Unicity (or not) of the solution of an integral equation Given the integral equation: $$\int_0^a f(x)\left[ \frac{d^2}{dx^2}f(x) \right]dx=a$$ with the condition: $$\lim_{x\to\infty}f(x)=0$$ how can I find its solution? Is the solution (if any) the only one possible?

Take any function $g$ defined on $[0,a]$, and write $$\int_0^a g(x)g''(x)\,dx=b.$$ All we require at this point is that $b>0$. Clearly, there is a huge number of such functions to choose from. Now define $$f(x)=\sqrt{\frac{a}{b}} g(x)\qquad\text{for } x\in[0,a],$$ and expand $f$ to $(a,\infty)$ in whatever way you like, just so long as $$\lim_{x\to\infty} f(x)=0.$$ Again, there is a huge range of possibilities.