In this paper some practical stability results for nonlinear differential equations with non-instantaneous impulses and state dependent delays are presented. The impulses start abruptly at some points and their action continue on given finite intervals. The delay depends on both the time and the state variable which is a generalization of time variable delay. Some sufficient conditions for practical stability and strong practical stability are obtained by the help with the appropriate modification of Razumikhin method and an appropriate definition of the derivative of the Lyapunov function. Examples are given to illustrate the results.

In this paper we obtain results on existence of standing waves in Discrete Nonlinear Shrödinger equation (DNLS) with saturable nonlinearity on a two-dimensional lattice. We consider two types of solutions: with periodic amplitude and vanishing at infinity (localized solution). Sufficient conditions for the existence of such solutions are obtained with the aid of Nehari manifold and periodic approximations.

We consider variations and generalizations of the initial Dirichlet problem for linear second order divergence form equations of parabolic type, with vanishing initial values and non-continuous lateral data, in the setting of Lipschitz cylinders. More precisely, lateral data in adequations of the Lebesgue classes $L^p$, and a family of Sobolev-type classes are considered. We also establish some basic connections between estimates related to solvability of each of these problems. This generalizes some of the well-known works for Laplace's equation, heat equation and some linear elliptic-type equations of second order.

Let $f$ be an element of the subspace $(L^{p},l^{q})^{\alpha}(\mathbb{R}^d)$ $(1\leq p \leq \alpha \leq q \leq 2)$ of the Wiener amalgam space $(L^{p},l^{q})(\mathbb{R}^{d})$. We give sufficient conditions for Lebesgue integrability of the Fourier transform of $f$. These conditions are in terms of the $(L^{p},l^{q})^{\alpha}(\mathbb{R}^{d})$ integral modulus of continuity of $f$. As an application, we obtain that if $1\leq\alpha\leq q\le 2$ with $\frac{1}{\alpha}-\frac{1}{q}\lt\frac{1}{d}$ and $N= [\frac{d}{\alpha}] + 1$, then the Fourier inversion theorem can be applied to the elements of the Sobolev space $W^{N}((L^{1},l^{q})^{\alpha}(\mathbb{R}^d))$.