We construct an example showing that there need not be a strictly positive solution of a competing species system with diffusion and Dirichlet boundary conditions in a case unresolved by earlier work.

We consider certain vector fields in the plane which possess a centre. The main result is that for Hamiltonian polynomial systems which are of even degree, which possess homogeneous nonlinearities, and which have a centre located at the origin, the period function is a strictly increasing function of the energy, throughout its interval of definition. It is also shown that for nonlinear homogeneous Hamiltonian polynomial vector fields of arbitrary degree which possess a centre, the period function is a strictly decreasing function of the energy. With appropriate modifications, this result is extended to arbitrary homogeneous vector fields which possess a centre, irrespective of their being Hamiltonian or polynomial; the period function is then strictly monotonic, except when the degree of homogeneity is one, when the systems are isochronous.

Systems of nonlinear elasticity described by Von Karman equations with nonlinear boundary dissipation are considered. Global existence, uniqueness of weakisolutions as well as the regularity of solutions with "smooth" data is established. Thus the paper solves, in particular, an outstanding problem of uniqueness of weak solutions to Von Karman system, which has been open in the literature even in the case of the homogeneous boundary data. This is accomplished by proving "sharp" regularity results of the Airy stress function.

We apply variants of the Mountain Pass Lemma without the (PS)-condition as developed by M. Schechter to the study of Semilinear Dirichlet Problems at Double Resonance.

The system $$ \begin{align} -\Delta u_{k} &= ( a_{k} -u_{k} - \sum ^{n}_{ j\neq k,\, j=1} \gamma _{kj} u_{j} ) u_{k} \quad \text{in} \; \Omega \\ u_{k} & =0 \quad \text{on} \; \partial \Omega ,\end{align} \tag {*} $$ $k=1,\ldots, n$, where $\Omega $ is a bounded domain in $\Bbb R^{n}$ and $a_{k} , \gamma _{kj}$ are positive parameters, determines the possible equilibrium configurations for a diffusive Lotka-Volterra competition model and is of interest in the study of the role of competition in structuring communities where space or resources are limited. The componentwise nonnegative solutions to $(*)$ can perhaps best be understood for fixed $\gamma _{kj}$ and varying $a_{k}$ as a subset of the Banach space $\Bbb R^{n} \times [C^{1}_{0} (\overline{\Omega })]^{n}$. The aims of this article are to enhance understanding of the structure of this set and to provide a firmer foundation for future analysis. We accomplish these aims through some new observations regarding the set of componentwise nonnegative solutions to $(*)$ which enable us to unify some preceding analyses.

The homogenization of first order ordinary differential equations in $\mathbb{R}^N$ and associated linear transport equations are studied. We prove the equivalence between $G$-convergence and strong $G$-convergence for the ordinary equations. We give a sufficient condition, which is also necessary in the autonomous case, for the weak homogenization of the linear transport equations. This condition is satisfied when div$_x f=0$.

If $a(t)$ and $b(t)$ are positive almost periodic functions, and $K(t)$ is nonnegative and piecewise continuous on $[0,\infty),$ conditions under which the equation $$ N'(t) = N(t)\Big( a(t) - b(t) \int_0^\infty K(s) N(t-s)\,ds \Big) $$ has a positive almost periodic solution $N^*(t)$ on $(-\infty,\infty)$ are given which attracts all other positive solutions as $t\to a.$ These conditions are quite explicit and apparently new.

Our first aim in this paper is to give a characterization of generators for the class of translation semigroups which are associated with a symbol $\phi$ and can brought into the form $u(t) = \phi (u_t).$ As applications, we consider two linear models from biology, a cell-cycle model based on unequal division and the Sharpe-Lotka age-dependent model. Our second aim is to show the relation of this last model to this class of semigroups in order to give a result on its asymptotic behavior. We show that the semigroup associated with the Sharpe-Lotka model is equivalent to a translation semigroup and both semigroups are essentially compact. It is also shown that the generators of the two semigroups have the same spectrum. We give properties of this spectrum and of its spectral bound.

The equation $ - \varDelta u = f(x, u)$ on $\Bbb R^N$ as a general equation of the Emden-Fowler type is considered. Under appropriate conditions, positive and multiple solutions are obtained.

The existence and uniqueness of a solution of a constrained system of integral equations is considered and the asymptotic behavior of this solution is studied.

A continuous-time, linear, time-invariant control system with a fixed time delay in the feedback loop is considered. We investigate the problem of feedback stabilization. Some easy-to-check criteria to the two-dimensional systems are presented.

We prove generation of analytic semigroups in $W^{k,p}(\Omega)$ and in $C^k(\bar \Omega)$, $ k \in N $ $ 1<p<\infty $, by non-variational elliptic operators under general boundary condition