In this section, we shall prove the existence of positive solutions of (2.1)-(2.2). Let
for any , .
Lemma 3.1 There exists a constant such that the following items hold.
For each , has two positive real roots .
If , then there exists such that and for any .
If , then for any .
The above result is clear and we omit the proof here. Using these constants, we can prove the following conclusion.
Theorem 3.2 Assume that and one of the following two items holds.
Then (2.1)-(2.2) has a monotone solution.
Proof Define continuous functions as follows:
Claim A: is an upper solution to (2.1).
Moreover, let hold and satisfy
Evidently, is a lower solution to (2.1) (for the existence of and , we refer to Pan et al. ). By Lemma 2.2, we see that (2.1)-(2.2) has a monotone solution . Now, it suffices to prove Claim A.
If or , the result is clear. If , then
which completes the proof on for .
We now consider with . If , then such that
Therefore, (3.1) leads to
If , then and (3.2) imply that
Therefore, Claim A is true. The proof is complete. □
Theorem 3.3 Assume that one of the following items holds.
Then (2.1)-(2.2) has a monotone solution with .
Proof If (3.3) or (3.4) holds, then there exists a decreasing sequence with , such that for each , (2.1)-(2.2) has a positive monotone solution . Note that a traveling wave solution is invariant in the sense of phase shift, so we can assume that
for any n. By the Ascoli-Arzela lemma and a standard nested subsequence argument (see, e.g., Thieme and Zhao ), there exists a subsequence of , which is still denoted by without confusion, such that converges uniformly on every bounded interval, and hence pointwise on ℝ to a continuous function . Moreover, for each , we have
and the convergence in s is uniform for . Letting and using the dominated convergence theorem in , we know that also satisfies (2.1) with . In addition, the following items are also clear.
(T1) (by (3.5));
(T2) , are nondecreasing in ξ;
(T3) , , .
The items (T1) to (T3) further indicate that exists for . Denote
From (T1), it is clear that
If , then the dominated convergence theorem in implies that
Using the dominated convergence theorem in for , we get the following possible conclusions:
If (L1) is true, then the dominated theorem in tells us
which implies a contradiction. If (L2) is true, then leads to
which is also a contradiction. What we have done implies that . Using the dominated convergence theorem in again, we see that and .
If , then a discussion similar to that on can be presented and we omit it here. Because , then the dominated convergence in as indicates that or . If is true, then holds and
has a monotone solution, which is impossible. Therefore, holds.
Thus, is a positive monotone solution of (2.1)-(2.2) with , the proof is complete. □