I have the following self-consistent integral equation
\begin{equation} f(x) = \int_{0}^{1-x}\frac{f(y)}{\sqrt{h^{2} + (f(y))^{2}}}dy, \end{equation} where $h$ is a real-valued constant.
I wish to find the real-valued function $f(x)$ over the domain $x \in [0,1]$ which satisfies this.
There is always the trivial solution $f(x) = 0$ for $h \neq 0$ but from my numerical calculations I believe that for $-2/3 \lessapprox h \lessapprox 2/3$ there exists a non-trivial function $f(x)$ that is continuous over the domain $x \in [0,1]$ and satisfies the above equation.
I was wondering if anyone had an idea on how to solve this? Or whether there exists some closed form analytical solution here.
Note I believe (correct me if wrong) this is equivalent to finding the non-trivial solution of the differential equation
\begin{equation} \frac{df(x)}{dx} =-\frac{f(1-x)}{\sqrt{h^{2} + (f(1-x))^{2}}} \end{equation}
over the domain $x \in [0,1]$.