Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 567319, 10 pages
doi:10.1155/2011/567319
Research Article
Positive Almost Periodic Solution on a Nonlinear
Differential Equation
Ni Hua, Tian Li-xin, and Liu Xun
Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China
Correspondence should be addressed to Ni Hua, nihua979@126.com
Received 17 March 2011; Accepted 10 July 2011
Academic Editor: Jitao Sun
Copyright q 2011 Ni Hua et al. 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 the following nonlinear equation dxt/dt xtat − btxα t − f t, xt gt,
by using fixed point theorem, the sufficient conditions of the existence of a unique positive
almost periodic solution for above system are obtained, by using the theories of stability, the
sufficient conditions which guarantee the stability of the unique positive almost periodic solution
are derived.
1. Introduction
Let us consider the following Logistic-type equation
dxt
atxt − btx2 t et.
dt
1.1
When the external perturbation et ≡ 0 and at, bt are positive constants, 1.1 is the
typical Logistic equation. It was firstly introduced as a mathematical model for studying
population dynamics and has become a classic topic in the textbooks on ordinary differential
equations and its quality theory see 1, 2. By using the method of separation of variables
and integration by partial fractions, we can get explicitly all the solutions of the typical
Logistic equation and completely analyze the behavior of all the solutions. But when at, bt
are no longer constants and there exists some perturbation, the problem is not so simple as
no explicit solutions can be found in general. In 2–6, the time-periodic case was considered.
In 7, the existence of positive almost periodic solutions was considered when at, bt
are almost periodic and there is no perturbation. When at, bt satisfy the assumption
α ≤ at ≤ A, β ≤ bt ≤ B, and there is no external perturbation et ≡ 0, but there
2
Mathematical Problems in Engineering
is a nonlinear perturbation qt, xx, Nkashama 8 obtained the existence of bounded and
positive almost periodic solutions. In 9, Zhu et al. considered the case that at, bt satisfy
the assumption α ≤ at ≤ A, β ≤ bt ≤ B and there exists an external perturbation et,
and they got the existence and uniqueness of positive periodic solutions and almost periodic
solutions of 1.1.
In this paper, we consider the following more complex system:
dxt
xt at − btxα t − ft, xt gt,
dt
1.2
where α > 0, t ∈ R, at, bt, gt are all continuous almost periodic functions, and ft, x is
almost periodic in t and uniformly with respect to x ∈ R.
In this paper, we use the fixed point theorem and get the existence and uniqueness of
positive almost periodic solution for 1.2, the stability of the unique positive almost periodic
solution of 1.2 is also discussed, and some new results are obtained.
2. The Existence and Uniqueness of Positive Almost Periodic Solution
Before we start with our main results, for the sake of convenience, suppose that f is a continuous bounded function, and we denote fM supt∈R ft and fL inft∈R ft; first, we introduce
some lemmas.
Lemma 2.1 see 10. Consider the following equation:
dx
atxt bt,
dt
2.1
where at, bt are continuous almost periodic functions; if Re mat / 0, then 2.1 exists a unique
almost periodic solution ηt, and ηt can be written as follows:
⎧ t
t
⎪
⎪
⎪
e s aτdτ bsds, Re mat < 0,
⎪
⎨ −∞
ηt ⎪ ∞ t
⎪
⎪
⎪
⎩−
e s aτdτ bsds, Re mat > 0,
2.2
t
where mat limT → ∞ 1/T
T
0
atdt, Re mat is the real part of mat.
Lemma 2.2. Consider 1.2; if gt ≥ 0; then the domain R {x | x > 0} is a positive invariant
with respect to 1.2.
Proof. Since gt ≥ 0, it follows that
dxt
xt at − btxα t − ft, xt gt
dt
≥ xt at − btxα t − ft, xt ,
2.3
Mathematical Problems in Engineering
3
thus we have
t
xt ≥ xt0 exp
α
as − bsx s − fs, xs ds ,
2.4
t0
the assertion is valid for all xt0 > 0. The proof is completed.
Theorem 2.3. Consider 1.2; α is a constant and 0 < α < bL /bM −bL , at, bt, gt are continuous almost periodic functions, ft, x is a continuous almost periodic function in t ∈ R and uniformly
with respect to x ∈ R, ft, 0 ≡ 0; if the following conditions hold:
1 aL > 0, bL > 0, gL ≥ 0,
2 |ft, x − ft, y| ≤ L|x − y|, for all t, x, y ∈ R, where L is a positive number,
3 Lγ /αaL β1α/α L/aL β1/α gM γ 1/α 1α/αaL < 1, where β 1αbL −αbM /1
αaM αaL α2 aL , γ 1 αbM aM αaL αaL bL /aL 1 αaM αaL α2 aL ,
then 1.2 exists a unique positive almost periodic solution φ∗ t, and 1/γ 1/α ≤ φ∗ t ≤ 1/β1/α .
Proof. Let ut 1/xα t, since we only consider positive solutions of 1.2, by Lemma 2.2, it
follows that xt u−1/α t, then 1.2 can be written as follows:
du
−αatut αbt αutf t, u−1/α t − αu1α/α tgt.
dt
2.5
B ϕt; ϕt ∈ AP R, β ≤ ϕt ≤ γ ,
2.6
Define
where AP R denotes the set of almost periodic functions in R, the norm is defined as ϕ supt∈R |ϕt|, thus B, · is a Banach space, given any ϕt ∈ B, consider the following
equation:
du
−αatut αbt αϕtf t, ϕ−1/α t − αϕ1α/α tgt,
dt
2.7
since aL > 0, it follows that mat > 0, thus m−αat < 0, from 2.7, by virtue of
Lemma 2.1, we can know that 2.7 has a unique almost periodic solution ηt, it can be
written as follows:
ηt α
t
−∞
e−α
t
s
aτdτ
bs ϕsf s, ϕ−1/α s − ϕ1α/α sgs ds.
2.8
4
Mathematical Problems in Engineering
Now, define an operator T as follows:
Tϕt α
t
−∞
e−α
t
s
aτdτ
bs ϕsf s, ϕ−1/α s − ϕ1α/α sgs ds,
2.9
note that
Tϕt ≤ α
≤α
≤α
t
−∞
t
−∞
t
−∞
e−αaL t−s bs ϕsf s, ϕ−1/α s ds
e−αaL t−s |bs| ϕsf s, ϕ−1/α s − fs, 0 ds
e−αaL t−s
|bs| Lϕsϕ−1/α s ds
2.10
1
bM γ Lβ−1/α .
≤
aL
Since the condition Lγ /αaL β1α/α L/aL β1/α gM γ 1/α 1 α/αaL < 1 holds, it follows that
Lγ < αaL β1α/α ; thus, we have
Tϕt ≤
1
bM αaL β γ.
aL
2.11
Also
Tϕt α
≥α
t
−∞
t
−∞
e−α
t
s
aτdτ
bs ϕsf s, ϕ−1/α s − ϕ1α/α sgs ds
e−αaM t−s bL − γ Lβ−1/α − γ 1α/α gs ds.
2.12
Since Lγ /αaL β1α/α L/aL β1/α gM γ 1/α 1α/αaL < 1, it follows that gt ≤ αaL /1αγ 1/α ;
thus, we have
Tϕt ≥ α
t
−∞
e−αaM t−s bL − γ Lβ−1/α − γ 1α/α gs ds
αaL γ 1 ≥
β.
bL − αβaL −
aM
1α
2.13
Mathematical Problems in Engineering
5
Hence Tϕt ∈ B; therefore, T : B → B. For any ϕt, ψt ∈ B, it follows that
t −α t aτdτ Tϕt − Tψt α
ϕsf s, ϕ−1/α s − ϕ1α/α sgs
e s
−∞
−ψsf s, ψ
−1/α
1α/α
s ψ
sgs ds
t
t
α
e−α s aτdτ ϕsf s, ϕ−1/α s − ψsf s, ψ −1/α s
−∞
ψsf s, ψ −1/α s − ψsf s, ψ −1/α s
1α/α
1α/α
gs ψ
s − ϕ
s ds
≤α
≤α
t
−∞
t
−∞
e−α
t
s
aτdτ
ϕsf s, ϕ−1/α s − f s, ψ −1/α s ϕs − ψsf s, ψ −1/α s gsϕ1α/α s − ψ 1α/α s ds
e−αaL t−s Lϕsϕ−1/α s − ψ −1/α s Lψ −1/α sϕs − ψs
gsϕ1α/α s − ψ 1α/α s ds.
2.14
According to mean value theorem, we can get
Tϕt − Tψt ≤ α
t
−∞
e
−αaL t−s
1 −1/α−1 ϕs − ψs Lψ −1/α ϕs − ψs
L ϕs − ξ
α
1 α 1/α ζ ϕs − ψs ds,
gs α
2.15
where ϕs < ξ, ζ < ψs or ψs < ξ, ζ < ϕs. Notice that ϕt ≤ γ , ζ1/α ≤ γ 1/α ,
ψ −1/α t ≤ β−1/α , ξ−1/α−1 ≤ β−1/α−1 ; it follows that
Tϕt − Tψt ≤ 1 Lγ 1 β−1/α−1 ϕ − ψ Lβ−1/α ϕ − ψ gM 1 α γ 1/α ϕ − ψ aL
α
α
1
1
1 α 1/α ϕ − ψ .
Lγ β−1/α−1 Lβ−1/α gM
γ
aL
α
α
2.16
6
Mathematical Problems in Engineering
Note that Lγ /αaL β1α/α L/aL β1/α gM γ 1/α 1 α/αaL < 1, and L is a positive number; it
follows that
0<
1
1
1 α 1/α
γ
Lγ β−1/α−1 Lβ−1/α gM
< 1,
aL
α
α
2.17
therefore, T is a contraction mapping on B, that is to say, T has a unique fixed point on B,
the unique fixed point is the unique positive almost periodic solution φt of 2.5, and β ≤
φt ≤ γ . Notice that ut 1/xα t, and, by Lemma 2.2, 1.2 has a unique positive almost
periodic solution φ∗ t φt−1/α , and 1/γ 1/α ≤ φ∗ t ≤ 1/β1/α . This completes the proof of
Theorem 2.3.
3. The Uniqueness of the Solution with Initial Value Problem
Consider 2.5; if given initial value ut0 u0 , and β ≤ u0 ≤ γ , then we have the following
theorem.
Theorem 3.1. Consider 2.5; α is a constant and 0 < α < bL /bM − bL , at, bt, gt are continuous functions, ft, x is a continuous function in t ∈ R and uniformly with respect to x ∈ R, ft, 0 ≡
0; if the following conditions hold:
1 aL > 0, bL > 0, gL ≥ 0,
2 |ft, x − ft, y| ≤ L|x − y|, for all t, x, y ∈ R, where L is a positive number,
3 Lγ /αaL β1α/α L/aL β1/α gM γ 1/α 1 α/αaL < 1, where β 1 αbL − αbM /1 αaM αaL α2 aL , γ 1 αbM aM αaL αaL bL /aL 1 αaM αaL α2 aL ,
then 2.5 exists a unique continuous solution ut with initial value ut0 u0 and β ≤ ut ≤ γ t ≥
t0 .
Proof. The initial value problem of 2.5 is equivalent to the solution of the following integral
equation:
t
ut e t0
−αaτdτ
u0 α
t
e−α
t
s
aτdτ
bs usf s, u−1/α s − u1α/α sgs ds.
t0
3.1
Define an operator T as follows:
Tut e
t
t0
−αaτdτ
u0 α
t
e−α
t
s
aτdτ
bs usf s, u−1/α s − u1α/α sgs ds.
t0
The following proof is similar as the proof of Theorem 2.3, so we omit it here.
3.2
Mathematical Problems in Engineering
7
4. The Stability of the Positive Almost Periodic Solution
Theorem 4.1. Consider 1.2; α is a constant and 0 < α < bL /bM − bL ,at, bt, gt are continuous almost periodic functions, ft, x is a continuous almost periodic function in t ∈ R and uniformly
with respect to x ∈ R; if the following conditions hold:
1 aL > 0, bL > 0, gL ≥ 0,
2 |ft, x − ft, y| ≤ L|x − y|, for all t, x, y ∈ R, where L is a positive number,
3 Lγ /αaL β1α/α L/aL β1/α gM γ 1/α 1α/αaL < 1, where β 1αbL −αbM /1
αaM αaL α2 aL , γ 1 αbM aM αaL αaL bL /aL 1 αaM αaL α2 aL ,
then there exists a region Ω {t, x; 1/γ 1/α ≤ x ≤ 1/β1/α , t ≥ t0 }, and, in this region, the unique
positive almost periodic solution of 1.2 is uniformly asymptotically stable.
Proof. From Theorem 3.1, we know when β ≤ u0 ≤ γ , 2.5 exists a unique solution ut with
initial value ut0 u0 , and β ≤ ut ≤ γ , since the transformation ut 1/xα t, it follows
that xt 1/u1/α t, so 1.2 exists a unique solution with initial value xt0 1/u1/α
0 , and
1/γ 1/α ≤ xt ≤ 1/β1/α . It is easy to know that, the conditions of Theorem 4.1 are met with
Theorem 2.3, so 2.5 exists a unique positive almost periodic solution φt; from 2.9, the
unique positive almost periodic solution φt of 2.5 can be expressed by integral equation
as follows:
φt α
α
t
−∞
t0
−∞
α
e−α
e−α
t
t
s
t
e−α
s
aτdτ
bs φsf s, φ−1/α s − φ1α/α sgs ds
aτdτ
bs φsf s, φ−1/α s − φ1α/α sgs ds
t
s
aτdτ
bs φsf s, φ−1/α s − φ1α/α sgs ds
t0
t
e t0
−αaτdτ
φt0 α
t
e−α
t
s
aτdτ
bs φsf s, φ−1/α s − φ1α/α sgs ds.
t0
4.1
The solution of 2.5 with initial value ut0 u0 is given as follows:
ut e
t
t0
−αaτdτ
u0 α
t
t0
e−α
t
s
aτdτ
bs usf s, u−1/α s − u1α/α sgs ds,
4.2
8
Mathematical Problems in Engineering
from 4.1 and 4.2, when t ≥ t0 , we have
t
ut − φt e t0 −αaτdτ ut0 − φt0 α
t
e−α
t
s
aτdτ
t0
f s, u−1/α s us − φs
φs f s, u−1/α s − f s, φ−1/α s
1α/α
1α/α
gs φ
s − u
s ds
t
≤ e t0
−αaτdτ α
t
e−α
t
s
ut0 − φt0 aτdτ
t0
f s, u−1/α s us − φs
φsf s, φ−1/α s − f s, u−1/α s gsφ1α/α s − u1α/α s ds
t
≤ e t0
−αaτdτ α
t
e−α
t
s
ut0 − φt0 aτdτ
t0
−1/α s us − φs Lφsφ−1/α s − u−1/α s
Lu
gsφ1α/α s − u1α/α s ds.
4.3
According to mean value theorem, we can get
t
ut − φt ≤ e t0 −αaτdτ ut0 − φt0 α
t
t0
e−α
t
s
aτdτ
1
−1/α s us − φs Lφs− ξ−1/α−1 φs − us Lu
α
1 α 1/α ζ
φs − us ds,
gs
α
4.4
where φs < ξ, ζ < us or us < ξ, ζ < φs. Notice that |φ| ≤ γ , |ξ−1−1/α | ≤ β−1−1/α ,
|u−1/α | ≤ β−1/α , |ζ1/α | ≤ γ 1/α ; it follows that
t
ut − φt ≤ e t0 −αaτdτ ut0 − φt0 t
1
e−αaL t−s Lβ−1/α us − φs Lγ β−1/α−1 us − φs
α
α
t0
Mathematical Problems in Engineering
9
1 α 1/α gs
γ us − φs ds
α
t
≤ e t0
−αaτdτ t
ut0 − φt0 1 α 1/α 1 −1/α−1
−1/α
us − φsds
γ
α
Lβ
e
Lγ β
gM
α
α
t0
≤ e−αaL t−t0 ut0 − φt0 t
1 α 1/α 1
us − φsds.
γ
α
e−αaL t−s Lβ−1/α Lγ β−1/α−1 gM
α
α
t0
−αaL t−s
4.5
Multiplying both sides of the above inequality by eαaL t−t0 , we have
eαaL t−t0 ut − φt ≤ ut0 − φt0 t
1 α 1/α 1
γ
us − φsds.
eαaL s−t0 Lβ−1/α Lγ β−1/α−1 gM
α
α
α
t0
4.6
According to Bellman’s inequality, we can obtain
t αLβ−1/α Lγ β−1/α−1 gM 1αγ 1/α ds
eαaL t−t0 ut − φt ≤ ut0 − φt0 e t0
.
4.7
Multiplied both sides of the above inequality by e−αaL t−t0 , we get
ut − φt ≤ ut0 − φt0 e−αaL αLβ−1/α Lγ β−1/α−1 gM 1αγ 1/α t−t0 .
4.8
Notice that ut 1/xα t, hence xt u−1/α t, and φ∗ t φ−1/α t is the unique positive
almost periodic solution of 1.2, so we have
xt − φ∗ t u−1/α t − φ−1/α t
1 −1/α−1 − ζ1
ut − φt
α
≤
1
αβ1α/α
4.9
ut0 − φt0 e−αaL αLβ−1/α Lγ β−1/α−1 gM 1αγ 1/α t−t0 ,
where ut < ζ1 < φt or φt < ζ1 < ut, note that the condition 3 holds, it follows that
−αaL αLβ−1/α Lγ β−1/α−1 gM 1 αγ 1/α < 0,
4.10
10
Mathematical Problems in Engineering
therefore, there must be a small positive constant δ such that the following equality holds:
xt − φ∗ t ≤
1
αβ1α/α
ut0 − φt0 e−δt−t0 .
4.11
From 4.11, we know that the unique positive almost periodic solution φ∗ t of 1.2 is uniformly asymptotically stable.
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