Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type
The paper is concerned with the existence and stability of an almost periodic solution of the system with deviating argument dx(t)/dt = A(t)x(t) + f(t, x(t), x(t − τ₁(t)), . . . , x(t − τk(t)). The Wexler inequality for the Cauchy matrix is used. Conditions for stability of the solution are given....
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irk-123456789-1770172021-02-10T01:25:37Z Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type Akhmet, M.U. Tleubergenova, M.A. The paper is concerned with the existence and stability of an almost periodic solution of the system with deviating argument dx(t)/dt = A(t)x(t) + f(t, x(t), x(t − τ₁(t)), . . . , x(t − τk(t)). The Wexler inequality for the Cauchy matrix is used. Conditions for stability of the solution are given. Розглядається проблема iснування та стабiльностi майже перiодичного розв’язку системи з вiдхиленням в аргументi dx(t)/dt = A(t)x(t) + f(t, x(t), x(t − τ₁(t)), . . . , x(t − τk(t)) Використовується нерiвнiсть Векслера. Наведено умови стабiльностi розв’язку. 2004 Article Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type / M.U. Akhmet, M.A. Tleubergenova // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 295-301. — Бібліогр.: 18 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177017 517.9 en Нелінійні коливання Інститут математики НАН України |
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The paper is concerned with the existence and stability of an almost periodic solution of the system with
deviating argument dx(t)/dt = A(t)x(t) + f(t, x(t), x(t − τ₁(t)), . . . , x(t − τk(t)).
The Wexler inequality for the Cauchy matrix is used. Conditions for stability of the solution are given. |
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Article |
| author |
Akhmet, M.U. Tleubergenova, M.A. |
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Akhmet, M.U. Tleubergenova, M.A. Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type Нелінійні коливання |
| author_facet |
Akhmet, M.U. Tleubergenova, M.A. |
| author_sort |
Akhmet, M.U. |
| title |
Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type |
| title_short |
Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type |
| title_full |
Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type |
| title_fullStr |
Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type |
| title_full_unstemmed |
Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type |
| title_sort |
wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type |
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Інститут математики НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/177017 |
| citation_txt |
Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type / M.U. Akhmet, M.A. Tleubergenova // Нелінійні коливання. — 2004. — Т. 7, № 3. — С. 295-301. — Бібліогр.: 18 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT akhmetmu wexlerinequalityandalmostperiodicsolutionsofdifferentialequationswithdeviatingargumentofmixedtype AT tleubergenovama wexlerinequalityandalmostperiodicsolutionsofdifferentialequationswithdeviatingargumentofmixedtype |
| first_indexed |
2025-07-15T14:58:38Z |
| last_indexed |
2025-07-15T14:58:38Z |
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UDC 517.9
WEXLER INEQUALITY AND ALMOST PERIODIC SOLUTIONS
OF DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT
OF MIXED TYPE
НЕРIВНIСТЬ ВЕКСЛЕРА I МАЙЖЕ ПЕРIОДИЧНI РОЗВ’ЯЗКИ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ З АРГУМЕНТОМ МIШАНОГО ТИПУ,
ЩО ВIДХИЛЯЄТЬСЯ
M. U. Akhmet
Middle East Techn. Univ.
06531 Ankara, Turkey
e-mail: marat@metu.edu.tr
M. A. Tleubergenova
Kazakh Nat. Techn. Univ.
463000 Aktobe, Mares’eva Str., 10
The paper is concerned with the existence and stability of an almost periodic solution of the system with
deviating argument
dx(t)
dt
= A(t)x(t) + f(t, x(t), x(t− τ1(t)), . . . , x(t− τk(t)).
The Wexler inequality for the Cauchy matrix is used. Conditions for stability of the solution are given.
Розглядається проблема iснування та стабiльностi майже перiодичного розв’язку системи з
вiдхиленням в аргументi
dx(t)
dt
= A(t)x(t) + f(t, x(t), x(t− τ1(t)), . . . , x(t− τk(t)).
Використовується нерiвнiсть Векслера. Наведено умови стабiльностi розв’язку.
1. Introduction and preliminaries. Different aspects of the theory of almost periodic solutions
of quasilinear differential equations with deviating argument, including applications, has been
investigated by many authors [1 – 7]. The problem of existence of almost periodic solutions
for functional differential equations was considered by J. Hale [2] for the case when the non
perturbed system is autonomous or periodic and the argument is retarded. The paper [4] deals
with the existence of almost periodic solutions of a system with unique and constant deviati-
on. The aim of the present paper is to investigate the problem for the system (1), where the
deviations and the matrix of coefficients are almost periodic functions. Moreover, we assume
that the equation is of mixed type [3], that is, the derivative of x depends on the past as well as
the future values of t. This type of equations, apparently, can be applied to problems of antici-
patory systems [8]. One should emphasize that the general theory has not been considered for
this type of equations as well as for the equations with retarded argument or for the systems of
the neutral type [3, 9].
c© M. U. Akhmet, M. A. Tleubergenova, 2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3 295
296 M. U. AKHMET, M. A. TLEUBERGENOVA
Let N,R be sets of all natural and real numbers respectively, || · || be the Euclidean norm
in Rn, n ∈ N. Let s ∈ R be a positive number. We denote Gs = {x ∈ Rn|||x|| ≤ s} and
Gk+1
s = Gs ×Gs × . . .×Gs (that is, Gk+1
s is the Cartesian product of k copies of Gs).
Let a set C0(R) (respectively, C0(R × Gk+1
H ) for a given H ∈ R,H > 0) be a set of all
functions that are bounded and uniformly continuous on R (respectively on R×Gk+1
H ).
Definition 1. For f ∈ C0(R) (respectively C0(R ×Gk+1
H )) and τ ∈ R the translation of f by
τ is the function Qτf = f(t+ τ), t ∈ R (respectively Qτf(t, z) = f(t+ τ, z), (z, t) ∈ R×Gk+1
H ).
Definition 2. A number τ ∈ R is called ε-translation number of a function f ∈ C0(R)(
C0(R×Gk+1
H )
)
if ||Qτf − f || < ε for every t ∈ R((t, z) ∈ R×Gk+1
H ).
Definition 3. A function f ∈ C0(R)
(
C0(R × Gk+1
H )
)
is called an almost periodic (almost
periodic in t uniformly with respect to z ∈ Gk+1
H ) if for every ε ∈ R, ε > 0, there exists a
corresponding dense set of ε-translations of f.
Denote by AP(R)
(
AP(R×Gk+1
H )
)
the set of all such functions [10, 11].
The aim of this paper is to investigate the problem of existence and exponential stability of
an almost periodic solution of a system of differential equations with deviating argument of the
form
dx
dt
= A(t)x + f(t, x(t), x(t− τ1(t)), . . . , x(t− τk(t)), (1)
where x ∈ Rn, t ∈ R. We will need the following assumptions:
(C1) A(t) ∈ AP(R) is an n× n matrix, τj ∈ AP(R), j = 1, k.
(C2) f ∈ AP(R×Gk+1
s ) for every s ∈ R, s ≥ 0.
Let X(t, s), X(s, s) = I , be a Cauchy matrix of the associate homogeneous linear system
dx
dt
= A(t)x. (2)
One of our basic assumptions is the following.
(C3) ∃{a, b} ⊂ R, b ≥ 1, a > 0, such that
||X(t, s)|| ≤ b exp(−a(t− s)), t ≥ s. (3)
Lemma 1 [12 – 14]. Let A(t) ∈ AP(R) and the condition (C3) be satisfied. Then
||X(t + τ, s + τ)−X(t, s)|| < ε exp
(
−a
2
(t− s)
)
, (4)
if τ is
aε
2b2
-translation number of A(t).
Lemma 2 [10]. If f ∈ AP(R × Gk+1
H ) and z ∈ AP(R), z : R → Gk+1
H . Then f(t, z(t)) ∈
∈ AP(R).
Remark 1. Lemma 1 is due to D. Wexler [12] and is of principal significance in the arti-
cle. It states that the matrix X(t, s) is "diagonal almost periodic". Information relevant to this
assertion can be found in the book by W. A. Coppel (see Proposition 4 in Lecture 8 of [15]).
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
WEXLER INEQUALITY AND ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS . . . 297
2. Main result. In this section we consider the problem of existence of an exponentially
stable almost periodic solution of (1).
Lemma 3. If φ ∈ AP(R), then φ(t− τj(t)) ∈ AP(R), j = 1, k.
Proof. Fix ε ∈ R, ε > 0. Since φ ∈ C0(R) there exists δ = δ
(ε
2
)
such that ||φ(t1)−φ(t2)|| <
<
ε
2
if |t1 − t2| < δ. Denote ε1 = min
(ε
2
, δ
)
. By using the method of common translati-
on numbers [10, 11] one can show that there exists a corresponding dense set of ε-translation
numbers of φ and τj(t). Let τ be one of them. Then
||φ(t + τ − τj(t + τ))− φ(t− τj(t))|| ≤ ||φ(t + τ − τj(t + τ))− φ(t + τ − τj(t))||+
+ ||φ(t + τ − τj(t))− φ(t− τj(t))|| <
ε
2
+
ε
2
= ε.
The lemma is proved.
Assume that the following conditions hold:
(C4) ∃l ∈ R, l > 0, such that
||f(t, z1)− f(t, z2)|| ≤ l
k∑
j=0
||zj
1 − zj
2||; (5)
where zi = (z0
i , . . . , zk
i ) ∈ Rn(k+1), i = 1, 2.
Fix σ ∈ R, 0 < σ < a. Denote
τ0 = max
{
0, sup
t
τj(t), j = 1, k
}
, m(l) = 1− exp(στ0)blk
a− σ
.
We assume in addition that l is sufficiently small so that
(C5) l <
bk
a
;
(C6) m(l) > 0.
Theorem 1. Assume that (C1)−(C5) hold. Then there exists a unique almost periodic solution
ξ0(t) of (1).
Proof. Let T be an operator on AP(R) such that
Tφ =
t∫
∞
X(t, s)Fφ(s)ds. (6)
Let φ ∈ AP(R).
By using Lemma 3 one can prove that Fφ(t) ∈ AP(R), and by Lemmas 1, 2 on the base of
the method of common translation numbers one can verify that there is a corresponding dense
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
298 M. U. AKHMET, M. A. TLEUBERGENOVA
set Γ ⊂ R of ε-translation numbers for Fφ such that if τ ∈ Γ then (4) is valid for t ≥ s. Then
||Tφ(t + τ)− Tφ(t)|| =
∥∥∥∥∥∥
t∫
∞
X(t + τ, s)Fφ(s)ds−
t∫
∞
X(t, s)Fφ(s)ds
∥∥∥∥∥∥ ≤
≤
t∫
∞
||X(t + τ, s + τ)|| ||Fφ(s + τ)− Fφ(s)||ds+
+
t∫
∞
||X(t + τ, s + τ)−X(t, s)|| ||Fφ(s)||ds ≤
≤
t∫
∞
b exp(−a(t− s))εds +
t∫
∞
ε exp
(
−a
a
2
(t− s)
)
Mφds =
b + 2Mφ
a
ε, (7)
where Mφ = supt ||Fφ||. Thus Tφ ∈ AP(R). Moreover if φ1, φ2 ∈ AP(R) then, for every
t ≥ t0,
||Tφ1(t)− Tφ2(t)|| =
∥∥∥∥∥∥
t∫
−∞
X(t, s)Fφ1(s)ds−
t∫
−∞
X(t, s)Fφ2(s)ds
∥∥∥∥∥∥ ≤
≤
t∫
−∞
||X(t, s)|| ||Fφ1(s)− Fφ2(s)||ds ≤
≤
t∫
−∞
b exp(−a(t− s))lk||φ1 − φ2||0ds =
bkl
a
||φ1 − φ2||0.
The last inequality implies that
||Tφ1(t)− Tφ2(t)||0 ≤
bkl
a
||φ1 − φ2||0, (8)
and the condition (C5) implies that the operator T : AP(R) → AP(R) is contractive. Thus
there exists a unique fixed point ξ0 ∈ AP(R) of T which is a solution of (1).
The theorem is proved.
Remark 2. Apparently, the general problem of existence of solutions for equations of mixed
type has not been considered yet. Even for the case of advanced argument there are certain
difficulties if we try to define a solution for inceasing t [16, 17]. J. Hale remarked in [3] that
"these equations seem to dictate that boundary conditions should be specified in order to obtain
a solution in the way as one does for elliptic partial differential equations". We regard the
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
WEXLER INEQUALITY AND ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS . . . 299
boundedness of the solution on R as a boundary condition in the proof of Theorem 1. Authors
of [18] used this method to prove existence of a bounded solution for the equation of advanced
type when t ≥ t0.
Formulation of stability conditions for the solution ξ0(t). Fix t0 ∈ R, denote C[t0 − τ0, t0]
the set of all initial functions. Let x(t) be a solution of (1) such that
(D1) x(t) =
π(t) ∈ C[t0 − τ0, t0], t0 − τ0 ≤ t ≤ t0, if τ0 > 0,
x0 ∈ Rn, if τ0 = 0.
(D2) x(t) satisfies the equation (1) for all t ≥ t0.
Definition 4. The solution ξ0(t) is called uniformly exponential stable if there exists a number
α ∈ R,α > 0, such that for every ε > 0 there exists a number δ = δ(ε) such that the inequality
maxt0−τ0≤t≤t0 ||π(t) − ξ0(t)|| < δ (or inequality ||x0 − φ(t0)|| < δ, if τ0 = 0) implies that
there exists a unique solution x(t) which satisfies conditions (D1), (D2) and ||x(t) − ξ0(t)|| <
< ε exp(−α(t− t0)) for all t ≥ t0.
Remark 3. Analyzing definitions of Lyapunov stability for different types of equations it is
possible to stress the following two generic conditions.The first one is the closeness of the initial
values (the initial functions) for a solution whose stability is tested and a neighbour solution.
The second one is the condition that the process after the initial moment of time should be
governed only by a differential equation.These circumstances are described by conditions (D1)
and (D2). And evidently, (D2) induces the condition of ignoring the negative values of τj(t)
for the construction of initial conditions.We decided to make this comment after Definition 4,
since the formulation of stability of solutions of the equations with mixed type of deviating
arguments has not been encountered in the literature previosly.
The following theorem holds.
Theorem 2. Assume that (C1) − (C6) hold. Then the almost periodic solution ξ0(t) of (1) is
uniformly exponential stable.
Proof. One can see that v(t) = x(t)− ξ0(t) is a solution of the equation
dv
dt
= A(t)v + w(t, v(t), v(t− τ1(t)), . . . , v(t− τk(t)), (9)
where
v(t) = θ(t)(θ = π(t)− ξ0(t)) on [t0 − τ0, t0],
w(t, v(t),v(t− τ1(t)), . . . , v(t− τk(t)) =
= f(t, ξ0(t) + v(t), . . . , ξ0(t− τk(t)) + v(t− τk(t)))− f(t, ξ0(t), . . . , ξ0(t− τk(t)),
and w satisfies
||w(t, v1)− w(t, v2)|| ≤ l
k∑
j=0
||vj
1 − vj
2||,
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
300 M. U. AKHMET, M. A. TLEUBERGENOVA
vi = (v0
i , . . . , v
k
i ) ∈ Rn(k+1), i = 1, 2. Thus we can reduce the problem of stability of ξ0(t)
to the problem of stability of the zero solution v = 0 of (9) . Let us fix ε > 0 and denote
K(l, δ) =
b
m(l)
δ, where δ ∈ R, δ > 0. Take δ so small that K(l, δ) < ε. Assume, without lost
of any generality, that t0 = 0, τ0 > 0 and let C[−τ0, 0] be the set of all initial functions. Fix
θ(t) ∈ C[−τ0, 0] such that max[−τ0,0] ||θ(t)|| < δ and let
Ψθ =
{
φ ∈ C0[−τ0,+∞)|φ(t) = θ(t) if − τ0 ≤ t ≤ 0, ||φ(t)||1 ≤ K(l, δ) exp(−σt)
}
,
where C0[−τ0,+∞) denotes the restrictions of all functions from C0(R) to [−τ0,+∞) and
||φ||1 = supt≥t0 ||φ(t)||. Define on Ψθ an operator Π (see [2]) such that if φ ∈ Ψθ then
Πφ =
θ(t) for all − τ0 ≤ t ≤ 0,
X(t, 0)θ(0) +
t∫
0
X(t, s)Fφ(s)ds, otherwise .
We shall show that Π : Ψθ → Ψθ. Indeed, for t ≥ 0 we have that
||Πφ|| ≤ b exp(−at)δ +
t∫
0
b exp
(
−a(t− s)
)
lK(l, δ)
k+1∑
j=0
exp
(
−σ(s− τj(s))
)
ds ≤
≤ exp(−σt)
[
bδ +
(k + 1) exp(στ0)blK(l, δ)
a− σ
]
= K(l, δ) exp(−σt).
Differentiating Πφ on [0,∞) it is easy to show that [Πφ]′ is a function uniformly bounded
on [0,∞) and, hence, Πφ is a uniformly continuous function.
Let φ1, φ2 ∈ Ψθ. Then
||Πφ1 −Πφ2|| ≤
t∫
0
b exp
(
−a(t− s)
)
lk||φ1 − φ2||1ds ≤ blk
a
||φ1 − φ2||1.
From (C5) it follows that there is a unique fixed point of the operator Π : Ψθ → Ψθ which
is a solution v(t) of (9) such that x(t) = v(t) + ξ0(t) is a unique solution of (1) which satisfies
conditions (D1), (D2).
The theorem is proved.
1. Gopalsamy K. On an almost periodic delay-differential system with almost periodic delays // Math. Jap. —
1985. — 30. — P. 849 – 856.
2. Hale J. Theory of functional differential equations. — New York etc.: Springer, 1977.
3. Hale J. Functional differential equations. — New York etc.: Springer, 1971.
4. Lin Qing Cong. Almost periodic solutions of a class of higher-dimensional almost periodic systems with
deviating arguments (Chinese. English, Chinese summary) // J. Zhangzhou Teach. Coll. Nat. Sci. — 1999. —
12. — P. 21 – 24.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
WEXLER INEQUALITY AND ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS . . . 301
5. Seifert G. Almost periodic solutions for undamped nonhomogeneous delay-differential equations // Proc.
Amer. Math. Soc. — 2002. — 130. — P. 2001 – 2005 (electronic).
6. Xu Jian Hua, Wang Zhi Cheng. Almost periodic solutions of a class of almost periodic predator-prey systems
with time delay // Acta math. appl. sinica. — 2000. — 23. — P. 394 – 402.
7. Zhang Shunian, Zheng Guang. Almost periodic solutions of delay difference systems // Appl. Math. Comput.
— 2002. — 131. — P. 497 – 516.
8. Corduneanu C. Some remarks on functional equations with advanced-delayed operations // Sixth Int. Conf.
Comput. Anticipatory Systems, HEC Liege, Belgium (August 11-16, 2003): Abstract book / Ed. D. M. Dubois.
— P. 4.
9. Driver R. D. Ordinary and delay differential equations // Appl. Math. Sci. — New York, Heidelberg: Springer,
1977. — 20.
10. Corduneanu C. Almost periodic functions. — New York etc.: Int. Publ., 1968.
11. Fink A. M. Almost periodic differential quations // Lect. Notes Math. — Berlin etc.: Springer, 1974.
12. Wexler D. Solutions périodiques et presque-périodiques des systémes d’équations différetielles aux impulsi-
ons // Rev. roum. math. pures et appl. — 1965. — 10. — P. 1163 – 1199.
13. Halanay A., Wexler D. Teoria calitativă a sistemelor cu impulsuri. — Bucuresti: Editura Acad. Republ. Soc.
România, 1968.
14. Samoilenko A. M., Perestyuk N. A. Impulsive differential equations. — Singapore: World Sci., 1995.
15. Coppel W. A. Dichotomies in stability theory. — Berlin etc.: Springer, 1978.
16. El’sgol’ts L. E. Introduction to the theory of differential equations with deviating arguments. — San Francisco
etc.: Holden-Day Inc., 1966.
17. Zverkin A. M., Kamenskii G. A., and Norkin S. B. On the formulation of initial conditions for differential
equations with advanced argument (Russian) // Uspekhi Mat. Nauk. — 1960. — 15. — P. 133 – 136.
18. Doss S., Nasr S. K. On the functional equation y
′
= f(x, y(x), y(x + h)), h > 0 // Amer. J. Math. — 1953. —
75. — P. 713 – 716.
Received 28.05.2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 3
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