Planar linear discrete systems with constant coefficients and delays $x(k+\mathrm{1})=Ax(k)+{\sum }_{l=\mathrm{1}}^n\mathrm{‍}{B}^l{x}_l(k-m_l)$ are considered where $k{\in \mathbb{Z}}_{\mathrm{0}}^{\mathrm{\infty }}:=\{\mathrm{0,1},\dots ,\mathrm{\infty }\}$, $m_{\mathrm{1}},m_{\mathrm{2}},\dots ,m_n$ are constant integer delays, , $A,B^{\mathrm{1}},\dots ,B^n$ are constant $\mathrm{2}\times \mathrm{2}$ matrices, and $x:{\mathbb{Z}}_{{-m}_n}^{\mathrm{\infty }}{\to ℝ}^{\mathrm{2}}$. It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $\mathrm{2}(m_n+\mathrm{1})$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

The purpose of this paper is to develop a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delays $\mathrm{\Delta }x(k)=Bx(k-m)+Cx(k-n)+f(k),$ where $m,n\in \mathbb{N}$, $m\ne \mathrm{n,}$ are fixed, $k=0,\dots ,\mathrm{\infty }$, $B=(b_{ij})$, $C=(c_{ij})$ are constant $r\times r$ matrices, $f$ is a given $r\times 1$ vector, and $x$ is an $r\times 1$ unknown vector. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential for two delays. Such approach results in a possibility to express an initial Cauchy problem in a closed form. Examples are shown illustrating the results obtained.

We study the spatial discretisation of the Cauchy problem for a multidimensional linear parabolic PDE of second order, with nondivergent operator and unbounded time- and space-dependent coefficients. The equation free term and the initial data are also allowed to grow. Under a nondegeneracy assumption, we consider the PDE solvability in the framework of the variational approach and approximate in space the PDE problem's generalised solution, with the use of finite-difference methods. The rate of convergence is estimated.

The problem of stability for nonlinear impulsive stochastic functional differential equations with delayed impulses is addressed in this paper. Based on the comparison principle and an impulsive delay differential inequality, some exponential stability and asymptotical stability criteria are derived, which show that the system will be stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous stochastic flows. The obtained results complement ones from some recent works. Two examples are discussed to illustrate the effectiveness and advantages of our results.

The paper deals with the local theory of internal symmetries of underdetermined systems of ordinary differential equations in full generality. The symmetries need not preserve the choice of the independent variable, the hierarchy of dependent variables, and the order of derivatives. Internal approach to the symmetries of one-dimensional constrained variational integrals is moreover proposed without the use of multipliers.

Some classes of systems of difference equations whose all well-defined solutions are periodic are presented in this note.

The following differential equation $u^{(n)}(t)+p(t)|u(\sigma (t)){|}^{\mu (t)}\text{\hspace{0.17em}sign}u(\sigma (t))=\mathrm{0}$ is considered. Here $p\in L_{\mathrm{\text{loc}}}(R_{+};R_{+})$, $\mu \in C(R_{+};(\mathrm{0},+\mathrm{\infty }))$, $\sigma \in C(R_{+};R_{+})$, $\sigma (t)\le t$, and ${\lim }_{t\to +\mathrm{\infty }}\sigma (t)=+\mathrm{\infty }$. We say that the equation is almost linear if the condition ${\lim }_{t\to +\mathrm{\infty }}\mu (t)=\mathrm{1}$ is fulfilled, while if ${\lim \sup }_{t\to +\mathrm{\infty }}\mu (t)\ne \mathrm{1}$ or ${\lim \inf }_{t\to +\mathrm{\infty }}\mu (t)\ne \mathrm{1}$, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying Property A for delay Emden-Fowler equations are obtained.

We study asymptotic behavior of solutions to a class of higher-order quasilinear neutral differential equations under the assumptions that allow applications to even- and odd-order differential equations with delayed and advanced arguments, as well as to functional differential equations with more complex arguments that may, for instance, alternate indefinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.