Vol 217, No 4 (2016)

We study the possibility of construction of the asymptotic solution of multipoint boundary-value problem for a linear singularly perturbed system of differential equations with identically degenerate matrix of derivatives in the case of multiple spectrum of the main operator. To this end, we use the results of asymptotic analysis for the general solution of a linear degenerate singularly perturbed system of differential equations. We establish the existence and uniqueness conditions for the solution of the analyzed boundary-value problem.

We establish sufficient conditions for the existence of a weak solution of the Neumann problem for the heat-conduction equation with random action described by an integral over the general stochastic measure.

The degenerate parabolic variational inequality with mixed boundary conditions and inhomogeneous initial conditions is studied in the case where the corresponding operator can lose the properties of coercivity and continuity in the corresponding Sobolev spaces. By using the Hardy–Poincaré inequality, we prove the unique solvability of the original evolutionary variational inequality under the condition that the degenerate weight function is a function of potential type.

We propose a new class of Lyapunov functions for the equations with fractional derivatives. The conditions of stability with respect to two measures are established for a given class of equations by using the generalized comparison principle and a vector-valued Lyapunov function.

We consider a problem of preservation of a piecewise continuous invariant toroidal set for a class of multifrequency systems with pulses at nonfixed times under perturbations of the right-hand side. New theorems that impose constraints on perturbation terms not in the entire phase space but only in a nonwandering set of the dynamical system guarantee the existence of exponentially stable invariant toroidal set.