I wish to solve the following integral equation that has popped up in my studies of focused light. If you notice, it looks almost like a homogenous Fredholm integral equation of the first kind. However, it involves the mag-square of the function to be found, and not the function itself. Is there a method to find the solution in general? Also, is it an ill-posed problem like the first-kind Fredholms?
$$\int_0^{2\pi}|F(p)|^2\cos (p) \cos^2\left (a + b*\sin[p] \right )dp =0 $$ Where the task is to find all $F(x):\mathbb{R}\rightarrow\mathbb{C}$ which satisfy the equation, and $a,b \in \mathbb{R}$. A few solutions are $F(x)=\phi$ for $\phi \in \mathbb{C}$ and $\cos^nx\sin^mx$ for $n,m\in \mathbb{\lbrace N,0 \rbrace }$. I am particularly interested in the solutions which are independent from $a$ and $b$ like the solutions I just mentioned.
Clearly it is an ill-posed problem for $|F(x)|^2$, but is it also ill-posed for $F(x)$? Also, is there an equivalent differential equation?