Oscillations in differential equations with state-dependent delays
In this paper we study the existence of oscillating solutions of a delay differential equations with the delay depending directly on the state. Necessary and sufficient conditions for oscillations are established.
Gespeichert in:
| Datum: | 2003 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2003
|
| Schriftenreihe: | Нелінійні коливання |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/176937 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Oscillations in differential equations with state-dependent delays / K. Niri // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 252-259. — Бібліогр.: 6 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-176937 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1769372021-02-10T01:26:28Z Oscillations in differential equations with state-dependent delays Niri, K. In this paper we study the existence of oscillating solutions of a delay differential equations with the delay depending directly on the state. Necessary and sufficient conditions for oscillations are established. Вивчається питання iснування коливних розв’язкiв диференцiальних рiвнянь iз загаюванням, що залежить безпосередньо вiд стану. Отримано необхiднi та достатнi умови iснування коливань. 2003 Article Oscillations in differential equations with state-dependent delays / K. Niri // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 252-259. — Бібліогр.: 6 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176937 517.929 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper we study the existence of oscillating solutions of a delay differential equations with the delay
depending directly on the state. Necessary and sufficient conditions for oscillations are established. |
| format |
Article |
| author |
Niri, K. |
| spellingShingle |
Niri, K. Oscillations in differential equations with state-dependent delays Нелінійні коливання |
| author_facet |
Niri, K. |
| author_sort |
Niri, K. |
| title |
Oscillations in differential equations with state-dependent delays |
| title_short |
Oscillations in differential equations with state-dependent delays |
| title_full |
Oscillations in differential equations with state-dependent delays |
| title_fullStr |
Oscillations in differential equations with state-dependent delays |
| title_full_unstemmed |
Oscillations in differential equations with state-dependent delays |
| title_sort |
oscillations in differential equations with state-dependent delays |
| publisher |
Інститут математики НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176937 |
| citation_txt |
Oscillations in differential equations with state-dependent delays / K. Niri // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 252-259. — Бібліогр.: 6 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT nirik oscillationsindifferentialequationswithstatedependentdelays |
| first_indexed |
2025-07-15T14:53:27Z |
| last_indexed |
2025-07-15T14:53:27Z |
| _version_ |
1837725078886809600 |
| fulltext |
UDC 517.929
OSCILLATIONS IN DIFFERENTIAL EQUATIONS
WITH STATE-DEPENDENT DELAYS
КОЛИВАННЯ В ДИФЕРЕНЦIАЛЬНИХ РIВНЯННЯХ
IЗ ЗАГАЮВАННЯМ, ЩО ЗАЛЕЖИТЬ ВIД СТАНУ
K. Niri
Université Hassan II-Aїn Chok, Casablanca, Maroc
In this paper we study the existence of oscillating solutions of a delay differential equations with the delay
depending directly on the state. Necessary and sufficient conditions for oscillations are established.
Вивчається питання iснування коливних розв’язкiв диференцiальних рiвнянь iз загаюванням,
що залежить безпосередньо вiд стану. Отримано необхiднi та достатнi умови iснування коли-
вань.
1. Introduction. We consider the differential equation with state-dependent delays,
x′(t) = −f
(
x(t− σ(xt))
)
, for t ≥ 0,
(1)
x0 = φ ∈ C0,
or more generally,
x′(t) = −f
0∫
−r
g
(
x(t− σ(xt) + θ))dθ
, (2)
where f is a function defined from R into R, σ is a function defined from C into [0,M ], σ(xt) ≥
≥ 0, xt : [−∞, 0] → R, xt ∈ BC0([−∞, 0],R) is defined by xt(θ) = x(t+ θ), for θ ∈ [−M, 0].
In many situations the time delay σ(xt) ≥ 0 and depends on the state xt via the threshold
condition
t∫
t−σ(xt)
k(x(s))ds = k0,
where k : R → R+ is continuous and nonnegative, and k(0) = k0. In this case the time delay
is an autonomous function σ(xt) > 0 with the derivative
∂
∂t
σ(xt) = 1− k(x(t))
k(x(t− σ(xt))
< 1. (3)
Throughout this paper, the following assumptions are imposed on f and σ:
uf(u) > 0, for u 6= 0, (4)
c© K. Niri, 2003
252 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
OSCILLATIONS IN DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAYS 253
lim
u→∞
inf
f(u)
u
≥ 1. (5)
There exists δ > 0 such that neither
f(u) ≤ u for 0 ≤ u ≤ δ,
(6)
f(u) ≥ u for − δ ≤ u ≤ 0;
k is uniformly positive, k ≥ k1 > 0.
2. Preliminary. Equations (l) and (2) are of the type
x′(t) = F (xt),
where
F (φ) = −f(φ(−σ(φ))
which is a nonlinear functional equation with values of x lying only in a bounded interval
completely contained in the past. Notice that the functional F is defined in C, but it is nei-
ther differentiable nor locally Lipschitz continuous, whatever the smootness of f and σ. Thus,
together with the differential equation (3) for σt, we can conclude the following.
Theorem 1. Let f : R → R be locally Lipschitzian, Lipschitz continuous uniformly on
bounded sets, k : R → R+ locally Lipschitz continuous. Then for each initial function φ ∈ C,
which is Lipschitz continuous on [−σ(φ), 0], there exists a unique continuously differentiable
solution x : [0,∞] → R equation (1) satisfying x0 = φ and depending only on values of φ in
[−σ(φ), 0].
A general theorem on existence of solutions was given in [1]. In this paper we present some
oscillation results for a class of nonlinear delay differential equations of the type (1). We begin
with a definition of the concept of oscillation.
Definition 1. Let x be a continuous function defined on some infinite interval a,∞]. The
function f is said to oscillate or to be oscillatory if x has arbitrarily large zeros. That is, for every
b > a there exists a point c > b such that x(c) = 0. Otherwise x is called nonoscillatory.
Definition 2. Let x be a continuous function defined on some infinite interval [a,∞]. The
function f is said to be eventually positive or eventually negative if there exists a T ∈ R such that
x(t) is positive for t ≥ T or is negative for t ≥ T .
Lemma 1. Every nonoscillatory solution of equation (1) tends to zero as t → +∞.
Proof. Let x(t) be a nonoscillatory solution of equation (1). We assume that x(t) is eventually
positive. Then eventually,
x′(t) = −f(x(t− σ(xt)) < 0
and so l = lim
t→+∞
x(t) exists and is a nonnegative number. We must show that l = 0. Otherwise,
l > 0 and so lim
t→+∞
x′(t) < 0, which implies that lim
t→+∞
x(t) = −∞. This is a contradiction and
the proof of Lemma 1 is complete.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
254 K. NIRI
3. Linearization of equation (1) at zero.
Lemma 2. Assume that f(0) = 0 and f ′(0) exists. Then the equation (1) is equivalent to a
nonlinear neutral functional at∞.
Let p = f ′(0) and σ0 = σ(φ). Using the Taylor formula for f in the equation (l), we obtain
x′(t) = −f(0)− px(t− σ(xt))− x(t− σ(xt))2f ′′(θ). (7)
We rewrite the term
x(t− σ(xt)) = x(t− σ0) +
∂
∂t
−σ(xt)∫
−σ0
x(t+ θ)dθ
+ x(t− σ(xt))
∂
∂t
σ(xt) (8)
and we put it in (7),
x′(t) = −px(t− σ0)− p ∂
∂t
−σ(xt)∫
−σ0
x(t+ θ)dθ
− px(t− σ(xt))2f ′′(θ)− x(t− σ(xt))
∂
∂t
σ(xt).
But with (3) we conclude that
∂
∂t
σ(xt) = ε
(
‖xt‖[−σ0,0]
)
(9)
so equation (1) is equivalent, at∞, to a nonlinear neutral functional differential equation
d
dt
[x(t) +G(xt)] = −px(t− σ0) +H(xt), (10)
where
G(ψ) = p
−σ(ψ)∫
−σ0
ψ(s)ds
and
H(ψ) = ε(ψ).
The linearized neutral functional differential equation associaded with equation (10) is of
the form
d
dt
[
x(t) +G(xt)
]
= −px(t− σ0),
(11)
x0 = φ.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
OSCILLATIONS IN DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAYS 255
Our first aim in this paper is to prove that equation (1) has the same oscillatory character as
the neutral equation (11).
In [2] and [3] the authors discuss existence of slowly oscillating periodic solutions for equati-
on of type (l). Necessary and sufficient conditions for oscillations of the solutions of equation
of neutral type in the case of a delay that depends on time are obtained in [4 – 6].
In this section we deal with the neutral equation in which the delay depends on the state
and give necessary and sufficient conditions for every solution of equation (11) to oscillate.
Before we present the oscillations results, we establish the following theorems.
Theorem 2. Every nonoscillatory solution of equation (11) goes to zero at infinity.
Proof. Let x(t) be a nonoscillatory solution of equation (11), and suppose that x(t) > 0.
Put z(t) = x(t) +G(xt). Then
z′(t) = −px(t− σ0), (12)
z′(t) < 0, so z(t) is a decreasing function. We claim that x(t) is bounded. Otherwise there exists
a sequence of points tn such that lim
n→∞
tn = ∞, lim
n→∞
x(tn) = ∞, and x(tn) = max
s≤tn
x(s).
Then
z(tn) = x(tn) +G(xtn) = x(tn) + p
−σ(xtn )∫
−σ0
x(tn + θ)dθ.
Because σ(xtn) ≤ σ0 and x(t) is decreasing,
z(tn) ≥ x(tn) + px(tn − σ0)(σ0 − σ(xtn)) ≥ xtn ,
which implies that z(tn) goes to∞ as n → ∞. This contradicts the fact that z(t) is decreasing.
Thus x(t) is bounded and so
l = lim
t→∞
z(t) ∈ R.
By integrating both sides of (12) from t1 to∞, with t sufficiently large, we find
p
∞∫
t1
x(s− σ0)ds = z(t1)− l.
This implies that lim
t→∞
inf x(t) = 0, if we write
z(t) = x(t) + P (t)x(t− σ0),
where
P (t) =
G(xt)
x(t− σ0)
=
p
−σ(xt)∫
−σ0
x(t+ θ)dθ
x(t− σ0)
.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
256 K. NIRI
We verify that P (t) ≤ p and show that l = 0. Consider a sequence of points tn such that
lim
n→∞
tn = ∞, and lim
n→∞
x(tn) = 0,
z(tn) = x(tn) + P (tn)x(tn − σ0),
z(tn + σ0)− z(tn) = x(tn + σ0) + P (tn + σ0)x(tn)− x(tn)− P (tn)x(tn − σ0) =
= x(tn + σ0) +
[
P (tn + σ0)− 1
]
x(tn)− P (tn)x(tn − σ0).
Thus
lim
t→∞
x(tn + σ0)− P (tn)x(tn − σ0) = 0, 0 < x(tn + σ0) < x(tn).
Hence, lim
t→∞
x(tn + σ0) = 0, and lim
t→∞
P (tn)x(tn − σ0) = 0, so
l = lim
n→∞
z(tn) = 0,
and thus lim
t→∞
(x(t) + G(xt)) = 0. Let ε > 0 be given. Then, for t sufficiently large, it follows
that 0 < x(t) < −G(xt) + ε < ε. We conclude that
lim
t→∞
x(t) = 0.
Theorem 3. Suppose that pσ0e > 1. Then the solution of equation (11) oscillates.
Proof. Assume that equation (11) has a nonoscillatory solution x(t). We assume that x(t) is
eventually positive. By Theorem 1, x(t) goes to 0 as t goes to∞ and G(xt) goes to zero too.
We integrate both sides of (11) from t to∞, for t sufficiently large. We have
x(t) +G(xt) = −p
∞∫
t
x(s− σ0)ds. (13)
Set
w(t) = p
∞∫
t
x(s− σ0)ds. (14)
Then
w′(t) = −px(t− σ0).
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
OSCILLATIONS IN DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAYS 257
Eventually w(t) > 0 and w′(t) < 0. We also have
−px(t) = w′(t+ σ0). (15)
By substituting (14) and (15) into (13) we obtain, for t sufficiently large,
w′(t+ σ0)− pG(xt) + pw(t) = 0.
Then
w′(t)− pG(xt−σ0)− pw(t− σ0) = 0.
G(xt + σ0) > 0
implies that
w′(t)− pw(t− σ0) ≥ 0, w(t) > 0. (16)
Then the differential inequality (16) has an eventually positive solution. We deduce that the
differential equation
w′(t)− pw(t− σ0) = 0 (17)
also has an eventually positive solution. This contradicts the fact that every solution of equation
(17) oscillates, because pσ0e > 1. The proof of Theorem 3 is complete.
4. Necessary and sufficient conditions for the oscillations of (1).
Theorem 4. Assume that (3) holds. Then every solution of equation (1) oscillates if every
solution of the linearized equation (11) oscillates.
Proof. Assume for the sake of contradiction, that equation (1) has a nonoscillatory solution
x(t). We assume that x(t) is eventually positive. By Lemma 1, we know that lim
t→+∞
x(t) = 0.
Thus by (5)
lim
u→+∞
inf
f(x(t− σ(xt))
x(t− σ(xt))
≥ 1.
Let ε ∈ (0, 1). Then there exists a T (ε) such that for every t > T (ε), x(t − σ(xt)) > 0 and
f(x(t− σ(xt)) ≥ (1− ε)x(t− σ(xt)).
Hence from equation (1),
x′(t) + (1− ε)x(t− σ(xt)) ≤ 0 for t ≥ T (ε). (18)
Substituting the term x(t− σ(xt)), by formula (8), we obtain
(
1− ∂
∂t
σ(xt)
)
x(t− σ(xt)) = x(t− σ0) +
∂
∂t
−σ(xt)∫
−σ0
x(t+ θ)dθ
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
258 K. NIRI
and so, by (8),
x(t− σ(xt)) =
(
1 +
∂
∂t
σ(xt) + o(t)
)
x(t− σ0) +
(
1 +
∂
∂t
σ(xt) + o(t)
)
1
p
∂
∂t
G(xt).
Then the inequality (18) becomes
x′(t) +
1− ε
p
(
1 +
∂
∂t
σ(xt) + o(t)
)
px(t− σ0) +
1− ε
p
(
1 +
∂
∂t
σ(xt) + o(t)
)
∂
∂t
G(xt) ≤ 0.
This inequality is valid for any ε ∈ (0, 1), and t > T (ε). Since xt is continuous,
∂
∂t
σ(xt) → 0
and [
x(t) +G(xt)
]′+px(t− σ0) ≤ 0 for t ≥ T (ε). (19)
Inequality (19) has an eventually positive solution. Then equation (9) also has an eventually
positive solution. This contradicts the hypothesis that every solution of equation (11) is osci-
llatory. This ends the proof.
Conversely we have the following.
Theorem 5. Suppose that (6) holds. If the solution of equation (l) is oscillatory, then
pσ0e > 1.
We assume that pσ0e > 1. Then equation y′(t) + py(t− σ0) = 0 has an eventually positive
solution y(t).
We know that lim
t→+∞
y(t) = 0 and so there exists a t0 such that 0 < y(t) < δ for t0 − σ0 <
< t < t0.
With the initial function equal to y(t) for t0 − σ0 < t < t0, equation (1) has a solution x(t)
which exists at least in some small heighbourhood to the right of t0. It suffices to show that for
as long as x(t) exists,
y(t) ≤ x(t) < δ
and then x(t) > 0, which is a contradiction.
For t0 < t < t0 + σ0, 0 ≤ x(t− σ(xt)) < δ, and so we have, by (6),
x′(t) = −f(x(t− σ(xt)) ≥ −px(t− σ(xt)) ≥ −px(t− σ0).
So
x′(t) + px(t− σ0) ≥ 0
and
y′(t) + py(t− σ0) = 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
OSCILLATIONS IN DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAYS 259
By a comparison result for the positive solution of the delay differential inequalities [6],
0 < y(t) ≤ x(t),
x(t) > 0, which is a contradiction to the fact that if x(t) oscillates, then pσ0e > 1.
Theorem 6. Suppose that (5), (6) hold. Then the solution of (1) is oscillating if and only if
pσ0e > 1.
By combining Theorem 3, Theorem 4, and Theorem 5, we have the needed result. This end
the proof.
We derive the following lemma.
Lemma 3. Assume that the conditions (5) and (6) are fulfilled, and σ is bounded by a positive
number τ > 0. Then, every solution of equation (1) is oscillating if and only if pτe > 1.
1. Hale J.K. Theory of functional differential equation // Appl. Math. Sci. — 1977. — 3.
2. Alt W. Periodic solutions of some autonomous differential equations with variable time delay // Funct. Di-
fferent. Equat. and Approxim. of Fixed Points. — 1978. — 730.
3. Arino O., Hadeler K.P., and Hbid M.L. Existence of periodic solutions for delay differential equations with
state dependent delay // J. Different. Equat. — 1998. — 144, № 2. — P. 263 – 301.
4. Arino O., Gyori I. Necessary and sufficient condition for oscillation of neutral differential system with several
delays // Ibid. — 1990. — 81. — P. 98 – 105.
5. Chuanxi Q., Ladas G. Linearized oscillations for odd-order neutral differential equations // Ibid. — 1990.
6. Gyori I., Ladas G. Oscillation theory of delay differential equations with applications: Oxford Math. Monogr.
— Oxford: Clarendon Press, 1991.
Received 27.02.2003
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 2
|