Nonlinear Noetherian boundary-value problems in the critical case

We establish necessary and sufficient conditions for the existence of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the critical case.     

Asymptotic Behavior of Solutions to a Model for the Flow of a Reacting Fluid

We give a complete analysis of solutions of a model for the flow of a multispecies reacting fluid occupying a thin cylinder whose walls may be semipermeable with respect to some or all of the chemical species. We prove the global existence of solutions and establish a number of time-independent a priori bounds sufficient to determine the corresponding time-asymptotic steady-state. We then derive necessary conditions and sufficient conditions ensuring that this steady-state reflects complete combustion, that is, that at least one of the reactant species is depleted.     

Absolute Exponential Stability of Solutions of Linear Parabolic Differential Equations with Delay

We find necessary and sufficient conditions for the absolute exponential stability of solutions of linear parabolic differential equations with delay in a pair of norms.     

On Solutions with Power Asymptotics for Differential Equations with Exponential Nonlinearity

We establish necessary and sufficient conditions for the existence of solutions with power asymptotics for two-term differential equations with exponential nonlinearity.     

Asymptotic Representations of Solutions of an Ordinary Differential Equation of the <Emphasis Type="Italic">n</Emphasis>th Order with Exponential Nonlinearity

We establish necessary and sufficient conditions for the existence of one class of solutions of a differential equation with exponential nonlinearity. Asymptotic representations of these solutions are obtained.__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 562–573, October–December, 2004.     

Stability of Solutions of a Degenerate Linear System of Differential Equations

We introduce the concept of stability of solutions of a system of linear differential equations with an identically degenerate matrix as the coefficient of the derivative. We find necessary and sufficient conditions for the stability of such systems. We generalize the Floquet–Lyapunov theory to systems of this type with periodic coefficients.     

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.     

On the beltram I flows

To illustrate the general properties of the Beltrami flows, a new sufficient and necessary condition of the general solution of the Beltrami flows has been given, on whose basis some sufficient and necessary conditions and sufficient conditions in literature are unified. The advantages of the new expressions are that one may get solutions of Beltrami flows with constant proportional factors which satisfy some given asymptotic decaying conditions at infinity. Finally, we find a class of spherical vortices of Beltrami flows, where the motion trajectories of the fluid particles are confined within a sphere or two adjacent concentric spherical shells. We find that the superpositions of the spherical vortices of the same order with different polar axes may produce chaotic trajectories. The project is supported by the National Foundamental Research Programme of State Commission of Sciences and Technology.     

Oscillation of Nonlinear Hyperbolic Differential Equations with Impulses

We study oscillatory properties of solutions of nonlinear impulsive hyperbolic differential equations and find new necessary and sufficient conditions for the existence of oscillations.__________Published in Neliniini Kolyvannya, Vol. 7, No. 4, pp. 439–445, October–December, 2004.     

Existence and Asymptotic Behavior of Positive Solutions of Neutral Impulsive Differential Equations

In this article, we consider first-order neutral impulsive differential equations with constant coefficients and constant delays. We study the asymptotic behavior of eventually positive solutions of these equations and establish necessary and sufficient conditions for the existence of such solutions. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 304–318, July–September, 2005.     

Motion of a nonisothermal multicomponent sorbable mixture in a porous medium

The numerical solutions of a system of quasilinear equations in partial derivatives, describing the motion of a nonisothermal multicomponent sorbable gas mixture (or mixture of liquids) through a porous saturated medium consisting of porous grains and incapable of undergoing deformation, are analyzed; the conditions for the convergence of the iteration process used in solving the difference scheme and for the stability of the numerical solutions are obtained; the necessary and sufficient conditions for the existence of solutions of the traveling-wave type, permitted by the system of equations of motion, are also analyzed.     

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

Summary We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.     

Exponential decay in a thermoelastic mixture of solids

In this paper, we investigate the asymptotic behaviour of solutions to the initial boundary value problem for a one-dimensional mixture of thermoelastic solids. Our main result is to establish a necessary and sufficient condition over the coefficients of the system to get the exponential stability of the corresponding semigroup. We also prove the impossibility of time localization of solutions.     

Global Solutions to a Model for Exothermically Reacting,Compressible Flows with Large Discontinuous Initial Data

 We prove the global existence of solutions of the Navier-Stokes equations describing the dynamic combustion of a compressible, exothermically reacting fluid, and we study the large-time behavior of solutions, giving necessary and sufficient conditions for complete combustion in certain cases. The adiabatic constants and specific heats of the burned (product) and unburned (reactant) fluids may differ, and the initial data may be large and discontinuous. (Accepted August 31, 2002) Published online January 9, 2003 Communicated by C. M. Dafermos     

冲击动力系统的鲁棒稳定性分析

考虑冲击动力系统的ｋ－ｐ周期运动的鲁棒稳定性问题。首先，根据微分方程的解、冲击条件和衔接条件，应用迭代法给出了系统存在ｋ－ｐ周期运动的充分必要条件，并利用稳定性的等价原理，通过周期运动的扰动差分方程导出其稳定条件；然后，着重对含有不确定参数的冲击动力系统的ｋ－ｐ周期运动的稳定性进行了分析，得出了鲁棒稳定的充分条件，文末给出了用于阐明理论结果的算例。     

Cauchy problem for one class of pseudodifferential systems with smooth symbols

For one class of pseudodifferential systems with smooth time-dependent symbols, we study properties of the fundamental matrix of solutions. We formulate sufficient and, for some systems, necessary conditions for the correct solvability of the Cauchy problem with generalized initial data. We also construct spaces of test and generalized functions that generalize certain classical spaces. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 502–524, October–December, 2006.     

Bounded solutions of weakly nonlinear differential equations in a Banach space

For a weakly nonlinear differential equation in a Banach space, we establish necessary and sufficient conditions for the existence of solutions bounded on the entire real axis under the assumption that the generating equation has bounded solutions and the corresponding homogeneous equation admits an exponential dichotomy on the semiaxes. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 151–159, April–June, 2008.     

Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems

In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.     

Existence of positive solutions of systems of Volterra nonlinear difference equations

In this paper, a classification scheme for eventually positive solutions of a class of two-dimensional Volterra nonlinear difference equations is given in terms of asymptotic magnitudes. Some necessary as well as sufficient conditions for the existence of such solutions are provided without any monotonicity conditions on the nonlinear term. Published in Neliniini Kolyvannya, Vol. 9, No. 1, pp. 37–47, January–March, 2006.     

Attractivity in non-autonomous systems

We discuss some sufficient conditions for attractivity of a closed set by using Liapunov functions and show how to apply our results to the study of asymptotic behaviors of solutions in nonautonomous systems. We also discuss the convergence of solutions by the same idea. These results are extendable to functional differential equations.