We construct nonanalytic solutions to the initial value problem for the KdV equation with analytic initial data in both the periodic and the nonperiodic cases.

A delayed discrete equation $\Delta u\left(k+n\right)=-p\left(k\right)u\left(k\right)$ with positive coefficient $p$ is considered. Sufficient conditions with respect to $p$ are formulated in order to guarantee the existence of positive solutions if $k\to \infty$. As a tool of the proof of corresponding result, the method described in the author's previous papers is used. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (if $k\to \infty$) with the speed not smaller than the speed characterized by the function $\sqrt{k}·{\left(n/\left(n+1\right)\right)}^k$. A comparison with the known results is given and some open questions are discussed.

We present several recent and novel results on the formulation and the analysis of the equations governing the evolution of electromagnetic fields in chiral media in the time domain. In particular, we present results concerning the well-posedness and the solvability of the problem for linear, time-dependent, and nonlocal media, and results concerning the validity of the local approximation of the nonlocal medium (optical response approximation). The paper concludes with the study of a class of nonlinear chiral media exhibiting Kerr-like nonlinearities, for which the existence of bright and dark solitary waves is shown.

We suggest some criteria for the stabilization of planar linear systems via linear hybrid feedback controls. The results are formulated in terms of the input matrices. For instance, this enables us to work out an algorithm which is directly suitable for a computer realization. At the same time, this algorithm helps to check easily if a given linear $2\times 2$ system can be stabilized (a) by a linear ordinary feedback control or (b) by a linear hybrid feedback control.

This paper deals with the problem $\Delta u=g$ on $G$ and $\partial u/\partial n+uf=L$ on $\partial G$. Here, $G\subset {ℝ}^m$, $m>2$, is a bounded domain with Lyapunov boundary, $f$ is a bounded nonnegative function on the boundary of $G$, $L$ is a bounded linear functional on $W^{1,2}\left(G\right)$ representable by a real measure $\mu$ on the boundary of $G$, and $g\in L_2\left(G\right)\cap L_p\left(G\right)$, $p>m/2$. It is shown that a weak solution of this problem is bounded in $G$ if and only if the Newtonian potential corresponding to the boundary condition $\mu$ is bounded in $G$.

The Darboux-Lamé equation is defined as the double Darboux transformation of the Lamé equation, and is studied from the viewpoint of the isomonodromic deformation theory. It is shown that the second-order ordinary differential equation of Fuchsian type on ${\mathbf{P}}_1$ corresponding to the second Darboux-Lamé equation is obtained as isomonodromic deformation of some specific Gauss' hypergeometric differential equation.

We investigate the existence of mild solutions on a compact interval to some classes of semilinear neutral functional differential inclusions. We will rely on a fixed-point theorem for contraction multivalued maps due to Covitz and Nadler and on Schaefer's fixed-point theorem combined with lower semicontinuous multivalued operators with decomposable values.