  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 $u\circ u\circ ... \circ u$ 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.
