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

Derfel, G.; Iserles, A.

J. Math. Anal. Appl. 213, No. 1, 117-132 (1997).

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)

MSC:

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

Keywords:

functional differential equations; pantograph equation; existence; uniqueness; asymptotic behavior; stability

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