Delay-Dependent Stability Conditions for Non-autonomous Functional Differential Equations with Several Delays in a Banach Space

Abstract

Abstract Let Bj (t) (j = 1,..., m) and B(t, τ) (t ≥ 0, 0 ≤ τ ≤ 1) be bounded variable operators in a Banach space. We consider the equation u ′ ( t ) = ∑ k = 1 m B k ( t ) u ( t - h k ( t ) ) + ∫ 0 1 B ( t , τ ) u ( t - h 0 ( τ ) ) d τ ( t ≥ 0 ) , u'\left( t \right) = \sum\limits_{k = 1}^m {{B_k}\left( t \right)u\left( {t - {h_k}\left( t \right)} \right)} + \int\limits_0^1 {B\left( {t,\tau } \right)u\left( {t - {h_0}\left( \tau \right)} \right)d\tau \,\,\,\,\left( {t \ge 0} \right),} where hk (t) (t ≥ 0; k = 1, ..., m) and h 0(τ) are continuous nonnegative bounded functions. Explicit delay-dependent exponential stability conditions for that equation are established. Applications to integro-differential equations with delay are also discussed