Existence of traveling wave solutions for some lattice differential equations is investigated. We prove that there exists c*>0 such that for each c≥c*, the systems under consideration admit monotonic nondecreasing traveling waves.
Consider the following lattice differential equation
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 -. 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
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.
Select positive constants such that, , and define operators by
Then, (1.2) can be rewritten as
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
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-
2. Upper-Lower Solutions of (1.2)
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,
are a pair of upper solutions and a pair of lower solutions of (1.2), respectively.
Since, there exists such that,. If, then,
. By, we get that
If, then. By, , and using
Lemma 2.1, we get that
Lemma 2.1 and yields
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-
Lemma 3.1 Assume that and hold. Then
and for if
are nondecreasing in if is nondecreasing in.
Proof. If such that and for , then by we have
where. Note that
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),
Similarly, is continuous.
By definition of, we have
If, it follows that
If, it follows that
Combining (3.5) and (3.6), we get that is continuous with respect to the norm. A Similar argument can be done for.
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 , 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).
This work was supported by the NNSF of China Grant 11571092.