Influence of a small perturbation on Poincare-Andronov operators with not well defined topological degree

Let $P_e\in C^0(R^n,R^n)$ be the Poincare-Andronov operator over period $T>0$ of the $T$-periodically perturbed autonomous system $x^{\mathrm{\prime }}=f(x)+eg(t,x,e),$ where $e>0$ is small. Assuming that for $e=0$ this system has a $T$-periodic limit cycle $x_0$ we evaluate the topological degree $d(I-P_e,U)$ of $I-P_e$ on an open bounded set $U$ whose boundary contains $x_0([0,T])$ and does not contain other fixed points of $P_0\mathrm{.}$ We give an explicit formula connecting $d(I-P_e,U)$ with topological indexes of zeros of the associated Malkin’s bifurcation function. The goal of the paper is to prove Mawhin’s conjecture which claims that $d(I-P_e,U)$ can be any integer in spite of the fact that the measure of the set of fixed points of $P_0$ on $\mathrm{\partial }U$ is zero.