Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 703474, 26 pages
doi:10.1155/2012/703474
Research Article
The Integrability and Existence of Periodic
Solutions on a First-Order Nonlinear Differential
Equation with a Polynomial Nonlinear Term
Ni Hua and Tian Li-Xin
Faculty of Science, Jiangsu University, Jiangsu, Zhenjiang 212013, China
Correspondence should be addressed to Ni Hua, nihua979@126.com
Received 18 July 2011; Revised 19 October 2011; Accepted 23 October 2011
Academic Editor: Oded Gottlieb
Copyright q 2012 N. Hua and T. Li-Xin. 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.
This paper deals with a first-order differential equation with a polynomial nonlinear term. The
integrability and existence of periodic solutions of the equation are obtained, and the stability of
periodic solutions of the equation is derived.
1. Introduction
Consider the following first-order nonlinear differential equation:
n
dx ak txn−k
dt k0
n ∈ N, n ≥ 2,
1.1
when n 2, 1.1 becomes Ricatti’s equation, when n 3, 1.1 becomes the following nonlinear Abel type differential equation:
dx
atx3 btx2 ctx dt.
dt
1.2
The nonlinear Abel type differential equation plays an important role in many physical
and technical applications 1–9. The mathematical properties of 1.2 have been intensively
investigated in the mathematical and physical literature. Matsuno 10 analyzed a twodimensional dynamical system associated with Abel nonlinear equation. Strobel and
2
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Reid 11, Reid and Strobel 12 have obtained superposition rules prescriptions for
combining a finite number of known particular solutions in such a way to obtain the general
solution to a system of differential equations without operation of integration for the
Abel type equation, involving four or two particular solutions. Mak et al. 13, Mak and
Harko 14 have presented a solution-generating technique for Abel type ordinary differential equation, both suppose that y y1 x is a particular solution of 1.2, by means of
the transformations methods, and present an alternative method of generating the general
solution of 1.2 from a particular one.
Zheltukhin and Trzetrzelewski 15 developed the geometric approach to study the
dynamics of U1-invariant membranes. The approach reveals an important role of the
Abel nonlinear differential equation of the first type with variable coefficients depending on
time and one of the membrane extendedness parameters. The general solution of the Abel
equation was constructed.
However, little work was done about the integrability and periodicity of 1.1. In this
paper, we discuss the integrability and the periodic solutions of 1.1; the sufficient conditions
which guarantee the integrability and the existence of the periodic solutions for 1.1 are
obtained, and the stability of the periodic solutions of 1.1 is discussed. To the best of authors’
knowledge, this is the first paper considering the three periodic solutions of 1.1, some new
results are obtained.
The present paper is organized as follows. In Section 2, we give three lemmas to be
used later. In Section 3, the integrability of 1.1 is derived. In Section 4, the existence and
stability of the periodic solutions of 1.1 are obtained. In Section 5, we conclude our results.
2. Preliminary Lemmas
For the sake of convenience, suppose that f is a continuous ω-periodic function defined on
ω
R, we denote fM supt∈0,ω ft and fL inft∈0,ω ft; mft 1/ω 0 ftdt.
Consider the following:
dx
x at btxn−1 ,
dt
2.1
n ∈ N, n ≥ 2, at, bt are continuous functions defined on R.
Lemma 2.1. The domain R {x | x > 0} is positive invariant with respect to 2.1.
Proof. By 2.1, it follows that
t
n−1
as bsx s ds ,
xt xt0 exp
t0
the assertion is valid for all xt0 > 0, t ≥ t0 . The proof is completed.
Lemma 2.2. The domain R− {x | x < 0} is negative invariant with respect to 2.1.
2.2
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3
Lemma 2.3. Consider the following:
dx
atx bt,
dt
2.3
at, bt are continuous ω-periodic functions, if mat /
0, then 2.3 exists a unique ω-periodic
solution ηt, and ηt can be written as follows:
tω
ηt t
t
bse s−ω aτdτ ds
ω
1−e0
aτdτ
,
mat /
0
2.4
or
⎧ t
t
⎪
⎪
⎪
s aτdτ bsds,
⎪
⎨ −∞ e
∞ ηt t
⎪
e s aτdτ bsds,
−
⎪
⎪
⎪
⎩ t
mat < 0,
mat > 0.
2.5
Proof. The proof of Lemma 2.3 is a few, such as that of the papers 16, 17 and others, But in
16, the author only proved the case of the almost periodic equation; in 17, the authors just
proved that Lotka-volterra equation has a unique globally attractive periodic solution, but
they did not give the very two expressions of the periodic solution as above. In order to make
the following proof clear, here we give our proof.
From 2.3, it is easy for us to get the unique solution xt with the initial value xt0 x0 which can be written as follows:
t
xt e t0
aτdτ
x0 t
t
2.6
bse s aτdτ ds,
t0
thus we have
xt ω e
tω
t0
aτdτ
x0 tω
tω
bse s
aτdτ
ds.
2.7
t0
Let t t0 , by 2.6 and 2.7, if xt is an ω-periodic function, then it follows that xt0 xt0 ω, thus we can get
t0 ω
x0 t0
t0 ω
bse s
ω
1−e0
aτdτ
aτdτ
ds
.
2.8
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Mathematical Problems in Engineering
Substitute 2.8 into 2.6, it follows that
t
xt e t0
t0 ω
aτdτ t0
t0 w
t0
t0 w
t0
tω
t
t0 ω
bse s
ω
1−e0
aτdτ
ds
aτdτ
t
bse s−ω aτdτ ds t
bse s−ω aτdτ ds t
t0
t
t
bse s aτdτ ds
t0
t
ω
bse s aτdτ ds − e 0
ω
tω
1−e0
aτdτ
t
ω
t0
t
bse s aτdτ ds
aτdτ
bse s−ω aτdτ ds −
t0 ω
1−e0
t
t
t
2.9
bse s−ω aτdτ ds
t0
aτdτ
t
bse s−ω aτdτ ds
ω
1−e0
aτdτ
.
When mat < 0, it follows that
tω
xt t
tω
−∞
t
−∞
t
−∞
t
−∞
t
bse s−ω aτdτ ds
ω
1−e0
aτdτ
t
bse s−ω aτdτ ds −
1−e
t
t
bse s−ω aτdτ ds
−∞
ω
aτdτ
0
bse s aτdτ ds −
1−e
t
t
t
bse s−ω aτdτ ds
−∞
ω
0 aτdτ
t
ω
bse s aτdτ ds − e 0
aτdτ
ω
1−e0
t
−∞
2.10
t
bse s aτdτ ds
aτdτ
t
bse s aτdτ ds,
thus we have
xt ω tω
tω
bse s
−∞
t
aτdτ
ds
t
−∞
bse s aτdτ ds
2.11
xt.
Hence, xt is a unique ω-periodic solution of 2.3; we rewrite it as follows:
ηt t
−∞
t
bse s aτdτ ds.
2.12
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5
Similarly, we can prove that when mat > 0, 2.3 has a unique ω-periodic solution
as follows:
ηt −
∞
t
2.13
bse s aτdτ ds.
t
Remark 2.4. Under the conditions of Lemma 2.3, all the infinite integrals above are convergent.
3. A Sufficient Condition of Integrability of the First-Order Nonlinear
Differential Equation
In this section, we discuss the integrability of 1.1
Theorem 3.1. Consider 1.1, where a0 t, an t are continuous functions defined on certain interval
I, if there is a constant γ such that the following conditions hold:
t
n−1
H1 an−1 t −1n−1 n − 1a0 t γ an sds ,
t
k
H2 ak t −1k Cnk a0 t γ an sds , k 1, 2, . . . , n − 2,
then
i the general solution of 1.1 can be written as follows:
−1/n−1 t
an sds,
xt ± e an−1 tdt C − n − 1 a0 te− an−1 tdt dt
3.1
γ
where C is an integration constant which makes the number of root greater than zero if n is an odd
number.
ii Equation 1.1 has a particular solution
ηt t
3.2
an sds.
γ
Proof. i Since the conditions H1 , H2 hold, 1.1 can be written as follows:
t
d x − γ an sds
dt
x−
t
an sds
γ
⎡
× ⎣−1n a0 t
t
r
n−1
an sds
a0 t x −
t
γ
n−1 ⎤
⎦.
an sds
3.3
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Mathematical Problems in Engineering
Let yt x −
t
γ
an sds, 3.3 can be changed as follows:
⎤
⎡
t
n−1
dy
y⎣−1n a0 t
an sds
a0 tyn−1 ⎦.
dt
r
3.4
This is the Bernoulli-type equation, set ut y1−n t, the Bernoulli-type equation 3.4
becomes
du
−1n−1 n − 1a0 t
dt
t
n−1
u − n − 1a0 t
an sds
3.5
r
an−1 tu − n − 1a0 t,
according to the formula of general solution of linear differential equation, we can get that
the general solution of 3.5 is given by
ut e
an−1 tdt
C − n − 1
a0 te
− an−1 tdt
dt .
3.6
By the variable transformation of ut y1−n t, it follows that
yt ± e
since yt x −
t
γ
an−1 tdt
C − n − 1
a0 te−
an−1 tdt
−1/n−1
dt
,
3.7
an sds, we can get
−1/n−1 t
xt ± e an−1 tdt C − n − 1 a0 te− an−1 tdt dt
an sds.
3.8
γ
Remark 3.2. If n is an even number, take “” in the front of above-corresponding formulas, if
n is an odd number, take “±”.
ii By 1.1, we set ft, x nk0 ak txn−k , thus ft, x is continuous on I × R, fx t, x
is also continuous on I × R, hence 1.1 satisfies the principle of existence and uniqueness of
differential equation, therefore, 1.1 has no singular solution.
0, but from 3.4, we
From the variable transformation of ut y1−n t, it requires y /
t
t
know y 0 is indeed a solution of 3.4. Since y x − γ an sds, substitute ηt γ an sds
into 1.1, it just satisfies 1.1, so it is a particular solution of 1.1.
This is the end of the proof of Theorem 3.1.
Corollary 3.3. Consider Abel type differential equation 1.2, at, dt are continuous functions
defined on certain interval I, if there is a constant γ such that the following conditions hold:
t
H1 bt −3at γ dsds,
t
H2 ct 2at γ dsds2 ,
Mathematical Problems in Engineering
7
then the general solution of 1.2 can be written as follows:
xt ± e
C−2
ctdt
ate
− ctdt
−1/2
dt
t
3.9
dsds,
γ
and
t
γ
dsds is a particular solution of 1.2.
Corollary 3.4. Consider the following Ricatti’s equation:
dx
atx2 btx ct,
dt
3.10
at, ct are continuous functions defined on certain interval I, if there is a constant γ such that the
following condition holds:
t
H1 bt −at γ csds,
then the general solution of 3.10 can be written as follows:
xt e
btdt
C−
ate
− btdt
−1
dt
t
3.11
csds,
γ
and ηt t
γ
csds is a particular solution of 3.10.
4. The Existence and Stability of Periodic Solutions of
the First-Order Nonlinear Differential Equation
Define
Ω1 x|x>
Ω3 x|x<
t0
an sds ,
γ
t0
Ω2 an sds ,
γ
x | x > inf
Ω4 t∈0,ω
x | x < sup
t∈0,ω
t
γ
t
an sds ,
4.1
an sds ,
γ
where an t, γ, ω are the same function and numbers as the following Theorem 4.1, and t0 is
any given initial time of 1.1.
Theorem 4.1. Consider 1.1, n is an odd number, and n > 2, a0 t, an t are continuous ω-periodic
functions defined on R, if there is a constant γ such that the following conditions hold:
t
H1 an−1 t n − 1a0 t γ an sdsn−1 ,
t
H2 ak t −1k Cnk a0 t γ an sdsk , k 1, 2, . . . , n − 2,
≡ 0,
H3 man t 0, an t /
H4 a0 t < 0,
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Mathematical Problems in Engineering
then
i equation 1.1 has three ω-periodic solutions ηt, φt, χt, and they can be written as
follows:
t
ηt an sds,
γ
φt −n − 1
t
χt − −n − 1
−∞
a0 se
t
−∞
−1/n−1
t
s an−1 τdτ
ds
t
an sds,
γ
a0 se
−1/n−1
t
s an−1 τdτ
ds
t
4.2
an sds;
γ
ii if given any initial value xt0 ∈ Ω1 , then the periodic solution φt of 1.1 is globally
attractive in Ω2 ;
iii if given any initial value xt0 ∈ Ω3 , then the periodic solution χt of 1.1 is globally
attractive in Ω4 .
Proof. i Since the conditions H1 , H2 hold, 1.1 can be written as follows:
t
d x − γ an sds
dt
x−
t
an sds
γ
⎡
t
× ⎣−a0 t
n−1
a0 t x −
an sds
r
Let yt x −
t
γ
t
n−1 ⎤
⎦.
an sds
4.3
γ
an sds, 4.3 can be changed as follows:
⎤
⎡
t
n−1
dy
y⎣−a0 t
an sds
a0 tyn−1 ⎦.
dt
r
4.4
This is the Bernoulli-type equation, set ut y1−n t, the Bernoulli-type equation 4.4
becomes
du
n − 1a0 t
dt
t
n−1
an sds
u − n − 1a0 t,
r
du
an−1 tu − n − 1a0 t.
dt
4.5
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9
By H3 , H4 , it follows that
tω
an sds t
γ
an sds tω
γ
t
an sds ω
γ
t
an sds
t
an sds
4.6
0
an sds,
γ
t
t
thus γ an sds is an ω-periodic function, therefore, n − 1a0 t γ an sdsn−1 is also an ωperiodic function and
t
an−1 t n − 1a0 t
n−1
an sds
≤ 0,
4.7
γ
0 not identically equal to zero, thus we have
since a0 t < 0, an t /
4.8
man−1 t < 0,
according to Lemma 2.3, it follows that 4.5 has a unique ω-periodic solution as follows
ϕt −n − 1
t
−∞
t
a0 se s an−1 τdτ ds,
4.9
from 4.9, it is easy to know that ϕt > 0, since ϕt is periodic on R, it is bounded on R, so it
is positive and bounded, since ut y1−n t, and by Lemmas 2.1 and 2.2, it follows that 4.4
has two ω-periodic solutions as follows:
ψt −n − 1
t
ψ ∗ t − −n − 1
−∞
a0 se
t
−∞
t
s an−1 τdτ
t
−1/n−1
ds
−1/n−1
a0 se s an−1 τdτ ds
ψt0 > 0 ,
4.10
ψ ∗ t0 < 0 ,
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Mathematical Problems in Engineering
thus ψt is also positive and bounded, and ψ ∗ t is negative and bounded, by virtue of yt t
x − γ an sds, 1.1 has two ω-periodic solutions as follows:
φt −n − 1
t
χt − −n − 1
−∞
a0 se
t
−∞
−1/n−1
t
s an−1 τdτ
ds
t
an sds,
γ
−1/n−1
t
a0 se s an−1 τdτ ds
t
4.11
an sds,
γ
and φt0 ∈ Ω1 , φt ∈ Ω2 , χt0 ∈ Ω3 , χt ∈ Ω4 .
By 1.1, we set ft, x nk0 ak txn−k , thus ft, x is continuous on R×R, and fx t, x
is also continuous on R × R, hence 1.1 satisfies the principle of existence and uniqueness of
differential equation, therefore, 1.1 has no singular solution.
0, but from 4.4, we
From the variable transformation of ut y1−n t, it requires y /
t
t
know y 0 is indeed a solution of 4.4, since yt x − γ an sds, hence ηt γ an sds
t
is a periodic solution of 1.1. Substitute ηt γ an sds into 1.1, it just satisfies 1.1, so
t
ηt γ an sds is indeed a periodic particular solution of 1.1, therefore, we have proved
that 1.1 has three ω-periodic solutions ηt, φt, and χt.
ii According to the theories of linear differential equation, we know that the unique
solution ut of 4.5 with positive initial value ut0 u0 is given by
t
ut u0 e t0
an−1 τdτ
− n − 1
t
t
a0 se s an−1 τdτ ds.
4.12
t0
Since
an−1 t n − 1a0 t
t
n−1
an τdτ
,
4.13
γ
thus an−1 t is an ω-periodic function on R, and by 4.7, we know an−1 t ≤ 0, thus an−1 L <
0.
Following we prove that there is a positive number T > t0 , such that
n − 1
a0M
2ωa0L
,
ω
1 − ε ≤ ut ≤ u0 −
an−1 L
1 − 0 an−1 τdτ
as t > T , here ε is any small positive number and ε < 1.
4.14
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11
From 4.12, given any positive number u0 > 0, by the condition a0 t < 0 of
Theorem 4.1, it follows that ut > 0; following we first prove that ut has upper bound
on t0 , ∞
ut u0 e
t
t0
an−1 τdτ
− n − 1
t
t
a0 se s an−1 τdτ ds
t0
≤ u0 − n − 1
t
−∞
a0 se
4.15
t
s an−1 τdτ
ds.
According to Lemma 2.3, it follows that
t
ut ≤ u0 − n − 1
u0 −
t
−∞
n − 1
tω
t
a0 se s an−1 τdτ ds
t
a0 se s−ω an−1 τdτ ds
ω
an−1 τdτ
1−e0
t
u0 −
ωn − 1a0 ξe ξ−ω an−1 τdτ
ω
1−e0
an−1 τdτ
t
u0 −
ω
ωn − 1a0 ξe ξ an−1 τdτ e 0
ω
1−e0
|an−1 τ|dτ
e
ω
0 an−1 τdτ
ωn − 1a0 ξe 0
1−e
an−1 τdτ
an−1 τdτ
ω
≤ u0 −
t < ξ < t ω
,
ω
0
4.16
an−1 τdτ
ω
u0 −
u0 −
≤ u0 −
ωn − 1a0 ξe 0 |an−1 τ|an−1 τdτ
ω
1−e0
an−1 τdτ
ωn − 1a0 ξ
ω
1−e0
an−1 τdτ
ωn − 1a0L
ω
1−e0
an−1 τdτ
.
So ut has upper bound on t0 , ∞.
Secondly, we prove that ut ≥ n − 1a0M /an−1 L 1 − ε as t > T
ut u0 e
t
t0
an−1 τdτ
− n − 1
t
t0
≥ −n − 1
t
t
a0 se s an−1 τdτ ds
t0
≥ −n − 1
t
t0
t
a0 se s an−1 τdτ ds
a0M ean−1 L t−s ds
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Mathematical Problems in Engineering
a0M
−n − 1 −
1 − ean−1 L t−t0 an−1 L
a0M 1 − ean−1 L t−t0 .
n − 1
an−1 L
4.17
Since limt → ∞ ean−1 L t−t0 0, forall ε > 0 ε < 1, there must be T > t0 , when t > T , we have
ean−1 L t−t0 < ε, hence when t > T , we can get
ut ≥ n − 1
a0M
1 − ε,
an−1 L
4.18
therefore, we have proved that there is a positive number T > t0 , as t > T , it follows that
n − 1
a0M
ωn − 1a0L
ω
,
1 − ε ≤ ut ≤ u0 −
an−1 L
1 − e 0 an−1 τdτ
under the transformation of ut y1−n t, yt xt −
If given the initial value
xt0 u0 −1/n−1 t0
t
t > T ,
4.19
an sds, we can get the following.
γ
4.20
an sds ∈ Ω1 ,
γ
then the unique solution xt of 1.1 is given by
xt u0 e
t
t0
an−1 τdτ
− n − 1
t
a0 se
−1/n−1
t
s an−1 τdτ
ds
t
t0
an sds,
4.21
γ
if given the initial value
xt0 −u0 −1/n−1 t0
4.22
an sds ∈ Ω3 ,
γ
then the unique solution x∗ t of 1.1 is given by
∗
x t − u0 e
t
t0
an−1 τdτ
− n − 1
t
a0 se
t
s an−1 τdτ
−1/n−1
ds
t0
t
an sds,
4.23
γ
and it is easy to know xt ∈ Ω2 and x∗ t ∈ Ω4 , respectively.
Define a Lyapunov function as follows:
2
V t, u, ϕ u − ϕ ,
4.24
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13
where ut is the unique solution with the positive initial value ut0 u0 of 4.5, and ϕ is
the unique positive ω-periodic solution of 4.5, differentiating both sides of 4.24 along the
solution of 4.5, we get
dV t, u, ϕ
du dϕ
2 u−ϕ
−
dt
dt
dt
2
2an−1 t u − ϕ ,
4.25
dV t, u, ϕ
2an−1 tdt,
V t, u, ϕ
4.26
t 2a sds
V t, u, ϕ V t0 , ut0 , ϕt0 e t0 n−1
,
4.27
2 t 2a sds
2 ,
u − ϕ ut0 − ϕt0 e t0 n−1
4.28
t
u − ϕ ut0 − ϕt0 e t0 an−1 sds ,
4.29
by 4.24, we have
thus
by 4.24, we can get
by 4.28, that it follows
since
xt u−1/n−1 t t
an sds,
φt ϕ−1/n−1 t γ
t
an sds,
4.30
γ
it follows that
ut 1
n−1 ,
xt − γ an sds
t
ϕt φt −
1
n−1 ,
a
sds
γ n
t
4.31
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Mathematical Problems in Engineering
substitute 4.31 into 4.29, it follows that
1
1
−
n−1
n−1 t
t
xt − γ an sds
φt − γ an sds
t
1
1
t0 an−1 sds
−
.
e
n−1
n−1
t0
t0
xt0 − γ an sds
φt0 − γ an sds
4.32
According to mean value theorem, we can get
t
−n − 1ξ−n xt − φt −n − 1ζ−n xt0 − φt0 e t0 an−1 sds ,
4.33
where
xt −
t
an sds < ξ < φt −
t
γ
γ
t
t
4.34
an sds
or
φt −
an sds < ξ < xt −
γ
xt0 −
t0
an sds,
γ
an sds < ζ < φt0 −
t0
γ
γ
t0
t0
4.35
an sds
or
φt0 −
an sds < ζ < xt0 −
γ
4.36
an sds.
γ
Since ϕt is positive and bounded, and from 4.19, 4.31, we know that
xt −
t
γ
an sds, φt −
t
an sds, xt0 −
γ
t0
an sds, φt0 −
t0
γ
an sds
4.37
γ
are positive and bounded as t > T , hence ξ, ζ are positive and bounded as t > T , so we have
xt − φt n
t a sds
ξ xt0 − φt0 e t0 n−1
,
ζ
4.38
Mathematical Problems in Engineering
15
by 4.8, it follows that
t
an−1 sds −→ −∞,
4.39
t0
as t → ∞, and by 4.38, we can get that the ω-periodic solution φt of 1.1 is globally attractive in Ω2 .
iii The proof of the periodic solution χt of 1.1 being globally attractive in Ω4 is
similar to that of the periodic solution φt of 1.1 being globally attractive in Ω2 , so we omit
it here.
This is the end of the proof of Theorem 4.1.
Remark 4.2. In Theorem 4.1, since the transformation of u y1−n , it follows u > 0, thus we
suppose the initial value ut0 > 0.
Theorem 4.3. Consider 1.1, n is an even number, and n ≥ 2, a0 t, an t are continuous ω-periodic
functions defined on R, if there is a constant γ such that the following conditions hold:
t
n−1
H1 an−1 t 1 − na0 t γ an sds ,
t
k
H2 ak t −1k Cnk a0 t γ an sds , k 1, 2, . . . , n − 2,
≡ 0,
H3 man t 0, an t /
t
H4 a0 t < 0, γ an sds ≤ 0,
then
i equation 1.1 has two ω-periodic solutions ηt, φt, and they can be written as follows:
ηt φt −n − 1
t
−∞
t
an sds,
γ
a0 se
t
s
an−1 τdτ
−1/n−1
ds
ii If given any initial value xt0 ∈ Ω1 {x | x >
t
4.40
an sds.
γ
t0
an sds}, then the periodic solution
t
φt of 1.1 is globally attractive in Ω2 {x | x > inft∈0,ω γ an sds}.
γ
Proof. The proof of Theorem 4.3 is similar to that of Theorem 4.1, so we omit it here.
Theorem 4.4. Consider 1.1, n is an even number, and n ≥ 2, a0 t, an t are continuous ω-periodic
functions defined on R, if there is a constant γ such that the following conditions hold:
t
H1 an−1 t 1 − na0 t γ an sdsn−1 ,
t
k
H2 ak t −1k Cnk a0 t γ an sds , k 1, 2, . . . , n − 2,
H3 man t 0, an t ≡
/ 0,
t
H4 a0 t > 0, γ an sds ≥ 0,
16
Mathematical Problems in Engineering
then
i equation 1.1 has two ω-periodic solutions ηt, φt, and they can be written as follows:
t
ηt an sds,
γ
φt −n − 1
t
−1/n−1
t
−∞
a0 se s an−1 τdτ ds
t
4.41
an sds.
γ
ii If given any initial value xt0 ∈ Ω1 {x | x <
t0
an sds}, then the periodic solution
t
φt of 1.1 is globally attractive in Ω2 {x | x < supt∈0,ω γ an sds}.
γ
Proof. The proof of Theorem 4.4 is similar to that of Theorem 4.1, so we omit it here.
From Theorem 4.1, if n 3, it is easy for us to draw the following corollary.
Define
Ω1 x|x>
t0
Ω2 dsds ,
x | x > inf
t∈0,ω
γ
Ω3 x|x<
t0
Ω4 dsds ,
x | x < sup
γ
t∈0,ω
t
dsds ,
γ
t
4.42
dsds ,
γ
where dt, γ, ω are the same function and numbers as the following Corollary 4.5, and t0 is
any given initial time of 1.2.
Corollary 4.5. Consider Abel type differential equation 1.2, at, dt are ω-periodic continuous
functions defined on R, if there is a constant γ such that the following conditions hold:
t
H1 bt −3at γ dsds;
t
H2 ct 2at γ dsds2 ;
≡ 0;
H3 mdt 0, dt /
H4 at < 0;
then
i equation 1.2 has three ω-periodic solutions ηt, φt, χt, and they can be written as
follows:
t
ηt φt −2
t
χt − −2
−∞
dsds,
γ
ase
t
s cτdτ
t
−∞
ase
t
−1/2
ds
s cτdτ
t
−1/2
ds
dsds,
γ
t
dsds.
γ
4.43
Mathematical Problems in Engineering
17
ii If given any initial value xt0 ∈ Ω1 , then the periodic solution φt of 1.2 is globally
attractive in Ω2 .
iii If given any initial value xt0 ∈ Ω3 , then the periodic solution χt of 1.2 is globally
attractive in Ω4 .
From Theorems 4.3 and 4.4, if n 2, it is easy for us to draw the following corollaries.
Corollary 4.6. Consider Ricatti’s equation 3.10, at, ct are ω-periodic continuous functions
defined on R, if there is a constant γ such that the following conditions hold:
t
H1 bt −at γ csds;
H2 mct 0, ct /
≡ 0;
t
H3 at < 0, γ csds ≤ 0;
then
i equation 3.10 has two ω-periodic solutions ηt, φt, and they can be written as follows:
ηt t
csds,
γ
−1 t
t
t
bτdτ
s
ase
ds
csds.
φt −
−∞
4.44
γ
ii If given any initial value xt0 ∈ Ω1 {x | x >
t0
csds}, then the periodic solution φt
t
of 3.10 is globally attractive in Ω2 {x | x > inft∈0,ω γ csds}.
γ
Corollary 4.7. Consider Ricatti’s equation 3.10 at, ct are ω-periodic continuous functions
defined on R, if there is a constant γ such that the following conditions hold:
t
H1 bt −at γ csds;
H2 mct 0, ct /
≡ 0;
t
H3 at > 0, γ csds ≥ 0;
then,
i equation 3.10 has two ω-periodic solutions ηt, φt, and they can be written as follows:
ηt t
csds,
γ
−1 t
t
t
bτdτ
s
ase
ds
csds.
φt −
−∞
ii If given any initial value xt0 ∈ Ω1 {x | x <
4.45
γ
t0
csds}, then the periodic solution φt
t
of 3.10 is globally attractive in Ω2 {x | x < supt∈0,ω γ csds}.
γ
18
Mathematical Problems in Engineering
Theorem 4.8. Consider 1.1, n is an odd number, and n > 2, a0 t, an t are continuous ω-periodic
functions defined on R, if there is a constant γ such that the following conditions hold:
t
H1 an−1 t n − 1a0 t γ an sdsn−1 ;
t
H2 ak t −1k Cnk a0 t γ an sdsk , k 1, 2, . . . , n − 2;
≡ 0;
H3 man t 0, an t /
H4 a0 t > 0;
then,
i equation 1.1 has three ω-periodic solutions ηt, φt, χt, and they can be written as
follows:
ηt t
an sds,
γ
φt n − 1
∞
−1/n−1 t
t
a0 se s an−1 τdτ ds
an sds,
t
4.46
γ
∞
−1/n−1 t
t
a0 se s an−1 τdτ ds
an sds.
χt − n − 1
t
γ
ii if given any initial value xt0 ∈ Ω1 ∪ Ω2 , then the periodic solution ηt of 1.1 is
globally attractive in Ω3 ∪ Ω4 , where
Ω1 x|
t0
an sds ≤ x < ϕt0 −1/n−1
t0
γ
Ω2 x|
t0
an sds ≥ x > − ϕt0 −1/n−1
t0
x | inf
t
t∈0,ω
x | sup
t
∞
an sds ≤ x < sup
γ
t∈0,ω
ϕt n − 1
an sds ,
γ
Ω4 an sds ,
γ
γ
Ω3 ϕt
−1/n−1
t
an sds ≥ x > inf
t∈0,ω
an sds
4.47
,
γ
t∈0,ω
γ
− ϕt
−1/n−1
t
an sds
,
γ
t
a0 se s an−1 τdτ ds.
t
Remark 4.9. Where ϕt above is the same function as that of the following formula 4.54,
and t0 is any given initial time of 1.1.
Mathematical Problems in Engineering
19
Proof. i Since the conditions H1 , H2 hold, 1.1 can be written as follows:
t
d x − γ an sds
dt
x−
t
an sds
γ
⎡
× ⎣−a0 t
t
n−1
a0 t x −
an sds
r
Let yt x −
t
γ
t
n−1 ⎤
⎦.
an sds
4.48
γ
an sds, 4.48 can be changed as follows:
⎤
⎡
t
n−1
dy
n−1
y⎣−a0 t
an sds
a0 ty ⎦.
dt
r
4.49
This is the Bernoulli-type equation, set ut y1−n t, the Bernoulli-type equation 4.49 becomes
du
n − 1a0 t
dt
t
n−1
u − n − 1a0 t,
an sds
r
4.50
du
an−1 tu − n − 1a0 t.
dt
By H3 , H4 , it follows that
tω
an sds γ
t
an sds γ
t
an sds
t
an sds γ
t
tω
ω
an sds
4.51
0
an sds,
γ
t
t
thus γ an sds is an ω-periodic function, therefore, n − 1a0 t γ an sdsn−1 is also an ωperiodic function and
t
an−1 t n − 1a0 t
n−1
an sds
≥ 0,
4.52
γ
since a0 t > 0, an t /
0 not identically equal to zero, thus we have
man−1 t > 0,
4.53
20
Mathematical Problems in Engineering
according to Lemma 2.3, it follows that 4.50 has a unique ω-periodic solution as follows:
ϕt n − 1
∞
t
a0 se s an−1 τdτ ds,
4.54
t
from 4.54, it is easy to know that ϕt > 0, since ϕt is periodic on R, it is bounded on R,
so it is positive and bounded, since ut y1−n t, and by Lemmas 2.1 and 2.2, it follows that
4.49 has two ω-periodic solutions as follows:
ψt n − 1
∞
−1/n−1
t
a0 se s an−1 τdτ ds
,
ψt0 > 0 ,
t
∞
−1/n−1
t
a0 se s an−1 τdτ ds
,
ψ ∗ t − n − 1
ψ ∗ t0 < 0 ,
4.55
t
thus ψt is also positive and bounded, and ψ ∗ t is negative and bounded, by virtue of yt t
x − γ an sds, 1.1 has two ω-periodic solutions as follows:
φt n − 1
∞
−1/n−1 t
t
a0 se s an−1 τdτ ds
an sds,
t
γ
∞
−1/n−1 t
t
an−1 τdτ
s
a0 se
ds
an sds.
χt − n − 1
t
4.56
γ
By 1.1, we set ft, x nk0 ak txn−k , thus ft, x is continuous on R × R, and fx t, x is
also continuous on R × R, hence 1.1 satisfies the principle of existence and uniqueness of
differential equation, therefore, 1.1 has no singular solution.
0, but from 4.49, we
From the variable transformation of ut y1−n t, it requires y /
t
t
know y 0 is indeed a solution of 4.49, since yt x − γ an sds, hence ηt γ an sds
t
is a periodic solution of 1.1. Substitute ηt γ an sds into 1.1, it just satisfies 1.1, so
t
ηt γ an sds is indeed a periodic particular solution of 1.1, therefore, we have proved
that 1.1 has three ω-periodic solutions ηt, φt and χt.
ii According to the theories of linear differential equation, we know that the unique
solution ut of 4.50 with the positive initial value ut0 u0 is given by
t
ut u0 e t0
an−1 τdτ
− n − 1
t
t
a0 se s an−1 τdτ ds
t0
t
a τdτ e t0 n−1
u0 − ϕt0 ϕt.
If given the initial value u0 ≥ ϕt0 , then ut ≥ ϕt > 0.
4.57
Mathematical Problems in Engineering
21
Remark 4.10. Since the transformation of ut y1−n t, it requires ut > 0, in addition,
t
e t0
an−1 τdτ
e
n−1
t
t0
τ
a0 τ
γ
n−1
an θdθ
dτ
−→ ∞
4.58
as t → ∞, that is to say, it requires u0 ≥ ϕt0 .
Under the transformation of ut y1−n t, yt xt −
following.
If given the initial value
xt0 u0 −1/n−1 t0
an sds < ϕt0 −1/n−1 γ
t0
t
γ
an sds, we can get the
an sds ∈ Ω1 ,
4.59
γ
then the unique solution xt of 1.1 is given by
xt u0 e
t
t0
an−1 τdτ
− n − 1
t
a0 se
−1/n−1
t
s an−1 τdτ
ds
t
t0
an sds,
4.60
γ
and it is easy for us to know that xt ∈ Ω3 .
If given the initial value
xt0 −u0 −1/n−1 t0
−1/n−1
an sds > − ϕt0 γ
t0
an sds ∈ Ω2 ;
4.61
γ
then the unique solution xt of 1.1 is given by
xt − u0 e
t
t0
an−1 τdτ
− n − 1
t
a0 se
t
s an−1 τdτ
−1/n−1
ds
t0
it is easy for us to know that xt ∈ Ω4 .
Therefore, we can draw the following conclusion.
If given the initial value xt0 ∈ Ω1 ∪ Ω2 , then xt ∈ Ω3 ∪ Ω4 .
t
γ
an sds
4.62
22
Mathematical Problems in Engineering
Next, we prove that the periodic solution ηt of 1.1 is globally attractive in Ω3 ∪ Ω4
if given the initial value xt0 ∈ Ω1 ∪ Ω2 .
−1/n−1 t
t
t
a
τdτ
n−1
a
τdτ
xt − ηt ± u0 e t0
− n − 1
a0 se s n−1
ds
t0
u0 e
t
t0
an−1 τdτ
− n − 1
t
a0 se
t
s an−1 τdτ
−1/n−1
ds
t0
t
e t0
an−1 τdτ u0 − ϕt0 ϕt
−1/n−1
4.63
xt0 − ηt0 t
xt0 − ηt0 n−1 e t0 an−1 τdτ u0 − ϕt0 ϕt
xt0 − ηt0 ,
t
a
τdτ
n−1
n−1 1/u e t0
u0 − ϕt0 ϕt
0
by 4.53, it follows that
t
an−1 sds −→ ∞,
4.64
t0
as t → ∞, from 4.63, we can get that the ω-periodic solution ηt of 1.1 is globally
attractive in Ω3 ∪ Ω4 .
This is the end of the proof of Theorem 4.8.
Theorem 4.11. Consider 1.1, n is an even number, and n ≥ 2, a0 t, an t are continuous ωperiodic functions defined on R, if there is a constant γ such that the following conditions hold:
t
H1 an−1 t 1 − na0 t γ an sdsn−1 ;
t
H2 ak t −1k Cnk a0 t γ an sdsk , k 1, 2, . . . , n − 2;
H3 man t 0, an t /
≡ 0;
t
H4 a0 t < 0, γ an sds ≥ 0;
then
i equation 1.1 has two ω-periodic solutions ηt, φt, and they can be written as follows:
t
ηt an sds,
γ
φt n − 1
∞
t
a0 se
t
s an−1 τdτ
−1/n−1 t
ds
an sds;
γ
4.65
Mathematical Problems in Engineering
23
ii if given any initial value xt0 ∈ Ω1 {x | x >
t0
an sds}, then the periodic solution
t
ηt of 1.1 is globally attractive in Ω2 {x | x > inft∈0,ω γ an sds}.
γ
Proof. The Proof of Theorem 4.11 is similar to that of Theorem 4.8, so we omit it here.
Theorem 4.12. Consider 1.1, n is an even number, and n ≥ 2, a0 t, an t are continuous ωperiodic functions defined on R, if there is a constant γ such that the following conditions hold:
t
H1 an−1 t 1 − na0 t γ an sdsn−1 ;
t
H2 ak t −1k Cnk a0 t γ an sdsk , k 1, 2, . . . , n − 2;
H3 man t 0, an t /
≡ 0;
t
H4 a0 t > 0, γ an sds ≤ 0;
then
i equation 1.1 has two ω-periodic solutions ηt, φt, and they can be written as follows:
ηt t
an sds,
γ
φt n − 1
∞
−1/n−1 t
an−1 τdτ
s
a0 se
ds
an sds;
4.66
t
t
ii if given any initial value xt0 ∈ Ω1 {x | x <
γ
t0
an sds}, then the periodic solution
t
ηt of 1.1 is globally attractive in Ω2 {x | x < supt∈0,ω γ an sds}.
γ
Proof. The Proof of Theorem 4.12 is similar to that of Theorem 4.8, so we omit it here.
From Theorem 4.8, if n 3, it is easy for us to draw the following corollary.
Corollary 4.13. Consider Abel type differential equation 1.2, at, dt are ω-periodic continuous
functions defined on R, if there is a constant γ such that the following conditions hold:
H1 bt −3at
t
γ
dsds;
t
H2 ct 2at γ dsds2 ;
H3 mdt 0, dt /
≡ 0;
H4 at > 0;
then
24
Mathematical Problems in Engineering
i equation 1.2 has three ω-periodic solutions ηt, φt, χt, and they can be written as
follows:
ηt t
dsds,
γ
−1/2 t
∞
t
cτdτ
s
ase
ds
dsds,
φt 2
t
4.67
γ
∞
−1/2 t
t
cτdτ
s
χt 2
ase
ds
dsds.
t
γ
ii If given any initial value xt0 ∈ Ω1 ∪ Ω2 , then the periodic solution ηt of 1.2 is globally attractive in Ω3 ∪ Ω4 , where
Ω1 x|
t0
dsds ≤ x < ϕt0 −1/2
dsds ,
γ
Ω2 x|
t0
γ
dsds ≥ x > − ϕt0 −1/2
γ
Ω3 x | inf
t∈0,ω
Ω4 x | sup
t∈0,ω
ϕt 2
∞
t0
t0
dsds ,
γ
t
dsds ≤ x < sup
γ
t
ϕt
−1/2
dsds ≥ x > inf
t∈0,ω
dsds
4.68
,
γ
t∈0,ω
γ
t
−1/2
− ϕt
t
dsds
,
γ
t
ase s cτdτ ds.
t
From Theorem 4.11 and 4.6, if n 2, it is easy for us to draw the following corollaries.
Corollary 4.14. Consider Riccati’s equation 3.10, at, ct are ω-periodic continuous functions
defined on R, if there is a constant γ such that the following conditions hold:
t
H1 bt −at γ csds;
H2 mct 0, ct /
≡ 0;
t
H3 at < 0, γ csds ≥ 0;
then
i equation 3.10 has two ω-periodic solutions ηt, φt, and they can be written as follows:
ηt t
csds,
γ
φt ∞
ase
t
t
s bτdτ
−1 t
ds
csds.
γ
4.69
Mathematical Problems in Engineering
25
ii If given the initial value xt0 ∈ Ω1 {x | x >
t0
csds}, then the periodic solution ηt
t
of 3.10 is globally attractive in Ω2 {x | x > inft∈0,ω γ csds}.
γ
Corollary 4.15. Consider Riccati’s equation 3.10, at, ct are ω-periodic continuous functions
defined on R, if there is a constant γ such that the following conditions hold:
t
H1 bt −at γ csds;
≡ 0;
H2 mct 0, ct /
t
H3 at > 0, γ csds ≤ 0;
then
i equation 3.10 has two ω-periodic solutions ηt, φt, and they can be written as follows:
ηt t
csds,
γ
φt ∞
−1 t
bτdτ
s
ase
ds
csds.
t
t
ii If given any initial value xt0 ∈ Ω1 {x | x <
4.70
γ
t0
csds}, then the periodic solution ηt
t
of 3.10 is globally attractive in Ω2 {x | x < supt∈0,ω γ csds}.
γ
5. Concluding Remarks
Generally, just as Ricatti’s equation, 1.1 is nonintegrable, however in the present paper,
we have got a sufficient condition which guarantees the integrability and have presented
some exact periodic solutions of nonlinear equation if the coefficients of 1.1 satisfy certain
conditions; moreover, we get that the periodic solutions are globally attractive in certain
conditions. These conclusions will have certain applications.
Acknowledgments
The authors are highly grateful to the reviewers for their constructive comments. This work
is supported by the National Nature Science Foundation of China no. 11171135 and Jiangsu
Province Innovation Project of Graduate Education 2011, no. 1221190037.
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International Journal of
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Volume 2014
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Discrete Dynamics in
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Volume 2014
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Applied Mathematics
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Function Spaces
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Volume 2014
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Optimization
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Volume 2014
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Volume 2014