Stability conditions for scalar delay differential equations with a nondelay term

The problem considered in the paper is exponential stability of linear equations and global attractivity of nonlinear non-autonomous equations which include a non-delay term and one or more delayed terms. First, we demonstrate that introducing a non-delay term with a non-negative coefficient can destroy stability of the delay equation. Next, sufficient exponential stability conditions for linear equations with concentrated or distributed delays and global attractivity conditions for nonlinear equations are obtained. The nonlinear results are applied tothe Mackey-Glass model of respiratory dynamics.Comment: 12 pages, 2 figures, published in 2015 in Applied Mathematics and Computatio

New Global Exponential Stability Criteria for Nonlinear Delay Differential Systems with Applications to BAM Neural Networks

We consider a nonlinear non-autonomous system with time-varying delays $$\dot{x_i}(t)=-a_i(t)x_{i}(h_i(t))+\sum_{j=1}^mF_{ij}(t,x_j(g_{ij}(t)))$$ which has a large number of applications in the theory of artificial neural networks. Via the M-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed by the system's parameters, is an $M$-matrix. Comparison with the tests obtained by K. Gopalsamy (2007) and B. Liu (2013) for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper.Comment: 15 page

On stability of linear neutral differential equations with variable delays

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$\dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0,$$ where $$|a(t)|<1,~ b(t)\geq 0, ~h(t)\leq t, ~g(t)\leq t,$$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.Comment: 28 pages. to appear in Czechoslovak Mathematical Journal, published onlin

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.Comment: 23 pages, 4 figure