Monotone 2-Transition Solutions

Abstract

The second multitransition case mentioned in Chapter 6 will be studied here. Proceeding somewhat more generally, suppose $v_0;w_0;\hat{v}_0;\hat{w}_0 \in \mathcal{M}0$, where $v_0 < w_0 \leq \hat{v}_0 < \hat{w}_0$ and the pairs v0;w0 and $\hat{v}_0; \hat{w}_0$ satisfy (*)0. The simplest special case is that of $\hat{v}_0 = v_0 + 1$ and $\hat{w}_0 = w_0 + 1$. The solutions constructed here will be monotone in x1 in the sense of Theorem 3.2, i.e., $u < \tau^1_{-1} u$. This allows us to work in a class of functions having this property and thereby use much less restrictive constraints than employed in Chapter 6 to get existence results.