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

R. G. Aliev

Matematičeskie zametki, Tome 13 (1973) no. 6, pp. 829-838.

Résumé

For the equation $$Lu=\frac1i\frac{du}{dt}-\sum_{j=0}^mA_ju(t-h_j^0-h_j^1(t))=f(t),$$ where $h_0^0=0$, $h_0^1\equiv0$, $h_j^1(t)$, $j=1,\dots,m$ are nonnegative continuously differentiable functions in $[0,\infty)$, $A_j$ are bounded linear operators, under conditions on the resolvent and on the right hand side $f(t)$, we have obtained an asymptotic formula for any solution $u(t)$ from $L_2$ in terms of the exponential solutions $u_k(t)$, $k=1,\dots,n$, of the equation $$\frac1i\frac{du}{dt}-A_0u-\sum_{j=1}^mA_ju(t-h_j^0)=0,$$ connected with the poles $\lambda_k$, $1,\dots,n$, of the resolvent $R_\lambda$ in a certain strip.