Iterative construction of Hermitian-Einstein metrics on stable bundles

Let $E$ be a stable holomorphic vector bundle over a compact Kähler (or Gauduchon) manifold $(M,\omega_g)$. We show that for any real number $\mu>0$ and any initial Hermitian metric $h_0$ on $E$, there exists a unique iteration sequence $\{h_m\}$ satisfying
$$
\Lambda_{\omega_g}\left(\sqrt{-1}R^{h_{m+1}}\right)
=(\lambda_E-\mu)h_{m+1}+\mu h_m,
$$
and $\{h_m\}$ converges smoothly to a Hermitian-Einstein metric $h_\infty$ on $E$ satisfying
$$
\Lambda_{\omega_g}\left(\sqrt{-1}R^{h_{\infty}}\right)
=\lambda_Eh_\infty,
$$
where $\lambda_E\in \mathbb R$ is the stability constant. A key feature of this proof is that it is independent of Donaldson's variational framework and applies to non-Kähler manifolds.