Evading the theoretical no-go theorem for nonsingular bounces in Horndeski/Galileon cosmology

We show that a nonsingular bounce, free of ghosts and gradient instabilities, can be realized in the framework of Horndeski or generalized Galileon cosmology. In particular, we first review that the theoretical no-go theorem, which states that the above is impossible, is based on two very strong assumptions, namely that a particular quantity cannot be discontinuous during the bounce, and that there is only one bounce. However, as we show in the present work, the first assumption not only can be violated in a general Horndeski/Galileon scenario, but also it is necessarily violated at the bounce point within the subclass of Horndeski/Galileon gravity in which $K(\phi,X)$ becomes zero at $X=0$. Additionally, concerning the second assumption, which is crucial in improved versions of the theorem which claim that even if a nonlinear free of pathologies can be realized it will lead to pathologies in the infinite past or infinite future, we show that if needed it can be evaded by considering cyclic cosmology, with an infinite sequence of nonsingular bounces free of pathologies, which forbids the universe to reach the "problematic" regime at infinite past or infinite future. Finally, in order to make the analysis more transparent we provide explicit examples where nonsingular bounces without theoretical pathologies can be achieved.