On semigroups of nonlinear operators and the solution of the functional differential equations (Q2720150)

The following nonlinear autonomous differential equation is studied NEWLINE\[NEWLINEx'(t)= F(x_t),\quad x_0= \varphi\in C([-r,0], \mathbb{R}^N),\tag{1}NEWLINE\]NEWLINE where \(x: [-r,T]\to\mathbb{R}^N\), \(0< r< \infty\) is the delay and \(x_t\in C([-r,0],\mathbb{R}^N)\) is the history of \(x\) at time \(t\) defined as usually by \(x_t(s)= x(t+ s)\) for \(s\in [-r,0]\). Moreover, \(F: C([-r,0],\mathbb{R}^N)\to \mathbb{R}^N\) is continuous and satisfies some extra conditions. Here, a few results concerning the existence, uniqueness and regularity of solutions to equation (1) are proved.