Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field

In earlier papers the author studied some classes of equations with Carlitz derivatives for ${\mathbb{F}}_q$-linear functions, which are the natural function field counterparts of linear ordinary differential equations. Here we consider equations containing self-compositions $u\circ u\circ \mathrm{.}\mathrm{.}\mathrm{.}\circ u$ of the unknown function. As an algebraic background, imbeddings of the composition ring of ${\mathbb{F}}_q$-linear holomorphic functions into skew fields are considered.