On Asymptotics of Solutions of Semiexplicit Systems of Differential Equations

We study the problem of the existence of analytic solutions of a certain semiexplicit system of differential equations and obtain sufficient conditions for the existence of analytic solutions of the Cauchy problem in the neighborhood of a singular point.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 132–144, January–March, 2005.     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

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 the Absolute Exponential Stability of Solutions of Systems of Linear Parabolic Differential Equations with One Delay

We investigate necessary conditions for the absolute exponential stability of a system of linear parabolic differential equations with one delay.     

On Aspects of Asymptotics for Plate Equations

In this paper some initial-boundary value problems for plate equations will be studied. These initial-boundary value problems can be regarded as simple models describing free oscillations of plates on elastic foundations or of plates to which elastic springs are attached on the boundary. It is assumed that the foundations and springs have a different behavior for compression and for extension. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For specific parameter values it turns out that complicated internal resonances occur.     

Iteration Method for the Construction of Solutions of Equations with Weak Nonlinearity and Restrictions

We consider an approach to the investigation of equations with weak nonlinearity and restrictions. The validity of the application of the iteration method to this problem is justified.     

Large-Amplitude Periodic Solutions for Differential Equations with Delayed Monotone Positive Feedback

The aim of this paper is to show that the structure of the global attractor for delayed monotone positive feedback can be more complicated than the union of spindle-like structures between consecutive stable equilibria with respect to the pointwise ordering. Large amplitude periodic orbits—in the sense that they are not between two consecutive stable equilibria—are constructed for nonlinearities close to a step function. For some nonlinearities there are exactly two large amplitude periodic orbits. By describing the unstable sets of these periodic orbits, a complete picture is obtained about the global attractor outside the spindle-like structures.     

Exponential Ordering for Neutral Functional Differential Equations With Non-Autonomous Linear D-Operator

We study neutral functional differential equations with stable linear non-autonomous D-operator. The operator of convolution [^(D)]{\widehat D} transforms BU into BU. We show that, if D is stable, then [^(D)]{\widehat D} is invertible and, besides, [^(D)]{\widehat D} and [^(D)]^-1{\widehat D^{-1}} are uniformly continuous for the compact-open topology on bounded sets. We introduce a new transformed exponential order and, under convenient assumptions, we deduce the 1-covering property of minimal sets. These conclusions are applied to describe the amount of material in a class of compartmental systems extensively studied in the literature.     

On Asymptotic Decomposition of Solutions of Systems of Differential Equations with Slowly Varying Coefficients

We propose an algorithm for the reduction of a singularly perturbed system of differential equations whose characteristic equation has multiple roots to a system with simple roots.     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

C 1 Solutions for Semi-Implicit Systems of Differential Equations

The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where

${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x ₀) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function,

${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function.

The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ^?1.     

Positive Solutions of Linear Impulsive Differential Equations

The paper deals with the existence of positive (nonnegative) solutions of linear homogeneous impulsive differential equations. The main result is also applied to the investigation of a similar problem for higher-order linear homogeneous impulsive differential equations. All results are formulated in terms of coefficients of the equations. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 291–297, July–September, 2005.     

On the Floquet Multipliers of Periodic Solutions to Non-linear Functional Differential Equations

For periodic solutions to the autonomous delay differential equation

On Periodic Solutions of a System of Nonlinear Functional Partial Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of a system of nonlinear functional partial differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 154–158, April–June, 2005.     

On Asymptotic Properties of Continuously Differentiable Solutions of Quasilinear Functional Differential Equations

We establish new properties of C ¹ [τ(1), + ∞)-solutions of the quasilinear functional differential equation

Periodic Solutions of Second Order Differential Equations with Discontinuous Nonlinearities

We consider the periodic solutions of

Behavior of Solutions of Second-Order Differential Equations with Sublinear Damping

We investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation

Exponential Dichotomy and Mild Solutions of Nonautonomous Equations in Banach Spaces

We prove that the exponential dichotomy of a strongly continuous evolution family on a Banach space is equivalent to the existence and uniqueness of continuous bounded mild solutions of the corresponding inhomogeneous equation. This result addresses nonautonomous abstract Cauchy problems with unbounded coefficients. The technique used involves evolution semigroups. Some applications are given to evolution families on scales of Banach spaces arising in center manifolds theory.     

On Solutions of Some Non-Linear Differential Equations Arising in Newtonian and Non-Newtonian Fluids

Some steady as well as unsteady solutions of the equations of motion for an incompressible Newtonian and non-Newtonian (second-grade) fluids are obtained by applying different methods including the Lie symmetry group method. The flows considered are axially symmetric with the swirling motion, and the governing equations for second-grade fluid flow have been modeled. Expressions for streamlines, velocity and vorticity components are constructed explicitly in each case. Exact analytical solutions in second-grade fluid are obtained and compared with the corresponding viscous solutions.