Asymptotic behavior of solutions of a scalar delay difference equations with continuous time

Ljiljana Petković, Miodrag Petković

Schröder's methods of the first and second kind for solving a nonlinear equation $f(x)=0$, originally derived in 1870, are of great importance in the theory and practice of iteration processes. They were rediscovered several times and expressed in different forms during the last 140 years. In this paper we consider the question of possible link between these two families of iteration methods, posed as an open problem by Steven Smale in 1994. We show that the method of the first kind (often called the Schröder-Euler basic sequence) is obtained from the method of the second kind (often called the Schröder-König method) \[ S_r(x)=x-\frac{u(1+a_1(x)u+\dots+a_{r-3}(x)u^{r-3})}{1+b_1(x)u+\dots+b_{r-2}(x)u^{r-2}},\quad u=f(x)/f'(x), \] by the development of the reciprocal of denominator into the power series and constructing a polynomial in $u$ of degree $r$ by neglecting the terms containing the powers of u higher than $u^{r-1}$.