Local invertible analytic solution of a functional differential equation with deviating arguments depending on the state derivative (Q860607)

The paper deals with the functional differential equation with deviating arguments depending on the state derivative \[ x'(z) = 1 / x(az + b x'(z)), \tag{1} \] where \(a\) and \(b\) are complex numbers. Reducing (1) by change of variables \(az + b x'(z) = g(\alpha g^{-1}(z))\) to some auxiliary equation with respect to \(g\), the authors prove the existence of local analytic solutions to (1) in the complex field. The complex parameter \(\alpha\) is located inside the unit circle \(S^1\) or \(\alpha \in S^1\) and satisfies some additional non-resonance or resonance conditions.