Consider the following equation
$$ H(y) = \int_{0}^{\infty} G\left(\frac{y-\phi_2(v)}{\phi_1(v)} \right) \exp(-v) ~\mathrm{d}v $$
where $\phi_i$ are well-behaved differentiable functions on the region of integration and G non-decreasing on the real line.
My question relates to whether there exists a family of solutions for $G$ written in terms of the function $H$ and if so what is its possible form?
