卷 236, 编号 3 (2019)

We show that the solution of a nonlocal problem multipoint in time for an evolutionary equation with differential operator of infinite order is stabilized to zero as t → + ∞ in the space of generalized functions of the form S′:

We establish new conditions for the existence of slowly varying solutions of a second-order differential binomial nonautonomous equation with rapidly varying nonlinearity. We also determine the asymptotic representations for these solutions and their first derivatives as t ↑ ω, ω ≤ + ∞.

We establish sufficient conditions for the bifurcation of solutions of the boundary-value problems for the Lyapunov equation in Hilbert spaces. The cases where the generating equation has or does not have solutions are analyzed. As an example, we consider the problem in the space l2 of sequences with matrices of countable dimensions.

We establish existence and uniqueness conditions for the solutions of difference equations with contracting compact operators in the metric space of two-sided sequences.

We study a class of evolutionary partial differential equations depending on a parameter τ (stemming from the problems of groundwater flows). The existence of an open interval ????⁰ of the parameter τ and of a function τ ⟼ Θ(τ), Θ: ????⁰ ⟼(0, + ∞), is proved with the property that any nonzero global solution u:ℝ⁺ × Ω → ℝ of the equation cannot remain nonnegative (nonpositive) throughout the set J × Ω; where J ⊂ ℝ⁺ is any interval whose length is greater than Θ (τ). In other words, these solutions are globally oscillatory and Θ (τ) is the uniform oscillatory time. The interval ????⁰ and the function Θ are explicitly determined.