Uniform exponential stability of first-order dynamic equations with several delays

This paper is concerned with the uniform exponential stability of ordinary and delay dynamic equations. After revealing the equivalence between various types of uniform exponential stability definitions on time scales with bounded graininess, and demonstrating their relation when the graininess is arbitrary, we confine our attention to the uniform exponential stability of ordinary dynamic equations. We introduce and prove the Bohl–Perron criterion for delay dynamic equations: if for any bounded right-hand side, the solution of the delay dynamic equation with bounded coefficients and delays is bounded, then the trivial solution of the equation is uniformly exponentially stable. We also obtain some corollaries of this criterion. Based on these results, explicit exponential stability tests are derived for delay dynamic equations with nonnegative coefficients, which are illustrated with an example on a nonstandard time scale.