Sergei Trofimchuk

Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima

We study the stability of periodic solutions of the scalar delay differential equation where f ( t ) is a periodic forcing term and δ, p ∈R. We study stability in the first approximation showing that the non-smooth equation (*) can be linearized along some ‘non-singular’ periodic solutions. Then the corresponding variational equation together with the Krasnoselskij index are used to prove the existence of multiple periodic solutions to (*). Finally, we apply a generalization of Halanays inequality to establish conditions for global attractivity in equations with maxima.

Mackey-Glass type delay differential equations near the boundary of absolute stability

For an equation x � (t) =− x(t) + ζf (x(t − h)), x ∈ R, f � (0) =− 1, ζ> 0, with C 3 nonlinearity f which has a negative Schwarzian derivative and satisfies xf (x) 0a nd h(ζ − 1) 1/8 are less than some constant (independent on h, ζ ). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey–Glass type delay differential equations.  2002 Elsevier Science (USA). All rights reserved.

Solvability of a multi-point boundary value problem of Neumann type

Let f:[0,1]×ℝ2→ℝ be a function satisfying Carathéodorys conditions and e(t)∈L1[0,1]. Let ξi∈(0,1),ai∈ℝ,i=1,2,…,m−2,0<ξ1<ξ2<⋯<ξm−2<1 be given. This paper is concerned with the problem of existence of a solution for the m-point boundary value problem x″(t)=f(t,x(t),x′(t))

Lp-Perturbations of Invariant Subbundles for Linear Systems

We use Riccatis equations and the ordinary and exponential dichotomies to get simple recurrent formulae for the asymptotic integration of linear systems subjected to Lp-perturbations with arbitrary p ≥ 1. Moreover, we establish conditions which are necessary and sufficient for the persistence of one-dimensional invariant subbundles for the linear system under Lp-perturbations. In this way, we prove the sharp nature of the well known Levinson and Hartman–Wintner asymptotic theorems.

Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.