The pantograph equation in the complex plane. (English) Zbl 0891.34072

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.
Reviewer: S.Yanchuk (Kyïv)


34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34M99 Ordinary differential equations in the complex domain
Full Text: DOI


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