I would sincerely appreciate if anyone can tell me how to solve g(x) defined by the following functional equation:

$h(t) = \int_0^t f(2t-x)g(x)dx$ for $0\leq t\leq \infty$?

where: f(x) is a known function (actually a probability density functions defined on $[0^+,\infty$) with finite first and second order moments. h(t) is a known function and g(x), the unknown function, is a pdf defined on $[0^+,\infty$ with finite first and second order moments.

Assume $f(x)=g(x)=h(x)=0$ when $x<0$ and $f(x) \geq 0, g(x)\geq 0, and h(x)\geq 0$ when $x \geq 0$.

I understand basic convolution stuff(Laplace transform etc) but have minimum explosures to functional analysis. The formulation looks somewhat similar to convolution but not exactly the same. Also it seems related to Wiener-Hopf integral equation but I am unfamiliar with it.

Numerical solution is OK. But any kind of analytic insight or closed-form solution under specific class of functions will be very beneficial. In particular I am interested in the case when f(t) follows a (truncated) Gaussian distribution.

Any textbook/web link/paper recommendation is highly appreciated.

Many thanks!