Том 53, № 12 (2017)

For families of n-dimensional linear differential systems (n ≥ 2) whose dependence on a parameter ranging in a metric space is continuous in the sense of the uniform topology on the half-line, we obtain a complete description of the ith Lyapunov exponent as a function of the parameter for each i = 1,..., n. As a corollary, we give a complete description of the Lebesgue sets and (in the case of a complete separable parameter space) the range of an individual Lyapunov exponent of such a family.

A comparison principle based on Minkowski mixed volumes is established for a family of differential equations with imprecise parameter values. Scalar and vector approaches are considered, and the basic inequalities of the comparison principle are established.

A singularly perturbed boundary value problem for a second-order quasilinear ordinary differential equation is studied. We consider a new class of problems in which the nonlinearities experience discontinuities, which leads to the appearance of sharp transition layers in a neighborhood of the points of discontinuity. The existence of solutions is proved, and their asymptotic expansion with an internal transition layer is constructed.

The behavior of solutions of the Poisson equation on noncompact Riemannian manifolds of a special form is studied. Sharp conditions for the unique solvability of the Dirichlet problem on the reconstruction of solutions of the Poisson equation from continuous boundary data at infinity are found.

The first boundary value problem for a multidimensional parabolic differential equation with a small parameter ε multiplying all derivatives is studied. A complete (i.e., of any order with respect to the parameter) regularized asymptotics of the solution is constructed, which contains a multidimensional boundary layer function that is bounded for x = (x¹, x₂) = 0 and tends to zero as ε → +0 for x ≠ 0. In addition, it contains corner boundary layer functions described by the product of a boundary layer function of the exponential type by a multidimensional parabolic boundary layer function.

The problem of constructing aggregated systems (quotient systems) of the simplest kind for nonlinear control systems is considered. With the help of this factorization, the original control system is reduced to a decomposition that allows one to reduce the dimension of control problems.

We study the general linear third-order differential equation with constant real coefficients in the case of two independent variables. Necessary and sufficient coefficient conditions for the hyperbolicity and strict hyperbolicity of the equation, as well as for the representability of the differential operator occurring in the equation in the form of a composition of first-order operators, are obtained.