Let $X$ be a random variable distributed according to the distribution function $P$ and density function $p$ with full support on compact interval $[a,b]\subset \mathbb{R}_+$.
I am trying to find a solution to the following integral equation:
$$ \int_a^b \left( g(x) - k y \right) f(x,y) p(x) dx = \int_a^b \left( g(x) - k d \right) f(x,d) p(x) dx + \int_y^d \int_a^b f(x,s) p(x) dx ds, \: (1) $$ for $y \in [c,d]\subset \mathbb{R}_+$, where $g:[a,b]\to\mathbb{R}$ is a given function that is continuously differentiable and increasing in $x$, and $ k \neq 0$ is a constant.
A point-wise solution exists – that is, a solution to $$(g(x)-ky)f(x,y)=\left( g(x) - k d \right) f(x,d) + \int_y^df(x,s)ds$$ is $$f(x,y)=h(x) \left( g(x) - k y \right)^{\frac{1-k}{k}} \qquad \qquad (2) $$ for any $h(x) \in \mathbb{R}$ (note that the trivial solution $f(x,y)=0$ for all $(x,y)\in[a,b]\times[c,d]$ is given by $h(x)=0$ for all $x\in[a,b]$). This solution also solves the integral equation (1) with expected values.
Are there any other solutions for (1)? When is (2) the unique solution?