We study a nonlinear integral equation arising from the parametric closure for the third spatial moment in the Dieckmann-Law model of stationary biological communities. The existence of a fixed point of the integral operator defined by this equation is analyzed. The noncompactness of the resulting operator is proved. Conditions are stated under which the equation in question has a nontrivial solution.

A nonautonomous version of the splitting method is presented and with its help an asymptotic solution of the nonautonomous gyroscopic system is constructed in the critical case.

We prove a comparison theorem for the solutions of Riccati matrix equations in which the diagonal entries of the matrix multiplying the linear term are perturbed by a bounded function. This theorem is used to study optimal trajectories in a pollution control problem stated in the form of a linear regulator over an infinite time horizon with a discount function of the general form.

For a certain class of two-dimensional autonomous systems of differential equations with an invariant curve that contains ovals, we indicate necessary and sufficient conditions for these ovals to be limit cycles of phase trajectories.

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.

The Cauchy problem for a linear homogeneous functional-differential equation of the pointwise type defined on a straight line is considered. Theorems on the existence and uniqueness of the solution in the class of functions with a given growth are formulated for the case of the one-dimensional equation. The study is performed using the group peculiarities of these equations and is based on the description of spectral properties of an operator that is induced by the right-hand side of the equation and acts in the scale of  spaces of infinite sequences.

A constructive method for qualitative analysis of initial and multipoint boundary value problems for non-autonomous systems of ordinary differential equations with a polynomial matrix is proposed. The Airy, Bessel, Hermite equations and a number of applied problems, for example, the equation of motion of a gyroscope at the stage of its acceleration, can be reduced to such systems. Effective criteria for the stability of these systems are obtained. The above results supplement or refine previously known.

We consider a multidimensional hyperbolic quasi-gasdynamic system of differential equations of the second order in time and space linearized at a constant solution (with an arbitrary velocity). For the linearized system with constant coefficients, we study an implicit three-level weighted difference scheme and an implicit two-level vector difference scheme. The important domination property of the operator of viscous terms over the operator of convective terms is derived. We apply this property to prove by the energy method that, regardless of the Mach number, our implicit schemes on a nonuniform rectangular mesh (without any conditions on the mesh steps) are stable with respect to the initial data and the right-hand side of the initial–boundary value problem uniformly in time and the relaxation parameter.

The Cauchy problem for the homogeneous linear functional-differential equation of a pointwise type, defined on the line, is considered. In the case of one-dimensional equation we formulated the theorem of existence and uniqueness of solutions with estimating of its order of growth. This research is carried out within the formalism based on group peculiarities of such equations. The main difficulty is associated with the description of the spectral properties of operator, induced by the right-hand side of this equation and acting in the scale of infinite sequences.

The paper is devoted to periodic solutions of a functional-differential equation of point type. Following the paper \ cite {Beklar_Belous}, in terms of the right-hand side of the original non-linear functional-differential equation of point type, easily verifiable conditions for the existence and uniqueness of the $ \ omega $ -periodic solution are formulated and an iterative process of constructing such a solution is described. In contrast to the scalar linearization considered in the article \ cite {Beklar_Belous}, a more complex matrix linearization is used here, which allows us to extend the class of equations for which the existence and uniqueness of the $ \ omega $ -periodic solution can be established in the framework of this approach.

For a certain class of two-dimensional autonomous systems of differential equations with an invariant curve that contains ovals, we indicate necessary and sufficient conditions for these ovals to be limit cycles of phase trajectories.