In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE
$$\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}-x y=2y |y|^\alpha, \quad -\infty<x<\infty, \quad \alpha>0. $$ In their studies, they formulate the following integral equation [Section 2]
$$y_k(x)=k \text{Ai}(x)+2\int_x^\infty \left\{ \text{Ai}(x) \text{Bi}(t)-\text{Bi}(x) \text{Ai}(t) \right\} y_k(t) \left| y_k(t) \right|^\alpha, $$
and write that the equation "can be solved (uniquely) *by iteration*, and this gives both $y_k$ and its continuous dependence on $k$.

I was trying to make the "by iteration" part more rigorous. Firstly, I've learned that the equation looks like a Volterra integral equation (although here the interval of integration is unbounded). I think that the formal way to prove that solutions exist and depend continuously over the parameter $k$ is to consider the right hand side of the integral equation is an integral operator, and then show that it is a contraction mapping under an appropriate choice of norms.

However, I don't really know how to choose the domain/codomain of this integral operator, nor the norm. Furthermore, I can't see how to get the continuous dependence on $k$.

If the authors' claim is a standard result, I'd like a reference. Otherwise, I'd really appreciate help in formalizing their proof.