The pantograph equation in the complex plane (Q1378398)

The subject matter is focused on two functional differential equations. First of them is the pantograph equation with involution on the complex plane: \[ y'(z)=\sum_{k=0}^{m-1} \left[ a_k y(\omega^k z) + b_k y(r \omega^k z) + c_k y'(r \omega^k z) \right] , \] where \(a_k, b_k, c_k \in \mathbb{C}, k= 0, 1, \dots , m-1,\) are given, \(r \in (0,1)\), and \(\omega\) is the primitive root of unity. The second one is the pantograph equation of the second type: \[ y(z)= \sum_{j=1}^{l} \sum_{k=1}^{n} a_{j,k} y^{(k)} (\omega_j z), \] \(a_{j,k}, \omega_j \in \mathbb{C},\) supplemented by appropriate initial conditions at the origin. The results concern the existence and uniqueness of solutions and their asymptotic behavior.