In this paper, we will establish several Lyapunov inequalities for linear Hamiltonian systems, which unite and generalize the most known ones. For planar linear Hamiltonian systems, the connection between Lyapunov inequalities and estimates of eigenvalues of stationary Dirac operators will be given, and some optimal stability criterion will be obtained.

In this paper we continue to study second-order linear measure differential equations (MDE), following the line of the recent work [13], where the dependence of solutions of MDE on measures with the weak* topology is studied. In this part, we consider the Dirichlet and the Neumann eigenvalues of MDE. It will be proved that these eigenvalues are continuous in measures with the weak* topology. Such a result extends recent works on eigenvalues of Sturm-Liouville operators with potentials or weights in [27]. As an application, we will give a natural, simple explanation to extremal problems of eigenvalues of Sturm-Liouville operators studied in [23, 28].

Measure differential equations (MDE) are frequently used to model non-classical prob-lems like the quantum effects. In Part I of these serial papers, we will rst use the Riemann-Stieltjes integrals to give an explanation to solutions of initial value problems of second-order linear MDE. Then we will present some deep results on dependence of solutions on measures. That is, solutions and some of their derivatives of MDE are continuously dependent on measures, considered in the weak* topology. Examples show that these continuity results are optimal. In Part II, these results will be used to prove the continuity of eigenvalues of MDE in measures with weak* topology.

In this paper, we will give, for the periodic solution of the scalar Newtonian equation, some twist criteria which can deal with the fourth order resonant case. These are established by developing some new estimates for the periodic solution of the Ermakov-Pinney equation, for which the associated Hill equation may across the fourth order resonances. As a concrete example, the least amplitude periodic solution of the forced pendulum is proved to be twist even when the frequency acroses the fourth order resonances. This improves the results in Lei et al. (2003). Twist character of the least amplitude periodic solution of the forced pendulm. SIAM J. Math. Anal. 35, 844-867.

In this paper we will show that the optimal bounds for certain static and dynamic bifurcation values of periodic solutions of some superlinear differential equations can be expressed explicitly using Sobolev constants.