We study local analytic solutions $f$ of the generalized Dhombres
equation $f(xf(x))=\varphi(f(x))$ with $f(0)=0$ in the complex domain.
We give an existence result, describe the structure of the set of all
local analytic solutions and solve the converse problem, i.e., we
characterize those local analytic functions which are solutions of a
generalized Dhombres equation. Connections of generalized Dhombers
equations with linear functional equations and generalized B\"ottcher
equations are used. Furthermore, we establish relations of generalized
Dhombres equations with Briont-Bouquet differential equations and with
iteration groups. Finally, as an application of B\"ottcher functions,
we describe the connections between two generalized Dhombres equations
and the representations of their solutions as infinite products.
Mathematics Subject Classification. Primary 30D05,
34M25, 39B12, 39B32; Secondary 30B10.