On the structure of the set of continuously differentiable solutions for systems of functional-differential equations in a neighborhood of a singular point (Q2459771)

The author investigates nonlinear functional-differential equations of the form \[ x'(qt)= x'(t)+ F(t, x(t),x(qt), x'(t)), \] where \(t\geq 0\), \(q\) a constant, \(F: \mathbb{R}^+\times\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}^n\). It is shown, under certain conditions on \(F\), that a continuously differentiable solution \(x(t)\), \(0\leq t\leq T\), can be represented as \[ x(t)= \omega(t)+ o(t),\quad t\downarrow 0, \] where \(\omega(qt)= q\omega(t)\) and \(\omega\) is continuously differentiable. It is also shown, under the same assumptions on \(F\), that for any continuously differentiable \(\omega(t)\) such that \(\omega(qt)= q\omega(t)\) there exists a unique continuously differentiable solution \(x(t)\) of the equation such that \(x(t)= \omega(t)+ o(t)\), \(t\downarrow 0\).