On exponential stability of linear delay equations with oscillatory coefficients and kernels

Abstract

. New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation where h k ( t ) ≤ t , g ( t ) ≤ t , a k ( · ) and the kernel K ( · , · ) are oscillatory and, generally, discontinuous functions. The proofs are based on establish-ing boundedness of solutions and later using the exponential dichotomy for linear equations stating that either the homogeneous equation is exponentially stable or a non-homogeneous equation has an unbounded solution for some bounded right-hand side. Explicit tests are applied to models of population dynamics, such as controlled Hutchinson and Mackey-Glass equations. The results are illustrated with numerical examples, and connection to known tests is discussed.