Analytic and asymptotic properties of solutions to a non-homogeneous functional differential equation

The paper investigates the analytic properties and asymptotic behaviors of solutions to the non-homogeneous functional differential equation y ′ (x)=ay(qx)+by(x)+1 1+x, where q is a constant satisfying 0<q<1, and a≠0, b≠0 are complex numbers, by following methods developed by J. P. Ramis for singular differential equations and q-difference equations. First, we study the existence and analytic properties of solutions expressed as series expansions around both zero and infinity. Next, by considering the equation as a perturbation of a differential equation, we derive a solution represented as a sum of integrals containing an infinite number of singularities. Finally, we establish a connection formula between the integral-sum solution and the series expansion at zero, which is crucial for determining the asymptotic behavior of the series solution as x→∞, given the initial conditions.