We deal with the Dirichlet, Neumann and periodic eigenvalue problems for the equation $$ \quad(|u'|^{p-2}u')' +\lambda |u|^{q-2}u=0, \text{\quad on~~}(0,T), $$ where $T>0,$ $\lambda>0,$ and $p,q>1.$ For those problems we obtain a complete description of the spectra and a closed form representation of the corresponding eigenfunctions. As an application of our results we present sharp Poincar\'e and Wirtinger inequalities for the imbeddings $W_0^{1,p}(0,T)$ into $L^q(0,T)$ and $W_T^{1,p}(0,T)$ into $L^q(0,T),$ respectively.

Under a non resonance condition, we establish the unique local existence results in weak function spaces for the initial value problem of the Zakharov system with the periodic boundary condition. The proof is based on the Fourier restriction norm method, which was developed by J. Bourgain and C.E. Kenig, G. Ponce and L. Vega.

In this paper we show that first order differential delay equations with negative feedback can possess asymptotically stable rapidly oscillating solutions. We construct an analytically tractable example in which the feedback is piecewise constant. In this case, the continuous-time dynamics on a proper subset of the phase space can be reduced exactly to a three-dimensional discrete-time map. The existence and stability properties of the delay equation's rapidly oscillating periodic solutions are given by the existence and stability of one of the fixed points of the corresponding map. When the feedback is smoothed appropriately, the stable rapidly oscillating periodic solution is shown to persist.

It is shown that predator-prey type reaction-diffusion systems on the whole real line have a unique coexistence state.

We consider a Hamilton-Jacobi equation with a measure in the Hamiltonian arising from the $\Gamma$-limit of some optimal control problems. We give a definition of viscosity solution in this case, by adapting the method of the reparametrization of Dal Maso and Rampazzo [9].

The main purpose of this paper is two-fold: to find sufficient conditions for the existence of entirely bounded solutions of Carathéodory quasi-linear differential inclusions and to show that, if the coefficients are specially constant and the right-hand sides are additionally Lipschitz continuous (with a sufficiently small Lipschitz constant) and almost-periodic in time, then these solutions become almost-periodic as well. The almost-periodicity is understood in the sense of H. Weyl and, because of set-valued analysis, we introduce for the first time the appropriately generalized concept. The related methods, including the fixed-point theorem for a class of $\mathcal{J}$-maps in locally convex topological vector spaces, are developed here too. In the single-valued case, the obtained criteria generalize those of the other authors.