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Abstract and Applied Analysis
Volume 2011, Article ID 843292, 11 pages
doi:10.1155/2011/843292
Research Article
One-Signed Periodic Solutions of First-Order
Functional Differential Equations with a Parameter
Ruyun Ma and Yanqiong Lu
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Ruyun Ma, ruyun ma@126.com
Received 8 June 2011; Accepted 29 August 2011
Academic Editor: Ferhan M. Atici
Copyright q 2011 R. Ma and Y. Lu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study one-signed periodic solutions of the first-order functional differential equation u t −atut λbtfut − τt, t ∈ R by using global
ω bifurcation techniques. Where a, b ∈
ω
CR, 0, ∞ are ω−periodic functions with 0 atdt > 0, 0 btdt > 0, τ is a continuous ω-periodic
function, and λ > 0 is a parameter. f ∈ CR,R and there exist two constants s2 < 0 < s1 such that
fs2 f0 fs1 0, fs > 0 for s ∈ 0, s1 ∪ s1 , ∞ and fs < 0 for s ∈ −∞, s2 ∪ s2 , 0.
1. Introduction
In recent years, there has been considerable interest in the existence of periodic solutions of
the following equation:
u t −atut λbtfut − τt,
1.1
ω
ω
where a, b ∈ CR, 0, ∞ are ω-periodic functions, and 0 atdt > 0, 0 btdt > 0, τ is a
continuous ω-periodic function, λ > 0 is a parameter. 1.1 has been proposed as a model
for a variety of physiological processes and conditions including production of blood cells,
respiration, and cardiac arrhythmias; see, for example, 1–12 and the references therein.
Roughly speaking, ut represents the number of adult sexually mature members in a
population at time t, at is the per capita death rate, and fut−τt is the rate at which new
members are recruited into the population at time t τ is the age at which members mature,
and it is assumed that the birth rate at a given time depends only on the adult population
size. The most famous models of this type are
2
Abstract and Applied Analysis
i the Nicholson’s blowflies equation proposed in 1 to explain the oscillatory
population fluctuations observed by A. J. Nicholson in 1957 in his studies of the
sheep blowfly Lucilia cuprina:
u t −aut p · ut − he−γut−h ,
1.2
a, p, γ, h > 0;
ii the model for blood cell populations proposed by Mackey and Glass in 2
u t −aut p
ut − h
,
1 ut − hn
1.3
a, p, γ, h > 0, n > 1;
iii the model for the survival of red blood cells in an animal proposed by WazewskaCzyzewska and Lasota in 3
u t −aut p · e−γut−h ,
1.4
a, p, γ, h > 0.
Recently, Cheng and Zhang 7 studied the existence of positive ω-periodic solutions
of the functional equation 1.1 under the assumptions:
H1 f ∈ C0, ∞, 0, ∞, and fs > 0 for s > 0;
H2 a, b ∈ CR, 0, ∞ are ω−periodic functions,
CR, R is a ω-periodic function;
ω
0
atdt > 0,
ω
0
btdt > 0, τ ∈
H3 there exist f0 , f∞ ∈ 0, ∞ such that
fs
,
|s| → 0 s
fs
.
|s| → ∞ s
f0 lim
1.5
f∞ lim
They proved the following.
Theorem A. Assume H1–H3hold. Then for each λ satisfying
1
1
<λ<
,
σBf∞
Af0
or
1
1
<λ<
,
σAf0
Bf∞
1.6
equation 1.1 has a positive periodic solution, where
A max
t∈0,ω
ω
Gt, sbsds,
0
B min
t∈0,ω
ω
Gt, sbsds,
ω
σe0
atdt
.
1.7
0
However, the condition used in 7 is not sharp, and the main results in 7 give no
any information about the global structure of the set of positive periodic solutions. Moreover,
f satisfied H1 in 7, so a natural question is what would happen if f is allowed to have
some zeros in R? The purpose of this work is to study the global behavior of the components
of one-signed solutions of 1.1 under the condition
Abstract and Applied Analysis
3
H4 f ∈ CR, R; there exist two constants s2 < 0 < s1 such that fs2 f0 fs1 0,
fs > 0 for s ∈ 0, s1 ∪ s1 , ∞, and fs < 0 for s ∈ −∞, s2 ∪ s2 , 0.
The rest of this paper is organized as follows. In Section 2, we give some notations and
the main results. Section 3 is devoted to proving the main results.
2. Statement of the Main Results
Let Y {u ∈ CR, R : ut ut ω} with the norm
u∞ max |ut|.
2.1
t∈0,ω
Then Y, · ∞ is a Banach space. Let
E u ∈ C1 R, R : ut ut ω
2.2
be the Banach space with the norm u max{u∞ , u ∞ }.
It is well known that 1.1 is equivalent to
ut λ
tω
Gt, sbsfus − τsds : Aut,
2.3
t
where
s
Gt, s Notice that
w
0
e t aθdθ
ω
e0
aθdθ
−1
,
s ∈ t, t ω.
2.4
atdt > 0, we have
1
σ
≤ Gt, s ≤
,
σ−1
σ−1
2.5
ω
where σ e 0 atdt , and 0 < 1/σ < 1.
Define that K is a cone in Y by
K
1
u ∈ Y : ut ≥ 0, ut ≥ u .
σ
2.6
It is not difficult to prove that AK ⊂ K and A : K → K is completely continuous.
Let us consider the spectrum of the linear eigenvalue problem
u t −atut λbtut − τt,
t ∈ R.
2.7
Lemma 2.1. Assume that H2 holds. Then the linear problem 2.7 has a unique eigenvalue λ1 ,
which is positive and simple, and the corresponding eigenfunction ϕ is of one sign.
4
Abstract and Applied Analysis
Proof. It is a direct consequence of the Krein-Rutman Theorem 13, Theorem 19.3.
In the rest of the paper, we always assume that
ϕ 1,
ϕt > 0, t ∈ R.
2.8
Define L : E → Y by setting
Lu : u t atut,
u ∈ E.
2.9
Then L−1 : Y → E is completely continuous.
Let ζ, ξ ∈ CR, R be such that
fs f0 s ζs,
fs f∞ s ξs.
2.10
ξs
0.
|s| → ∞ s
2.11
Clearly,
ζs
0,
|s| → 0 s
lim
lim
Let us consider
Lut − λbtf0 ut − τt λbtζut − τt
2.12
as a bifurcation problem from the trivial solution u ≡ 0 and
Lut − λbtf∞ ut − τt λbtξut − τt
2.13
as a bifurcation problem from infinity. We note that 2.12 and 2.13 are the same and each
of them is equivalent to 1.1.
Let E R × E under the product topology. We add the points {λ, ∞ | λ ∈ R} to our
space E. Let S denote the set of positive functions in E and S− −S , and S S− ∪ S . They
are disjoint and open in E. Finally, let Φ± R × S± and Φ R × S.
Remark 2.2. It is worth remaking that if u is a nontrivial solution of 1.1 and a, b, and f satisfy
H2–H4, then u ∈ Sν for some ν {, −}. To see this, define
⎧
⎪
⎨ fut , ut / 0,
ut
qt ⎪
⎩f0 ,
ut 0.
2.14
Thus 1.1 is equivalent to
u t −atut λbtqt − τtut − τt,
t ∈ R.
2.15
Abstract and Applied Analysis
5
Obviously, b·q· − τ· satisfies H2. From Lemma 2.1, the nontrivial solution u ∈ Sν for
some ν ∈ {, −}.
The result of Rabinowitz 14 for 2.12 can be stated as follows: for each ν ∈ {, −},
there exists a continuum Cν of solutions of 2.12 joining λ1 /f0 , 0 to infinity, and Cν \
{λ1 /f0 , 0} ⊂ Φν .
The result of Rabinowitz 15 for 2.13 can be stated as follows: for each ν ∈
{, −}, there exists a continuum Dν of solutions of 2.13 meeting λ1 /f∞ , ∞, and Dν \
{λ1 /f∞ , ∞} ⊂ Φν .
Our main result is the following.
Theorem 2.3. Assume H2–H4 hold. Moreover, suppose that
H5 f satisfies the Lipschitz condition in s2 , s1 .
Then
i for λ, u ∈ C ∪ C− ,
s2 < ut < s1 ,
t ∈ 0, ω;
2.16
ii for λ, u ∈ D ∪ D− , we have that either
max ut > s1
2.17
min ut < s2 .
2.18
t∈0,ω
or
t∈0,ω
Corollary 2.4. Let H2–H5 hold. Then
i if λ ∈ λ1 /f∞ , λ1 /f0 , then 1.1 has at least two solutions u∞ and u−∞ , such that u∞ is
positive on 0, ω and u−∞ is negative on 0, ω;
ii if λ ∈ λ1 /f0 , ∞, then 1.1 has at least four solutions u∞ , u−∞ , u0 , and u−0 , such that u∞ ,
u0 are positive on 0, ω and u−∞ , u−0 are negative on 0, ω.
Corollary 2.5. Let H2–H5 hold. Then
i if λ ∈ λ1 /f0 , λ1 /f∞ , then 1.1 has at least two solutions u0 and u−0 , such that u0 is
positive on 0, ω and u−0 is negative on 0, ω;
ii if λ ∈ λ1 /f∞ , ∞, then 1.1 has at least four solutions u∞ , u−∞ , u0 , and u−0 , such that u∞ ,
u0 are positive on 0, ω and u−∞ , u−0 are negative on 0, ω.
6
Abstract and Applied Analysis
3. Proof of the Main Results
To prove Theorem 2.3, we give a Proposition.
Proposition 3.1. i The first-order boundary value problem
u t atut ht,
u0 uω
3.1
ω
has a unique solution for all h ∈ L1 0, ω if and only if 0 asds /
0.
ii Assume that u is a solution of 3.1. If h ≥ 0 and h· /
≡ 0 on any subinterval of 0, ω,
ω
then ut 0 asds > 0 on 0, ω.
t
Proof. i The equation u t atut 0 has a solution ut Ce − 0 asds , where C is a
ω
If ut is a nontrivial solution, then by u0 C, uω Ce− 0 asds , we can get that
constant.
ω
asds 0.
0
ω
On the other hand, from 0 asds 0, we can get that u t atut 0 has a
t
nontrivial solution ut Ce− 0 asds , where C ∈ R \ {0}.
ii We claim that ut /
0, t ∈ 0, ω. Suppose on the contrary that there exists t0 ∈
0, ω, such that ut0 0; it is not difficult to compute that
u t atut ht,
ut0 u0
3.2
has a solution
ut t
s
hse t aτdτ ds.
3.3
t0
Since h ≥ 0, we have
u0 ≤ ut0 ≤ uω.
3.4
exists a neighborhood Ut ⊂ 0, t0 of t, such that
If ht > 0, t ∈ 0, t0 , then there
s
0
aτdτ
ds < 0; this contradicts with u0 uω.
ht > 0 on Ut. Thus, u0 t0 hse 0
If ht > 0, t ∈ t0 , ω, then there exists a neighborhood Ut ⊂ t0 , ω of t, such that
ht > 0 on Ut. By using a similar way, we can prove that uω > 0, which also contradicts
with u0 uω.
Hence ut /
0 on 0, ω. Moreover, it follows that
ω
0
u t
dt ut
ω
0
atdt ω
0
ht
dt,
ut
3.5
Abstract and Applied Analysis
7
that is,
ln ut|ω
0
Thus
ω
0
atdt ω
0
ω
atdt ω
0
ht/utdt, that is u
ω
0
0
ht
dt.
ut
3.6
asds > 0.
Next, we prove Theorem 2.3 and Corollaries 2.4 and 2.5.
Proof of Theorem 2.3. Suppose on the contrary that there exists λ, u ∈ C ∪ C− ∪ D ∪ D− such
that either
max{ut | t ∈ 0, ω} s1
3.7
min{ut | t ∈ 0, ω} s2 .
3.8
or
We divide the proof into two cases.
Case 1 max{ut | t ∈ 0, ω} s1 . In this case, we know that
0 ≤ ut ≤ s1 ,
0 ≤ ut − τt ≤ s1 ,
t ∈ 0, ω.
3.9
t ∈ R.
3.10
Let us consider the functional differential equation
u t atut λbtfut − τt,
By H2, H4 and H5, there exists m ≥ 0 such that btfs ms is strictly increasing on s
for s ∈ s2 , s1 . Then 3.10 can be rewritten to the form
Lu λmut − τt λ btfut − τt mut − τt ,
3.11
and since Ls1 − ats1 0 fs1 ,
Ls1 − ats1 λms1 λ btfs1 ms1 .
3.12
Ls1 − u λms1 − ut − τt − ats1 ≥ 0.
3.13
Subtracting, we get
That is,
Ls1 − u λms1 ≥ 0,
t ∈ 0, ω,
s1 − u0 s1 − uω > 0.
3.14
8
Abstract and Applied Analysis
From Proposition 3.1, we deduce that s1 > ut, t ∈ 0, ω, which contradicts with that
max{ut | t ∈ 0, ω} s1 . Hence,
ut < s1 ,
t ∈ 0, ω.
3.15
Case 2 min{ut | t ∈ 0, ω} s2 . In this case, we know that
s2 ≤ ut ≤ 0,
s2 ≤ ut − τt ≤ 0,
t ∈ 0, ω.
3.16
Let us consider 3.10; by H2, H4, and H5, there exists m ≥ 0 such that btfs ms is
strictly increasing in s for s ∈ s2 , s1 . Then
Lu λmut − τt λ btfut − τt mut − τt
3.17
and since Ls2 − ats2 0 fs2 ,
Ls2 − ats2 λms2 λ btfs2 ms2 .
3.18
Ls2 − u λms2 − ut − τt − ats2 ≤ 0.
3.19
Subtracting, we get
That is
Ls2 − u λms2 ≤ 0,
t ∈ 0, ω,
s2 − u0 s2 − uω < 0.
3.20
From Proposition 3.1, we deduce that s2 − ut < 0, t ∈ 0, ω, this contradicts with that
min{ut | t ∈ 0, ω} s2 . Therefore,
s2 < ut,
t ∈ 0, ω.
3.21
Proof of Corollaries 2.4 and 2.5. Since boundary value problem
u t atut 0,
u0 uω
3.22
has a unique solution u ≡ 0, we get
C ∪ C− ∪ D ∪ D− ⊂ {λ, u ∈ R × E | λ ≥ 0}.
3.23
Abstract and Applied Analysis
9
Take Λ ∈ R as an interval such that Λ ∩ {λ1 /f∞ } {λ1 /f∞ } and M as a neighborhood of
λ1 /f∞ , ∞ whose projection on R lies in Λ and whose projection on E is bounded away from
0. Then by 15, Theorem 1.6, and Corollary 1.8, we have that for each ν ∈ {, −}, either
1 Dν \ M is bounded in R × E in which case Dν \ M meets {λ, 0 | λ ∈ R}, or
2 Dν \ M is unbounded.
Moreover, if 1 occurs and Dν \ M has a bounded projection on R, then Dν \ M meets
λ1 is another eigenvalue of 2.7.
λk /f∞ , ∞, where λk /
Obviously, Theorem 2.3 ii implies that 1 does not occur. So D \ M is unbounded.
Remark 2.2 guarantees that D is a component of solutions of 2.12 in S which meets
λ1 /f∞ , ∞, and consequently ProjR D \ M is unbounded. Thus
ProjR D ⊃
λ1
, ∞ .
f∞
3.24
λ1
, ∞ .
f∞
3.25
Similarly, we get
ProjR D
−
⊃
By Theorem 2.3, for any λ, u ∈ C ∪ C− ,
u∞ < max{s1 , |s2 |} : s∗ .
3.26
u < max s , a∞ s λb∞ max∗ fs ,
3.27
3.26 and 2.12 imply that
∗
∗
|s|≤s
which means that the sets {λ, u ∈ C | λ ∈ 0, d} and {λ, u ∈ C− | λ ∈ 0, d} are bounded
for any fixed d ∈ 0, ∞. This together with the fact that C and C− join λ1 /f0 , 0 to infinity
yields, respectively, that
λ1
, ∞ ,
ProjR C ⊃
f0
−
λ1
ProjR C ⊃
, ∞ .
f0
3.28
Combining 3.24, 3.25, and 3.28, we conclude the desired results.
Remark 3.2. The methods used in the proof of Theorem 2.3, Corollaries 2.4, and 2.5 have been
used in the study of other kinds of boundary value problems; see 16–18 and the references
therein.
10
Abstract and Applied Analysis
Remark 3.3. The conditions in Corollaries 2.4 and 2.5 are sharp. Let us take
at ≡ a > 0,
λ a,
bt 1,
fs s hs,
τt ≡ 0.
3.29
Let
⎧
2s
⎪
⎪
,
⎨− 2
s 1
hs 3
⎪
2s
⎪
⎩−
,
2
s 1
s ∈ −∞, −1 ∪ 1, ∞,
s ∈ −1, 1,
3.30
and consider problem
u t −atut aut hut,
t ∈ 0, ω, u0 uω.
3.31
It is easy to see that λ1 a, f0 f∞ 1. Since
λ1
λ1
a ,
f∞
f0
3.32
the conditions of Corollaries 2.4 and 2.5 are not valid. In this case, 3.31 has no nontrivial
solution. In fact, if u is a nontrivial solution of 3.31, then
0
ω
0
u tdt a
ω
0
hutdt /
0,
3.33
which is a contradiction.
Acknowledgment
The paper is supported by the NSFC no. 11061030, the Fundamental Research Funds for the
Gansu Universities.
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