On solutions of two-point boundary value problems inside isolating segments (Q1809484)

The problem under consideration reads \[ x'= f(t,x),\quad x(a)= g(x(b)), \] where \(f\) is continuous on a smooth manifold. Assuming that \(f\) properly behaves on the boundary of a certain isolating segment, a fixed point index formula is given for the composition of \(g\) with the associated translation operator. This formula is applied for obtaining existence results. A special attention is paid to some planar problem related to a complex scalar differential equation.