Second order nonlinear differential equations with an infinite set of periodic solutions
For the differential equation u′′ = f(t, u, u′), where the function f : R × R² → R is periodic in the first argument and f(t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Для диференцiального рiвняння u′′ = f(t, u, u′), де функцiя f...
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Kiguradze, I.T. 2021-02-18T08:11:12Z 2021-02-18T08:11:12Z 2008 Second order nonlinear differential equations with an infinite set of periodic solutions / I.T. Kiguradze // Нелінійні коливання. — 2008. — Т. 11, № 4. — С. 495-500. — Бібліогр.: 16 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178191 517.9 For the differential equation u′′ = f(t, u, u′), where the function f : R × R² → R is periodic in the first argument and f(t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Для диференцiального рiвняння u′′ = f(t, u, u′), де функцiя f : R × R² → R є перiодичною за першим аргументом i f(t, x, 0) ≡ 0, знайдено необхiднi умови для iснування континууму перiодичних розв’язкiв, що не є сталими. This work is supported by the Georgian National Science Foundation (Project № GNSF/ST06/3-002). en Інститут математики НАН України Нелінійні коливання Second order nonlinear differential equations with an infinite set of periodic solutions Нелінійні диференціальні рівняння другого порядку з нескінченною множиною періодичних розв’язків Нелинейные дифференциальные уравнения второго порядка с бесконечным множеством периодических решений Article published earlier |
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Second order nonlinear differential equations with an infinite set of periodic solutions |
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Second order nonlinear differential equations with an infinite set of periodic solutions Kiguradze, I.T. |
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Second order nonlinear differential equations with an infinite set of periodic solutions |
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Second order nonlinear differential equations with an infinite set of periodic solutions |
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Second order nonlinear differential equations with an infinite set of periodic solutions |
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Second order nonlinear differential equations with an infinite set of periodic solutions |
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second order nonlinear differential equations with an infinite set of periodic solutions |
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Kiguradze, I.T. |
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Kiguradze, I.T. |
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2008 |
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English |
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Нелінійні коливання |
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Інститут математики НАН України |
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Нелінійні диференціальні рівняння другого порядку з нескінченною множиною періодичних розв’язків Нелинейные дифференциальные уравнения второго порядка с бесконечным множеством периодических решений |
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For the differential equation u′′ = f(t, u, u′), where the function f : R × R² → R is periodic in the first
argument and f(t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic
solutions are found.
Для диференцiального рiвняння u′′ = f(t, u, u′), де функцiя f : R × R² → R є перiодичною за
першим аргументом i f(t, x, 0) ≡ 0, знайдено необхiднi умови для iснування континууму перiодичних розв’язкiв, що не є сталими.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/178191 |
| citation_txt |
Second order nonlinear differential equations with an infinite set of periodic solutions / I.T. Kiguradze // Нелінійні коливання. — 2008. — Т. 11, № 4. — С. 495-500. — Бібліогр.: 16 назв. — англ. |
| work_keys_str_mv |
AT kiguradzeit secondordernonlineardifferentialequationswithaninfinitesetofperiodicsolutions AT kiguradzeit nelíníinídiferencíalʹnírívnânnâdrugogoporâdkuzneskínčennoûmnožinoûperíodičnihrozvâzkív AT kiguradzeit nelineinyedifferencialʹnyeuravneniâvtorogoporâdkasbeskonečnymmnožestvomperiodičeskihrešenii |
| first_indexed |
2025-11-24T21:53:31Z |
| last_indexed |
2025-11-24T21:53:31Z |
| _version_ |
1850498878433918976 |
| fulltext |
UDC 517.9
SECOND ORDER
NONLINEAR DIFFERENTIAL EQUATIONS
WITH AN INFINITE SET OF PERIODIC SOLUTIONS*
НЕЛIНIЙНI ДИФЕРЕНЦIАЛЬНI РIВНЯННЯ ДРУГОГО ПОРЯДКУ
З НЕСКIНЧЕННОЮ МНОЖИНОЮ ПЕРIОДИЧНИХ РОЗВ’ЯЗКIВ
I. Kiguradze
A. Razmadze Math. Inst.
1, M. Aleksidze St., Tbilisi 0193, Georgia
e-mail: kig@rmi.acnet.ge
For the differential equation u′′ = f(t, u, u′), where the function f : R × R2 → R is periodic in the first
argument and f(t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic
solutions are found.
Для диференцiального рiвняння u′′ = f(t, u, u′), де функцiя f : R × R2 → R є перiодичною за
першим аргументом i f(t, x, 0) ≡ 0, знайдено необхiднi умови для iснування континууму перiо-
дичних розв’язкiв, що не є сталими.
The problems on the existence, uniqueness and non-uniqueness of periodic solutions of nonli-
near differential equations and systems attract attention of many mathematicians and are the
subject of numerous investigations (see, e.g, [1 – 16] and the references therein). Nevertheless,
the description of classes of equations having a continuum of periodic solutions is far from
being complete. The goal of the present paper is to fill this gap to a certain extent.
Below we consider the differential equation
u′′ = f(t, u, u′), (1)
where the function f : R × R2 → R satisfies the local Carathéodory conditions, i.e., f(t, ·, ·) :
R2 → R is continuous for almost all t ∈ R, f(·, x, y) : R → R is measurable for all (x, y) ∈ R2,
and for an arbitrary ρ > 0 the function fρ, given by
fρ(t) = max {|f(t, x, y)| : |x| + |y| ≤ ρ} for t ∈ R,
is Lebesgue integrable on every finite interval.
We are interested in the case, where the equalities
f(t+ ω, x, y) = f(t, x, y), f(−t, x,−y) = f(t, x, y),
(2)
f(t,−x,−y) = −f(t, x, y),
f(t, x, 0) = 0 (3)
∗ This work is supported by the Georgian National Science Foundation (Project № GNSF/ST06/3-002).
c© I. Kiguradze, 2008
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 4 495
496 I. KIGURADZE
are fulfilled on R×R2; here ω is a positive constant.
In view of (3), equation (1) has a continuum of constant solutions. There naturally arises the
question whether equation (1) under the conditions (2) and (3) may have nonconstant periodic
solutions. As is stated in the proven below Theorem 1, the answer is positive.
Let R+ = [0,+∞), Lω be the space of ω-periodic and Lebesgue integrable on [0, ω] real
functions, and Mω the set of functions ϕ : R × R+ → R+ such that ϕ(·, x) ∈ Lω for arbitrary
x ∈ R+, ϕ(t, ·) : R+ → R a continuous nondecreasing function for almost all t ∈ R, ϕ(t, 0) ≡
≡ 0 and
ω
∫
0
ϕ(t, x) dt > 0 for x > 0. (4)
Theorem 1. Let conditions (2), (3) be fulfilled and
f(t, x, y) ≤ −ϕ(t, x)ψ(y) for t ∈ R+, x ∈ R+, 0 ≤ y ≤ r, (5)
where r > 0, ϕ ∈ Mω, and ψ : [0, r] → R+ is a continuous function such that
ψ(0) = 0, ψ(y) > 0 for 0 < y ≤ r,
r
∫
0
dy
ψ(y)
< +∞. (6)
Then equation (1) has a continuum of nonconstant periodic solutions.
To prove the theorem, we will need the following lemma.
Lemma 1. Let inequality (5) be fulfilled, where ϕ ∈ Mω, and ψ : [0, r] → R+ is a continuous
function satisfying condition (6). Then for an arbitrary c ∈ (0, r), there exists tc ∈ (0,+∞) such
that equation (1) on the interval [0, tc] has a solution uc satisfying the conditions
uc(0) = 0, u′c(0) = c, (7)
uc(t) > 0, 0 < u′c(t) < r for 0 < t < tc, u′c(tc) = 0. (8)
Proof. Let uc be a maximally extended to the right solution of problem (1), (7). Then either
uc is defined on R+, and
uc(t) > 0, 0 < u′c(t) < r for t ∈ R+, (9)
or there exists tc ∈ (0,+∞) such that
uc(t) > 0, 0 < u′c(t) < r for 0 < t < tc (10)
and
u′c(tc) ∈ {0, r}. (11)
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SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS . . . 497
First we assume that condition (9) is fulfilled. Then in view of (5), for an arbitrarily fixed
a > 0 almost everywhere on [a,+∞) the inequality
ϕ(t, x) ≤ −
u′′c (t)
ψ(u′c(t))
is fulfilled, where x = uc(a) > 0. Integrating this inequality from a to a + kω, where k is an
arbitrary natural number, due to the ω-periodicity of ϕ(·, x) and condition (6) we find
k
a+ω
∫
a
ϕ(t, x) dt ≤
u′
c(a)
∫
u′
c(a+kω)
dy
ψ(y)
< ρ,
where
ρ =
r
∫
0
dy
ψ(y)
< +∞.
Consequently,
ω
∫
0
ϕ(t, x) dt =
a+ω
∫
a
ϕ(t, x) dt ≤
ρ
k
→ 0 as k → +∞,
which contradicts condition (4). The obtained contradiction proves that the function uc does
not satisfy inequalities (9). Hence for some tc ∈ (0,+∞), conditions (10) and (11) are fulfilled.
According to (5) and (10), almost everywhere on (0, tc) the inequality
u′′c (t) ≤ 0
is satisfied. Therefore u′c(tc) ≤ c < r, whence by virtue of (11) it follows that uc(tc) = 0. Thus
condition (8) holds.
The lemma is proved.
Lemma 2. Let on R × R2 equalities (2) be fulfilled and let the function u be a solution of
equation (1) on some interval [0, t0] ⊂ R+. Then for an arbitrary natural k the function v, given
by the equality
v(t) = u(kω − t) for kω − t0 ≤ t ≤ kω,
is a solution of equation (1) on [kω − t0, kω].
Proof. Indeed,
v′′(t) = u′′(kω − t) = f
(
kω − t, u(kω − t), u′(kω − t)
)
=
= f
(
kω − t, v(t),−v′(t)
)
almost everywhere on [kω − t, kω].
Thus according to (2), we find
v′′(t) = f
(
−t, v(t),−v′(t)
)
=
= f
(
t, v(t), v′(t)
)
almost everywhere on [kω − t, kω].
The lemma is proved.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 4
498 I. KIGURADZE
Proof of the theorem. Owing to Lemma 1, for an arbitrary c ∈ (0, r) there exists tc ∈
∈ (0,+∞) such that equation (1) on [0, tc] has a solution u satisfying conditions (7) and (8). We
choose a natural number k so that
kω ≥ 2tc
and extend uc on R in the following manner:
uc(t) =
{
uc(tc) for tc ≤ t ≤ kω − tc,
uc(kω − t) for kω − tc ≤ t ≤ kω,
uc(t+ kω) = −uc(t) for t ∈ R.
By conditions (2), (3) and Lemma 2, the function uc is a 2kω-periodic solution of equation
(1). On the other hand, it is evident that
uc1(t) 6≡ uc2(t) 6≡ const for 0 < c1 < c2 < r.
Consequently, if c runs through the interval (0, r), we obtain a continuum of periodic nonconstant
solutions of equation (1).
The theorem is proved.
As an example, we consider the generalized Emden – Fowler equation
u′′ =
m
∑
k=1
pk(t)|u
′|µk |u|λksgnu, (12)
where
λk > 0, µk > 0 pk ∈ Lω,
(13)
pk(−t) = pk(t) ≤ 0 for t ∈ R,
ω
∫
0
pk(t) dt < 0, k = 1, . . . ,m. (14)
The following proposition holds.
Corollary. Let conditions (13) and (14) be fulfilled. Then for the existence of a continuum of
periodic solutions of equation (12) it is necessary and sufficient that
min{µ1, . . . , µn} < 1. (15)
Proof. Assume first that along with (13) and (14) condition (15) is fulfilled. Then without
loss of generality we can assume that µ1 < 1. Due to condition (13), the function f, given by
the equality
f(t, x, y) =
m
∑
k=1
pk(t)|y|
µk |x|λksgnx,
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 4
SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS . . . 499
satisfies conditions (2) and (3). On the other hand, for an arbitrary r > 0 inequality (5) is
fulfilled, where
ϕ(t, x) = |p1(t)|x
λ1 , ψ(y) = yµ1 .
Moreover, ϕ ∈ Mω, and ψ satisfies condition (6) since
ω
∫
0
|p1(t)| dt > 0 and 0 < µ1 < 1.
Consequently, all the conditions of the above-given theorem are fulfilled which guarantees
the existence of a continuum of nonconstant ω-periodic solutions of equation (12).
It remains to state that if
µk ≥ 1, k = 1, . . . ,m, (16)
then an arbitrary periodic solution u of equation (12) is constant. Indeed, almost everywhere
on R the equality
u′′(t) = p(t)u′(t) (17)
is fulfilled, where
p(t) =
m
∑
k=1
pk(t)|u
′(t)|µk−1|u(t)|λksgn (u(t)u′(t));
in addition, in view of (16), we have
p ∈ Lω. (18)
On the other hand, owing to the ω-periodicity of u, there exists t0 ∈ R such that
u′(t0) = 0.
Thus it follows from (17) and (18) that u′(t) ≡ 0, i.e., u(t) ≡ const.
The corollary is proved.
Remark. If pk(t) ≡ 0, k = 1, . . . ,m, then equation (12) has no nonconstant periodic
solution. Consequently, condition (14) in the above-given corollary is essential and it cannot
be weakened.
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Received 03.10.08
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