We prove a number of asymptotic stability theorems of Lyapunov type for first order ordinary differential systems, extending and generalizing previous work on the subject.

Of concern is a quantum mechanical system having $N_{1}$ (resp. $N_{2}$) spin up (resp. spin down) electrons, in the presence of a potential $V$ and a magnetic field $B.$ When the Fermi-Amaldi correction is incorporated into the Thomas-Fermi energy functional, convexity is lost and the computation of the ground state spin up and down electron densities becomes nontrivial. We discuss the existence of these densities and various approximation procedures for them, via variational calculus, differential equations, and numerical procedures.

Several works, quoted in the introduction, provide criteria which establish global existence in the future of all solutions of nonlinear second order differential systems. The aim of the present paper is to establish further sufficient conditions for global existence in the future for submanifolds of solutions.

We study the one-dimensional system of Ginzburg-Landau equations that models a thin film of superconductor subjected to a tangential magnetic field. We prove that the bifurcation curve for the symmetric problem is the graph of a continuous function of the supremum of the order parameter. We also prove the existence of a critical magnetic field. In general, there is more than one positive solution to the symmetric boundary value problem. Our numerical experiments have shown cases with three solutions. It is still an open question whether only one of these corresponds to the physical solution that minimizes the Gibbs free energy. We establish uniqueness for a related boundary value problem.

We study the equation $$ x'(t) = g(x(t-1)) \tag{$g$} $$ for smooth functions $g:\mathbb{R} \rightarrow \mathbb{R} $ satisfying $ \xi g(\xi)<0 $ for $ \xi \neq 0, $ and the equation $$ x'(t) = b(t)x(t-1) \tag {$b$} $$ with a periodic coefficient $b:\mathbb{R} \rightarrow (-\infty,0)$. Equation $(b)$ generalizes variational equations along periodic solutions $y$ of equation $(g)$ in case $g'(\xi) < 0$ for all $\xi \in y(\mathbb{R} )$. We investigate the largest Floquet multipliers of equation $(b)$ and derive a characterization of vectors transversal to stable manifolds of Poincar\'e maps associated with slowly oscillating periodic solutions of equation $(g)$. The criterion is used in Part II of the paper in order to find $g$ and $y$ so that a Poincar\'e map has a transversal homoclinic trajectory, and a hyperbolic set on which the dynamics are chaotic.

For an arbitrary bounded linear operator $C$, on a Banach space, and a closable linear operator $A$, we introduce a $C$-regularized semigroup for $A$. We present equivalences between $A$ having a $C$-regularized semigroup, the corresponding abstract Cauchy problem, well-posedness on a continuously embedded subspace, and (for exponentially bounded $C$-regularized semigroups) the Laplace transform.

A model for nonisothermal flow in double-porosity medium is constructed by the technique of homogenization. The coupling between microscopic and macroscopic equations is discussed, with numerical examples given. The model may be imbedded in a framework common for the models corresponding to different memory effects exhibited by the flow which are related to the hierarchical structure of the medium. For this framework results of numerical experiments are presented.

Existence of solutions to initial value problems involving a discontinuous differential equation is studied. Under fairly weak conditions, existence of Carathéodory solutions is proven. Comparisons to theorems in the literature are given and motivation for the result is also discussed.

Several well-known existence theorems and several new existence theorems for semilinear elliptic boundary value problems at resonance are shown to be corollaries of a single abstract result. The proofs rely on Leray-Schauder degree theory and the approximate domain method.

We consider the quasilinear hyperbolic equation $$ u_{tt}-M\Bigl( \int_\Omega|\text{ grad }u|^2\,dx\Bigr)\,\Delta u=0, \tag1 $$ where $x\in\Omega=\Bbb R^n$, $t$ denotes time and $M(s)$ is a smooth function satisfying $M(s)>0$ for all $s\ge 0$. We prove that there are no non-trivial ``breathers" for equation (1). Here, a "breather" means a time periodic solution which is "small" as $|x|\to +\infty$. We also present a simpler proof of the so-called Pohozaev's second conservation law for (1) solving the global Cauchy problem for "non-physical" nonlinearity $M$ arising from this conservation law.