On the unique solvability of a nonautonomous functional-differential equation of neutral type (Q2759328)

The author considers the functional-differential equation of neutral type NEWLINE\[NEWLINE\frac{d}{dt}[x(t)-G(t,x_t)]=F(t,x_t),NEWLINE\]NEWLINE with \(F:\Gamma\to\mathbb{R}^n,\) \(G:\Gamma\to\mathbb{R}^n,\) \(\Gamma\) an open set of \(\mathbb{R}^+\times C_{[-h, 0]}\). Under the Lipschitz condition \(|G(t,\varphi)-G(t,\psi)|\leq \lambda \|\varphi-\psi\|\), \(\lambda\in(0,1),\) an existence theorem is proved by means of the principle of \(L\)-contraction mapping, where \(L\) is the Kuratowsky measure of noncompactness in a Banach space. The continuous dependence of the solution on the initial value under the Lipschitz condition \(|G(t_2, \varphi_2)-G(t_1,\varphi_1)|\leq N|t_2-t_1 |+ \lambda\|\varphi_2-\varphi_1\|,\) \(\lambda\in (0, 1)\), is investigated. The uniqueness of the solution is proved, and conditions for the existence of a solution to the problem on a maximal interval are derived.