  In this paper, we study Hom-Lie bialgebras by a new notion of the dual
representation of a representation of a Hom-Lie algebra. Motivated by the
essential connection between Lie bialgebras and Manin triples, we introduce the
notion of a Hom-Lie bialgebra with emphasis on its compatibility with a Manin
triple of Hom-Lie algebras associated to a nondegenerate symmetric bilinear
form satisfying a new invariance condition. With this notion, coboundary
Hom-Lie bialgebras can be studied without a skew-symmetric condition of
$r\in\mathfrak{g}\otimes \mathfrak{g}$, naturally leading to the classical
Hom-Yang-Baxter equation whose solutions are used to construct coboundary
Hom-Lie bialgebras. In particular, they are used to obtain a canonical Hom-Lie
bialgebra structure on the double space of a Hom-Lie bialgebra. We also derive
solutions of the classical Hom-Yang-Baxter equation from
$\mathcal{O}$-operators and Hom-left-symmetric algebras.
