Massera type theorem for almost automorphic solutions of functional differential equations of neutral type (Q819748)

The authors consider the functional differential equation of neutral type \[ \frac{dD(x_t)}{dt}=L(x_t)+f(t),\qquad t\geq\sigma, \qquad x_{\sigma}=\varphi,\tag{1} \] where \(D\) and \(L\) are bounded linear operators from \(C([-r,0];\mathbb R^n)\) into \(\mathbb R^n\), \(f:[\sigma,+\infty[\) \(\to \mathbb R^n\) is a continuous function, \(\varphi\in C([-r,0];\mathbb R^n)\), and \(x_t(\theta)=x(t+\theta)\) for \(\theta\in[-r,0]\). There is proved that, under some natural conditions on \(D\) and \(f\), the existence of a bounded solution to \((1)\) on \(\mathbb R^+\) implies the existence of an almost automorphic solution to \((1)\).