I am considering the following 2nd order PDE :
On a domain $R$, for some $r > 0$, \begin{equation*} \frac{1}{2}\sum_{i, j = 1}^{K}\gamma_{i}\gamma_{j}U_{x_{i}x_{j}}(x) + \sum_{i = 1}^{K}\frac{\gamma_{0}\gamma_{i}}{x_{0}}U_{x_{i}}(x) - rU(x) = -rg\,\,\,\,\,\,\,(1) \end{equation*} \begin{align} \gamma_{i} &= \gamma_{i}(a, b, x)\,\,\textrm{is Lipschitz continuous}\,\,\,\textrm{for each}\,\,i \in \{0,1,...,K\}\\ g &= g(a, b)\,\,\,\textrm{is a Lipschitz continuous function} \end{align} where, with $DU(x) = \big(U_{x_{1}}(x),...,U_{x_{K}}(x)\big)$, \begin{align*} a &= a\big(x, DU(x)\big)\\ b &= b\big(x, DU(x)\big) \end{align*}
I am considering the following iteration to show an exitence of $U$ to the above PDE:
Pick any continuous functions $a_{0}(x)$ and $b_{0}(x)$ on $R$
Plug them into $\gamma_{i}(a_{0}(x), b_{0}(x), x) = \gamma_{i}^{0}(x)$ and $g(a_{0}(x), b_{0}(x)) = g_{0}(x)$
Solve the linear elliptic 2nd order PDE (1) on $R$ with $\gamma_{i}^{0}(x)$ and $g_{0}(x)$. Let the solution $U_{0}(x)$
Let $a_{1}(x) = a(x, DU_{0}(x))$ and $b_{1}(x) = b(x, DU_{0}(x))$ be continuous functions w.r.t $x \in R$
Plug them into $\gamma_{i}(a_{1}(x), b_{1}(x), x) = \gamma_{i}^{1}(x)$ and $g(a_{1}(x), b_{1}(x)) = g_{1}(x)$
Solve the linear elliptic 2nd order PDE (1) on $R$ with $\gamma_{i}^{1}(x)$ and $g_{1}(x)$. Let the solution $U_{1}(x)$
Iterate...
[QUESTION]
For the convergence of $\{U_{n}\}$ on $R$ to a limit,
I want to show something like a continuity property of the linear solution $U_{n}$ on the coefficients $\{\gamma^{n}_{i}, g_{n}\}_{i=0}^{K}$.
Could anyone suggest me some references for this question?
