Stable non-singular solutions in beyond Horndeski theories

The theory of interest is Generalized Galileon or Horndeski theory (and beyond). We start with motivation to study these theories. Then we discuss the stability issue in two different types of scenarios: cosmological and static, spherically-symmetric case. We show that it leads to the no-go theorems for non-singular solutions in Horndeski theory. Finally, we show the possibility to obtain stable solutions in beyond Horndeski theory and discuss further issues.