I've been reading this paper on the discrete-time version of the Markus-Yamabe conjecture. While the conjecture is false in general, Theorem A in this paper proves the result for systems which are "lower triangular". I'm confused by two parts of the proof though, and any help would be appreciated.
Below Equation (3), there is an expansion of the function $F_s$ in terms of a sum of integrals. It looks like the standard mean-value theorem-type expansion of a function, except some of the partial derivatives are being evaluated at zero; I can't quite parse why this is valid.
Below Equation (4), they seem to be saying that all off-diagonal Jacobian elements are less than one. I can't tell why this must be true; the induction hypothesis is that the diagonal elements are less than one.
Solved, the author was kind enough to reply to an email:
It is not a standard expansion as it turns out, but its a clever construction. The series as written telescopes, leaving only $F_s(x)$.
This is a mistake in the paper. The off-diagonal elements just need to be bounded by some constant (which by the induction hypothesis they are), and the rest of the proof goes through.