I am having a trouble with applying the fixed point theorem. For instance, suppose I have three value functions.
Two of them are linear differential equations where $r \in (0,1)$ is a discount factor.
$rJ(x)= f(V(x),J'(x))$
$rH(x)= g(V(x),H'(x))$
$rV(x) = z(J(x),H(x))$
I am finding $V(x)$ which solves the above three equations at the same time, and suppose I have all necessary initial values.
In my real problem, I could find one, and now I want to show this solution is indeed unique.
I would appreciate to hear some suggestions.
I think the best option is to deal only with the differential equations. substituting $V$ in the first two equations, you get something of the type $$ \begin{cases} F(J(x),J'(x),H(x)) = 0\\ G(J(x),H(x),H'(x)) = 0, \end{cases} $$
which is a system of first order differential equations. Although most of the existence and uniqueness results are for differential equations where the derivative appears in an explicit way, you may explore/assume properties of $F,G$ to argue that the system can be written in the form
$$ \begin{cases} J'(x) = \tilde F(J(x),H(x))\\ H'(x) = \tilde G(J(x),H(x)), \end{cases} $$
at this point you can use the Picard-Lindelof theorem.