Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 91, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
POSITIVE ALMOST PERIODIC SOLUTIONS FOR
STATE-DEPENDENT DELAY LOTKA-VOLTERRA
COMPETITION SYSTEMS
YONGKUN LI, CHAO WANG
Abstract. In this article, using Mawhin’s continuation theorem of coincidence degree theory, we obtain sufficient conditions for the existence of positive
almost periodic solutions for the system of equations
n
h
X
`
´i
u̇i (t) = ui (t) ri (t) − aii (t)ui (t) −
aij (t)uj t − τj (t, u1 (t), . . . , un (t)) ,
j=1,j6=i
where ri , aii > 0, aij ≥ 0(j 6= i, i, j = 1, 2, . . . , n) are almost periodic functions, τi ∈ C(Rn+1 , R), and τi (i = 1, 2, . . . , n) are almost periodic in t uniformly for (u1 , . . . , un )T ∈ Rn . An example and its simulation figure illustrate
our results.
1. Introduction
Proposed by Lotka [14] and Volterra [18], the well-known Lotka-Volterra models
concerning ecological population modeling have been extensively investigated in
the literature. When two or more species live in proximity and share the same
basic requirements, they usually compete for resources, food, habitat, or territory.
In recent years, it has also been found with successful and interesting applications
in epidemiology, physics, chemistry, economics, biological science and other areas
(see [3, 5, 6]). Owing to their theoretical and practical significance, the LotkaVolterra systems have been studied extensively [7, 8, 9, 10, 15]. To consider periodic
environmental factors, it is reasonable to study the Lotka-Volterra system with both
the periodically changing environment and the effects of time delays. Li [11] studied
the state dependent delay Lotka-Volterra competition system by using coincidence
degree theory:
n
h
X
i
u̇i (t) = ui (t) ri (t) − aii (t)ui (t) −
aij (t)uj t − τj (t, u1 (t), . . . , un (t)) , (1.1)
j=1,j6=i
2000 Mathematics Subject Classification. 34K14, 92D25.
Key words and phrases. Lotka-Volterra competition system; almost periodic solutions;
coincidence degree; state dependent delays.
c
2012
Texas State University - San Marcos.
Submitted April 12, 2012. Published June 7, 2012.
Supported by grant 10971183 from the National Natural Sciences Foundation of China.
1
2
Y. LI, C. WANG
EJDE-2012/91
where i = 1, 2, . . . , n, ui (t) stands for the ith species population density at time t,
ri (t) is the natural reproduction rate for the ith species, aij represents the effect of
interspecific (if i 6= j) or intraspecific (if i = j) interaction.
Virtually all biological systems exist in environments which vary with time, frequently in a periodic way. Ecosystem effects and environmental variability are very
important factors and mathematical models cannot ignore, for example, year-toyear changes in weather, habitat destruction and exploitation, the expanding food
surplus, and other factors that affect the population growth.
Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment are considered as
important selective forces on systems in a fluctuating environment. Therefore, on
the one hand, models should take into account the seasonality of the periodically
changing environment. However, on the other hand, in fact, it is more realistic
to consider almost periodic system than periodic system. Recently, there are two
main approaches to obtain sufficient conditions for the existence and stability of the
almost periodic solutions of biological models: One is by using the fixed point theorem, Lyapunov functional method and differential inequality techniques (see [2, 12]);
the other is by using functional hull theory and Lyapunov functional method (see
[16, 17]). To the best of our knowledge, there are few papers published on the
existence of almost periodic solutions to almost periodic differential equations done
by the method of coincidence degree theory [16-18] and no published papers considering the almost periodic solutions for non-autonomous Lotka-Volterra competitive
system with time delay by applying the method of coincidence degree theory.
Motivated above, we apply the coincidence degree theory to study the existence
of positive almost periodic solutions for the state dependent delay Lotka-Volterra
competition system (1.1) under the following assumptions:
(H1) ri , aii > 0, aij ≥ 0(j 6= i, i, j = 1, 2, . . . , n) are almost periodic functions,
τi ∈ C(Rn+1 , R), and τi (t, u1 , . . . , un ) (i = 1, 2, . . . , n) are bounded and
almost periodic in t uniformly for (u1 , . . . , un )T ∈ Rn .
The result obtained in this paper is new, and our method can be used to study
other population models.
2. Preliminaries
Let X, Y be normed vector spaces, L : Dom L ⊂ X → Y be a linear mapping and
N : X → Y be a continuous mapping. The mapping L will be called a Fredholm
mapping of index zero if dim ker L = codim Im L < +∞ and Im L is closed in Y .
If L is a Fredholm mapping of index zero and there exists continuous projectors
P : X → X and Q : Y → Y such that Im L = ker L, ker Q = Im L = Im(I − Q),
it follows that the mapping LDom L∩ker P : (I − P )X → Im L is invertible. We
denote the inverse of that mapping by KP . If Ω is an open bounded subset of
X, then the mapping N will be called L-compact on Ω̄ if QN (Ω̄) is bounded and
KP (I − Q)N : Ω̄ → X is compact. Since Im Q is isomorphic to ker L, there exists
an isomorphism J : Im Q → ker L.
For convenience, we introduce the Mawhin’s continuation theorem [4] as follows.
Lemma 2.1 ([4]). Let Ω ⊂ X be an open bounded set and let N : X → Y be a
continuous operator which is L-compact on Ω̄. Assume that
(1) Ly 6= λN y for every y ∈ ∂Ω ∩ Dom L and λ ∈ (0, 1);
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3
(2) QN y 6= 0 for every y ∈ ∂Ω ∩ ker L;
(3) deg{JQN, Ω ∩ ker L, 0} =
6 0.
Then Ly = N y has at least one solution in Dom L ∩ Ω̄.
For f ∈ AP (R, Rn ) we denote by
Z
n
o
1 T
Λ(f ) = λ ∈ R : lim
f (s)e−iλs ds 6= 0
T →∞ T 0
and
mod(f ) =
m
nX
nj λj : nj ∈ Z, m ∈ N, λj ∈ Λ(f ), j = 1, 2, . . . , m
o
j=1
the set of Fourier exponents and the module of f , respectively.
Suppose that f (t, φ) is almost periodic in t, uniformly with respect to φ ∈
S. E{f, ε, S} denotes the set of ε-almost periods for f with respect to S ⊂
C([−σ, 0], Rn ), l(ε, S) denotes the length of the inclusion interval and m[f ] =
RT
limT →∞ T1 0 f (s) ds denote the mean value of f . Set
X = Y = V1 ⊕ V2 ,
where
V1 = y = (x1 , x2 , . . . , xn )T ∈ AP (R, Rn ) : mod(y) ⊂ mod(F ) ∀µ0 ∈ Λ(y) satisfies
|µ0 | > α
and
V2 = y = (x1 (t), . . . , xn (t))T ≡ (k1 , . . . , kn )T , (k1 , . . . , kn )T ∈ Rn ,
where F = (F1 , F2 , . . . , Fn )T . For i = 1, 2, . . . , n,
Fi (t, ϕ) = ri (t) − aii (t) exp{ϕi (0)}
−
n
X
aij (t) exp ϕj − τj (t, ϕ1 (0), . . . , ϕn (0)) ,
j=1,j6=i
ϕ = (ϕ1 , ϕ2 , . . . , ϕn )T ∈ C([−σ, 0], Rn ), σ = max1≤j≤n sup(t,u)∈R×Rn {τj (t, u)} and
α is a given positive constant. Define the norm
kyk = sup |y(t)| = sup max {|xi (t)|},
t∈R
t∈R 1≤i≤n
y ∈ X(or Y).
The following lemma will play an important role in the proof of our main result.
Lemma 2.2. If f ∈ C(R, R) is almost periodic, t0 ∈ R. For any ε > 0 and
inclusion length l(ε), for all t1 , t2 ∈ [t0 , t0 + l(ε)] := Il(ε) . Then for all t ∈ R, the
following two inequalities hold
Z t0 +l(ε)
f (t) ≤ f (t1 ) +
|f 0 (s)| ds + ε
(2.1)
Z
t0
t0 +l(ε)
f (t) ≥ f (t2 ) −
t0
|f 0 (s)| ds − ε.
(2.2)
4
Y. LI, C. WANG
EJDE-2012/91
Proof. For any t ∈ R, there exists τ ∈ E{f, ε} such that t ∈ [t0 − τ, t0 − τ + l(ε)].
Thus, t + τ ∈ [t0 , t0 + l(ε)]. So we can obtain
Z t
Z t+τ
Z t
f (t) − f (t1 ) =
f 0 (s) ds =
f 0 (s)ds +
f 0 (s) ds
t1
t+τ
Z
t1
t+τ
|f 0 (s)| ds + |f (t + τ ) − f (t)|
≤
t1
Z t0 +l(ε)
≤
|f 0 (s)| ds + ε.
t0
Hence, (2.1) holds.
Similarly, we have
Z
t
Z
0
f (t) − f (t2 ) =
t+τ
f (s) ds =
t2
0
t
f (s)ds +
t2
Z
Z
f 0 (s) ds
t+τ
t+τ
|f 0 (s)| ds − |f (t + τ ) − f (t)|
≥−
t2
Z t0 +l(ε)
≥−
|f 0 (s)| ds − ε.
t0
Thus, (2.2) holds. The proof is complete.
3. Main results
By making the substitution
ui (t) = exp{xi (t)},
i = 1, 2, . . . , n,
Equation (1.1) is reformulated as
ẋi (t) = ri (t) − aii (t) exp{xi (t)}
−
n
X
aij (t) exp xj t − τj (t, exp{x1 (t)}, . . . , exp{xn (t)}) .
(3.1)
j=1,j6=i
Lemma 3.1. X and Y are Banach spaces endowed with the norm k · k.
Proof. If {yn } ⊂ V1 and yn converges to y0 , then it is easy to show that y0 ∈
AP (R, Rn ) with mod(y0 ) ⊂ mod(F ). Indeed, for all |λ| ≤ α we have
Z
1 T
yn (s)e−iλs ds = 0.
lim
T →∞ T 0
Thus
Z
1 T
y0 e−iλs ds = 0,
T →∞ T 0
which implies that y0 ∈ V1 . One can easily see that V1 is a Banach space endowed
with the norm k · k. The same can be concluded for the spaces X and Y. The proof
is complete.
lim
Lemma 3.2. Let L : X → Y such that Ly = ẏ. Then L is a Fredholm mapping of
index zero.
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5
Proof. Clearly, ker L = V2 . It remains to prove that Im L = V1 . Suppose that
(1)
(2)
(n)
φ ∈ Im L ⊂ Y. Then, there exist φV1 = (φ1 , φ1 , . . . , φ1 )T ∈ V1 and φV2 =
(1)
(2)
(n)
(φ2 , φ2 , . . . , φ2 )T ∈ V2 such that
φ = φV1 + φV2 .
Rt
Rt
From the definitions of φ(t) and φV1 (t), we deduce that
φ(s) ds and
φV1 (s) ds
are almost periodic functions and thus φV2 (t) ≡ (0, 0, . . . , 0)T := 0, which implies
that φ(t) ∈ V1 . This tells us that
Im L ⊂ V1 .
Rt
On the other hand, if ϕ(t) = (ϕ1 (t), . . . , ϕn (t))T ∈ V1 \{0} then we have 0 ϕ(s) ds ∈
AP (R, Rn ). Indeed, if λ 6= 0 then we obtain
Z
Z
Z
i
1
1 T
1 Th t
ϕ(s) ds e−iλt dt =
lim
ϕ(s)e−iλt ds.
lim
T →∞ T 0
iλ T →∞ T 0
0
It follows that
i
Z t
hZ t
ϕ(s) ds = Λ(ϕ).
ϕ(s) ds − m
Λ
0
0
Thus
t
ϕ(s) ds − m
ϕ(s) ds ∈ V1 ⊂ X.
0
0
Rt
Rt
Note that 0 ϕ(s) ds − m( 0 ϕ(s) ds) is the primitive of ϕ(t) in X, so we have
ϕ(t) ∈ Im L. Hence, we deduce that V1 ⊂ Im L, which completes the proof of our
claim. Therefore, Im L = V1 .
Furthermore, one can easily show that Im L is closed in Y and
Z
t
Z
dim ker L = n = codim Im L.
Therefore, L is a Fredholm mapping of index zero. The proof is complete.
Lemma 3.3. Let N : X → Y, P : X → X, Q : Y → Y such that N y =
(G1 y, G2 y, . . . , Gn y)T , y = (x1 , x2 , . . . , xn )T ∈ X, where, for i = 1, 2, . . . , n, t ∈ R,
Gi y(t) = ri (t) − aii (t) exp{xi (t)}
−
n
X
aij (t) exp xj t − τj (t, exp{x1 (t)}, . . . , exp{xn (t)}) ,
j=1,j6=i
P y = m(y), y ∈ X, Qz = m(z), and z ∈ Y. Then N is L-compact on Ω̄, where Ω
is any open bounded subset of X.
Proof. The projections P and Q are continuous such that Im P = ker L and Im L =
ker Q. It is clear that
(I − Q)V2 = {0}
and (I − Q)V1 = V1 .
Therefore, Im(I − Q) = V1 = Im L. In view of
Im P = ker L and
Im L = ker Q = Im(I − Q),
we can conclude that the generalized inverse (of L) KP : Im L → ker P ∩ Dom L
exists and is given by
Z t
hZ t
i
KP (z) =
z(s) ds − m
z(s) ds .
0
0
6
Y. LI, C. WANG
EJDE-2012/91
Thus
QN y = (H1 y, H2 y, . . . , Hn y)T ,
KP (I − Q)N y = f [y(t)] − Qf [y(t)],
where
Z
t
[N y(s) − QN y(s)] ds
f [y(t)] =
0
and
h
Hi y = m[Gi y] = m ri (t) − aii (t) exp{xi (t)}
−
n
X
i
aij (t) exp xj t − τj (t, exp{x1 (t)}, . . . , exp{xn (t)})
j=1,j6=i
for i = 1, 2, . . . , n. QN and (I − Q)N are obviously continuous. Now we claim
that KP is also continuous. By our hypothesis, for any ε < 1 and any compact
set S ⊂ C([−σ, 0], Rn ), let l(ε, S) be the inclusion interval of E{F, ε, S}. Suppose that {zn (t)} ⊂ Im L = V1 and zn (t) uniformly converges to z0 (t). Since
Rt
z (s) ds ∈ Y (n = 0, 1, 2, . . . ), there exists ρ, (0 < ρ < ε) such that E{F, ρ, S} ⊂
0 Rn
t
E{ 0 zn (s) ds, ε}. Let l(ρ, S) be the inclusion interval of E{F, ρ, S} and
l = max{l(ρ, S), l(ε, S)}.
It is easy to see that l is the inclusion interval of both E{F, ε, S} and E{F, ρ, S}.
Rt
Hence, for all t 6∈ [0, l], there exists τt ∈ E{F, ρ, S} ⊂ E{ 0 zn (s) ds, ε} such that
t + τt ∈ [0, l]. Therefore, by the definition of almost periodic functions we observe
that
Z
t
zn (s) ds
0
Z t
= sup zn (s) ds
t∈R
0
Z
Z t
Z t
≤ sup zn (s) ds + sup zn (s) ds −
t+τt
Z
Z t
Z t
≤ 2 sup zn (s) ds + sup zn (s) ds −
t+τt
t∈[0,l]
t∈[0,l]
Z
≤2
t6∈[0,l]
0
0
t6∈[0,l]
0
0
Z
zn (s) ds +
0
0
t+τt
0
zn (s) ds
zn (s) ds
t
|zn (s)| ds + ε.
0
(3.2)
Rt
By applying (3.2), we conclude that 0 z(s) ds (z ∈ Im L) is continuous and consequently KP and KP (I − Q)N y are also continuous.
Rt
From (3.2), we also have that 0 z(s) ds and KP (I −Q)N y are uniformly bounded
in Ω̄. In addition, we can easily conclude that QN (Ω̄) is bounded and KP (I −Q)N y
is equicontinuous in Ω̄. Hence by the Arzelà-Ascoli theorem, we can immediately
conclude that KP (I − Q)N (Ω̄) is compact. Thus N is L-compact on Ω̄. The proof
is complete.
Theorem 3.4. If (H1) holds and the following conditions are satisfied:
(H2) m[ri ] > 0, i = 1, 2, . . . , n.
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7
Pn
(H3)
j=1 m[aij ] > 0, i = 1, 2, . . . , n.
(H4) The system of linear algebraic equations
m[ri ] =
n
X
m[aij ]vj ,
i = 1, 2, . . . , n
(3.3)
j=1
has a unique solution (v1∗ , v2∗ , . . . , vn∗ )T ∈ Rn with vi∗ > 0, i = 1, 2, . . . , n.
Then (1.1) has at least one positive almost periodic solution.
Proof. To apply the continuation theorem of coincidence degree theory, we set the
Banach spaces X and Y the same as those in Lemma 3.1 and the mappings L, N, P, Q
the same as those defined in Lemmas 3.2 and 3.3, respectively. Thus, we can obtain
that L is a Fredholm mapping of index zero and N is a continuous operator which is
L-compact on Ω̄. It remains to search for an appropriate open and bounded subset
Ω.
Corresponding to the operator equation Ly = λN y, λ ∈ (0, 1), where y =
(x1 , x2 , . . . , xn )T , we obtain, for i = 1, 2, . . . , n,
n
h
X
aij (t)
ẋi (t) = λ ri (t) − aii (t) exp{xi (t)} −
j=1,j6=i
(3.4)
i
.
× exp xj t − τj (t, exp{x1 (t)}, . . . , exp{xn (t)})
Suppose that y ∈ X is a solution of (3.4) for a certain λ ∈ (0, 1). For any t0 ∈ R,
we can choose a point τ̃ − t0 ∈ [l, 2l] ∩ E{F, ρ, S), where ρ (0 < ρ < ε) satisfies
E{F, ρ} ⊂ E{y, ε}. Integrating (3.4) from t0 to τ̃ , we obtain
Z τ̃ h
λ
aii (s) exp{xi (s)}
t0
n
X
+
i
aij (s) exp xj s − τj (s, exp{x1 (s)}, . . . , exp{xn (s)})
ds
(3.5)
j=1,j6=i
Z
τ̃
≤λ
t0
Z
ri (s) ds + τ̃
t0
Z
ẋi (s) ds ≤ λ
τ̃
ri (s) ds + ε,
i = 1, 2, . . . , n.
t0
Hence, from (3.4) and (3.5), we obtain
Z τ̃
Z τ̃ h
Z τ̃
|ẋi (s)| ds ≤ λ
ri (s) ds + λ
aii (s) exp{xi (s)}
t0
t0
t0
n
X
+
i
aij (s) exp xj s − τj (s, exp{x1 (s)}, . . . , exp{xn (s)})
ds
j=1,j6=i
Z
τ̃
≤ 2λ
Z
τ̃
ri (s) ds + ε ≤ 2λ
t0
ri (s) ds + 1 := Ai ,
i = 1, 2, . . . , n.
t0
Therefore, for τ̃ ≥ t0 + l, we can easily have
Z t0 +l
|ẋi (t)| dt ≤ Ai , i = 1, 2, . . . , n.
t0
Denote
θ̄ = max sup xi (t),
1≤i≤n t∈R
θ = min inf xi (t),
1≤i≤n t∈R
i = 1, 2, . . . , n.
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Y. LI, C. WANG
EJDE-2012/91
In view of (3.4), for i = 1, 2, . . . , n, we obtain
h
m[ri ] = m aii (t) exp{xi (t)}
+
n
X
i
aij (t) exp xj t − τj (t, exp{x1 (t)}, . . . , exp{xn (t)})
.
(3.6)
j=1,j6=i
From (3.6), one has
m[ri ] ≥
n
X
m[aij ] exp{θ},
i = 1, 2, . . . , n,
j=1
or
m[ri ]
,
j=1 m[aij ]
θ ≤ ln Pn
i = 1, 2, . . . , n.
Consequently, by Lemma 2.2, for any ε > 0, there exists a ξεi such that
Z t0 +l
xi (t) ≤ xi (ξεi ) +
|ẋi (t)| dt < (θ + ε) + Ai
t0
m[ri ]
< ln Pn
+ 1 + Ai ,
j=1 m[aij ]
(3.7)
i = 1, 2, . . . , n.
Similarly, we obtain
m[ri ] ≤
n
nX
o
m[aij ] exp{θ̄},
i = 1, 2, . . . , n,
j=1
so
m[ri ]
,
j=1 m[aij ]
θ̄ ≥ ln Pn
i = 1, 2, . . . , n.
By Lemma 2.2, for any ε > 0, there exists a ηεi such that
Z t0 +l
xi (t) ≥ xi (ηεi ) −
|ẋ1 (t)| dt > (θ̄ − ε) − Ai
t0
m[ri ]
≥ ln Pn
− Ai − 1,
j=1 m[aij ]
(3.8)
i = 1, 2, . . . , n.
It follows from (3.7) and (3.8) that for i = 1, 2, . . . , n,
sup |xi (t)|
t∈R
o
n
(3.9)
m[ri ]
m[ri ]
≤ max ln Pn
+ (Ai + 1), ln Pn
− (Ai + 1) := Mi .
j=1 m[aij ]
j=1 m[aij ]
Clearly, Mi (i = 1, 2, . . . , n)) are independent of the choice of λ. Take M =
max1≤i≤n {Mi } + K, where K > 0 is taken sufficiently large such that the unique
solution (v1∗ , v2∗ , . . . , vn∗ )T of system (3.3) satisfies k(v1∗ , v2∗ , . . . , vn∗ )T k < M . Next,
take
Ω = y(t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ X : kxk < M ,
then it is clear that Ω satisfies condition (1) of Lemma 2.2. When y ∈ ∂Ω ∩ ker L,
then y is a constant vector with kyk = M . Hence
QN y = (H1 y, H2 y, . . . , Hn y)T 6= 0,
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POSITIVE ALMOST PERIODIC SOLUTIONS
9
where
n
X
Hi y = m[Gi y] = m ri ] −
m[aij ] exp{xj }, i = 1, 2, . . . , n,
j=1
which implies that condition (2) of Lemma 2.1 is satisfied. Furthermore, take
J : Im Q → ker L such that J(z) = z for z ∈ Y. In view of (H4), by a straightforward
computation, we find
n
∗
deg{JQN, Ω ∩ ker L, 0} = sgn{(−1)n [det(m(aij ))]eΣi=1 vi } =
6 0.
Therefore, condition (3) of Lemma 2.1 holds. Hence, Ly = N y has at least
one solution in Dom L ∩ Ω̄. In other words, (3.1) has at least one almost periodic solution x(t), that is, (1.1) has at least one positive almost periodic solution
(u1 (t), . . . , un (t))T . The proof is complete.
4. An example and simulation
Consider the Lotka-Volterra system
√
u̇(t) = u(t) 3 − cos 2t − (3 − cos t)u(t) − (2 + sin t)v t − τ1 (t, u(t), v(t)) ,
√
√
v̇(t) = v(t) 2 − sin 3t − (1 − sin t)v(t) − (3 − cos 2t)u(t − τ2 (t, u(t), v(t)) ,
where τi ∈ C(R3 , R) (i = 1, 2) are almost periodic in t uniformly for (u, v)T ∈ R2 .
One can calculate that m[r1 ] = 3, m[r2 ] = 2, m[a11 ] = 3, m[a22 ] = 1, m[a12 ] =
2 m[a21 ] = 3. It is easy to check that (H1)–(H4) are satisfied. By Theorem
T
3.4, Equation (1.1) has at least one
√ periodic solution (u(t), v(t)) .
√ positive almost
2v(t) + cos 3u(t)} cos t and τ2 (t, u(t), v(t)) =
We take√τ1 (t, u(t), v(t)) = exp{sin
√
exp{sin 3v(t) + cos u(t)} sin 2t. Figure 1 shows the numerical simulation which
illustrates the effectiveness of our results.
9
u
v
8
7
6
5
4
3
2
1
0
−1
0
20
40
60
80
100
time t
Figure 1. Population density for the two species u, v
10
Y. LI, C. WANG
EJDE-2012/91
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Yongkun Li
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
E-mail address: yklie@ynu.edu.cn
Chao Wang
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
E-mail address: super2003050239@163.com