Optimal existence theory for single and multiple positive periodic solutions to functional differential equations
This paper deals with a new optimal existence theory for single and multiple positive periodic solutions to functional differential equations by employing a fixed point theorem in cones. We illustrate our theory by examining several biomathematical models. The paper improves and extends previous r...
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nasplib_isofts_kiev_ua-123456789-1769432025-02-09T23:24:37Z Optimal existence theory for single and multiple positive periodic solutions to functional differential equations Оптимальна теорія існування єдиного і кратних додатних періодичних розв'язків функціонально-диференціальних рівнянь Оптимальная теория существования единственного и кратных положительных периодических решений функционально-дифференциальных уравнений Jiang, D. O'Regan, D. Agarwal, R.P. This paper deals with a new optimal existence theory for single and multiple positive periodic solutions to functional differential equations by employing a fixed point theorem in cones. We illustrate our theory by examining several biomathematical models. The paper improves and extends previous results in the literature З використанням теореми про нерухому точку в конусах розглянуто оптимальну теорiю iснування єдиного i кратних додатних перiодичних розв’язкiв функцiонально-диференцiальних рiвнянь. Теорiю проiлюстровано прикладами кiлькох математичних моделей, що використовуються в бiологiї. Отриманi результати покращують i узагальнюють попереднi результати. The work was supported by NNSF of China (project 10171010). 2003 Article Optimal existence theory for single and multiple positive periodic solutions to functional differential equations / D. Jiang, D. O'Regan, R.P. Agarwal // Нелінійні коливання. — 2003. — Т. 6, № 3. — С. 334-345. — Бібліогр.: 12 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/176943 517.9 en Нелінійні коливання application/pdf Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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This paper deals with a new optimal existence theory for single and multiple positive periodic solutions
to functional differential equations by employing a fixed point theorem in cones. We illustrate our theory
by examining several biomathematical models. The paper improves and extends previous results in the
literature |
| format |
Article |
| author |
Jiang, D. O'Regan, D. Agarwal, R.P. |
| spellingShingle |
Jiang, D. O'Regan, D. Agarwal, R.P. Optimal existence theory for single and multiple positive periodic solutions to functional differential equations Нелінійні коливання |
| author_facet |
Jiang, D. O'Regan, D. Agarwal, R.P. |
| author_sort |
Jiang, D. |
| title |
Optimal existence theory for single and multiple positive periodic solutions to functional differential equations |
| title_short |
Optimal existence theory for single and multiple positive periodic solutions to functional differential equations |
| title_full |
Optimal existence theory for single and multiple positive periodic solutions to functional differential equations |
| title_fullStr |
Optimal existence theory for single and multiple positive periodic solutions to functional differential equations |
| title_full_unstemmed |
Optimal existence theory for single and multiple positive periodic solutions to functional differential equations |
| title_sort |
optimal existence theory for single and multiple positive periodic solutions to functional differential equations |
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Інститут математики НАН України |
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2003 |
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https://nasplib.isofts.kiev.ua/handle/123456789/176943 |
| citation_txt |
Optimal existence theory for single and multiple positive periodic solutions to functional differential equations / D. Jiang, D. O'Regan, R.P. Agarwal // Нелінійні коливання. — 2003. — Т. 6, № 3. — С. 334-345. — Бібліогр.: 12 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT jiangd optimalexistencetheoryforsingleandmultiplepositiveperiodicsolutionstofunctionaldifferentialequations AT oregand optimalexistencetheoryforsingleandmultiplepositiveperiodicsolutionstofunctionaldifferentialequations AT agarwalrp optimalexistencetheoryforsingleandmultiplepositiveperiodicsolutionstofunctionaldifferentialequations AT jiangd optimalʹnateoríâísnuvannâêdinogoíkratnihdodatnihperíodičnihrozvâzkívfunkcíonalʹnodiferencíalʹnihrívnânʹ AT oregand optimalʹnateoríâísnuvannâêdinogoíkratnihdodatnihperíodičnihrozvâzkívfunkcíonalʹnodiferencíalʹnihrívnânʹ AT agarwalrp optimalʹnateoríâísnuvannâêdinogoíkratnihdodatnihperíodičnihrozvâzkívfunkcíonalʹnodiferencíalʹnihrívnânʹ AT jiangd optimalʹnaâteoriâsuŝestvovaniâedinstvennogoikratnyhpoložitelʹnyhperiodičeskihrešeniifunkcionalʹnodifferencialʹnyhuravnenii AT oregand optimalʹnaâteoriâsuŝestvovaniâedinstvennogoikratnyhpoložitelʹnyhperiodičeskihrešeniifunkcionalʹnodifferencialʹnyhuravnenii AT agarwalrp optimalʹnaâteoriâsuŝestvovaniâedinstvennogoikratnyhpoložitelʹnyhperiodičeskihrešeniifunkcionalʹnodifferencialʹnyhuravnenii |
| first_indexed |
2025-12-01T17:02:26Z |
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2025-12-01T17:02:26Z |
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1850326172593815552 |
| fulltext |
UDC 517.9
OPTIMAL EXISTENCE THEORY FOR SINGLE
AND MULTIPLE POSITIVE PERIODIC SOLUTIONS
TO FUNCTIONAL DIFFERENTIAL EQUATIONS*
ОПТИМАЛЬНА ТЕОРIЯ IСНУВАННЯ ЄДИНОГО
I КРАТНИХ ДОДАТНИХ ПЕРIОДИЧНИХ РОЗВ’ЯЗКIВ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
D. Jiang
Northeast Normal Univ.
Changchun 130024, P. R. China
e-mail: daqingjiang@vip.163.com.
D. O’Regan
Nat. Univ. Ireland
Galway, Ireland
e-mail: donal.oregan@nuigalway.ie
R. P. Agarwal
Florida Inst. Technology
Melbourne, Florida 32901-6975, USA
This paper deals with a new optimal existence theory for single and multiple positive periodic solutions
to functional differential equations by employing a fixed point theorem in cones. We illustrate our theory
by examining several biomathematical models. The paper improves and extends previous results in the
literature.
З використанням теореми про нерухому точку в конусах розглянуто оптимальну теорiю iсну-
вання єдиного i кратних додатних перiодичних розв’язкiв функцiонально-диференцiальних рiв-
нянь. Теорiю проiлюстровано прикладами кiлькох математичних моделей, що використову-
ються в бiологiї. Отриманi результати покращують i узагальнюють попереднi результати.
1. Introduction. The purpose of the present paper is to present optimal existence conditions for
single and multiple positive periodic solutions for the general functional differential equation
·
y (t) = −a(t)y(t) + g(t, y(t− τ(t))) (1.1)
where a(t) ∈ C(R, (0,∞)), τ(t) ∈ C(R,R), g ∈ C(R × [0,∞), [0,∞)), and a(t), τ(t), g(t, y)
are all ω-periodic functions; here ω > 0 is a constant.
It is well known that the functional differential equation (1.1) includes many mathematical
ecological equations. For example, see the Hematopoiesis model [1 – 3]
·
y (t) = −a(t)y(t) + b(t)e−β(t)y(t−τ(t)); (1.2)
* The work was supported by NNSF of China (project 10171010).
c© D. Jiang, D. O’Regan, and R. P. Agarwal, 2003
334 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
OPTIMAL EXISTENCE THEORY FOR SINGLE AND MULTIPLE POSITIVE PERIODIC SOLUTIONS . . . 335
and the more general model of blood cell production [1, 3 – 5]
·
y (t) = −a(t)y(t) + b(t)
1
1 + y(t− τ(t))n
, n > 0, (1.3)
·
y (t) = −a(t)y(t) + b(t)
y(t− τ(t))
1 + y(t− τ(t))n
, n > 0; (1.4)
and also the more general Nicholson’s blowflies model [1, 3, 6 – 8]
·
y (t) = −a(t)y(t) + b(t)y(t− τ(t))e−β(t)y(t−τ(t)). (1.5)
To our knowledge, there are only a few papers on the existence of positive periodic solutions
for Eq. (1.1), even for (1.2) – (1.5). The systems (1.2), (1.3) and (1.5) have been investigated in [2,
4, 6]. In these papers estimates of solutions are obtained and also it is shown that the solutions
are uniformly bounded and uniformly-ultimately bounded. In addition a group of conditions
are given to guarantee the existence of one positive ω-periodic solution for Eq. (1.2), (1.3) and
(1.5) by applying the Yoshizawa theorem [9].
Very recently, the authors in [3] have considered the existence of one positive periodic
solution for the general functional differential equation
·
y (t) = −a(t)y(t) + b(t)f(t, y(t− τ(t))) (1.6)
where a(t), b(t) ∈ C(R, (0,∞)), τ(t) ∈ C(R,R), f ∈ C(R×[0,∞), [0,∞)), and a(t), b(t), τ(t),
f(t, y) are all ω-periodic functions; here ω > 0 is a constant. The main results in [3] are as
follows.
Theorem A. Eq. (1.6) has at least one ω-periodic positive solution, provided the following
condition holds:
lim
u↓0
min
t∈[0,ω]
f(t, u)
u
= ∞ and lim
u↑∞
max
t∈[0,ω]
f(t, u)
u
= 0 (sublinear).
Theorem B. Assume that
B1) mint∈[0,ω]{b(t)− a(t)} > 0;
B2) there exists a ε0 > 0 such that f(t, u) is increasing in 0 ≤ u ≤ ε0. Then Eq. (1.6) has at
least one ω-periodic positive solution, provided the following condition holds:
lim
u↓0
min
t∈[0,ω]
f(t, u)
u
= 1 and lim
u↑∞
max
t∈[0,ω]
f(t, u)
u
= 0.
The proofs of Theorems A and B are based on an application of the norm-type compression
theorem in cones due to Krasnoselskii (see [10, 11]).
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
336 D. JIANG, D. O’REGAN, AND R. P. AGARWAL
Motivated by the work above, in this paper we shall present a new optimal existence theory
for single and multiple positive periodic solutions of Eq. (1.1).
Let
σ = e
−
ω∫
0
a(ξ)dξ
. (1.7)
In this paper, we have the following hypotheses:
H1) lim infu↓0
g(t, u)
u
> a(t) and lim inf
u↑∞
g(t, u)
u
> a(t);
H2) lim supu↓0
g(t, u)
u
< a(t) and lim sup
u↑∞
g(t, u)
u
< a(t);
H3) there is a p > 0 such that σp ≤ u ≤ p implies
g(t, u) < a(t)p, 0 ≤ t ≤ ω;
H4) there is a p > 0 such that σp ≤ u ≤ p implies
g(t, u) > a(t)u, 0 ≤ t ≤ ω.
Remark 1. If there is a p > 0 such that σp ≤ u ≤ p implies
g(t, u) < a(t)u, 0 ≤ t ≤ ω,
then H3) holds.
2. Main results. First of all, notice that to find a ω-periodic solution of Eq. (1.1) is equivalent
to finding a ω-periodic solution of the integral equation
y(t) =
t+ω∫
t
G(t, s)g(s, y(s− τ(s)))ds, (2.1)
where
G(t, s) :=
exp
s∫
t
a(ξ)dξ
exp
ω∫
0
a(ξ)dξ
− 1
. (2.2)
One can see, for s ∈ [t, t+ ω], that
A
df
= G(t, t) ≤ G(t, s) ≤ G(t, t+ ω)
df
= B, (2.3)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
OPTIMAL EXISTENCE THEORY FOR SINGLE AND MULTIPLE POSITIVE PERIODIC SOLUTIONS . . . 337
where
A =
1
exp
ω∫
0
a(ξ)dξ
− 1
, B =
exp
ω∫
0
a(ξ)dξ
exp
ω∫
0
a(ξ)dξ
− 1
.
Thus σ = A/B, where σ is as in (1.7).
Let
X = {y(t) : y(t) ∈ C(R,R), y(t+ ω) = y(t)}, (2.4)
and define
‖y‖ = sup
t∈[0,ω]
{|y(t)| : y ∈ X}.
Then X with the norm ‖ · ‖ is a Banach space.
By using (2.1), (2.2), we know for every positive ω-periodic solution of Eq. (1.1), one has
‖y‖ ≤ B
ω∫
0
g(s, y(s− τ(s)))ds,
and
y(t) ≥ A
ω∫
0
g(s, y(s− τ(s)))ds,
so we have
y(t) ≥ A
B
‖y‖ = σ‖y‖. (2.5)
The following theorems are our main results.
Theorem 1. Assume that H1) and H3) are satisfied. Then Eq. (1.1) has at least two ω-periodic
positive solutions y1 and y2 such that
0 < ‖y1‖ < p < ‖y2‖.
Corollary 1. The conclusion of Theorem 1 remains valid if H1), is replaced by:
H∗1) lim infu↓0
g(t, u)
u
= ∞ and lim inf
u↑∞
g(t, u)
u
= ∞.
Theorem 2. Assume that H2) and H4) are satisfied. Then Eq. (1.1) has at least two ω-periodic
positive solutions y1 and y2 such that
0 < ‖y1‖ < p < ‖y2‖.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
338 D. JIANG, D. O’REGAN, AND R. P. AGARWAL
Corollary 2. The conclusion of Theorem 2 remains valid if H2) is replaced by:
H∗2) lim supu↓0
g(t, u)
u
= 0 and lim sup
u↑∞
g(t, u)
u
= 0.
Theorem 3. Eq. (1.1) has at least one ω-periodic positive solution, provided one of the
following conditions holds:
i) lim infu↓0
g(t, u)
u
> a(t) and lim sup
u↑∞
disp
g(t, u)
u
< a(t);
ii) lim supu↓0
g(t, u)
u
< a(t) and lim inf
u↑∞
g(t, u)
u
> a(t).
Corollary 3. Eq. (1.1) has at least one ω-periodic positive solution, provided one of the
following conditions holds:
i) lim infu↓0
g(t, u)
u
= ∞ and lim sup
u↑∞
g(t, u)
u
= 0 (sublinear);
ii) lim supu↓0
g(t, u)
u
= 0 and lim inf
u↑∞
g(t, u)
u
= ∞ (superlinear).
Remark 2. Theorem 3 extends and improves Theorems A and B in [3].
Remark 3. Note that if g(t, u) = a(t)u, then the existence of positive ω-periodic solutions
for linear problem
·
y (t) = −a(t)y(t) + a(t)y(t− τ(t))
cannot be guaranteed. As a result the conditions in Theorems 1 – 3 are optimal.
3. Proof of main results. First, we state the fixed point theorem in cones which will be used
in this section.
Lemma 1[10]. Let X = (X, || · ||) be a Banach space and let K be a cone in X . Also, r,R
are constants with 0 < r < R. Suppose Φ : Ω̄R ∩K → K( here ΩR = {x ∈ X, ||x|| < R}) be
a continuous and completely continuous operator such that
i) x 6= λΦx, for λ ∈ [0, 1] and x ∈ K ∩ ∂Ωr,
and
ii) there exists ψ ∈ K\{0} such that x 6= Φx+ δψ for x ∈ K ∩ ∂ΩR and δ ≥ 0.
Then Φ has a fixed point in K ∩ {x ∈ X : r < ||x|| < R}.
Remark 4. In Theorem 1, if i) and ii) are replaced by
i) ∗ x 6= λΦx, for λ ∈ [0, 1] and x ∈ K ∩ ∂ΩR,
and
ii) ∗ there exists ψ ∈ K\{0} such that x 6= Φx+ δψ for x ∈ K ∩ ∂Ωr and δ ≥ 0.
Then Φ has a fixed point in K ∩ {x ∈ X : r < ||x|| < R}.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
OPTIMAL EXISTENCE THEORY FOR SINGLE AND MULTIPLE POSITIVE PERIODIC SOLUTIONS . . . 339
Let X be as in (2.4). Define an operator on X as follows:
y = Φy (3.1)
where Φ is defined by
(Φy)(t) =
t+ω∫
t
G(t, s)g(s, y(s− τ(s)))ds, (3.2)
for y ∈ X . Clearly, Φ is a continuous and completely continuous operator on X .
Let
K = {y ∈ X : y(t) ≥ 0 and y(t) ≥ σ‖y‖};
here σ is as in (1.7). It is not difficult to verify that K is a cone in X .
Lemma 2. Φ(K) ⊆ K.
Proof. For any y ∈ K, we have
‖Φy‖ ≤ B
ω∫
0
g(s, y(s− τ(s)))ds,
and
(Φy)(t) ≥ A
ω∫
0
g(s, y(s− τ(s)))ds.
Thus we have
(Φy)(t) ≥ A
B
‖Φy‖ = σ‖Φy‖,
i. e., Φy ∈ K. This completes the proof of Lemma 2.
Proof of Theorem 1. Suppose that H1) and H3) hold. By using the first inequality of H1),
i.e., lim infu↓0
g(t, u)
u
> a(t), one can find 0 < r < p and ε > 0 such that
g(t, u) ≥ a(t)(1 + ε)u, whenever 0 ≤ u ≤ r. (3.3)
Thus, if y ∈ K with ‖y‖ = r, then y(t) ≥ σr.
Let ψ ≡ 1 and we now prove that
y 6= Φy + δψ for y ∈ K ∩ ∂Ωr and δ ≥ 0. (3.4)
If not, there exist y0 ∈ K ∩ ∂Ωr and δ0 ≥ 0 such that
y0 = Φy0 + δ0ψ.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
340 D. JIANG, D. O’REGAN, AND R. P. AGARWAL
Let µ = mint∈R y0(t). Then for t ∈ R we have
y0(t) = (Φy0)(t) + δ0 =
=
t+ω∫
t
G(t, s)g(s, y0(s− τ(s)))ds+ δ0 ≥
≥
t+ω∫
t
G(t, s)a(s)(1 + ε)y0(s− τ(s))ds ≥
≥ µ(1 + ε)
t+ω∫
t
G(t, s)a(s)ds = µ(1 + ε),
and this implies µ ≥ µ(1 + ε), a contradiction.
Next, by using the inequality in H3), we prove that
y 6= λΦy for y ∈ K ∩ ∂Ωp and 0 ≤ λ ≤ 1. (3.5)
If not, there exist y0 ∈ K ∩ ∂Ωp and 0 ≤ λ0 ≤ 1 such that
y0 = λ0Φy0.
Clearly, λ0 > 0. Thus, ‖y0‖ = p and σp ≤ y0(t) ≤ p for t ∈ R, so we have
g(t, y0(t− τ(t))) < a(t)p, t ∈ R. (3.6)
Then we obtain
y0(t) = λ0(Φy0)(t) =
= λ0
t+ω∫
t
G(t, s)g(s, y0(s− τ(s)))ds <
<
t+ω∫
t
G(t, s)a(s)pds = p,
and this implies ||y0|| = p < p, a contradiction.
In view of (3.4) and (3.5), by Lemma 1, we see that Φ has a fixed point y1 ∈ K and r <
< ‖y1‖ < p. Thus y1(t) ≥ σr > 0, which means that y1(t) is a ω-periodic positive solution
of (1.1).
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
OPTIMAL EXISTENCE THEORY FOR SINGLE AND MULTIPLE POSITIVE PERIODIC SOLUTIONS . . . 341
Next, by using the second inequality of H1), i. e., lim infu↑∞
g(t, u)
u
> a(t), one can find
r1 > p and ε > 0 such that
g(t, u) ≥ a(t)(1 + ε)u, whenever u ≥ r1. (3.7)
Let R =
r1
σ
, so we have
u(t) ≥ σ‖u‖ = σR = r1 for u ∈ K ∩ ∂ΩR. (3.8)
Thus, if y ∈ K with ‖y‖ = R, then y(t) ≥ σR = r1.
Let ψ ≡ 1 and we now prove that
y 6= Φy + δψ for y ∈ K ∩ ∂ΩR and δ ≥ 0. (3.9)
If not, there exist y0 ∈ K ∩ ∂ΩR and δ0 ≥ 0 such that
y0 = Φy0 + δ0ψ.
Let µ = mint∈R y0(t). Then for t ∈ R we have
y0(t) = (Φy0)(t) + δ0 =
=
t+ω∫
t
G(t, s)g(s, y0(s− τ(s)))ds+ δ0 ≥
≥
t+ω∫
t
G(t, s)a(s)(1 + ε)y0(s− τ(s))ds ≥
≥ µ(1 + ε)
t+ω∫
t
G(t, s)a(s)ds = µ(1 + ε),
and this implies µ ≥ µ(1 + ε), a contradiction.
In view of (3.5) and (3.9), by Lemma 1, we see that Φ has a fixed point y2 ∈ K and p <
< ‖y2‖ < R. Thus y2(t) ≥ σp > 0, which means that y2(t) is a ω-periodic positive solution
of (1.1).
This completes the proof of Theorem 1.
Proof of Theorem 2. Suppose that H2) and H4) hold. By using the first inequality of H2),
i.e., lim supu↓0
g(t, u)
u
< a(t), one can find 0 < r < p and 0 < ε < 1 such that
g(t, u) ≤ a(t)(1− ε)u, whenever 0 ≤ u ≤ r. (3.10)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
342 D. JIANG, D. O’REGAN, AND R. P. AGARWAL
Thus, if y ∈ K with ‖y‖ = r, then y(t) ≥ σr. We want to show that
y 6= λΦy for y ∈ K ∩ ∂Ωr and 0 ≤ λ ≤ 1. (3.11)
If not, there exist y0 ∈ K ∩ ∂Ωr and 0 ≤ λ0 ≤ 1 such that
y0 = λ0Φy0.
Clearly, λ0 > 0. Then we have
y0(t) = λ0(Φy0)(t) =
= λ0
t+ω∫
t
G(t, s)g(s, y0(s− τ(s)))ds ≤
≤
t+ω∫
t
G(t, s)a(s)(1− ε)y0(s− τ(s))ds ≤
≤ (1− ε)||y0||
t+ω∫
t
G(t, s)a(s)ds =
= (1− ε)||y0||,
and this implies ||y0|| ≤ (1− ε)||y0||, a contradiction.
Next, by using the inequality in H4), and letting ψ ≡ 1 we now prove that
y 6= Φy + δψ for y ∈ K ∩ ∂Ωp and δ ≥ 0. (3.12)
If not, there exist y0 ∈ K ∩ ∂Ωp and δ0 ≥ 0 such that
y0 = Φy0 + δ0ψ.
Thus, ‖y0‖ = p and σp ≤ y0(t) ≤ p for t ∈ R, so we have
g(t, y0(t− τ(t))) > a(t)y0(t− τ(t)), t ∈ R. (3.13)
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OPTIMAL EXISTENCE THEORY FOR SINGLE AND MULTIPLE POSITIVE PERIODIC SOLUTIONS . . . 343
Let µ = mint∈R y0(t). Then for t ∈ R we have
y0(t) = (Φy0)(t) + δ0 =
=
t+ω∫
t
G(t, s)g(s, y0(s− τ(s)))ds+ δ0 >
>
t+ω∫
t
G(t, s)a(s)y0(s− τ(s))ds ≥
≥ µ
t+ω∫
t
G(t, s)a(s)ds = µ,
and this implies µ > µ, a contradiction.
In view of (3.11) and (3.12), by Lemma 1, we see that Φ has a fixed point y1 ∈ K and
r < ‖y1‖ < p. Thus y1(t) ≥ σr > 0, which means that y1(t) is a ω-periodic positive solution
of (1.1).
Next, by using the second inequality of H2), i.e., lim supu↑∞
g(t, u)
u
< a(t), one can find
r1 > p and 0 < ε < 1 such that
g(t, u) ≤ a(t)(1 + ε)u, whenever u ≥ r1. (3.14)
Let R =
r1
σ
, so we have,
u(t) ≥ σ‖u‖ = σR = r1 for u ∈ K ∩ ∂ΩR. (3.15)
Thus, if y ∈ K with ‖y‖ = R, then y(t) ≥ σR = r1.
Essentially the same reasoning as before (the details are left to the reader) yields
y 6= λΦy for y ∈ K ∩ ∂ΩR and 0 ≤ λ ≤ 1. (3.16)
In view of (3.12) and (3.16), by Lemma 1, we see that Φ has a fixed point y2 ∈ K and
p < ‖y2‖ < R. Thus y2(t) ≥ σp > 0, which means that y2(t) is a ω-periodic positive solution
of (1.1).
This completes the proof Theorem 2.
Proof of Theorem 3. Essentially the same reasoning as in the proof of Theorems 1 and 2
establishes the result.
Remark 5. Essentially the same reasoning as in this paper establishes (the details are left
to the reader) the existence of single and multiple positive periodic solutions for the general
Volterra integro-differential equation (see [12], which has results similar to those in [3])
·
y (t) = −a(t)y(t) +
0∫
−∞
K(r)g(t, y(t+ r))dr
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344 D. JIANG, D. O’REGAN, AND R. P. AGARWAL
where a(t) ∈ C(R, (0,∞)), g ∈ C(R× [0,∞), [0,∞)), and a(t), g(t, y) are all ω-periodic functi-
ons; here ω > 0 is a constant. Moreover, K(r) ∈ C((−∞, 0], [0,∞)) and
∫ 0
−∞
K(r)dr = 1.
4. Examples. In this section, we apply the main result obtained in the previous section to
examples modelling biological phenomena.
It follows from Theorem 3 and Corollary 3 that the following results hold.
Corollary 4. Assume that
H1) a(t), b(t) ∈ C(R, (0,∞)), β(t) ∈ C(R, (0,∞)), τ(t) ∈ C(R,R), a(t), b(t), τ(t) and β(t)
are all ω-periodic functions; here ω > 0 is a constant.
Then Eq(1.2) has at least one ω-periodic positive solution.
Corollary 5. Assume that
H1) a(t), b(t) ∈ C(R, (0,∞)), τ(t) ∈ C(R,R), a(t), b(t) and τ(t) are all ω-periodic functi-
ons; here ω > 0 is a constant.
Then Eq. (1.3) has at least one ω-periodic positive solution.
Corollary 6. Assume that
H1) a(t), b(t) ∈ C(R, (0,∞)), τ(t) ∈ C(R,R), a(t), b(t) and τ(t) are all ω-periodic functi-
ons; here ω > 0 is a constant;
H2) b(t) > a(t) for t ∈ [0, ω].
Then Eq. (1.4) has at least one ω-periodic positive solution.
Corollary 7. Assume that
H1) a(t), b(t) ∈ C(R, (0,∞)), β(t) ∈ C(R, (0,∞)), τ(t) ∈ C(R,R), a(t), b(t), τ(t) and
β(t) are all ω-periodic functions; here ω > 0 is a constant;
H2) b(t) > a(t) for t ∈ [0, ω].
Then Eq. (1.5) has at least one ω-periodic positive solution.
Corollary 4 and Corollary 5 can be checked easily. For Corollary 6 and Corollary 7, notice
lim
u↓0
g(t, u)
u
= b(t) > a(t) and lim
u↑∞
g(t, u)
u
= 0 < a(t),
so the result follows from Theorem 3.
Example 1. Consider the equation
·
y (t) = −a(t)y(t) + b(t)[ya(t− τ(t)) + yb(t− τ(t))], 0 < a < 1 < b, (4.1)
where a(t), b(t) ∈ C(R, (0,∞)), τ(t) ∈ C(R,R), and a(t), b(t), τ(t) are all ω-periodic functi-
ons; here ω > 0 is a constant.
Applying Theorem 1, we will show that Eq. (4.1) has two ω-periodic positive solutions provi-
ded
max
t∈[0,ω]
b(t)
a(t)
< sup
x∈(0,∞)
x
xa + xb
. (4.2)
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OPTIMAL EXISTENCE THEORY FOR SINGLE AND MULTIPLE POSITIVE PERIODIC SOLUTIONS . . . 345
Set g(t, u) = b(t)(ua + ub), then
lim
u↓0
g(t, u)
u
= ∞ and lim
u↑∞
g(t, u)
u
= ∞,
so H1) holds. Set
T (x) :=
x
xa + xb
, x > 0,
then T (0+) = 0, T (∞) = 0, and
T (p) = sup
x∈(0,∞)
T (x), p =
(
1− a
b− 1
) 1
b−a
.
Then for σp ≤ u ≤ p, we have
g(t, u) ≤ b(t)(pa + pb) ≤
≤ a(t)(pa + pb) max
t∈[0,ω]
b(t)
a(t)
<
< a(t)(pa + pb)T (p) = a(t)p,
so H3) holds. Then the result follows from Theorem 1 (or Corollary 1).
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Received 10.06.2003
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