1 - 10 of 14 articles
We introduce a new mathematical model of a circular neural network with unidirectional chemical bonds. The model is a singularly perturbed system of delay differential-difference equations. We study the existence and stability of relaxation periodic motions in the system. It is proved that the...
For the lower sigma-exponent of the linear differential system ẋ = A(t)x, x ∈ R
, t ≥ 0, defined by the formula Δσ(A) ≡ infλ[Q]≤-σ
1(A + Q), σ > 0, on the basis of the lower characteristic exponents λ
1(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ −σ < 0 of perturbations...
We consider a classical spectral problem that arises when studying the natural vibrations of a loaded rectangular membrane. We establish conditions ensuring the uniform convergence of spectral expansions in the selected Riesz basis and in the entire system of eigen-functions.
We study the spectral properties of a second-order differential operator with regular but not strongly regular boundary conditions. We show that the system of root functions of this operator contains infinitely many associated functions. We prove that a specially chosen system of root functions...
We introduce the notions of equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables for solutions with partly controlled initial conditions. We obtain sufficient conditions for the equiultimate boundedness and uniform ultimate boundedness with respect to...
For the set of equations of perturbed motion whose solutions satisfy interval initial conditions, we obtain sufficient conditions for the Lyapunov stability and the practical stability of these solutions. The analysis is performed on the basis of locally large scalar Lyapunov functions. As...
For a singularly perturbed parabolic equation, we construct and justify the asymptotics of the classical solution of an initial-boundary value problem in the case of a double root of the degenerate equation. This case substantially differs from the case of a simple root in that the scales of the...
We consider the Gellerstedt problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator in the leading part and with advanced-retarded multiple deviations of the argument in the derivatives and the function. We prove the uniqueness theorem for the problem without restrictions on...
In the spaces L
), we study complex powers of a nonelliptic differential operator D = I +Δ
x′ represented in the form of partial acoustic potentials (in the variable x′). By using the method of approximative inverse operators, we construct the inversion of the potentials A
We obtain conditions under which a totally conservative solution of the Cauchy problem for a stochastic partial differential equation of parabolic type with nonlinearities of power-law type can only be the identically zero solution.
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