆˆπΌπ‘Žπ‘˜(𝑑)<1.(3.36) By Lemma 2.9, (3.33) is exponentially stable. Hence for any solution π‘₯(𝑑) of (1.1) we have limπ‘‘β†’βˆžπ‘₯(𝑑)=limπ‘ β†’βˆžπ‘¦(𝑠)=0. The end of the proof is similar to the proof of Theorem 3.9. In particular, to remove the condition βˆ‘π‘˜βˆˆπΌπ‘Žπ‘˜(𝑑)β‰ 0 we rewrite the equation by adding the term π‘’βˆ’π‘‘ to one of π‘Žπ‘˜(𝑑),π‘˜βˆˆπΌ.

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

Corollary 3.13. Suppose ξ€œπ‘Ž(𝑑)β‰₯0,∞0π‘Ž(𝑑)𝑑𝑑=∞,limsupπ‘‘β†’βˆžξ€œπ‘‘π‘”π‘˜(𝑑)π‘Ž(𝑠)𝑑𝑠<∞,π‘›ξ“π‘˜=1||π‘π‘˜||(𝑑)=0,π‘‘βˆˆπΈ,limsupπ‘‘β†’βˆž,π‘‘βˆ‰πΈβˆ‘π‘›π‘˜=1||π‘π‘˜||(𝑑)π‘Ž(𝑑)<1,(3.37) where 𝐸={𝑑β‰₯0,π‘Ž(𝑑)=0}. Then the equation Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(𝑑)+π‘›ξ“π‘˜=1π‘π‘˜ξ€·π‘”(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=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 limsupπ‘‘β†’βˆž(π‘‘βˆ’π‘”π‘˜(𝑑))<∞ then (3.38) is exponentially stable.

Theorem 3.14. Suppose ∫∞0βˆ‘π‘šπ‘˜=1|π‘Žπ‘˜(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 ||||ξƒ―ξ€œπ‘‹(𝑑,𝑠)≀expπ‘‘π‘ π‘šξ“π‘˜=1||π‘Žπ‘˜||ξƒ°.(𝜏)π‘‘πœ(3.39) Then by solution representation formula (2.5) for any solution π‘₯(𝑑) of (1.1) we have ||||ξƒ―ξ€œπ‘₯(𝑑)≀exp𝑑𝑑0π‘šξ“π‘˜=1||π‘Žπ‘˜||ξƒ°||π‘₯𝑑(𝑠)𝑑𝑠0ξ€Έ||+ξ€œπ‘‘π‘‘0ξƒ―ξ€œexpπ‘‘π‘ π‘šξ“π‘˜=1||π‘Žπ‘˜||ξƒ°(𝜏)π‘‘πœπ‘šξ“π‘˜=1||π‘Žπ‘˜||||πœ‘ξ€·β„Ž(𝑠)π‘˜ξ€Έ||ξƒ―ξ€œ(𝑠)𝑑𝑠≀expβˆžπ‘‘0π‘šξ“π‘˜=1||π‘Žπ‘˜||||π‘₯𝑑(𝑠)𝑑𝑠0ξ€Έ||+ξ€œβˆžπ‘‘0π‘šξ“π‘˜=1||π‘Žπ‘˜||ξƒͺ,(𝑠)π‘‘π‘ β€–πœ‘β€–(3.40) where β€–πœ‘β€–=max𝑑<0|πœ‘(𝑑)|. Then π‘₯(𝑑) is a bounded function.
Moreover, ∫|𝑋(𝑑,𝑠)|β‰€π΄βˆΆ=exp{∞0βˆ‘π‘šπ‘˜=1|π‘Žπ‘˜(𝑠)|𝑑𝑠}, 𝑑β‰₯𝑠β‰₯0. Let us choose 𝑑0β‰₯0 such that βˆ«βˆžπ‘‘0βˆ‘π‘šπ‘˜=1|π‘Žπ‘˜(𝑠)|𝑑𝑠<1/(2𝐴), then π‘‹ξ…žπ‘‘(𝑑,𝑑0βˆ‘)+π‘šπ‘˜=1π‘Žπ‘˜(𝑑)𝑋(β„Žπ‘˜(𝑑),𝑑0)=0, 𝑋(𝑑0,𝑑0)=1 implies 𝑋(𝑑,𝑑0∫)β‰₯1βˆ’βˆžπ‘‘0βˆ‘π‘šπ‘˜=1|π‘Žπ‘˜(𝑠)|𝐴𝑑𝑠>1βˆ’π΄(1/(2𝐴))=1/2, thus 𝑋(𝑑,𝑑0) 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 π‘Žπ‘˜(𝑑)β‰₯0, there exists a set of indices πΌβŠ‚{1,…,π‘š} such that condition (3.30) and the second condition in (3.29) hold. Then all solutions of (1.1) are bounded.

Proof. If ∫∞0βˆ‘π‘˜βˆˆπΌ|π‘Žπ‘˜(𝑑)|𝑑𝑑=∞, then all solutions of (1.1) are bounded by Theorem 3.11. Let ∫∞0βˆ‘π‘˜βˆˆπΌ|π‘Žπ‘˜(𝑑)|𝑑𝑑<∞. By (3.30) we have ∫∞0βˆ‘π‘˜βˆ‰πΌ|π‘Žπ‘˜βˆ«(𝑑)|π‘‘π‘‘β‰€βˆž0βˆ‘π‘˜βˆˆπΌ|π‘Žπ‘˜(𝑑)|𝑑𝑑<∞. Then ∫∞0βˆ‘π‘šπ‘˜=1|π‘Žπ‘˜(𝑑)|𝑑𝑑<∞. By Theorem 3.14 all solutions of (1.1) are bounded.

Theorem 3.16. Suppose π‘Žπ‘˜(𝑑)β‰₯0. If (1.1) is asymptotically stable, then the ordinary differential equation ̇π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜ξƒͺ(𝑑)π‘₯(𝑑)=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 π‘₯(𝑑0)=π‘₯0, can be presented as π‘₯(𝑑)=π‘₯0∫exp{βˆ’π‘‘π‘‘0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑠)𝑑𝑠}, so (3.41) is asymptotically stable, as far as ξ€œβˆž0π‘šξ“π‘˜=1π‘Žπ‘˜(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 {𝑑𝑛}, π‘‘π‘›β†’βˆž, such thatξ€œπ‘‘π‘›π‘‘+π‘›π‘›π‘šξ“π‘˜=1π‘Žπ‘˜1(𝜏)π‘‘πœ<𝑛<1𝑒,𝑛β‰₯3.(3.43) By (2.12) there exists 𝑛0>3 such that π‘‘βˆ’β„Žπ‘˜(𝑑)≀𝑛0, π‘˜=1,…,π‘š. Lemma 2.7 implies 𝑋(𝑑,𝑠)>0, 𝑑𝑛≀𝑠≀𝑑≀𝑑𝑛+𝑛, 𝑛β‰₯𝑛0. Similar to the proof of Theorem 3.14 and using the inequality 1βˆ’π‘₯β‰₯π‘’βˆ’π‘₯, π‘₯>0, we obtain 𝑋𝑑𝑛,π‘‘π‘›ξ€Έξ€œ+𝑛β‰₯1βˆ’π‘‘π‘›π‘‘+π‘›π‘›π‘šξ“π‘˜=1π‘Žπ‘˜ξƒ―βˆ’ξ€œ(𝜏)π‘‘πœβ‰₯exp𝑑𝑛𝑑+π‘›π‘›π‘šξ“π‘˜=1π‘Žπ‘˜ξƒ°(𝜏)π‘‘πœ>π‘’βˆ’1/𝑛.(3.44) Inequality (2.7) implies |𝑋(𝑑𝑛+𝑛,𝑑𝑛)|β‰€πΎπ‘’βˆ’πœ†π‘›. Hence πΎπ‘’βˆ’πœ†π‘›β‰₯π‘’βˆ’1/𝑛, 𝑛β‰₯𝑛0, or 𝐾>π‘’πœ†π‘›βˆ’1/3 for any 𝑛β‰₯𝑛0. The contradiction proves the theorem.

Theorems 3.11 and 3.16 imply the following statement.

Corollary 3.17. Suppose π‘Žπ‘˜(𝑑)β‰₯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=1.3678β‹―<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 π‘Ž(𝑑)β‰₯0, β„Ž(𝑑)≀𝑑 are continuous functions and limsupπ‘‘β†’βˆžξ€œπ‘‘β„Ž(𝑑)3π‘Ž(𝑠)𝑑𝑠≀2.(4.1) Then all solutions of (3.28) are bounded.
If in additionliminfπ‘‘β†’βˆžξ€œπ‘‘β„Ž(𝑑)π‘Ž(𝑠)𝑑𝑠>0,(4.2) and the strict inequality in (4.1) holds then (3.28) is exponentially stable.

Statement 2 (see [7]). Suppose π‘Ž(𝑑)β‰₯0, β„Ž(𝑑)≀𝑑 are continuous functions, the strict inequality (4.1) holds and ∫∞0π‘Ž(𝑠)𝑑𝑠=∞. Then all solutions of (3.28) tend to zero as π‘‘β†’βˆž.

Statement 3 (see [9, 10]). Suppose π‘Ž(𝑑)β‰₯0, β„Ž(𝑑)≀𝑑 are measurable functions, ∫∞0π‘Ž(𝑠)𝑑𝑠=∞, ∫𝐴(𝑑)=𝑑0π‘Ž(𝑠)𝑑𝑠 is a strictly monotone increasing function and limsupπ‘‘β†’βˆžξ€œπ‘‘β„Ž(𝑑)π‘Ž(𝑠)𝑑𝑠<sup0<πœ”<πœ‹/2ξ‚΅1πœ”+ξ‚ΆΞ¦(πœ”)β‰ˆ1.45…,(4.3)∫Φ(πœ”)=∞0𝑒(𝑑,πœ”)𝑑𝑑, where 𝑒(𝑑,πœ”) is a solution of the initial value problem ̇𝑦(𝑑)+𝑦(π‘‘βˆ’πœ”)=0,𝑦(𝑑)=0,𝑑<0,𝑦(0)=1.(4.4) Then (3.28) is asymptotically stable.

Note that instead of the equation ̇𝑦(𝑑)+𝑦(π‘‘βˆ’πœ”)=0 with a constant delay, the equatioṅ𝑦(𝑑)+𝑦(π‘‘βˆ’πœ(𝑑))=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(𝑑)βˆ’πœ”|β‰€π‘˜/πœ’(πœ”), where π‘˜βˆˆ[0,πœ”), 0β‰€πœ”<πœ‹/2 and βˆ«πœ’(πœ”)=∞0|𝑒(𝑑,πœ”)|𝑑𝑑.

Obviously in this statement the delay can exceed 2.

Statement 5 (see [10]). Let 𝜏(𝑑)β‰€π‘˜+πœ”{𝑑/πœ”}, where π‘˜βˆˆ(0,1), 0<πœ”<1, {π‘ž} is the fractional part of π‘ž. Then (4.5) is exponentially stable.

Here the delay 𝜏(𝑑) can be in the neighbourhood of πœ” which is close to 1.

Example 4.1. Consider the equation ξ€·||||ξ€Έπ‘₯Μ‡π‘₯(𝑑)+𝛼sinπ‘‘βˆ’sin𝑑(β„Ž(𝑑))=0,β„Ž(𝑑)≀𝑑,(4.6) where β„Ž(𝑑) is an arbitrary measurable function such that π‘‘βˆ’β„Ž(𝑑)β‰€πœ‹ and 𝛼>0.

This equation has the form (3.28) where π‘Ž(𝑑)=𝛼(|sin𝑑|βˆ’sin𝑑). Let us check that the conditions of Corollary 3.10 hold. It is evident that ∫∞0π‘Ž(𝑠)𝑑𝑠=∞. We have limsupπ‘‘β†’βˆžξ€œπ‘‘β„Ž(𝑑)π‘Ž(𝑠)𝑑𝑠≀limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘βˆ’πœ‹ξ€œπ‘Ž(𝑠)π‘‘π‘ β‰€βˆ’π›Όπœ‹2πœ‹2sin𝑠𝑑𝑠=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 π‘‘β†’βˆž.

Statements 1–3 fail for this equation. In Statements 1 and 2 the delay should be continuous. In Statement 3 function ∫𝐴(𝑑)=𝑑0π‘Ž(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 π‘Žπ‘˜(𝑑)β‰₯0, β„Žπ‘˜(𝑑)≀𝑑 are continuous functions and limsupπ‘‘β†’βˆžπ‘Žπ‘˜(𝑑)limsupπ‘‘β†’βˆžξ€·π‘‘βˆ’β„Žπ‘˜ξ€Έ(𝑑)≀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 βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)>0 and the strict inequality in (4.8) is valid then all solutions of (1.1) tend to zero as π‘‘β†’βˆž.
If π‘Žπ‘˜(𝑑) are constants then in (4.8) the number 1 can be replaced by 3/2.

Statement 7 (see [7]). Suppose π‘Žπ‘˜(𝑑)β‰₯0, β„Žπ‘˜(𝑑)≀𝑑 are continuous, β„Ž1(𝑑)β‰€β„Ž2(𝑑)β‰€β‹―β‰€β„Žπ‘š(𝑑) and limsupπ‘‘β†’βˆžξ€œπ‘‘β„Ž1π‘š(𝑑)ξ“π‘˜=1π‘Žπ‘˜3(𝑠)𝑑𝑠≀2.(4.9) Then any solution of (1.1) tends to a constant as π‘‘β†’βˆž.
If in addition ∫∞0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑠)𝑑𝑠=∞, then all solutions of (1.1) tend to zero as π‘‘β†’βˆž.

Example 4.2. Consider the equation 𝛼̇π‘₯(𝑑)+𝑑π‘₯𝑑2+π›½βˆ’sin𝑑𝑑π‘₯𝑑2=0,𝑑β‰₯𝑑0>0,(4.10) where 𝛼>0, 𝛽>0. Denote 𝑝(𝑑)=1/𝑑, β„Ž(𝑑)=𝑑/2βˆ’sin𝑑, 𝑔(𝑑)=𝑑/2.

We apply Corollary 3.6. Since limπ‘‘β†’βˆž[ln(𝑑/2)βˆ’ln(𝑑/2βˆ’sin𝑑)]=0, then limsupπ‘‘β†’βˆžξ‚΅π›Όξ€œπ‘‘β„Ž(𝑑)ξ€œπ‘(𝑠)𝑑𝑠+𝛽𝑑𝑔(𝑑)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 β„Ž(𝑑)≀𝑔(𝑑) nor 𝑔(𝑑)β‰€β„Ž(𝑑) 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 π‘Ž(𝑑), π‘π‘˜(𝑑), π‘‘βˆ’β„Žπ‘˜(𝑑) are bounded continuous functions, there exist 𝛿, π‘˜, 𝛿>0, 0<π‘˜<1, such that π‘Ž(𝑑)β‰₯𝛿 and βˆ‘π‘šπ‘˜=1|π‘π‘˜(𝑑)|<π‘˜π›Ώ. Then the equation Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(𝑑)+π‘šξ“π‘˜=1π‘π‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=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 1Μ‡π‘₯(𝑑)+𝑑𝛼π‘₯(𝑑)+𝑑π‘₯𝑑2=0,𝑑β‰₯𝑑0>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 GyΕ‘ri [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 π‘Ž(𝑑) 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 limsupπ‘šπ‘‘β†’βˆžξ“π‘˜=1||π‘Žπ‘˜||(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–ξ€œ(𝑑)π‘‘β„Žπ‘˜π‘š(𝑑)𝑖=1π‘Žπ‘–1(𝑠)𝑑𝑠<1+𝑒.(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 π‘Žπ‘˜(𝑑)β‰₯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.

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