Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows:
$$ 1-t= \int\limits_0^\infty p(x)\Phi\left(\frac{x+t}{\sqrt{h(t)}}\right)dx - \int\limits_0^t\Phi\left(\frac{t-u}{\sqrt{h(t)-h(u)}}\right)du,\quad \forall t\in [0,1),\label{1}\tag{$\star$} $$ where $p: \mathbb R_+\to\mathbb R_+$ is a probability density as good as possible and $\Phi$ denotes the cumulative distribution function of a standard normal distribution.
Questions
Has the equation of this form been studied in the literature? If the solution to \eqref{1} exists in $M$, can we prove its uniqueness?
PS : If needed, we may restrict to a smaller set $M_1\subset M$ that consists of differentiable functions on $[0,1)$. With this additional condition, can we show the uniqueness of the solution to $(\star)$?
