zbMATH — the first resource for mathematics

Fundamental solutions for linear retarded functional differential equations in Banach space. (English) Zbl 0771.34060
Let \(X\) be a Banach space, and let \(A:D(A)\subset X\to X\) denote the infinitesimal generator of an analytic semigroup. In addition let \(A_ 1\) and \(A_ 2\) be closed linear operators in \(X\) with domains containing \(D(A)\). The author constructs a fundamental solution to the linear retarded functional differential equation (E) \(u'(t)=Au(t)+A_ 1u(t- r)+\int^ 0_{-r}a(s)A_ 2u(t+s)ds=0\) \((t\geq 0)\) where \(r>0\) and \(a:[- r,0]\to\mathbb{R}\) is Hölder continuous. The relationship between the fundamental solution and the solvability of an initial value problem for a nonhomogeneous version of (E) is also discussed.

34K30 Functional-differential equations in abstract spaces
34K05 General theory of functional-differential equations
34G10 Linear differential equations in abstract spaces
45J05 Integro-ordinary differential equations