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
Highlights
This paper is devoted to a class of linear nonautonomous functional di erential equations in a Banach space with several variable delays, whose coe cients are bounded operators
That method has been extended to functional di erential equations in a Hilbert space, cf. [10, 11, 14, 16, 18] and references given therein
To the best of our knowledge, the stability of nonautonomous equations in a Banach space with several delays are not investigated in the available literature
Summary
This paper is devoted to a class of linear nonautonomous functional di erential equations in a Banach space with several variable delays, whose coe cients are bounded operators. Such equations include integrodi erential equations with delay. To the best of our knowledge, the stability of nonautonomous equations in a Banach space with several delays are not investigated in the available literature. We obtain delay-dependent exponential stability conditions for nonautonomous functionaldi erential equations in a Banach space with several delays. The literature on the delay-dependent stability criteria is rather rich, but mainly equations in a nite dimensional space are considered, cf [7, 8, 15].
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