Vol 236, No 3 (2019)
- Year: 2019
- Articles: 10
- URL: https://journals.rcsi.science/1072-3374/issue/view/14980
Article
Stabilization of Solutions of a Nonlocal Problem Multipoint in Time for One Class of Evolutionary Pseudodifferential Equations
Abstract
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′:
Asymptotic Behavior of Slowly Varying Solutions of Second-Order Ordinary Binomial Differential Equations with Rapidly Varying Nonlinearity
Abstract
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 ≤ + ∞.
Global Attractor of an Impulsive Dynamical System Generated by The wave Equation
Abstract
We prove the existence and invariance of a global attractor for a discontinuous system generated by the wave equation such that solutions of the system undergo impulsive perturbations when they reach a certain fixed subset of the phase space.
Bifurcation Conditions for the Solutions of the Lyapunov Equation in a Hilbert Space
Abstract
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.
Continuability and Boundedness of Solutions for a Kind of Nonlinear Delay Integrodifferential Equations of the Third Order
Abstract
In the paper, we consider a nonlinear integrodifferential equation of the third order with delay. We establish sufficient conditions guaranteeing the global existence and boundedness of the solutions of the analyzed equation. We use the Lyapunov second method to prove the main result. An example is also given to illustrate the applicability of our result. The result of this paper is new and improves previously known results.
Oscillatory Solutions of Some Autonomous Partial Differential Equations with a Parameter
Abstract
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.