Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

Abstract

For the equation

$$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$

whereh ^o₀ =0,h ¹₀ =0 (t) ≡ 0,h ^o_j = const > 0,h ^j₁ (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation

$$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$

connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.