I want to solve the following integral equation
$$\mathbf v(\mathbf r)=\int_V \mathbf v(\mathbf r^\prime)\cdot \mathbf F(\mathbf r, \mathbf r^\prime) \mathrm ~d^3r^\prime +\mathbf g(\mathbf r)$$
using a self consistency iterative method. Namely,
$$\mathbf v_0=\mathbf g,$$ $$\mathbf v_1=\int_V \mathbf v_0\cdot \mathbf F \mathrm ~d^3r^\prime +\mathbf g,$$ $$\mathbf v_2=\int_V \mathbf v_1\cdot \mathbf F \mathrm ~d^3r^\prime+\mathbf g,$$ $$\vdots$$
Is there a condition that can be applied to determine if $$\lim_{n\to\infty}\mathbf v_n(\mathbf r)=\mathbf v(\mathbf r)~\text{?}$$
EDIT: I don't know if it's relevant, but note that $\mathbf F$ is a rank 2 tensor, and $\mathbf v$ and $\mathbf g$ are vectors.