Analytic solutions of an iterative functional differential equation (Q5953083)

Several mathematicians have been focused their study on functional equations with state dependent delay.NEWLINENEWLINENEWLINEHere, the authors develop a new method in order to obtain analytic solutions to the following iterative functional-differential equation NEWLINE\[NEWLINEx'(x^{[r]}(z))= c_0 z+ c_1 x(z)+ c_2 x(x(z))+\cdots+ c_m x^{[m]}(z),NEWLINE\]NEWLINE where \(r\) and \(m\) are nonnegative integers and \(x^{[0]}(z)=z,x^{[1]}= x(z)\), \(x^{[2]}(z)= x(x(z))\), etc. are the iterates of the function \(x(z)\) and \(\sum^m_{j=0} c_j\neq 0\).NEWLINENEWLINENEWLINEThe used technique is a rather simple one and also a very accurate one. The authors search for an analytic solution \(y(z)\) to the initial value problem NEWLINE\[NEWLINE\alpha y'(\alpha^{r+1} z)= y'(\alpha^r z) \sum^m_{j=0} c_j y(\alpha^j z),\quad y(0)= \alpha\Biggl/\sum^m_{j=0} c_j.NEWLINE\]NEWLINE After that, they show that \(x(z)= y(\alpha y^{-1}(z))\) is an analytic solution to the studied equation in a neighborhood of \(\alpha/\sum^m_{j=0} c_j\).NEWLINENEWLINENEWLINEUsing four lemmas and a theorem, they finally show how they succeeded to derive an explicit power series solution.