On fundamental solution of differential equation with time delay in Banach space
This paper is concerned with the construction of the fundamental solution in the sense of \textit{S. Nakagiri}, J. Differ. Equations 41, 349-368 (1981; Zbl 0441.35068) to the linear differential equation with time delay in a Banach space: \[ du(t)/dt=Au(t)+A_ 1u(t-r)+\int^{0}_{- r}a(s)A_ 2u(t+s)ds \] of the type investigated by \textit{G. Di Blasio}, \textit{K. Kunisch} and \textit{E. Sinestrari} [J. Math. Anal. Appl. Vol. 102, 38-57 (1984; Zbl 0538.45007), Israel J. Math. 50, 231-263 (1985; Zbl 0594.45014), and \textit{E. Sinestrari}, Math. Z. 186, 223-246 (1984; Zbl 0521.35044)]. Here, A is the infinitesimal generator of an analytic semigroup such that D(A) is dense, \(A_ 1\) and \(A_ 2\) are operators such that \(D(A_ 1)\supset D(A)\), \(D(A_ 2)\supset D(A)\) and a(s) is a real valued Hölder continuous function in the interval [-h,0]. It is shown that the fundamental solution W(t) exists and satisfies \(\| dW(t)/dt\| \leq C_ n/(t-nr),\quad \| AW(t)\| \leq C_ n/(t-nr)\) in the interval \((nr,(n+1)r]\).
scientific article; zbMATH DE number 4075596