Searcharxiv⌕ Search

arXiv subjects

Elena Braverman

Publications and source records attributed to Elena Braverman.

60 records · Page 4Linked to original sources
arXiv2009

New Stability Conditions for Linear Difference Equations using Bohl-Perron Type Theorems

The Bohl-Perron result on exponential dichotomy for a linear difference equation $$ x(n+1)-x(n) + \sum_{l=1}^m a_l(n)x(h_l(n))=0, h_l(n)\leq n, $$ states (under some natural conditions) that if all solutions of the non-homogeneous equation with a bounded right hand side are bounded, then the relevant homogeneous equation is exponentially stable. According to its corollary, if a given equation is {\em close} to an exponentially stable comparison equation (the norm of some operator is less than one), then the considered equation is exponentially stable. For a difference equation with several variable delays and coefficients we obtain new exponential stability tests using the above results, representation of solutions and comparison equations with a positive fundamental function.

math.DS↗
arXiv2009

Absolute and Delay-Dependent Stability of Equations with a Distributed Delay: a Bridge from Nonlinear Differential to Difference Equations

We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Such models frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delay-independent attracting set and also demonstrate that the equation with a distributed delay is stable for small enough delays.

math.DS↗
arXiv2008

Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term

In this paper we consider the linear ordinary equation of the second order $$ L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the corresponding homogeneous equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0. \eqno{(2)} $$ Note that $[α,β]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis - nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxis and apply them to propose new results on asymptotic properties and the exponential stability of the second order equation (2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of equation (2).

math.DS↗