Existence results for partial neutral functional differential equations with unbounded delay (Q1269079)

The authors prove existence of mild solutions of the Cauchy problem \[ {d\over dt} (x(t)+ F(t, x_t))= Ax(t)+ G(t, x_t),\quad t\geq\sigma,\quad x(\sigma)= \varphi\in \Omega,\tag{1} \] where \(\Omega\) is an open subset of the phase space, \(F,G: [0,a]\times \Omega\mapsto X\) are continuous functions, \(0\leq\sigma< a\) and \(A\) is the infinitesimal generator of an analytic semigroup \(T(.)\) of bounded linear operators on \(X\). In the case \(\sigma= 0\) the existence of strong solutions of the Cauchy problem (1) is proved.