R E S E A R C H
Open Access
W-transform for exponential stability of
second order delay differential equations
without damping terms
Alexander Domoshnitsky
1*, Abraham Maghakyan
1and Leonid Berezansky
2*Correspondence: [email protected] 1Department of Mathematics, Ariel University, Ariel, Israel
Full list of author information is available at the end of the article
Abstract
In this paper a method for studying stability of the equationx(t) +mi=1ai(t)x(t–
τ
i(t)) = 0 not including explicitly the first derivative is proposed. We demonstrate thatalthough the corresponding ordinary differential equationx(t) +mi=1ai(t)x(t) = 0 is
not exponentially stable, the delay equation can be exponentially stable.
MSC: 34K20; 34K06; 34K25
Keywords: second order delay differential equations; W-method; exponential stability
1 Introduction
In this paper we apply the W-method to obtain explicit conditions for exponential stability of linear second order delay differential equations. The method consists in a transforma-tion of a given differential equatransforma-tion to an operator equatransforma-tion by the substitutransforma-tion
x(t) =
t
W(t,s)z(s)ds,
whereW(t,s) is the Cauchy function for some known exponentially stable equation. The main object of this paper is the second order delay differential equation
x(t) +
m
i= ai(t)x
t–τi(t)
–
m
i= bi(t)x
t–θi(t)
=f(t), t∈[, +∞), (.)
with a corresponding initial function defining what should be put in the equation instead ofx(t–τi(t)) whent–τi(t) < orx(t–θi(t)) whent–θi(t) < . For simplicity and without
loss of generality we can consider the zero initial function
x(ξ) = , forξ< . (.)
Concerning the coefficients, delays, and functionf we assume that f, ai, bi, ϕ, τi, θi
(i= , . . . ,m) are measurable essentially bounded functions [, +∞)→(–∞, +∞), and τi(t)≥,θi(t)≥ fort≥. We understand by a solution of equation (.), (.) a function
x: [, +∞)→(–∞, +∞) with absolutely continuous on every finite interval derivativex and essentially bounded second derivativexwhich satisfies this equation almost every-where.
Various applications of equation (.) and its generalizations can be found, for example, in the theory of self-excited oscillations, in oscillation processes in a vacuum tube, in dy-namics of an auto-generator, in description of processes of in-feed grinding and cutting (see []); on position control in mechanical engineering, on electromechanical systems, and on combustion engines []. The problem of stabilizing the rolling of a ship by the ac-tivated tanks method in which ballast water is pumped from one position to another was reduced in [] to analysis of stability of the second order delay equation.
Asymptotic properties of second order equations without damping term were studied in ([], Chapter III, Section , pp.-), where instability of the equation
x(t) +bx(t–τ) = ,
for every pair of positive constantsbandτ was obtained. Conditions of unboundedness of solutions to the equation
x(t) +
m
i= bi(t)x
t–τi(t)
= ,
with variable coefficients and delays were obtained in []. The condition∞τ(t)dt<∞is necessary and sufficient for the boundedness of all solutions to the equation
x(t) +bxt–τ(t)=
(see []). Results as regards the boundedness of solutions for vanishing delays (τi(t)→
fort→ ∞) and as regards asymptotic representations of solutions were obtained in [, ], see also [], Chapter III, Section . Boundedness of solutions for equations with advanced arguments (τi(t)≤) was studied in []. First results on the exponential stability for the
equationx(t) +ax(t) –bx(t–τ) = with constant coefficients and delay were obtained in [–]. First results on the exponential stability of the second order equation (.) without damping term and with variable coefficients and delays were obtained recently in []. All results of [] concern only the case of a non-oscillation equation,
x(t) +
m
i= ai(t)x
t–τi(t)
–
m
i= bi(t)x
t–θi(t)
= , t∈[, +∞),
x(ξ) = , forξ<
(.)
Lyapunov functions was used in the works [–] and the technique of the fixed point theorems in []. The technique of non-oscillation and positivity of the Cauchy functions for studying stability of delay equations was proposed [] and then developed in [].
The general solution of equation (.), (.) can be represented in the form []
x(t) =
t
C(t,s)f(s)ds+x(t)x() +x(t)x(), (.)
where x(t), x(t) are two solutions of homogeneous equation (.), (.) satisfying the
conditions
x() = , x() = , x() = , x() = , (.)
the kernelC(t,s) in this representation is called the Cauchy function (fundamental func-tion in other terminology) of equafunc-tion (.).
Consider equation (.) with the following initial conditions:
x(ξ) =ϕ(ξ), ξ<t, x(t) =x, x(t) =x. (.)
Let us formulate several definitions concerning stability.
Definition Equation (.) is uniformly exponentially stable if there existN> andα> , such that the solution of (.), (.) satisfies the estimates
x(t)≤Ne–α(t–t)sup
t<t
ϕ(ξ), x(t)≤Ne–α(t–t)sup
t<t
ϕ(ξ), t≤t< +∞, (.)
whereNandαdo not depend ontand initial functionϕ.
Definition We say that the Cauchy functionC(t,s) of equation (.) satisfies the expo-nential estimate if there exist positiveNandαsuch that
C(t,s)≤Ne–α(t–s), Ct(t,s)≤Ne–α(t–s), ≤s≤t< +∞. (.)
It is well known that for equation (.) with bounded delays these two definitions are equivalent [].
The paper is built as follows. In the first section we describe known results on asymptotic properties of second order delay equations. In Section , we formulate the main results of the paper and compare them with known results. Auxiliary assertions can be found in Section . The proofs of the assertions, formulated in Section , can be found in Section .
2 Formulation of main results
Let us consider the following ordinary differential equation:
x(t) +Ax(t) +Bx(t) = , t∈[, +∞), (.)
exponential stability of delay differential equation (.). Ifxis a solution of equation (.) satisfying (.), we can write the equality
x(t) +
m
i= Ai(t)x
t–ηi(t)
+
m
i= Bi(t)x
t–τi(t)
=f(t),
x(ξ) =x(ξ) = , ξ< ,
(.)
where ηi are corresponding measurable functions satisfying inequalitiesθi(t)≥ηi(t)≥
τi(t),
Ai(t) =bi(t)
θi(t) –τi(t)
, Bi(t) =ai(t) –bi(t). (.)
Let us denote by
Ai =lim sup
t→∞
t
t
Ai(s)ds, Bi =lim sup t→∞
t
t
Bi(s)ds
the average values ofAi(t) andBi(t), respectively,
Ai(t) =Ai –Ai(t), Bi(t) =Bi –Bi(t),
θi∗=esssup
t≥
θi(t), θ∗= max i=,...,mθ
∗
i, (.)
τi∗=esssup
t≥
τi(t), τ∗= max i=,...,mτ
∗ i.
To connect (.) and (.) we suppose that the constantsAandBare such that
A=
m
i=
Ai, B=
m
i=
Bi. (.)
Theorem Let A> ,B> ,A> B,τ
i(t)≤θi(t)for i= , . . . ,m,and the following
in-equality be fulfilled:
m
i=
Aiθi∗ A+√A– B
A–√A– B A+√A– B
A–√√A–B A–B
+
+
m
i=
Ai(t)
A+√A– B
A–√A– B A+√A– B
A–√A–B √A–B
+
m
i=
Biτi∗ A+√A– B
A–√A– B A+√A– B
A–√A–B √A–B
+
m
i=
Bi(t)
B< , (.)
Theorem Let A> ,B> ,A= B,τ
i(t)≤θi(t)for i= , . . . ,m,and the following
in-equality be fulfilled:
m
i=
Aiθi∗ +A –
–A
e
+
m
i=
Ai(t)
Ae
+
m
i=
Biτi∗ Ae+
m
i=
Bi(t)
B< , (.)
then the Cauchy function C(t,s)of equation(.)and the fundamental system x(t),x(t) of equation(.), (.)satisfy an exponential estimate.
Theorem Let <A< B,τ
i(t)≤θi(t)for i= , . . . ,m,and the following inequality be
fulfilled:
m
i=
Aiθi∗
exp[–√ A
B–A(π+ arctan
√
B–A
A )]
–exp[–√ A
B–Aπ]
+
+
m
i=
Ai(t)√ B
exp[–√ A
B–A(π+arctan
√
B–A
A )]
–exp[–√ A
B–Aπ]
+
m
i=
Biτi∗√ B
exp[–√ A
B–A(π+arctan
√
B–A
A )]
–exp[–√ A
B–Aπ]
+
m
i=
Bi(t)
B
+exp(–√πA
B–A)
–exp(–√πA
B–A)
< , (.)
then the Cauchy function C(t,s)of equation(.)and the fundamental system x(t),x(t) of equation(.), (.)satisfy an exponential estimate.
Let us formulate corollaries for the equation
x(t) +ax(t–τ) –bx(t–θ) =f(t), t∈[, +∞), (.)
where
x(ξ) = , ξ< , (.)
f is a measurable essentially bounded function.
Corollary Let a> ,b> ,b(θ–τ)> (a–b),τ<θ,
b(θ–τ)θ +
b(θ–τ)+b(θ–τ)– (a–b)
× b(θ–τ) –
b(θ–τ)– (a–b) b(θ–τ) +b(θ–τ)– (a–b)
b(θ–√τ)–√b(θ–τ)–(a–b) b(θ–τ)–(a–b)
+ (a–b)τ
b(θ–τ) +b(θ–τ)– (a–b)
× b(θ–τ) –
b(θ–τ)– (a–b) b(θ–τ) +b(θ–τ)– (a–b)
b(θ–τ)–√b(θ–τ)–B √b(θ–τ)–B
< , (.)
then the Cauchy function C(t,s)of equation(.)and the fundamental system x(t),x(t) of equation(.), (.)satisfy an exponential estimate.
Example Let us seta=√ + .,b=√,θ–τ= .. In this case we haveb(θ–τ)>
(a–b) and condition (.) is fulfilled ifθ< ..
Corollary Let a> ,b> ,b(θ–τ)= (a–b),τ<θ,
b(θ–τ)θ +b(θ–τ)
–
–b(θ–τ)
e
+τ (a–b)
b(θ–τ)e< , (.)
then the Cauchy function C(t,s)of equation(.)and the fundamental system x(t),x(t) of equation(.), (.)satisfy an exponential estimate.
Example Let us seta= .,b= ,θ–τ= .. In this case we haveb(θ–τ)= (a–b)
and condition (.) is fulfilled ifθ< .
Corollary Let a> ,b> ,b(θ–τ)< (a–b),τ<θ,
bθ(θ–τ)
exp[–√ b(θ–τ)
(a–b)–b(θ–τ)(π+ arctan
√
(a–b)–b(θ–τ)
b(θ–τ) )]
–exp[–√ b
(a–b)–b(θ–τ)π]
+
+ (a–b)τ
exp[–√ b(θ–τ)
(a–b)–b(θ–τ)(π+arctan
√
(a–b)–b(θ–τ)
b(θ–τ) )]
–exp[–√ b(θ–τ)
(a–b)–b(θ–τ)π]
< , (.)
then the Cauchy function C(t,s)of equation(.)and the fundamental system x(t),x(t) of equation(.), (.)satisfy an exponential estimate.
Example Let us seta= ,b= .,θ–τ = .. In this case we haveb(θ–τ)< (a–b)
and condition (.) is fulfilled ifθ< ..
Remark Results as regards an exponential stability of equation (.), only in the case τ = , were obtained in [–], but as far as we know, there are no other results in the case of positive delayτ and oscillating solutions to equation (.).
3 Estimates of integrals of the Cauchy functions for auxiliary equations
Let us consider the following auxiliary equation:
whereA,Bare positive constants. Denote byW(t,s) the Cauchy function of equation (.) according to the following rule: for every fixedsthe functionW(t,s), as a function of the variablet, satisfies the equation (.) and the initial conditions
x(s) = , x(s) = . (.)
For equation (.) with constant coefficients, we can construct the Cauchy functionW(t,s) and get the following integrals:
t
W(t,s)ds,
t
Wt(t,s)ds and
t
Wtt(t,s)ds, ≤t< +∞. (.)
Consider all possible cases: ()A> B, ()A= B, ()A< B.
Lemma Let A> ,B> ,A> B,then
lim sup
t→∞ t
W(t,s)ds=
B, (.)
lim sup
t→∞ t
Wt(t,s)ds= A+√A– B
A–√A– B A+√A– B
A–√A–B √A–B
, (.)
lim sup
t→∞ t
Wtt(t,s)ds= A+√A– B
A–√A– B
A+√A– B
A–√√A–B A–B
. (.)
Lemma Let A> ,B> ,A= B,then
lim sup
t→∞ t
W(t,s)ds=
B, (.)
lim sup
t→∞ t
Wt(t,s)ds=
Ae, (.)
lim sup
t→∞ t
Wtt(t,s)ds= + A –
–A
e. (.)
Lemma Let A> ,B> ,A< B,then
lim sup
t→∞ t
W(t,s)ds= B
+exp(–√πA
B–A)
–exp(–√πA
B–A)
, (.)
lim sup
t→∞ t
Wt(t,s)ds=√ B
exp[–√ A
B–A(π+arctan
√
B–A
A )]
–exp[–√ A
B–Aπ]
, (.)
lim sup
t→∞ t
Wtt(t,s)ds=
exp[–√ A
B–A(π+ arctan
√
B–A
A )]
–exp[–√ A
B–Aπ]
. (.)
The proofs of these lemmas can be found in [], Lemmas .-..
4 Proofs of main theorems
Lemma Suppose the solution of equation(.)with the zero initial conditions,
x(ξ) = , ξ≤,
is bounded on[,∞)for any essentially bounded on the semi-axes right-hand side f.Then equation(.)is uniformly exponentially stable.
Let us consider the ordinary differential equation
x(t) +Ax(t) +Bx(t) =z(t), t∈[, +∞), (.)
with constant positive coefficientsAandB.
It is clear that the solution of equation (.) which satisfies the initial conditions
x() = , x() = , (.)
can be written in the form
x(t) =
t
W(t,s)z(s)ds, (.)
whereW(t,s) is the Cauchy function of equation (.). Its derivatives are the following:
x(t) =
t
Wt(t,s)z(s)ds, x(t) =
t
Wtt(t,s)z(s)ds+z(t). (.)
Let us denote
W=lim sup
t→∞ t
W(t,s)ds, (.)
Wt=lim sup
t→∞ t
Wt(t,s)ds, Wtt=lim sup
t→∞ t
Wtt(t,s)ds. (.)
Theorem Let A> ,B> ,τi(t)≤θi(t)and the following inequality be fulfilled:
m
i=
Aiθi∗Wtt+ +
m
i=
Ai(t)Wt
+
m
i=
Biτi∗Wt+
m
i=
Bi(t)W< , (.)
then the Cauchy function C(t,s)of equation(.)and the fundamental system x(t),x(t) of equation(.), (.)satisfy an exponential estimate.
Remark It is clear from inequality (.) that in the case, when the coefficientsai(t) –bi(t)
andbi(t)(θi(t) –τi(t)) ( = , . . . ,m) are close to constants, the second and fourth terms are
small, and in the case of small delaysθi(t) andτi(t) ( = , . . . ,m), the first and the third
Proof Consider the equation
x(t) +
m
i= ai(t)x
t–τi(t)
–
m
i= bi(t)x
t–θi(t)
=f(t), t∈[, +∞), (.)
x(ξ) = , forξ< . (.)
Let us rewrite equation (.) in the following forms:
x(t) +
m
i=
ai(t) –bi(t)
xt–τi(t)
+
m
i= bi(t)
xt–τi(t)
–xt–θi(t)
=f(t),
t∈[, +∞), (.)
and
x(t) +
m
i= bi(t)
t–τi(t)
t–θi(t)
x(s)ds+
m
i=
ai(t) –bi(t)
xt–τi(t)
=f(t),
t∈[, +∞), (.)
where
x(ξ) =x(ξ) = , forξ< . (.)
We have to prove that, for every essentially bounded functionf(t), the solutionx(t) is also bounded on the semiaxist∈[, +∞). To prove exponential stability of (.) we as-sume the existence of an unbounded solutionx(t) and demonstrate that this is impossible. There exist measurable delay functionsηi,τi(t)≤ηi(t)≤θi(t) such that equality (.)
can be written as
x(t) +
m
i= bi(t)
θi(t) –τi(t)
xt–ηi(t) + m i=
ai(t) –bi(t)
xt–τi(t)
=f(t),
t∈[, +∞), (.)
and then in the form
x(t) +
m
i= Ai(t)x
t–ηi(t)
+
m
i= Bi(t)x
t–τi(t)
=f(t), (.)
whereAi(t) =bi(t)[θi(t) –τi(t)] andBi(t) =ai(t) –bi(t). Let us rewrite it in the form
x(t) +Ax(t) –Ax(t) +
m
i= Ai(t)x
t–ηi(t)
+Bx(t) –Bx(t) +
m
i= Bi(t)x
t–τi(t)
hence
x(t) +Ax(t) +Bx(t) =
m
i= Ai
t
t–ηi(t)
x(s)ds+
m
i=
Ai(t)x
t–ηi(t)
+
m
i= Bi
t
t–τi(t)
x(s)ds+
m
i=
Bi(t)x
t–τi(t)
+f(t), t∈[,∞), (.)
where
x(ξ) =x(ξ) =x(ξ) = , forξ< , (.)
andAi(t),Ai,Bi(t),Bi are defined by equations (.)-(.).
Let us make a so-calledW-transform, substitutingx(t) =tW(t,s)z(s)ds, wherez∈L∞
(L∞ is the space of essentially bounded functionsz: [,∞)→(–∞, +∞)), into equation
(.). It is clear that the derivativesx(t) andx(t) are defined by equalities (.). We get the following equation:
z(t) = (Kz)(t) +f(t), (.)
where the operatorK:L∞→L∞is defined by the equality
(Kz)(t) =
m
i=
Aiσt–ηi(t) t
t–ηi(t)
s
Wss(s,ξ)z(ξ)dξ+z(s)
ds
+
m
i=
Ai(t)σ
t–ηi(t)
t–ηi(t)
Wtt–ηi(t),s
z(s)ds
+
m
i=
Biσt–τi(t) t
t–τi(t)
s
Ws(s,ξ)z(ξ)dξds
+
m
i=
Bi(t)σ
t–τi(t) t–τi(t)
Wt–τi(t),s
z(s)ds, (.)
and
σ(t) =
⎧ ⎨ ⎩
, t≥,
, t< . (.)
The inequality (.) implies that the normKof the operatorK:L∞→L∞is less than
one. In this case, there exists the bounded operator (I–K)–. For every bounded
right-hand side the solutionzof equation (.) is bounded.
In the caseA> andB> , the Cauchy functionW(t,s) and its derivativeWt(t,s) sat-isfy exponential estimates. The boundedness of the solution xof equation (.) and its derivativexfollow now from the boundedness ofz.
Thus solutionsx(t) are bounded on the semiaxis for a bounded right-hand side f(t). Then by Lemma the Cauchy functionC(t,s) of equation (.) and the solutionsxand xsatisfy the exponential estimates.
This completes the proof of Theorem .
To prove Theorems - we set the norms ofW,WtandWttobtained in Lem-mas - into Theorem .
The proofs of Corollaries - are results of substitution ofW,WtandWttinto Theorems -, when we take into account thatA=b(θ–τ),B=a–b,A=B= .
5 Conclusion
In this paper to obtain exponential stability conditions we use the substitution
x(t) = (Wz)(t) :=
t
t
Y(t,s)z(s)ds,
where Y(t,s) is the fundamental function of exponentially stable autonomous ordinary differential equation of the second order and then analyze the operator equation
z=Tz+f
in some functional Banach spaces on semi-axes. This method is usually called the W-method and used in many problems for FDE such as stability, oscillation and non-oscillation, and boundary value problems. To apply this method in our research we need integral estimations of a fundamental function and its first and second derivative. Such estimates were obtained recently and allow us to obtain here new stability results for a wide class of delay differential equations of the second order. In particular we obtain new stability results for delay equations with positive and negative coefficients. Such equations without delay are unstable so the results obtained here one can use to stabilize mathemat-ical models described by ordinary differential equations of the second order.
6 Discussion and some topics for future research
The idea of applications of W-transform for second order delay differential equations first appeared in paper []. Lemmas and were taken from []. All other results in the paper are new and have not been published before.
Explicit integral estimates of the fundamental function and its derivatives we obtain here only for the simplest equation: ordinary differential equation with constant coefficients. It is interesting to obtain such estimates for the ordinary differential equation with variable coefficients or for delay differential equation with constant coefficients. It will allow one to improve the results obtained in this paper.
We did not study here nonlinear equations. It is interesting to consider the W-transformation method for nonlinear equations such that
¨
x(t) +ft,x˙r(t)+Gt,xh(t)= .
We suppose that the W-transformation method can also be applied for a vector delay differential equation
¨
X(t) +A(t)X˙g(t)+B(t)Xh(t)= ,
whereAandBaren×nmatrix-functions. To apply this method it is necessary to obtain explicit integral estimations for the fundamental matrix and its derivative of the following vector ordinary differential equation with constant coefficients:
¨
X(t) +AX˙(t) +BX(t) = .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors worked and obtained the results together.
Author details
1Department of Mathematics, Ariel University, Ariel, Israel.2Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel.
Acknowledgements
The authors express their sincere gratitude to the referees for careful reading of the manuscript and valuable suggestions, which helped to improve the paper.
Received: 16 November 2016 Accepted: 4 January 2017 References
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