On solutions of two-point boundary value problems inside isolating segments

Abstract

We consider a two-point boundary value problem
$$
\dot x=f(t,x), \quad x(a)=g(x(b)).
$$
We assume that in the extended space of the equation
there exist an isolating segment,
a set such that $f$ properly behaves on its boundary. We give
a formula for the fixed point index of the
composition of $g$ with the translation operator in a neighbourhood of
the set of the initial points of solutions contained
in the isolating segment. We apply that formula to results
on existence of solutions of some planar boundary value problem
associated to equations of the form $\dot z=\overline z^q+\ldots$
and $\dot z=e^{it}\overline z^q+\ldots$.