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

DOI: http://dx.doi.org/10.12775/TMNA.1999.004

#### 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 $\dot z=e^{it}\overline z^q+\ldots$.

$$

\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 $\dot z=e^{it}\overline z^q+\ldots$.

#### Keywords

Boundary value problem; isolating segment; Lefschetz number; fixed point index

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