We obtain the maximum principles for the first-order neutral functional differential equation Mxt ≡ x t − Sx are linear continuous operators, A and B are positive operators, C 0,ω is the space of continuous functions, and L ∞ 0,ω is the space of essentially bounded functions defined on 0, ω. New tests on positivity of the Cauchy function and its derivative… (More)
For the delay differential equations ¨ x(t) + a(t) ˙ x(g(t)) + b(t)x(h(t)) = 0, g(t) ≤ t, h(t) ≤ t, and¨x(t) + a(t) ˙ x(t) + b(t)x(t) + a 1 (t) ˙ x(g(t)) + b 1 (t)x(h(t)) = 0 explicit exponential stability conditions are obtained.
A boundary value problem is considered for an N-th order functional diierential equation with impulses. It is reduced to the same boundary value problem for another equation of the same order without impulses. The reduction is based on constructing of an isomorphism between the space of the functions which are piece-wise absolutely continuous up to the (N… (More)
In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet… (More)
The state-dependent delay differential equation ˙ x(t) + m i=1 p i (t)x t − (H i x)(t) = f (t), t ∈ [0, ∞), x(ξ) = ϕ(ξ), ξ < 0, with state-dependent impulses is under consideration. Sufficient conditions for positivity of solutions to the Cauchy and periodic problems as well as conditions for positivity of solutions to the problem with a condition on the… (More)
In this paper, distribution of zeros of solutions to functional equations is studied. It will be demonstrated that oscillation properties of functional equations are determined by the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Distances between zeros of solutions are estimated. On this basis, zones of… (More)
We consider the following second order differential equation with delay (Lx)(t) ≡ x (t) + p ∑ j=1 a j (t)x (t − τ j (t)) + p ∑ j=1 b j (t)x(t − θ j (t)) = f (t), t ∈ [0, ω] x(t k) = γ k x(t k − 0), x (t k) = δ k x In this paper we find sufficient conditions of positivity of Green's functions for this impulsive equation coupled with two-point… (More)