Limit differential inclusions and the invariance principle for nonautonomous systems

Considering nonautonomous differential inclusions we introduce the concept of limit differential inclusions, study their properties and invariance-type properties of the $\omega$-limit sets of solutions, and establish an analog of La Salle's invariance principle using Lyapunov functions with the derivatives of constant sign. The method is equally applicable to differential equations and, under appropriate assumptions, yields some previously-available results.