Vol 222, No 3 (2017)

We study a two-point boundary-value problem for singularly perturbed third-order integrodifferential equations with small parameters at two higher derivatives. The asymptotic convergence of the solution of this problem to the solution of a certain degenerate boundary-value problem is established.

For a broad class of systems of nonlinear functional equations, we establish conditions for the existence and uniqueness of continuous solutions. A method for the construction of solutions of this kind and investigation of their properties as ε → 0 is proposed.

We establish necessary and sufficient conditions for solvability and determine the general form of the solutions of a linear integral equation (with square summable kernel) and a boundary-value problem for equations of this kind.

We study the problem of motion of a cylindrical reservoir (on a pendulum suspension) partially filled with an ideal liquid. At the initial time, the deviation of reservoir from the equilibrium position is described by a certain angle. We analyze the redistribution of kinetic energy in the system and explain the decay of the surface wave motion. The accumulated results qualitatively agree with the available experimental data.

We present additional information on the dynamics of a fast-slow system with dynamic bifurcation of multifrequency oscillations. It is shown that, parallel with an asymptotically stable invariant torus, the system also possesses hyperbolic invariant tori of lower dimensions located within a small neighborhood of its invariant manifold of slow motions.

We deduce representations for the solutions of initial-value problems for n-dimensional differential equations of the second order with delays:

\( x^{{\prime\prime} }(t)=2 Ax^{\prime}\left( t-\tau \right)-\left({A}^2+{B}^2\right) x\left( t-2\tau \right) \)

and

\( x^{{\prime\prime} }(t)=\left( A+ B\right) x^{\prime}\left( t-\tau \right)- A B x\left( t-2\tau \right) \)

by using special delay matrix functions. Here, A and B are commuting (n × n)-matrices and τ > 0. Moreover, a formula connecting the delay matrix exponential function with delayed matrix sine and delayed matrix cosine is obtained. We also discuss common features of the considered equations.