(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, is odd, for some odd integer and there exists some with such that ; and.
(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, is even, for some odd integer and there exists some with such that ; (ii)furthermore, every such solution is of the form (5.35).
(i)(4.4) has a -periodic traveling wave solution with velocity if, and only if, is even, for some odd integer and there exists some such that and either (a) or (b) is odd and for any with one has either or ; (ii)furthermore, if, every such solution is of the form.
Suppose (i) where is even, with or (ii) where is even, with odd and even and for any with one has either or Then (6.1) has at least one -periodic traveling wave solution with velocity for arbitrary and which satisfy (2.16), is odd, and for some odd integer.
If is even, then for some odd integer and is odd.
If is odd, then is even and for some odd integer.
Since for some odd integer and is odd, by (5.32), we have (5.33).
Let be a -periodic traveling wave solution with velocity Since is odd, by Lemma 4.4 ii), we have that is even and for some odd integer From Lemma 4.3 iii), we also have (5.24).
If d n = 0 for some odd integer n, then g has a fixed point.
By Lemma 4.4 ii),a necessary condition for the existence of such solutions is for some odd integer Hence the fact that implies is odd.
It is also clear that is even, for some odd integer, and By Theorem 5.7(i), 6.3) has -periodic traveling wave solution with velocity By Theorem 5.7 ii), any such solution of (6.3) is of the form (6.5).
