Structural operators for linear delay-differential equations in Hilbert space
Let \[ (1)\quad du(t)/dt=A_ 0u(t)+A_ 1u(t-h)+\int^{0}_{- h}a(s)A_ 2u(t+s)ds \] be a linear delay-differential equation in a Hilbert space H. Let V be another Hilbert space such that \(V\subset H\subset V^*\). \(A_ 0\) is a linear operator associated with a strongly elliptic sesquilinear form defined on \(V\times V\). \(A_ i\), \(i=1,2\), are bounded linear operators from V to \(V^*\) such that \(A_ iA_ 0^{- 1}\), \(i=1,2\), map H into itself. If we apply the result of \textit{G. Di Blasio}, \textit{K. Kunisch} and \textit{E. Sinestrari} [J. Math. Anal. Appl. 102, 38-57 (1984; Zbl 0538.45007)] to (1) considering it as an equation in \(V^*\), then the associated solution semigroup S(t) is a \(C_ 0\)- semigroup in the space \(Z=H\times L^ 2(-h,0;V)\). Similarly, the solution semigroup \(S_ T(t)\) is defined for the equation (1) with \(A_ i\), \(i=0,1,2\), replaced by their adjoints. It is shown that \(FS(t)=S^*_ T(t)F\) and \(S^*(t)F^*=F^*S_ T(t)\) holds for \(t>0\), where F: \(V\to V^*\) is the structural operator for (1).
scientific article; zbMATH DE number 4094142