Let $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the $p$-adic Nevanlinna theory to functional equations of the form $g=R\circ f$, where $R\in K\left(x\right)$, $f,g$ are meromorphic functions in $K$, or in an “open disk”, $g$ satisfying conditions on the order of its zeros and poles. In various cases we show that $f$ and $g$ must be constant when they are meromorphic in all $K$, or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular,...

We discuss the asymptotic behaviour of all solutions of the functional differential equation $$y^{\text{'}}\left(x\right)=\sum _{i=1}^m{a}_i\left(x\right)y\left({\tau }_i\left(x\right)\right)+b\left(x\right)y\left(x\right),$$ where $b\left(x\right)<0$. The asymptotic bounds are given in terms of a solution of the functional nondifferential equation $$\sum _{i=1}^m\left|a_i\left(x\right)\right|\omega \left({\tau }_i\left(x\right)\right)+b\left(x\right)\omega \left(x\right)=0.$$