# 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.

##### MSC:
 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