Let $p:\mathbb R^n\to\mathbb R^n$ be a continuous function, and $v(z,y):\mathbb R^n\times\mathbb R^n\to\mathbb R$ be a continuous bounded function so that
$$v(z,y)=\max_{c,x}\biggr(U(c)+\beta\int v(x,y')dF(y',y)\biggr)$$ subject to $$c+p(y)\cdot x\le y\cdot z+p(y)\cdot z,$$ where $F:\mathbb R^n\times\mathbb R^n\to\mathbb R$ is continuous and is a distribution function for each fixed $y$, $U:\mathbb R\to\mathbb R$ is continuously differentiable, bounded, increasing, and strictly concave.
How can we construct an operator $T_p$ on the functions $v(z,y)$ so that $T_pv=v$ iff $v$ solves the maximization problem above? I'd also appreciate some references. I suspect this relates to functional analysis and convex optimization (possibly topology since we're talking about fixed points?)
Context: Robert Lucas makes this claim in his 1978 paper Asset Prices in an Exchange Economy.