I'm studying a simple dynamic programming problem whose solution is a system of Bellman equations, and am running into some issues trying to prove the Contraction Mapping Theorem for the system as a whole.
The problem is a labor search problem: an unemployed worker gets benefit $b$ each period and meets a firm with probability $\lambda$, represented as a wage draw from from some known distribution $F$ with finite support $[0,\overline{B}]$, and can choose to accept or reject and keep searching. An worker employed at wage $w$ earns $w$ each period, and with probability $\delta$ becomes unemployed.
The problem is to show that the system of Bellman equations that represent this problem satisfy the contraction mapping property, and hence that we have a guaranteed (unique) convergent solution.
What I've tried
The set-up is straightforward, and the problem can be represented as a system of two Bellman equations.
$$ U = b + \beta \left\lbrace \lambda \int_0^\overline{B} \,\max_{A,R} \lbrace V(w^\prime),U\rbrace \,\text{d} F(w^\prime) + (1-\lambda) U \right\rbrace $$
$$ V(w) = w + \beta \lbrace \delta U + (1-\delta) V(w)\rbrace $$
I'd like to show that the system of Bellman equations is a contraction mapping, and hence that there is a unique solution that we are guaranteed to converge to.
By the Contraction Mapping Theorem (Stokey, Lucas, Prescott, 1989, Thm. 3.2) if my system exhibits the contraction mapping property, then I have a guaranteed solution.
My difficulty is that I don't know how to verify the contraction property for the system as a whole. If I am allowed to treat $U$ as a fixed number, then the problem is relatively straightforward: I can consider an operator $T f(w) = f(w)$ and show that the operator satisfies Blackwell's sufficient conditions (monotonicity and discounting). But $U$ is a function of the continuation value $V(w^\prime)$, and so any attempts to verify the contraction property start to fall apart under Blackwell, since those conditions only apply to a single-valued function.
Other approaches
It is fairly straightforward to show that $U$ and $V(w)$ are both bounded functions, and of each other.
I also know that there will be an optimal policy rule to the $Accept, Reject$ problem of the unemployed worker, represented by a threshold wage $w^*$ above which I will accept any offer and below which I will reject any offer.
It is also possible to combine the two equations, but given the recursive nature of the problem I cannot figure out a way to eliminate either $U$ or $V(w)$.
My main question
Is there a way to apply Blackwell's sufficient conditions to a system of two Bellman equations to verify the contraction mapping property? Or is there an alternative to Bellman's sufficient conditions that applies to systems of multiple Bellman equations?