arXiv subjects

L. Berezansky

Publications and source records attributed to L. Berezansky.

9 recordsLinked to original sources

arXiv2011

Stability of Hahnfeldt Angiogenesis Models with Time Lags

Mathematical models of angiogenesis, pioneered by P. Hahnfeldt, are under study. To enrich the dynamics of three models, we introduced biologically motivated time-varying delays. All models under study belong to a special class of nonlinear nonautonomous systems with delays. Explicit conditions for the existence of positive global solutions and the equilibria solutions were obtained. Based on a notion of an M-matrix, new results are presented for the global stability of the system and were used to prove local stability of one model. For a local stability of a second model, the recent result for a Lienard-type second-order differential equation with delays was used. It was shown that models with delays produce a complex and nontrivial dynamics. Some open problems are presented for further studies.

math.DS↗

arXiv2006

Population Models With Delay in Dynamic Environment

We study the combined effects of periodically varying carrying capacity and survival rates on the fish population in the ocean (sea). We introduce the Getz type delay differential equation model with a control parameter which describes how fish are harvested. We will modify and extend harvesting model of an exploited fish population to include periodic and rotational harvesting rates. We study the existence of global solutions for the initial value problem, extinction and persistence conditions, and the existence of periodic solutions.

math.DS↗

arXiv2006

Stability of a Time-varying Fishing Model with Delay

We introduce a delay nonlinear differential equation model which describes how fish are harvested. In our previous studies we investigated the persistence of that equation and existence of a periodic solution for this equation. Here we study the stability (local and global) of the periodic solutions of that equation.

math.DS↗

arXiv1995

Impulsive Stabilization of Linear Delay Differential Equations

The paper is concerned with stabilization of a scalar delay differention equation $$ {\dot x}(t) - \sum_{k=1}^m A_k(t)x[h_k(t)] = 0,~t\geq 0,~ x(ξ)=φ(ξ), ξ<0, $$ by introducing impulses in certain moments of time $$ x(τ_j) = B_j x(τ_j -0), ~j=1,2, \dots ~. $$ Explicit stability results are presented both for the equation with positive coefficients and for the equation with $A_k$ being of arbitrary sign.

funct-an↗

arXiv1995

Oscillation of a Linear Delay Impulsive Differential Equation

The main result of the paper is that the oscillation (non-oscillation) of the impulsive delay differential equation $\dot {x}(t)+\sum_{k=1}^m A_k(t)x[h_k(t)]=0,~~t\geq 0$, $x(τ_j)=B_jx(τ_j-0), \lim τ_j = \infty$ is equivalent to the oscillation (non-oscillation) of the equation without impulses $\dot {x}(t)=\sum_{k=1}^m A_k(t) \prod_{h_k(t)<τ_j\leq t} B_j^{-1}x[h_k(t)]=0, t \geq 0$. Explicit oscillation results are presented.

funct-an↗

arXiv1994

On Integrable Solutions of Impulsive Delay Differential Equations

The connection of function properties of solutions with exponential stability of linear impulsive differential equation $$\dot{x} (t) - \sum_{k=1}^m {A_k (t) x[h_k(t)]} = r(t),~ t \geq 0, x(ξ) = φ(ξ),~ ξ< 0,$$ $$x(τ_j) = B_j x(τ_j - 0) , ~j=1,2, \dots .$$ The explicit stability results and sufficient conditions for existence of integrable solutions are presented.

funct-an↗

arXiv1994

Boundedness and Stability of Impulsively Perturbed Delay Differential Equations

Suppose any solution of a linear impulsive delay differential equation $$ \dot{x} (t) + \sum_{i=1}^m A_i (t) x[h_i (t)] = 0,~t \geq 0, x(s) = 0, s < 0, $$ $$ x(τ_j +0) = B_j x(τ_j -0) + α_j, ~j=1,2, ... ,$$ is bounded for any bounded sequence $\{ α_i \}$. The conditions ensuring exponential stability of this equation are presented. The behavior of solutions of the non-homogeneous equation is analyzed.

funct-an↗

arXiv1993

Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

Consider a linear impulsive equation in a Banach space $$\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0,$$ $$x(τ_i +0)= B_i x(τ_i -0) + α_i,$$ with $\lim_{i \rightarrow \infty} τ_i = \infty $. Suppose each solution of the corresponding semi-homogeneous equation $$\dot{x}(t)+A(t)x(t) = 0,$$ (2) is bounded for any bounded sequence $\{ α_i \}$. The conditions are determined ensuring (a) the solution of the corresponding homogeneous equation has an exponential estimate; (b) each solution of (1),(2) is bounded on the half-line for any bounded $f$ and bounded sequence $\{ α_i \}$ ; (c) $\lim_{t \rightarrow \infty}x(t)=0$ for any $f, α_i$ tending to zero; (d) exponential estimate of $f$ implies a similar estimate for $x$.

funct-an↗

arXiv1993

Exponential Stability of Linear Delay Impulsive Differential Equations

For ordinary differential equations and functional differential equations the following result is well known. Suppose any solution is bounded on the half-line for each bounded on the half-line right-hand side. Then under certain conditions the equations is exponentially stable. We prove the same result for a delay differential equation $$ \dot{x}(t) + \sum_{i=1}^k A_i (t)x[h_i(t)] = f(t), $$ with impulses $$ x(τ_i + 0) = B_i x(τ_i - 0) $$ at fixed moments $τ_i$. The proof is based on a solution representation formula obtained here.

funct-an↗