\(\mathcal{BC}\)-total stability and almost periodicity for some partial functional differential equations with infinite delay (Q2498181)

The authors deal with the existence of an almost periodic solution of the following partial functional differential equation with infinite delay \[ {du\over dt}= Au(t)+ L(t, u_t)+ f(t),\quad t\in\mathbb{R}, \] where \(A: D(A)\subset X\to X\) is a nondensely defined linear operator on a Banach space \(X\). They prove that if the null solution of the homogeneous equation is in some sense stable (more precisely \({\mathcal B}{\mathcal C}\)-total stable) then the nonhomogeneous equation has an almost periodic solution.