  This paper studies operator forms of the nonhomogeneous associative classical
Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the
classical Yang-Baxter equation and associative Yang-Baxter equation that can be
tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on
Rota-Baxter Lie algebras and $\mathcal{O}$-operators. In general, solutions of
the nhacYBe are characterized in terms of generalized $\mathcal{O}$-operators.
The characterization can be given by the classical $\mathcal{O}$-operators
precisely when the solutions satisfy an invariant condition. When the invariant
condition is compatible with a Frobenius algebra, such solutions have close
relationships with Rota-Baxter operators on the Frobenius algebra. In general,
solutions of the nhacYBe can be produced from Rota-Baxter operators, and then
from $\mathcal{O}$-operators when the solutions are taken in semi-direct
product algebras. In the other direction, Rota-Baxter operators can be obtained
from solutions of the nhacYBe in unitizations of algebras. Finally a
classifications of solutions of the nhacYBe satisfying the mentioned invariant
condition in all unital complex algebras of dimensions two and three are
obtained. All these solutions are shown to come from Rota-Baxter operators.
