Electronic Journal of Differential Equations, Vol. 2010(2010), No. 103, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
TRAVELLING WAVES IN THE LATTICE EPIDEMIC MODEL
ZHIXIAN YU, RONG YUAN
Abstract. In this article, we establish the existence and nonexistence of travelling waves for a lattice non-monotone integral equation which is an epidemic
model. Moreover, the wave is either convergent to the positive equilibrium or
oscillating on the positive equilibrium at positive infinity, and has the exponential asymptotic behavior at negative infinity. For the non-monotone case, the
asymptotic speed of propagation also coincides with the minimal wave speed.
1. Introduction
One of the epidemic models on the lattice Z is
Z tX
un (t) =
An−j (t−τ )g(uj (τ ))dτ +fn (t), t ≥ 0, n ∈ Z = {0, ±1, . . . }. (1.1)
0 j∈ Z
The existence of nonnegative solution and the asymptotic speed of propagation of
(1.1) have been studied in [24] under some suitable assumptions on the functions
An , g, and fn .
Equation (1.1) corresponds to an initial value problem (the history up to t = 0
is prescribed; in fact it is incorporated in the function fn ). On the other hand, if
one wants to describe an epidemic which has been evolving from the beginning of
time then one arrives at the time-translation invariant homogeneous equation
Z t X
un (t) =
An−j (t − τ )g(uj (τ ))dτ, t ∈ R := (−∞, ∞), n ∈ Z,
(1.2)
−∞ j∈ Z
which is investigated in [24] about the existence and nonexistence of the travelling
wave (i.e. solution of the form un (t) = u(n + ct), where c > 0) when g is a nondecreasing function. Furthermore, authors in [24] showed that the asymptotic spread
speed coincides with the minimal wave speed in the case that g is a nondecreasing
function.
For a continuous analogue of (1.1), one may consider the integral equation
Z t
Z
u(t, x) =
A(t − τ )
g(u(τ, ξ))V (x − ξ)dξdτ + f (t, x), t ≥ 0, x ∈ R. (1.3)
0
Rn
2000 Mathematics Subject Classification. 35C07, 35R09, 34A33.
Key words and phrases. Travelling waves; Schauder’s fixed point theorem;
non-monotone integral equation; oscillatory solution.
c
2010
Texas State University - San Marcos.
Submitted May 10, 2010. Published July 28, 2010.
R. Yuan was supported by the National Natural Science Foundation of China and RFDP.
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Z. YU, R. YUAN
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The behaviors of (1.3) and its corresponding homogeneous equation were investigated in [4, 5, 6], such as the existence of travelling waves, the minimal wave speed,
the asymptotic speed of propagation, etc. There have been extensive investigations on travelling waves for reaction-diffusion equations [7, 8, 12, 13, 15, 16, 17,
21, 20, 25, 26, 27, 28, 29], integral equations [4, 5, 6, 18, 19], and lattice equations
[3, 9, 10, 14, 22]. The concept of the asymptotic speed of spread was first introduced
by Aronson & Weinberger [1, 2], (see also [9, 10, 11, 14, 18, 19, 22, 23, 24] and the
references therein).
Note that authors in [24] established the asymptotic speed of propagation for
(1.1) when g may be a non-monotone function, but the existence of travelling
waves for (1.2) was admitted only when g is nondecreasing. To our knowledge, the
existence of travelling wave solutions for (1.2) with a non-monotone function g is
still an open problem, let alone the relation between the spreading speed and the
minimal wave speed for (1.2) with a non-monotone function g. In this paper, we
will give an affirmative answer. More precisely, we will be mainly concerned with
the existence and nonexistence of travelling waves, and the relation between the
spreading speed and the minimal wave speed for (1.2) when g is a non-monotone
function. Moreover, the wave is either convergent to the positive equilibrium or
oscillating on the positive equilibrium at positive infinity.
We use the notation: Z = {0, ±1, ±2, . . . }, N+ = {1, 2, . . . }, R+ = [0, +∞). We
shall assume that the following hold through this article.
P
(A1) For any j ∈ Z, Aj ∈ C(R+ ) and Aj (t) = A−j (t) ≥ 0; j∈ Z Aj ∈ L1 (R+ ) ∩
R
∞P
L∞ (R+ ) with 0
j∈ Z Aj (τ )dτ = 1; there exists a λ̄ : 0 < λ̄ ≤ ∞ such
that
Z ∞X
Aj (τ )e−λj dτ < ∞
0
j∈ Z
for λ ∈ [0, λ̄).
(G1) g ∈ C(R+ , R+ ) and |g(u) − g(v)| ≤ g 0 (0)|u − v| for u, v ∈ R+ ;
(G2) g(0) = 0; there exists K > 0 such that g(K) = K, g(u) > u for 0 < u < K
and g(u) < u for u > K;
(G3) g 0 (0)u ≥ g(u) for u ∈ R+ and there exists K ∗ ≥ K such that g(u) ≤ K ∗
for all u ∈ [0, K ∗ ].
(G4) There exists σ ∈ (0, 1] such that
lim sup[g 0 (0) − g(u)/u]u−σ < ∞.
u→0+
(G5) g(u) < 2K − u for u ∈ [0, K) and g(u) > 2K − u for u ∈ (K, K ∗ ].
2. Main results
In this section, we first establish the existence of travelling waves for the system
(1.2) by using the Schauder’s fixed point theorem. The key idea is to construct two
monotone functions to squeeze g. This approach was initially used in [13]. We will
further establish the nonexistence of travelling wave solutions.
A travelling wave of (1.2) is a special translation invariant solution of (1.2) with
the form un (t) = φ(n + ct), where c > 0 is the wave speed. Letting ξ = n + ct, it
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TRAVELLING WAVES
follows that φ must be a solution of the following wave profile equation
Z ∞X
φ(ξ) =
Aj (τ )g(φ(ξ − j − cτ ))dτ, ξ ∈ R.
0
3
(2.1)
j∈ Z
Under the assumptions (G1)–(G2), it is easily seen that 0 and K are the only two
equilibria of (2.1). We will find a solution φ of (2.1) with the boundary conditions
lim φ(ξ) = 0,
ξ→−∞
lim inf φ(ξ) > 0.
ξ→∞
(2.2)
Define
g − (u) =
inf
u≤v≤K ∗
g + (u) = min{g 0 (0)u, K ∗ } for u ∈ [0, K ∗ ].
{g(v)},
Similar to the proof of [13, Lemma 3.1], it is easily seen that the following lemma
holds.
Lemma 2.1. Assume that (A1), (G1)–(G4) hold. Then the following assertions
hold.
(i) g − (u) and g + (u) are nondecreasing on [0, K ∗ ] and Lispchitz continuous on
[0, K ∗ ]; that is, |g ± (u) − g ± (v)| ≤ g 0 (0)|u − v| for u, v ∈ [0, K ∗ ];
(ii) g − (u) ≤ g(u) ≤ g + (u), for u ∈ [0, K ∗ ];
(iii) g 0 (0)u ≥ g − (u) > 0 and g 0 (0)u ≥ g + (u) > 0 for all u ∈ (0, K ∗ ];
(iv) There exists 0 < K∗ ≤ K such that g − (K∗ ) = K∗ . Moreover, g − (0) =
g − (K∗ ) − K∗ = 0 and g + (0) = g + (K ∗ ) − K ∗ = 0;
(v) lim supu→0+ [g 0 (0) − g − (u)/u]u−σ < ∞.
Let
∆c (λ) = g 0 (0)
Z
0
∞
e−λcτ
X
Aj (τ )e−λj dτ.
j∈ Z
According to Assumption (A1), for any c > 0, ∆c (λ) is well defined on [0, λ̄). We
have the following lemma.
Lemma 2.2 (Lemma 3.1, [24]). Assume that (A1) holds and g 0 (0) > 1. Then,
there exist c∗ > 0 and λ∗ > 0 such that the following assertions hold.
c (λ)
(i) ∆c∗ (λ∗ ) = 1, ∂∆∂λ
|c=c∗ ,λ=λ∗ = 0; i.e, λ∗ is the minimal zero point of
∆c∗ (λ) = 1;
(ii) For any c ∈ (0, c∗ ) and λ ∈ [0, λ̄), ∆c (λ) > 1;
(iii) For any c > c∗ , the equation ∆c (λ) = 1 has two real roots λ1 and λ2 ; that
is, ∆c (λ1 ) = ∆c (λ2 ) = 1: 0 < λ1 < λ∗ < λ2 < λ̄. Moreover, ∆c (λ) < 1 for
any λ ∈ (λ1 , λ2 ).
Now we are in a position to state our main results about the existence and
nonexistence of travelling waves.
Theorem 2.3 (Existence). Assume that (A1), (G1)–(G4) hold, and that g 0 (0) > 1.
Then we have
(i) For c > c∗ , the system (1.2) has a travelling wave φ ∈ C(R, [0, K ∗ ]) satisfying
lim φ(ξ)e−λ1 ξ = 1,
ξ→−∞
K∗ ≤ α := lim inf φ(ξ) ≤ K ≤ β := lim sup φ(ξ) ≤ K ∗
ξ→∞
ξ→∞
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Moreover, either α = β = K or K∗ < α < K < β; that is, the wave is
either convergent to the positive equilibrium or oscillating on the positive
equilibrium at positive infinity.
(ii) If, in addition, either g(u) is nondecreasing in u ∈ [0, K] or (G5) holds,
then α = β = K; that is, the wave is convergent to the positive equilibrium.
(iii) For c = c∗ , the system (1.2) has a travelling wave φ(n+c∗ t) ∈ C(R, [0, K ∗ ])\
{0, K}. Moreover, if g(u) is nondecreasing in u ∈ [0, K], then the travelling
wave φ(n + c∗ t) satisfies the asymptotic behavior
lim φ(ξ) = 0,
ξ→−∞
lim φ(ξ) = K.
ξ→∞
Theorem 2.4 (Non-existence). Assume that (A1), (G1)–(G4) hold. Then we have
the following assertions:
(i) If g 0 (0) < 1, for any c > 0, the system (1.2) has no nonnegative bounded
travelling wave solution satisfying (2.2);
(ii) If g 0 (0) > 1, for 0 < c < c∗ , the system (1.2) has also no nonnegative
bounded travelling wave satisfying (2.2).
Remark 2.5. If g(u) is nondecreasing in u ∈ [0, K], letting K ∗ = K, then Theorem
2.3 in this present paper reduces to [24, Theorem 4.1]. The nonexistence, (ii) of
Theorem 2.4, is a consequence of the result that c∗ is the spreading speed which
is established in [24, Theorem 3.2]. Note that the non-existence result of travelling
waves still has not been reported in [24] when g 0 (0) < 1.
Remark 2.6. By Theorems 2.3 and 2.4, it is easily seen that c∗ is the minimal
wave speed. According to the results in [24], c∗ is also the asymptotic speed of propagation for the model (1.1) without presupposing that the function g be monotone.
Thus we can also conclude that the minimal wave speed for the model (1.2) coincides with the asymptotic speed of propagation for the model (1.1) without the
monotonicity of g. We can further obtain some good properties of travelling waves
at positive infinity for the model (1.2) with non-monotone function g.
According to the above remarks, we need to prove only Theorem 2.3 with nonmonotonicity of g and Theorem 2.4(i). To complete our main results, we need to
make some preparations. Define an operator T : C(R, [0, K ∗ ]) → C(R, R+ ) by
Z ∞X
T (φ)(ξ) =
Aj (τ )g(φ(ξ − j − cτ ))dτ, ξ ∈ R, φ ∈ C(R, [0, K ∗ ]). (2.3)
0
j∈ Z
It is obvious that T is well defined and a fixed point of T is a solution of (2.1),
which is a travelling wave of (1.2). Let T ± be as in (2.3) with g replaced by g ± .
By Lemma 2.1, it is easily seen that T ± are nondecreasing on C(R, [0, K ∗ ]) and
T − (φ) ≤ T (φ) ≤ T + (φ),
for φ ∈ C(R, [0, K ∗ ]).
(2.4)
Lemma 2.7. Assume (A1), (G1)–(G4), and that g 0 (0) > 1. Then T + (φ̄+ )(ξ) ≤
φ̄+ (ξ), where
φ̄+ (ξ) =: min{K ∗ , eλ1 ξ }, for ξ ∈ R.
(2.5)
Proof. Since g + (u) is nondecreasing in u and φ̄+ (ξ) ≤ K ∗ , for ξ ∈ R, we can obtain
Z ∞X
T + (φ̄+ )(ξ) ≤
Aj (τ )g + (K ∗ )dτ = g + (K ∗ ) = K ∗ , ξ ∈ R.
(2.6)
0
j∈ Z
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5
Note that g + (u) ≤ g 0 (0)u, for u ∈ [0, K ∗ ], and φ̄+ (ξ) ≤ eλ1 ξ for ξ ∈ R. It follows
that
Z ∞X
+ +
T (φ̄ )(ξ) ≤
Aj (τ )g 0 (0)φ̄+ (ξ − cτ − j)dτ
0
Z
j∈ Z
∞
≤
0
λ1 ξ
=e
X
Aj (τ )g 0 (0)eλ1 (ξ−cτ −j) dτ
(2.7)
j∈ Z
∆c (λ1 ) = eλ1 ξ .
According to the definition of φ̄+ and (2.6)-(2.7), we have T + (φ̄+ )(ξ) ≤ φ̄+ (ξ) for
ξ ∈ R. This completes the proof.
Similar to the proof of Lemma 2.7, it is easily seen that the following lemma
holds.
Lemma 2.8. Assume (A1), (G1)–(G4), and that g 0 (0) > 1. Then T − (φ̄− )(ξ) ≤
φ̄− (ξ), where
φ̄− (ξ) =: min{K∗ , eλ1 ξ },
for ξ ∈ R.
Lemma 2.9. Assume that (A1), (G1)–(G4) hold, and g 0 (0) > 1. Then for every
γ ∈ (1, min{1 + σ, λλ12 }), there exists large enough Q(γ) > 1 such that for any
q ≥ Q(γ), we have T − (φ− )(ξ) ≥ φ− (ξ), where
φ− (ξ) =: max{0, eλ1 ξ − qeγλ1 ξ },
for ξ ∈ R.
(2.8)
q
Proof. Let ξ0 = − λ1ln
(γ−1) . Then we have
φ− (ξ) = 0
for ξ ≥ ξ0 ,
φ− (ξ) = eλ1 ξ − qeγλ1 ξ
for ξ ≤ ξ0 .
Obviously, it follows that
T − (φ− )(ξ) ≥ 0
for ξ ∈ R.
(2.9)
By the definition of g − (u) and Lemma 2.1(v), there exist δ > 0 and 0 < M < ∞
such that
g(u) ≥ g − (u) ≥ g 0 (0)u − M u1+σ
for u ∈ [0, δ].
(2.10)
It is easily seen that there exists Q1 (γ) > 1 such that eλ1 ξ −qeγλ1 ξ ≤ δ for q ≥ Q1 (γ).
Therefore,
0 ≤ φ− (ξ) ≤ δ
for ξ ∈ R.
(2.11)
Since ξ0 < 0 and 1 + σ > γ, it is easy to see that
φ̄+ (ξ) ≥ φ− (ξ) ≥ eλ1 ξ − qeγλ1 ξ
−
[φ (ξ)]1+σ ≤ eγλ1 ξ
for ξ ∈ R,
for ξ ∈ R.
Thus, according to (2.10)-(2.13), we have
g 0 (0)φ− (ξ) − M [φ− (ξ)]1+σ ≥ g 0 (0)eλ1 ξ − g 0 (0)qeγλ1 ξ − M eγλ1 ξ
(2.12)
(2.13)
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and
T − (φ− )(ξ)
Z ∞X
Aj (τ )[g 0 (0)φ− (ξ − cτ − j) − M [φ− (ξ − cτ − j)]1+σ dτ
≥
0
Z
≥
0
j∈ Z
∞
X
h
i
Aj (τ ) g 0 (0)eλ1 (ξ−cτ −j) − g 0 (0)qeγλ1 (ξ−cτ −j) − M eγλ1 (ξ−cτ −j) dτ
j∈ Z
Z
h
i
M ∞X
= eλ1 ξ ∆c (λ1 ) − qeγλ1 ξ ∆c (γλ1 ) +
Aj (τ )e−γλ1 (j+cτ ) dτ.
q 0
j∈ Z
(2.14)
Since λ1 < γλ1 < λ2 , we have
∆c (γλ1 ) < 1.
Therefore, there exists
Z
∞
Q(γ) ≥ max{1, Q1 (γ), M
0
X
Aj (τ )e−γλ1 (j+cτ ) dτ },
j∈ Z
large enough, such that for q ≥ Q(γ), it follows
Z
M ∞X
∆c (γλ1 ) +
Aj (τ )e−γλ1 (j+cτ ) dτ ≤ 1.
q 0
(2.15)
j∈ Z
Hence, by (2.14)-(2.15), we have
T − (φ− )(ξ) ≥ eλ1 ξ − qeγλ1 ξ
for ξ ∈ R.
(2.16)
According to the definition of φ− (ξ), (2.9) and (2.16), it is easy to see that
T − (φ− )(ξ) ≥ φ− (ξ)
for ξ ∈ R.
This completes the proof.
−
Remark 2.10. The construction of φ̄+ , φ̄− and φ
that for monotone case in [5] and [22].
in (2.5) and (2.8) is due to
Applying the iteration monotone scheme
φn+1 (ξ) = T − (φn )(ξ)
φ0 (ξ) = φ̄− (ξ),
by Lemmas 2.8 and 2.9, we can easily obtain the following result.
Proposition 2.11. Assume that (A1), (G1)–(G4) hold, and g 0 (0) > 1. Then, for
any c > c∗ , there exists a nondecreasing fixed point φ− of T − such that T − (φ− )(ξ) =
φ− (ξ) and φ− (ξ) ≤ φ̄− (ξ) for all ξ ∈ R. Moreover,
lim φ− (ξ)e−λ1 ξ = 1
ξ→−∞
and
lim φ− (ξ) = K∗ .
ξ→∞
For a given number λ > 0, let
Xλ := {φ ∈ C(R, R) : sup |φ(ξ)|e−λξ < ∞}
ξ∈R
−λξ
and kφkλ = supξ∈R |φ(ξ)|e
space.
. Then it is easy to see that (Xλ , k · kλ ) is a Banach
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TRAVELLING WAVES
7
By Lemmas 2.7–2.9 and Proposition 2.11, we have
φ− (ξ) = T − (φ− )(ξ) ≤ T − (φ̄− )(ξ) ≤ T + (φ̄− )(ξ) ≤ T + (φ̄+ )(ξ) ≤ φ̄+ (ξ).
Now fix a number λ ∈ (0, λ1 ). Clearly, both φ− and φ̄+ are elements in Xλ . Thus,
the set
Γ := {φ ∈ Xλ : φ− (ξ) ≤ φ(ξ) ≤ φ̄+ (ξ), for ξ ∈ R}
is a nonempty, closed and convex subset of Xλ . For the operator T defined by (2.3),
we have the following lemmas.
Lemma 2.12. Assume that (A1), (G1)–(G4) hold, and g 0 (0) > 1. The following
assertions hold.
(i) T (Γ) ⊂ Γ;
(ii) T : Γ → Γ is continuous with respect to the norm k · kλ in Xλ .
Proof. For any φ ∈ Γ, it is obvious that
φ− = T − (φ− ) ≤ T − (φ) ≤ T (φ) ≤ T + (φ̄+ ) ≤ φ̄+ ,
which implies that T (Γ) ⊂ Γ. For any φ, ψ ∈ Γ, we have
kT (φ) − T (ψ)kλ = sup |T (φ)(ξ) − T (ψ)(ξ)|e−λξ
ξ∈R
Z
∞
≤ sup
ξ∈R
0
Z
Z
≤ g 0 (0)
X
j∈ Z
∞X
0
≤ g 0 (0)
Aj (τ )g 0 (0)|φ(ξ − cτ − j) − ψ(ξ − cτ − j)|e−λξ dτ
0
Aj (τ )e−λ(cτ +j) dτ kφ − ψkλ
j∈Z
∞
X
Aj (τ )e−λj dτ kφ − ψkλ .
j∈ Z
Thus, T : Γ → Γ is continuous with respect to the norm k·kλ in Xλ . This completes
the proof.
Lemma 2.13. Assume that (A1), (G1)–(G4) hold, and g 0 (0) > 1. Then T : Γ → Γ
is compact.
Proof. Since 0 ≤ φ− ≤ T (φ) ≤ φ̄+ ≤ K ∗ for φ ∈ Γ, it is obvious that the family of
functions {T (φ)(ξ) : φ ∈ Γ} is uniformly bounded in ξ ∈ R. On the other hand, it
follows
P from (A1) and (G1) that g(u) ≤ Lu for u ∈ R+ and there is M1 > 0 such
that j∈ Z Aj (ξ) ≤ M1 for ξ ∈ R. Therefore, for ξ1 > ξ2 , we have
|T (φ)(ξ1 ) − T (φ)(ξ2 )|
Z
Z ∞ X
=
Aj (τ )g(φ(ξ1 − cτ − j))dτ −
0
0
j∈ Z
∞
X
Aj (τ )g(φ(ξ2 − cτ − j))dτ j∈ Z
1 Z ξ1 X
1
Aj ( (ξ1 − τ ))g(φ(τ − j))dτ
=
c −∞
c
j∈ Z
Z ξ2 X
1
1
−
Aj ( (ξ2 − τ ))g(φ(τ − j))dτ c −∞
c
j∈ Z
Z ξ2 X
X
1
1
1
≤
Aj ( (ξ2 − τ )) −
Aj ( (ξ1 − τ ))g(φ(τ − j))dτ
c −∞
c
c
j∈ Z
j∈ Z
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Z. YU, R. YUAN
+
1
c
Z
ξ1
ξ2
0
X
j∈Z
1
Aj ( (ξ1 − τ ))g(φ(τ − j))dτ
c
ξ2
X
X
1
g 0 (0)K ∗ M1
1
Aj ( (ξ2 − τ )) −
Aj ( (ξ1 − τ ))dτ +
|ξ1 − ξ2 |
c
c
c
−∞ j∈ Z
j∈ Z
Z ∞X
X
g 0 (0)K ∗ M1
1
= g 0 (0)K ∗
Aj ( (ξ1 − ξ2 ) + τ ) −
Aj (τ )dτ +
|ξ1 − ξ2 |
c
c
0
≤
g (0)K
c
∗
EJDE-2010/103
Z
j∈ Z
j∈ Z
= g 0 (0)K ∗ A(ξ1 − ξ2 ) +
where
Z
A(ξ) =
0
0
∗
g (0)K M1
|ξ1 − ξ2 |,
c
∞
X
X
1
Aj ( ξ + τ ) −
Aj (τ )dτ
c
j∈ Z
for ξ ∈ R.
(2.17)
j∈ Z
Since limξ→0 A(ξ) = 0, it follows from the above inequality that the family of functions {T (φ)(ξ) : φ ∈ Γ} is equicontinuous in ξ ∈ R. Thus, by Arzera-Ascoli theorem,
for any given sequence {ψn }n≥1 in T (Γ), there exist a subsequence {ψnk }k≥1 and
ψ ∈ C(R, R) such that limk→∞ ψnk (ξ) = ψ(ξ) uniformly for ξ in any compact subset of R. Since φ− (ξ) ≤ ψnk (ξ) ≤ φ̄+ (ξ) for ξ ∈ R, we have φ− (ξ) ≤ ψ(ξ) ≤ φ̄+ (ξ)
for ξ ∈ R and ψ ∈ Γ. Since λ ∈ (0, λ1 ), it is easy to see that
lim (φ̄+ (ξ) − φ− (ξ))e−λξ = 0,
ξ→∞
lim (φ̄+ (ξ) − φ− (ξ))e−λξ = 0
ξ→−∞
which imply that for any > 0, there exists B > 0 such that
0 ≤ (φ̄+ (ξ) − φ− (ξ))e−λξ ≤ ,
for |ξ| ≥ B.
(2.18)
−λξ
Since limk→∞ (ψnk (ξ) − ψ(ξ))e
= 0 uniformly for ξ ∈ [−B, B], there exists a
k 0 ∈ N+ such that
|ψnk (ξ) − ψ(ξ)|e−λξ < ,
(2.19)
0
for ξ ∈ [−B, B], k ≥ k . It follows from (2.18) and (2.19) that
kψnk (ξ) − ψ(ξ)kλ = sup |ψnk (ξ) − ψ(ξ)|e−λξ ≤ ,
for k ≥ k 0 .
ξ∈R
Thus, limk→∞ ψnk = ψ in Xλ . This completes the proof.
Proof of Theorem 2.3(i). By Lemmas 2.9 and 2.12, the Schauder’s fixed point theorem implies that there exists φ ∈ Γ such that φ = T (φ) and hence φ is a travelling
wave of (1.2). Since 0 ≤ φ− (ξ) ≤ φ(ξ) ≤ φ̄+ (ξ) ≤ K ∗ for ξ ∈ R, we can obtain
lim φ(ξ) = 0,
ξ→−∞
lim φ(ξ)e−λ1 ξ = 1.
ξ→−∞
Moreover,
K∗ = lim φ− (ξ) ≤ lim inf φ(ξ) ≤ lim sup φ(ξ) ≤ lim φ̄+ (ξ) = K ∗ ,
ξ→∞
ξ→∞
ξ→∞
ξ→∞
(2.20)
that is, K∗ ≤ α ≤ β ≤ K ∗ . Next, we shall show that α ≤ K ≤ β. Indeed, if α = β,
then limξ→∞ φ(ξ) = α exists. Taking ξ → ∞ and applying L’Hospital’s rule to
(2.1), we can obtain α = g(α) which yields α = β = K.
Now we consider that α < β. If there is a large number M2 > 0 such that φ0 > 0
on [M2 , ∞) or φ0 < 0 on [M2 , ∞), then limξ→∞ φ(ξ) exists and hence α = β, which
is impossible. Thus, φ(ξ) is oscillating on positive infinity and then there exist two
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TRAVELLING WAVES
9
sequences {ξj }j∈N with ξj → ∞ and {yi }i∈N with yi → ∞ as i → ∞ such that
φ(ξj ) → α as j → ∞, and φ(yi ) → β as i → ∞.
By (A1), for any > 0, there are sufficiently large numbers τ0 > 0 and J0 > 0
such that
Z ∞X
Z τ0 X
∗
∗
Aj (τ )dτ <
K
Aj (τ )dτ < .
and K
3
3
τ0
0
j∈ Z
|j|≥J0
Since g is continuous, there exists δ > 0 such that
max{g(u) : u ∈ [α − δ, β + δ]} < max{g(u) : u ∈ [α, β]} + .
3
For such a δ > 0, there exists a large enough number M3 > 0 such that
φ(ξ) ∈ [α − δ, β + δ],
for all ξ ≥ M3 .
We take a large enough integer M4 > 0 such that ξm ≥ M3 + J0 + cτ0 for all
m ≥ M4 . Thus, for m ≥ M4 , we have
φ(ξm )
Z ∞X
=
Aj (τ )g(φ(ξm − j − cτ ))dτ
0
Z
j∈ Z
∞
=
τ0
X
Aj (τ )g(φ(ξm − j − cτ ))dτ +
j∈ Z
τ0 X
Z
+
0
≤ K∗
Z
Z
|j|<J0
∞X
τ0
0
τ0
X
Aj (τ )g(φ(ξm − j − cτ ))dτ
|j|≥J0
Aj (τ )g(φ(ξm − j − cτ ))dτ
Aj (τ )dτ + K ∗
Z
0
j∈ Z
τ0
X
Aj (τ )dτ + max{g(u) : u ∈ [α − δ, β + δ]}
|j|≥J0
< max{g(u) : u ∈ [α, β]} + .
Taking the limit as m → ∞, we have
β ≤ max{g(u) : u ∈ [α, β]} + .
+
Thus, letting → 0 , it follows that
β ≤ max{g(u) : u ∈ [α, β]}.
(2.21)
α ≥ min{g(u) : u ∈ [α, β]}.
(2.22)
Similarly, it follows that
If α < β ≤ K, then (2.22) and (G2) imply that α ≥ min{g(u) : α ≤ u ≤ β} > α,
which is a contradiction. If K ≤ α < β, then (2.21) and (G2) also imply that
β ≤ max{g(u) : α ≤ u ≤ β} < β, a contradiction. Hence we conclude that
α < K < β.
(ii). Let u1 , u2 ∈ [α, β] such that g(u1 ) = max{g(u) : α ≤ u ≤ β} and g(u2 ) =
min{g(u) : α ≤ u ≤ β}. Then it is split into three cases.
Case 1. K ≤ u1 ≤ β. If u1 = β, according to (2.21), β ≤ g(β), which is invalid
since β > K. Therefore u1 < β. By (2.21) and u1 ≥ K, we have β ≤ g(u1 ) ≤ u1 <
β, which is contradiction.
Case 2. α ≤ u2 ≤ K. If u2 = α, according to (2.22), α ≥ g(α), which is
impossible since α < K. Therefore α < u2 . By (2.22) and u2 ≤ K, we have
α ≥ g(u2 ) ≥ u2 > α, which is also contradiction.
10
Z. YU, R. YUAN
EJDE-2010/103
Case 3. u1 < K < u2 . By (2.21)-(2.22) and (G5), we have
β − α ≤ g(u1 ) − g(u2 ) < (2K − u1 ) − (2K − u2 ) = u2 − u1 ,
which is impossible.
Thus, α = β and hence the limit limξ→∞ φ(y) = α ∈ [K∗ , K ∗ ] exists. Taking
y → ∞ and applying L’Hospital’s rule to (2.1), we can obtain α = g(α) which
yields α = K.
(iii). In the case where c = c∗ , we can obtain the existence of travelling waves
by using a limiting argument similar to [9, 19, 24]. Let {cm } ⊂ (c∗ , c∗ + 1] with
limm→∞ cm = c∗ . Since cm > c∗ , (2.1) with c = cm admits a solution φm ∈
C(R, [0, K ∗ ]) such that
lim φm (ξ) = 0,
ξ→−∞
lim inf φm (ξ) ≥ K∗ .
ξ→∞
Without loss of generality, we may assume that φm (0) = 12 K∗ for m ∈ N+ . Note
that
Z ∞X
φm (ξ) =
Aj (τ )g(φm (ξ − j − cm τ ))dτ, for ξ ∈ R, m ∈ N+ .
(2.23)
0
j∈ Z
It is easy to see that
g 0 (0)K ∗ M1
|ξ1 − ξ2 |,
cm
where A(ξ) and M1 are defined in the proof of Lemma 2.12. Therefore, {φm (ξ)} is
uniformly bounded and equicontinuous in ξ ∈ R. Using Arzera-Ascoli theorem, we
can obtain a subsequence {φmk } and φ such that limk→∞ φmk (ξ) = φ(ξ) uniformly
for ξ in any compact subset of R. Clearly, φ(0) = 21 K∗ and φ ∈ C(R, [0, K ∗ ]). By
the dominated convergence theorem and (2.21), it follows that
Z ∞X
φ(ξ) =
Aj (τ )g(φ(ξ − j − c∗ τ ))dτ, for ξ ∈ R.
|φm (ξ1 ) − φm (ξ2 )| ≤ g 0 (0)K ∗ A(ξ1 − ξ2 ) +
0
j∈ Z
and hence, φ(n + c∗ t) is a travelling wave of (1.2). This completes the proof.
Remark 2.14. Note that Γ in the present paper and the set Y in [9] are different.
We can obtain (2.20) by the construction of a suitable set Γ. The conclusion in [9]
similar to (2.20) must be obtained by the property of the spreading speed (see, [9,
Theorem 2.2]) since it could not be obtained by the construction of Y .
Proof of Theorem 2.4(i). Assume that (1.2) has a nonnegative bounded solution φ
with (2.2). It is obvious that φ 6≡ constant. Letting kφk∞ = supξ∈R |φ(ξ)| > 0 and
according to (2.1), we have
kφk∞ = sup |φ(ξ)|
ξ∈R
Z
∞
≤ sup
ξ∈R
Z
<
0
∞
X
Aj (τ )g 0 (0)φ(ξ − j − cτ )dτ
0
j∈ Z
X
Aj (τ )kφk∞ dτ
j∈ Z
= kφk∞
which is a contradiction. This completes the proof.
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TRAVELLING WAVES
11
Acknowledgements. We would like to thank the anonymous referee for his/her
valuable suggestions and comments which led to an improvement of our original
manuscript.
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Zhixian Yu
College of Science, University of Shanghai for Science and Technology, Shanghai
200093, China
E-mail address: yuzx@mail.bnu.edu.cn
Rong Yuan
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
E-mail address: ryuan@bnu.edu.cn