The converse Lyapunov's theorem for ODEs of Carathéodory type

The converse problem of Lyapunov stability for systems of ODEs of Caratheodory type is considered. It is proved that if the right hand side of an ODE satisfies only the Osgood condition, the uniform stability of the origin is sufficient to the existence of a locally Lipschitz continuous Lyapunov function. Actually, the uniform stability is equivalent to the robust stability in this case. Moreover, as an auxiliary result, the generalization of the famous Gronwall-Bellman-Bihari inequality is also proved.