Electronic Journal of Differential Equations, Vol. 2009(2009), No. 100, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
POSITIVE ALMOST PERIODIC SOLUTIONS OF
NON-AUTONOMOUS DELAY COMPETITIVE SYSTEMS WITH
WEAK ALLEE EFFECT
YONGKUN LI, KAIHONG ZHAO
Abstract. By using Mawhin’s continuation theorem of coincidence degree
theory, we obtain sufficient conditions for the existence of positive almost periodic solutions for a non-autonomous delay competitive system with weak
Allee effect.
1. Introduction
The Lotka-Volterra type systems have been studied in various fields of epidemiology, chemistry, economics and biological science. In the past few years, there
has been increasing interest in studying dynamical characteristics such as stability, persistence and periodicity of Lotka-Volterra type systems. There have been
considerable works on the qualitative analysis of Lotka-Volterra type systems with
delays; see [5, 6, 7, 8, 10, 18, 17]. Naturally, the study of almost periodic solutions
for Lotka-Volterra type systems is important and of great interest.
There are two main approaches to obtain sufficient conditions for the existence
and stability of the almost periodic solutions of biological models: One is using
the fixed point theorem, Lyapunov functional method and differential inequality
techniques [1, 9, 19]; the other is using functional hull theory and Lyapunov functional method [11, 12, 13]. However, to the best of our knowledge, there are very
few published papers considering the almost periodic solutions for non-autonomous
Lotka-Volterra type systems by applying the method of coincidence degree theory.
Motivated by this, in this paper, we apply the coincidence theory to study the existence of positive almost periodic solutions for the following non-autonomous delay
competitive systems with weak Allee effect
n
h
X
u̇i (t) = ui (t) Fi (t, ui (t − τii (t))) −
bij (t)uj (t)
j=1
−
n
X
cij (t)uj (t − τij (t)) −
n Z
X
j=1
j=1,j6=i
0
(1.1)
i
µij (t, s)uj (t + s) ds ,
−σij
2000 Mathematics Subject Classification. 34K14, 92D25.
Key words and phrases. Positive almost periodic solution; coincidence degree; delay;
non-autonomous competitive systems.
c
2009
Texas State University - San Marcos.
Submitted April 6, 2009. Published August 19, 2009.
Supported by grant 04Y239A from the Natural Sciences Foundation of Yunnan Province.
1
2
Y. LI, K. ZHAO
EJDE-2009/100
where i = 1, 2, . . . , n, ui (t) stands for the ith species population density at time
t ∈ R, bij (t) ≥ 0, cij (t) ≥ 0, τij (t) are continuous almost periodic functions on R,
µij (t, s) are positive almost periodic functions on R × [−σij , 0], continuous with
respect to t ∈ R and integrable with respect to s ∈ [−σij , 0], where σij are nonnegR0
ative constants, moreover −σij µij (t, s) ds = 1, i, j = 1, 2, . . . , n. The per capita
growth rate Fi ∈ C(R2 , R) is defined by the form for each i = 1, 2, . . . , n,
Fi (t, x) = ri (t) − fi (t, x)x.
(1.2)
In this definition, ri is the natural reproduction rate and fi represents the innerspecific competition, cij in (1.1) represents the interspecific competition. In addition, fi satisfies the following condition for each i = 1, 2, . . . , n,
∂fi (t, x)
> 0 and fi (t, x) are almost periodic in t,
∂x
for each t ∈ R, there exists a constant αi > 0 such that
fi (t, αi ) = 0.
∂fi
∂x
(1.3)
(1.4)
∂fi
∂x
> 0 and
< 0 are called the weak Allee effect
The situations formulated by
and the strong Allee effect respectively. Details about the Allee effect can be found
in [14, 15, 16]. The initial condition for (1.1) is
ui (s) = φi (s),
i = 1, 2, . . . , n,
(1.5)
where φi (s) are positive bounded continuous function on [−τ, 0], i = 1, 2, . . . , n and
τ = max1≤i,j≤n {maxt∈R |τij (t)|, σij }.
The organization of the rest of this paper is as follows. In Section 2, we introduce some preliminary results which are needed in later sections. In Section 3, we
establish our main results for the existence of almost periodic solutions of (1.1).
Finally, we make the conclusion in Section 4.
2. Preliminaries
To obtain the existence of an almost periodic solution of (1.1), we firstly make
the following preparations.
Definition 2.1 ([3]). Let u(t) : R → R be continuous in t. u(t) is said to be
almost periodic on R, if, for any > 0, the set K(u, ) = {δ : |u(t + δ) − u(t)| <
, for any t ∈ R} is relatively dense, that is for any > 0, it is possible to find a real
number l() > 0, for any interval with length l(), there exists a number δ = δ()
in this interval such that |u(t + δ) − u(t)| < , for any t ∈ R.
Definition 2.2. A solution u(t) = (u1 (t), u2 (t), . . . , un (t))T of (1.1) is called an
almost periodic solution if and only if for each i = 1, 2, . . . , n, ui (t) is almost periodic.
For convenience, we denote AP (R) is the set of all real valued, almost periodic
functions on R and let
Z T
n
o
e
e ∈ R : lim 1
∧(fj ) = λ
fj (s)e−iλs ds 6= 0 , j = 1, 2, . . . , n,
T →∞ T 0
N
nX
o
mod(fj ) =
ni λei : ni ∈ Z, N ∈ N + , λei ∈ ∧(fj ) , j = 1, 2, . . . , n
i=1
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POSITIVE ALMOST PERIODIC SOLUTIONS
3
be the set of Fourier exponents and the module of fj , respectively, where fj (·) is
almost periodic. Suppose fj (t, φj ) is almost periodic in t, uniformly with respect to
φj ∈ C([−τ, 0], R). Kj (fj , , φj (s)) = {δ : |fj (t+δ, φj (s))−fj (t, φj (s))| < , ∀t ∈ R}
denote the set of -almost periods uniformly with respect to Φj (s) ∈ C([−τ, 0], R).
RT
lj () denote the length of inclusion interval. m(fj ) = T1 0 fj (s) ds be the mean
value of fj on interval [0, T ], where T > 0 is a constant. Clearly, m(fj ) depends on
RT
T . m[fj ] = limT →∞ T1 0 fj (s) ds.
Lemma 2.3 ([3]). Suppose that f and g are almost periodic. Then the following
statements are equivalent
(i) mod(f ) ⊃ mod(g),
(ii) for any sequence {t∗n }, if limn→∞ f (t + t∗n ) = f (t) for each t ∈ R, then
there exists a subsequence {tn } ⊆ {t∗n } such that limn→∞ g(t + tn ) = f (t)
for each t ∈ R.
Lemma 2.4 ([2]). Let u ∈ AP (R). Then
Rt
t−τ
u(s) ds is almost periodic.
Let X and Z be Banach spaces. A linear mapping L : dom(L) ⊂ X → Z is called
Fredholm mapping if its kernel, denoted by ker(L) = {X ∈ dom(L) : Lx = 0},
has finite dimension and its range, denoted by Im(L) = {Lx : x ∈ dom(L)},
is closed and has finite codimension. The index of L is defined by the integer
dim K(L) − codim dom(L). If L is a Fredholm mapping with index 0, then there
exists continuous projections P : X → X and Q : Z → Z such that Im(P ) = ker(L)
and ker(Q) = Im(L). Then L|dom(L)∩ker(P ) : Im(L) ∩ ker(P ) → Im(L) is bijective,
and its inverse mapping is denoted by KP : Im(L) → dom(L)∩ker(P ). Since ker(L)
is isomorphic to Im(Q), there exists a bijection J : ker(L) → Im(Q). Let Ω be a
bounded open subset of X and let N : X → Z be a continuous mapping. If QN (Ω)
is bounded and KP (I − Q)N : Ω → X is compact, then N is called L-compact on
Ω, where I is the identity.
Let L be a Fredholm linear mapping with index 0 and let N be a L-compact
mapping on Ω. Define mapping F : dom(L) ∩ Ω → Z by F = L − N . If Lx 6= N x
for all x ∈ dom(L) ∩ ∂Ω, then by using P, Q, KP , J defined above, the coincidence
degree of F in Ω with respect to L is defined by
degL (F, Ω) = deg(I − P − (J −1 Q + KP (I − Q))N, Ω, 0),
where deg(g, Γ, p) is the Leray-Schauder degree of g at p relative to Γ.
Then The Mawhin’s continuous theorem is given as follows:
Lemma 2.5 ([4]). Let Ω ⊂ X be an open bounded set and let N : X → Z be a
continuous operator which is L-compact on Ω. Assume
(a) for each λ ∈ (0, 1), x ∈ ∂Ω ∩ dom(L), Lx 6= λN x;
(b) for each x ∈ ∂Ω ∩ L, QN x 6= 0;
(c) deg(JN Q, Ω ∩ ker(L), 0) 6= 0.
Then Lx = N x has at least one solution in Ω ∩ dom(L).
4
Y. LI, K. ZHAO
EJDE-2009/100
3. Main Result
In this section, we state and prove the main results of this paper. By making
the substitution ui (t) = exp{yi (t)}, i = 1, 2, . . . , n, (1.1) can be reformulated as
ẏi (t) = ri (t) − fi t, exp{yi (t − τii (t))} exp yi (t − τii (t))
−
n
X
n
X
bij (t) exp yj (t) −
cij (t) exp yj (t − τij (t))
j=1
−
(3.1)
j=1,i6=j
n Z 0
X
exp yj (t + s) ds,
i = 1, 2, . . . , n.
−σij
j=1
The initial condition (1.5) can be rewritten as
yi (s) = ln φi (s) =: ψi (s),
i = 1, 2, . . . , n
(3.2)
Set X = Z = V1 ⊕ V2 , where
n
V1 = y(t) = (y1 (t), y2 (t), . . . , yn (t))T ∈ C(R, Rn ) : yi (t) ∈ AP (R),
o
mod(yi (t)) ⊂ mod(Hi (t)), ∀λei ∈ ∧(yi (t)) satisfies |λei | > β, i = 1, 2, . . . , n ,
V2 = {y(t) ≡ (h1 , h2 , . . . , hn )T ∈ Rn },
Hi (t) = ri (t) − fi t, exp{ψi (−τii (t))} exp ψi (−τii (t))
−
n
X
n
X
cij (t) exp ψj (−τij (0))
bij (t) exp ψj (0) −
j=1
−
j=1,i6=j
n Z
X
j=1
0
µij (t, s) exp ψj (s) ds
−σij
and ψi (·) is defined as (3.2), i = 1, 2, . . . , n. β is a given constant. For y =
(y1 , y2 , . . . , yn )T ∈ Z, define kyk = max1≤i≤n supt∈R |yi (t)|.
Lemma 3.1. Z is a Banach space equipped with the norm k · k.
{k}
{k}
{k}
Proof. If y {k} ⊂ V1 and y {k} = (y1 , y2 , . . . , yn )T converges to
{k}
y = (y 1 , y 2 , . . . , y n )T , that is yj → y j , as k → ∞, j = 1, 2, . . . , n. Then it is easy
ej ≤ β, we have
to show that y ∈ AP (R) and mod(y ) ∈ mod(Hj ). For any λ
j
j
1
T →∞ T
Z
T
lim
{k}
yj
(t)e−iλj t dt = 0,
e
j = 1, 2, . . . , n;
0
therefore,
1
T →∞ T
Z
lim
T
y j (t)e−iλj t dt = 0,
e
j = 1, 2, . . . , n,
0
which implies y ∈ V1 . Then it is not difficult to see that V1 is a Banach space
equipped with the norm k · k. Thus, we can easily verify that x and Z are Banach
spaces equipped with the norm k · k. The proof is complete.
Lemma 3.2. Let L : X → Z, Ly = ẏ, then L is a Fredholm mapping of index 0.
EJDE-2009/100
POSITIVE ALMOST PERIODIC SOLUTIONS
5
Proof. Clearly, L is a linear operator and ker(L) = V2 . We claim that Im(L) = V1 .
Firstly, we suppose that z(t) = (z1 (t), z2 (t), . . . , zn (t))T ∈ Im(L) ⊂ Z. Then there
{1}
{1}
{1}
exist z {1} (t) = (z1 (t), z2 (t), . . . , zn (t))T ∈ V1 and constant vector z {2} =
{2} {2}
{2}
(z1 , z2 , . . . , zn )T ∈ V2 such that
z(t) = z {1} (t) + z {2} ;
that is,
{1}
zi (t) = zi
{2}
(t) + zi
,
i = 1, 2, . . . , n.
{1}
zi (t),
Rt
From the definition of zi (t) and
we can easily see that t−τ zi (s) ds and
Rt
{1}
{2}
z (s) ds are almost periodic function. So we have zi ≡ 0, i = 1, 2, . . . , n,
t−τ i
then z {2} ≡ (0, 0, . . . , 0)T , which implies z(t) ∈ V1 , that is Im(L) ⊂ V1 .
On the other hand, if u(t) = (u1 (t), u2 (t), . . . , un (t))T ∈ V1 \{0}, then we have
Rt
ej 6= 0, then we obtain
u (s) ds ∈ AP (R), j = 1, 2, . . . , n. If λ
0 j
Z
Z
Z
1
1 T
1 T t
e
e
uj (s)ds e−iλj t dt =
uj (t)e−iλj t dt,
lim
lim
e
T
→∞
T →∞ T 0
T
iλj
0
0
j = 1, 2, . . . , n. It follows that
hZ t
Z t
i
∧
uj (s) ds − m
uj (s) ds = ∧(uj (t)),
0
j = 1, 2, . . . , n,
0
hence
t
Z
u(s) ds − m
0
Z
t
u(s) ds ∈ V1 ⊂ X
0
Rt
Rt
Note that 0 u(s) ds − m( 0 u(s) ds) is the primitive of u(t) in X, we have u(t) ∈
Im(L), that is V1 ⊂ Im(L). Therefore, Im(L) = V1 .
Furthermore, one can easily show that Im(L) is closed in Z and
dim ker(L) = n = codim Im(L);
therefore, L is a Fredholm mapping of index 0. The proof is complete.
Lemma 3.3. Let N : X → Z, N y = (Gy1 , Gy2 , . . . , Gyn )T , where
Gyi = ri (t) − fi t, exp{yi (t − τii (t))} exp yi (t − τii (t))
−
n
X
n
X
bij (t) exp yj (t) −
cij (t) exp yj (t − τij (t))
j=1
−
j=1,i6=j
n Z
X
j=1
0
exp yj (t + s) ds, i = 1, 2, . . . , n.
−σij
Set
T
P y = m(y1 ), m(y2 ), . . . , m(yn ) ,
T
Q : Z → Z, Qz = m[z1 ], m[z2 ], . . . , m[zn ] .
P : X → Z,
Then N is L-compact on Ω, where Ω is an open bounded subset of X.
6
Y. LI, K. ZHAO
EJDE-2009/100
Proof. Obviously, P and Q are continuous projectors such that
Im P = ker(L), Im(L) = ker(Q).
It is clear that (I − Q)V2 = {0}, (I − Q)V1 = V1 . Hence
Im(I − Q) = V1 = Im(L).
Then in view of
Im(P ) = ker(L), Im(L) = ker(Q) = Im(I − Q),
we obtain that the inverse KP : Im(L) → ker(P ) ∩ dom(L) of LP exists and is given
by
Z
KP (z) =
t
hZ t
i
z(s) ds − m
z(s) ds .
0
0
Thus,
T
QN y = m[Gy1 ], m[Gy2 ], . . . , m[Gyn ] ,
T
KP (I − Q)N y = f (y1 ) − Q(f (y1 )), f (y2 ) − Q(f (y2 )), . . . , f (yn ) − Q(f (yn )) ,
where
Z
f (yi ) =
t
Gyi − m[Gyi ] ds,
i = 1, 2, . . . , n.
0
Clearly, QN and (I −Q)N are continuous. Now we will show that KP is also con
tinuous. By assumptions, for any 0 < < 1 and any compact set φi ⊂ C [−τ, 0], R ,
let li (i ) be the length of the inclusion interval of Ki (Hi , i , φi ), i = 1, 2, . . . , n. Suppose that {z k (t)} ⊂ Im(L) = V1 and z k (t) = (z1k (t), z2k (t), . . . , znk (t))T uniformly
converges to z(t) = (z 1 (t), z 2 (t), . . . , z n (t))T , that is zik → z i , as k → ∞, i =
Rt
1, 2, . . . , n. Because of 0 z k (s) ds ∈ Z, k = 1, 2, . . . , n, there exists σi (0 < σi < i )
Rt
such that Ki (Hi , σi , φi ) ⊂ Ki ( 0 zik (s)ds, σi , φi ), i = 1, 2, . . . , n. Let li (σi ) be the
length of the inclusion interval of Ki (Hi , σi , φi ) and
li = max li (i ), li (σi ) ,
i = 1, 2, . . . , n.
It is easy to see that li is the length of the inclusion interval of Ki (Hi , σi , φi )
and Ki (Hi , i , φi ), i = 1, 2, . . . , n. Hence, for any t ∈
/ [0, li ], there exists ξt ∈
Rt
Ki (Hi , σi , φi ) ⊂ Ki ( 0 zik (s) ds, σi , φi ) such that t + ξt ∈ [0, li ], i = 1, 2, . . . , n.
EJDE-2009/100
POSITIVE ALMOST PERIODIC SOLUTIONS
7
Hence, by the definition of almost periodic function we have
Z
t k
z (s) ds
0
Z t
zik (s) ds
= max sup 1≤i≤n t∈R
0
Z
Z t
Z t
k
k
zi (s) ds −
zi (s) ds + max sup ≤ max sup 1≤i≤n t∈[0,li ]
Z
1≤i≤n t∈[0,l
/
i]
0
0
t+ξt
zik (s) ds
0
t+ξt
+
zik (s) ds
0
Z
Z t
Z t
k
k
zi (s) ds −
zi (s) ds + max sup ≤ 2 max sup 1≤i≤n t∈[0,li ]
Z
≤ 2 max 1≤i≤n
0
li
1≤i≤n t∈[0,l
/
i]
0
0
0
t+ξt
zik (s) ds
zik (s) ds + max i .
1≤i≤n
(3.3)
Rt
From this inequality, we can conclude that 0 z(s)ds is continuous, where z(t) =
(z1 (t), z2 (t), . . . , zn (t))T ∈ Im(L). Consequently, KP and KP (I − Q)N y are continuous.
Rt
From (3.3), we also have 0 z(s) ds and KP (I −Q)N y also are uniformly bounded
in Ω. Further, it is not difficult to verify that QN (Ω) is bounded and KP (I − Q)N y
is equicontinuous in Ω. By the Arzela-Ascoli theorm, we have immediately conclude
that KP (I − Q)N (Ω) is compact. Thus N is L-compact on Ω. The proof is
complete.
By (1.3), fi (t, x) can be represented as a power-series at αi of x, in form of
∂fi fi (t, x) = fi (t, αi ) +
x + o(x), i = 1, 2, . . . , n,
∂x (t,αi )
where o(x) is a higher-order infinitely small quantity of x. By (1.4),
we conclude
i
that fi (t, αi ) = 0, i = 1, 2, . . . , n. For convenience, we denote ∂f
∂x (t,αi ) := cii (t),
i = 1, 2, . . . , n. By (1.3), cii (t) > 0.
Theorem 3.4. Assume that
1
T →∞ T
Z
m[ri (t)] = lim
T
ri (t) dt > 0,
0
Z
n
n
hX
i
1 TX
(bij (t) + cij (t)) dt > 0 .
m
(bij (t) + cij (t)) = lim
T →∞ T 0
j=1
j=1
Then (1.1) has at least one positive almost periodic solution.
Proof. To use the continuation theorem of coincidence degree theorem to establish
the existence of a solution of (3.1), we set Banach space X and Z the same as those
in Lemma 3.1 and set mappings L, N, P, Q the same as those in Lemma 3.2 and
Lemma 3.3, respectively. Then we can obtain that L is a Fredholm mapping of
index 0 and N is a continuous operator which is L-compact on Ω.
Now, we are in the position of searching for an appropriate open, bounded subset
Ω for the application of the continuation theorem. Corresponding to the operator
8
Y. LI, K. ZHAO
EJDE-2009/100
equation
Ly = λN y, λ ∈ (0, 1),
we obtain
h
ẏi (t) = λ ri (t) − cii (t) exp{yi (t − τii (t))} − o(exp{2yi (t − τii (t))})
−
n
X
bij (t) exp{yj (t)} −
j=1
−
cij (t) exp{yj (t − τij (t))}
j=1,i6=j
n Z
X
j=1
n
X
0
i
µij (t, s) exp{yj (t + s)} ds ,
(3.4)
i = 1, 2, . . . , n.
−σij
Assume that y(t) = (y1 (t), y2 (t), . . . , yn (t))T ∈ X is a solution of (3.4) for some
λ ∈ (0, 1). Denote
M1 = max sup{yi (t)},
1≤i≤n t∈R
M2 = min inf {yi (t)},
1≤i≤n t∈R
by (3.4), we derive
h
m[ri (t)] = m cii (t) exp{yi (t − τii (t))} + o(exp{2yi (t − τii (t))})
+
n
X
bij (t) exp{yj (t)} +
j=1
+
cij (t) exp{yj (t − τij (t))}
j=1,i6=j
n Z
X
j=1
n
X
0
i
µij (t, s) exp{yj (t + s)} ds ,
i = 1, 2, . . . , n
−σij
and consequently
n
i
o
n hX
(bij (t) + cij (t)) + n + 1 ,
m[ri (t)] ≤ exp{M1 } m
i = 1, 2, . . . , n;
j=1
that is,
M1 ≥ ln
m[ri (t)]
Pn
,
m[ j=1 (bij (t) + cij (t))] + n + 1
i = 1, 2, . . . , n.
(3.5)
i = 1, 2, . . . , n.
(3.6)
Similarly, we can get
M2 ≤ ln
m[ri (t)]
Pn
,
m[ j=1 (bij (t) + cij (t))] + n − 1
By (3.5) and (3.6), we find that there exist ti1 ∈ R, i = 1, 2, . . . , n such that
yi (ti1 ) ≤ R1 , where
R1 = max ln
1≤i≤n
m[ri (t)]
Pn
+ 1.
m[ j=1 (bij (t) + cij (t))] + n − 1
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POSITIVE ALMOST PERIODIC SOLUTIONS
9
Furthermore, we have
y ≤ max |yi (ti1 )| + max sup 1≤i≤n
Z
1≤i≤n t∈R
≤ R1 + 2 max
sup
1≤i≤n t∈[ti ,ti +l ]
i
1 1
Z
≤ R1 + 2 max 1≤i≤n
ti1 +li
ti1
t
ti1
ẏi (s) ds
Z t
ẏi (s) ds + max i
ti1
1≤i≤n
(3.7)
ẏi (s) ds + 1.
Choose a point τei such that τei − ti1 ∈ [li , 2li ] ∩ Ki (Hi , σi , φi ), where σi (0 < σi < i )
satisfies Ki (Hi , σi , φi ) ⊂ Ki (yi , i , φi ), i = 1, 2, . . . , n. Integrating (3.4) from ti1 to
τei , we get
Z τei h
n
X
bij (t) exp{yj (t)}
cii (t) exp{yi (t − τii (t))} + o(exp{2yi (t − τii (t))}) +
λ
ti1
j=1
n
X
+
cij (t) exp{yj (t − τij (t))} +
j=1
j=1,i6=j
Z
=λ
τei
Z
ri (t) dt −
ti1
Z
≤λ
n Z
X
τei
0
i
µij (t, s) exp{yj (t + s)} ds dt
−σij
ẏi (t) dt
ti1
τei
|ri (t)| dt + i ,
i = 1, 2, . . . , n.
ti1
From the above inequality and (3.4), we obtain
Z τei
|ẏi (t)| dt
ti1
Z
≤λ
+
τei
ti1
n
X
Z
|ri (t)| dt + λ
bij (t) exp{yj (t)} +
j=1
≤2
τei
cii (t) exp{yi (t − τii (t))} + o(exp{2yi (t − τii (t))})
n
X
cij (t) exp{yj (t − τij (t))}
j=1,i6=j
n Z
X
Z
h
ti1
j=1
+
τei
0
i
µij (t, s) exp{yj (t + s)} ds dt
−σij
|ri (t)|dt + i
ti1
Z
≤2
τei
|ri (t)| dt + 1,
i = 1, 2, . . . , n,
ti1
which together with (3.7) and τe ≥ ti1 + li , i = 1, 2, . . . , n, we have kyk ≤ R, where
Z τe
R = R1 + 4 max
|ri (t)|dt + 3.
1≤i≤n
0
Clearly, R is independent of λ. Take
Ω = {y = (y1 , y2 , . . . , yn )T ∈ X : kyk < R + 1}.
10
Y. LI, K. ZHAO
EJDE-2009/100
It is clear that Ω satisfies the requirement (a) in Lemma 2.5. when y ∈ ∂Ω ∩ ker(L),
y = (y1 , y2 , . . . , yn )T is a constant vector in Rn with kyk = R + 1. Then
T
QN y = m[G1 ], m[G2 ], . . . , m[Gn ] , y ∈ X
where
Gi = ri (t) − fi (t, exp{yi }) exp{yi }
−
n
X
bij (t) exp{yj } −
j=1
−
cij (t) exp{yj }
j=1,i6=j
n Z
X
j=1
n
X
0
µij (t, s) exp{yj } ds,
i = 1, 2, . . . , n,
−σij
thus QN y 6= 0, which implies that the requirement (b) in Lemma 2.5 is satisfied.
Furthermore, take the isomorphism J : Im(Q) → ker(L), Jz ≡ z and let Φ(γ; y) =
−γy + (1 − γ)JQN y, then for any y ∈ ∂Ω ∩ ker(L), y T Φ(γ; y) < 0, we have
deg{JQN, Ω ∩ ker(L), 0} = deg{−y, Ω ∩ ker(L), 0} =
6 0.
So, the requirement (c) in Lemma 2.5 is satisfied. Hence, (3.1) has at least one
almost periodic solution in Ω, that is (1.1) has at least one positive almost periodic
solution. The proof is complete.
We remark that when n = 1 in (1.1), if F1 (t, x) is a linear function and µ11 (t) ≡ 0,
then Theorem 3.4 is the same as [13, Theorem 3.1].
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Yongkun Li
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
E-mail address: yklie@ynu.edu.cn
Kaihong Zhao
Department of Mathematics, Yuxi Normal University, Yuxi, Yunnan 653100, China
E-mail address: zhaokaihongs@126.com