Integraded semigroups and integrodifferential equations
We consider the integral equation \[ (1)\quad du/dt=A(u(t))+\int^{t}_{0}B(t-s)u(s)ds+f(t) \] and, as a corollary, the abstract Cauchy problem \[ (2)\quad v^{(n+1)}(t)=A(v^{(n)}(t))+\sum^{n}_{j=1}B_ jv^{(n- j)}(t)+g(t), \] where the domain of A is contained in a domain of B(r) (r\(\geq 0)\) and \(B_ j\) (1\(\leq j\leq n)\), all operators are linear, on a Banach space, and A generates a k-times integrated semigroup, for sme \(k\in {\mathbb{N}}\cup \{0\}\). We use C-semigroups to give sufficient conditions for (1) and (2) to be well-posed, and construct the solutions.
scientific article; zbMATH DE number 4117279
scientific article