Inhomogeneous degenerate Sobolev type equations with delay
The aim of the paper is to study the existence of a unique mild solution to the delayed Sobolev type operator equation \[ L\dot{u}(t) = Mu(t) + \Phi u_t + f(t), \qquad t\in [0, \infty), \] where \(L: X\to Y\) is a continuous linear operator between Banach spaces \(X\) and \(Y\), \(M : D(M)\to Y\) is a linear, closed and densely defined operator on \(X\), and the delay operator \(\Phi : {\mathcal C}([-r, 0];X)\to Y\) is linear and continuous. For this, the authors use the methods of the theory of degenerate semigroups which are developed for studying Sobolev type equations, along with a semigroup approach to deal with delay equations.
scientific article; zbMATH DE number 6105684