Journal of Applied Mathematics and Physics
Vol.04 No.07(2016), Article ID:68833,6 pages
10.4236/jamp.2016.47128
Existence of Traveling Waves in Lattice Dynamical Systems
Xiaojun Li, Yong Jiang, Ziming Du
School of Science, Hohai University, Nanjing, China
Received 28 December 2015; accepted 12 July 2016; published 15 July 2016
ABSTRACT
Existence of traveling wave solutions for some lattice differential equations is investigated. We prove that there exists such that for each
, the systems under consideration admit monotonic nondecreasing traveling waves.
Keywords:
Traveling Wave, Lattice Dynamical Systems, Schauder’s Fixed Point Theorem
1. Introduction
Consider the following lattice differential equation
(1.1)
where,
are positive constants,
,
is a
-function, and
.
Lattice dynamical systems occur in a wide variety of applications, and a lot of studies have been done, e.g., see [1]-[4]. A pair of solutions,
of (1.1) is called a traveling wave solution with wave
speed if there exist functions
such that
,
with
and
. Let
, note that (1.1) has a pair of traveling wave solutions if and only if
,
satisfy the functional differential equation
(1.2)
Without loss of generality, we can impose (1.1) with asymptotic boundary conditions
,
,
,
. (1.3)
By the property of equation, we can assume that. In the following, we give some assumptions on nonlinear function
:
,
,
.
There exists a positive-value continuous function
such that
,
.
,
,
for any
,
,
where,
is given in Lemma 2.1.
for any
.
Select positive constants such that
,
, and define operators
by
. (1.4)
Then, (1.2) can be rewritten as
,
. (1.5)
Define the operators by
.
Note that satisfy
and a fixed point of
is a solution of (1.2). Denote
the Euclidean norm in
. Define
,
where. Note that
is a Banach space.
Definition 1.1. If the continuous functions are differentiable almost everywhere and satisfy
(1.6)
Then, is called an upper solution of (1.2).
Similarity, we can define a lower solution of (1.2). The main result of this paper is
Theorem 1.1. Assume that hold. Then there exists
such that for every
, (1.2) admits a traveling wave solution
connecting
and
. Moreover, each component of traveling wave solution is monotonically nondecreasing in
, and for each
,
,
also
satisfy,
, where
is the smallest solution of the eq-
uation
.
2. Upper-Lower Solutions of (1.2)
Set.
Lemma 2.1. Assume that holds. Then there exists a unique
such that
if
, then there exist two positive numbers
and
with
such that
,
in
, and
in
;
if
, then
for all
;
if
, then
, and
.
Proof. Using assumption, we can get the result directly.
Lemma 2.2. Assume that,
and
hold. Let
,
, and
be defined as in
Lemma 2.1, and be any number. Then for every
and
, there exists
such that for any
,
,
and
are a pair of upper solutions and a pair of lower solutions of (1.2), respectively.
Proof. Let
(2.1)
. (2.2)
Since, there exists
such that
,
. If
, then
,
. By
, we get that
,
.
If, then
. By
,
, and using
Lemma 2.1, we get that
(2.3)
Lemma 2.1 and yields
. (2.4)
Thus,
Therefore, is an upper solution of (1.2). Similarly, we can prove that
is a lower solution.
3. Existence of Traveling Wave
Let,
. We have the fol-
lowing result.
Lemma 3.1 Assume that and
hold. Then
and
for
if
satisfy
,
for
;
are nondecreasing in
if
is nondecreasing in
.
Proof. If such that
and
for
, then by
we have
(3.1)
where. Note that
(3.2)
Thus, from (3.1)-(3.2), we have
which implies that. A similar argument can be done for
. Thus, we can get the desired results.
Lemma 3.2. Assume that and
hold. Then
is continuous
with respect to the norm with
.
Proof. We first prove that are continuous. Denote
. For any
, choose
, where
. If
and
satisfy
, then by (3.1),
(3.3)
Similarly, is continuous.
By definition of, we have
(3.4)
If, it follows that
. (3.5)
If, it follows that
(3.6)
Combining (3.5) and (3.6), we get that is continuous with respect to the norm
. A Similar argument can be done for
.
Define
It is easy to verify that is nonempty, convex and compact in
. As the proof of Claim 2 in the proof of Theorem A in [5], we have
Lemma 3.3. Assume that hold. Then
.
Proof of Theorem 1.1. By the definition of, Lemma 3.2-3.3 and Schauder’s fixed point theorem, we get that there exists a fixed point
. Note that
is nondecreasing in
, as-
sumption and Lemma 2.2 imply that
. There-
fore, is a traveling wave solution of (1.1).
Acknowledgements
This work was supported by the NNSF of China Grant 11571092.
Cite this paper
Xiaojun Li,Yong Jiang,Ziming Du, (2016) Existence of Traveling Waves in Lattice Dynamical Systems. Journal of Applied Mathematics and Physics,04,1231-1236. doi: 10.4236/jamp.2016.47128
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