Analytic solutions of a second-order nonautonomous iterative functional differential equation. (English) Zbl 1083.34060

The second-order iterative functional-differential equation \[ x''(t)=f\left[\sum_{j=0}^{m}c_jx^{j}(z)\right]+G(z)\tag{1} \] is considered, where \(f\) and \(G\) are given analytic functions on the complex domain \(| z | <\sigma, x^{j}(z)=x(x^{j-1})(z).\) By the Schröder transformation \(x(z)=y(\alpha y^{-1}(z))\), equation (1) is reduced to an auxiliary equation in which the iterates of the unknown function are not involved and the existence of its local analytic solutions are proved for various cases with respect to \(\alpha.\) By this way, theorems on the existence of analytic solutions are proved for equation (1).


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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