On the Asymptotic Properties of Solutions of Linear Functional Differential Equations with Constant Coefficients and Linearly Transformed Argument

We establish new properties of C ¹(0, +)-solutions of the linear functional differential equation in the neighborhood of the singular point t = +.     

Asymptotic Properties of Global Solutions of Systems of Functional Differential Equations of Neutral Type with Nonlinear Deviations of an Argument

For a system of functional differential equations of the neutral type with nonlinear deviations of an argument dependent on an unknown function, we establish sufficient conditions for the existence of a solution continuously differentiable and bounded for t and study its properties.     

On the Asymptotic Behavior of Solutions of a System of Nonlinear Differential Equations

We investigate the asymptotic behavior of a system of nonlinear differential equations of a special form at infinity. We also propose a method for the reduction of more general systems of nonlinear differential equations to this form, which enables one to study their asymptotic properties.     

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.     

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

For periodic solutions to the autonomous delay differential equation

Generic Properties of Solutions to Partial Differential Equations

(Accepted July 1, 1996)     

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.     

Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations

Initial value problems for quasilinear parabolic equations having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon measures of the initial data. We call such solutions measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the measure-valued solution of problem (P) is, in fact, function-valued.     

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.     

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.     

Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable,Hyperbolic Case

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}(t)&=f(\theta ,x(t),\epsilon x(t-\tau (x(t))))\\ \dot{\theta }(t)&=\omega . \end{aligned} \right. \end{aligned}$$

Under the assumption of exponential dichotomies for the \(\epsilon =0\) case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc.

     

Impulsive Semilinear Functional Differential Equations

In this paper the Leray–Schauder nonlinear alternative combined with semigroup theory is used to investigate the existence of mild solutions for first-order impulsive semilinear functional differential equations in Banach spaces.     

Asymptotic Stability of Differential Equations on Convex Sets

We show a sufficient condition for the asymptotic stability of solutions of an abstract evolutionary equation with the right-hand side defined on a convex set. This condition is applied to a generalized version of the Tjon-Wu equation and to the Boltzmann-Kac equation.     

Optimal Existence Theory for Single and Multiple Positive Periodic Solutions of Functional Differential Equations

This paper deals with a new optimal existence theory for single and multiple positive periodic solutions of functional differential equations using a fixed-point theorem in cones. We illustrate our theory by examining several biomathematical models. The paper improves and extends previous results in the literature.     

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.     

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.     

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.     

Neutral Mixed Type Functional Differential Equations

We extend the existing Fredholm theory for mixed type functional differential equations developed by Mallet-Paret (J Dyn Differ Equ 11:1–47, 1999) to the case of implicitly defined mixed type functional differential equations. We present analogous results for the Fredholm alternative theorem, the cocycle property, and spectral flow property. In particular, we apply the theory to examples, one of which arises from modeling signal propagation in nerve fibers, and show the existence of traveling wave solutions via local continuation.     

Periodic Solutions of Delay Differential Equations with a Small Parameter: Existence,Stability and Asymptotic Expansion

We consider a scalar delay differential equation with a small parameter, and employ Walthers method to obtain a result on the existence and stability of a slowly oscillatory periodic solution that represents a refinement of the estimate for the Lipschitz constant of a returning map. We also develop a matching method and obtain asymptotic expansions of the slowly oscillatory periodic solution and its minimal period.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthdayAMS subject classifications: 34K15; 34K20; 34C25.