We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$\dot{x}\left(t\right)-a\left(t\right)\dot{x}\left(g\left(t\right)\right)+b\left(t\right)x\left(h\left(t\right)\right)=0,$$ where $$\left|a\right(t\left)\right|<1,b\left(t\right)\ge 0,h\left(t\right)\le t,g\left(t\right)\le t,$$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

The article is a survey on problem of the theorem of Hurwitz. The starting point of explanations is Schur's decomposition theorem for polynomials. It is showed how to obtain the well-known criteria on the distribution of roots of polynomials. The theorem on uniqueness of constants in Schur's decomposition seems to be new.

We show that the holomorphic functions on polysectors whose derivatives remain bounded on proper subpolysectors are precisely those strongly asymptotically developable in the sense of Majima. This fact allows us to solve two Borel-Ritt type interpolation problems from a functional-analytic viewpoint.

M. Hirsch's famous theorem on strongly monotone flows generated by autonomous systems u'(t) = f(u(t)) is generalized to the case where f depends also on t, satisfies Carathéodory hypotheses and is only locally Lipschitz continuous in u. The main result is a corresponding Comparison Theorem, where f(t,u) is quasimonotone increasing in u; it describes precisely for which components equality or strict inequality holds.

We investigate the nonautonomous periodic system of ODE’s of the form $\dot{x}=\overrightarrow{v}\left(x\right)+r_p\left(t\right)(\overrightarrow{w}\left(x\right)-\overrightarrow{v}\left(x\right))$, where $r_p\left(t\right)$ is a $2p$-periodic function defined by $r_p\left(t\right)=0$ for $t\in \langle 0,p\rangle$, $r_p\left(t\right)=1$ for $t\in (p,2p)$ and the vector fields $\overrightarrow{v}$ and $\overrightarrow{w}$ are related by an involutive diffeomorphism.

Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti $u\in 𝐂([0,T],E)$ dell'equazione di evoluzione non autonoma $u^{\prime }(t)-A(t)u(t)=f(t)$ con il dato iniziale $u(0)=x$, in uno spazio di Banach $E$. Gli operatori $A(t)$ sono generatori infinitesimali di semi-gruppi analitici ed hanno dominio indipendente da $t$ e non necessariamente denso in $E$. Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità hölderiana della soluzione e della sua derivata.