Structural operators and eigenmanifold decomposition for functional differential equations in Hilbert spaces
The authors study spectral properties of linear retarded functional differential equations \[ {du(t)\over dt}= A_0u(t)+ A_1u(t- h)+ \int^0_{-h} a(s)A_2u(t+ s)ds+ f(t)\quad\text{a.e. }t\geq 0, \] \[ u(0)= g^0,\quad u(s)= g^1(s)\quad\text{a.e. }s\in [-h, 0), \] and its transposed equation \[ {dv(t)\over dt}= A^*_0v(t)+ A^*_1 v(t- h)+ \int^0_{-h} a(s) A^*_2v(t+ s)ds+ h(t)\quad\text{a.e. }t\geq 0, \] \[ v(0)= \varphi^0,\quad v(s)= \varphi^1(s)\quad\text{a.e. }s\in [-h,0), \] in Hilbert spaces with the emphasis on their relations to structural operators. The results extend those by \textit{S. Nakagiri} [Osaka J. Math. 25, No. 2, 353-398 (1988; Zbl 0713.34069)] to the above equations considered by \textit{J.-M. Jeong}, \textit{S. Nakagiri} and \textit{H. Tanabe} [Osaka J. Math. 30, No. 3, 365-395 (1993; Zbl 0818.47039)].
linear retarded functional differential equations
scientific article; zbMATH DE number 1005653