I need to find the family of real-valued single variable functions $F:\, (0,1) \to [0,1]$ that satisfy the following integral equation: $$\int_{\theta = 0}^{\pi} F\big(~ x\, \sin\theta ~\big)\text{d} \theta + \int_{\theta = \pi}^{2\pi} \bigg[ 1 - F\big(~ 1 + x\, \sin\theta ~\big) \bigg] \text{d} \theta = Ax $$ or equivalently $$\int_{\theta = 0}^{\pi} \bigg[ F\big(~ x\, \sin\theta ~\big) + 1 - F\big(~ 1 - x\, \sin\theta ~\big) \bigg]\text{d} \theta = Ax $$ where $0 < x < 1$ and $0 < A < 1$, with $F$ required to be non-decreasing and differentiable (just once is needed). The open- or closed-ness of the domain and range of $F$ is not important to me.
The idea is that the integrations yield a RHS that is directly proportional to the coefficient of the argument in the integrand.
My Questions:
- How can this be solved? I'm open to additional assumptions (necessary or not) or restrictions to the solution $F$.
- If this kind of problem is hard to tackle in general, what material (like a particular topic, or some textbooks) should I familiarize myself with to even begin considering this task?
So far I know that linear $F(u) = u$ and quadratic $F(u) = u^2$ both satisfy the equation. The cubic $F(u) = u^3$ is not a solution.
Due to the some possible skew-symmetry with respect to $\theta = \pi$ and $\frac12$ for the argument of $F$, it's tempting to guess that rotational symmetric function (odd functions) like $F(u) = \int_{0}^{u} [1+8(s-\frac12)^3 ]\;\text{d}s$, which is a quartic polynomial, might do the job, but it doesn't. Nor do $F(u) = \frac14 \int_{0}^{u} [1 - \cos( \pi s) ]\;\text{d}s = \frac14\left[u- \frac1{\pi} \sin(\pi u) \right]$ or the several sinusoidal functions I looked at.
Taking the derivative $\partial_x$ to the whole equation isn't very helpful. If I'm not mistaken the equation is sensitive to the range of $\theta$ angle integration as well as the "$1-$" in the 2nd integral and the "$+1$" in $F$, as seen in the first format.
Thank you.
P.S.
One might notice that $F$ can be a probability CDF. This is indeed the context within which this integral equation emerged. As of now I don't see much reason to provide the context, which might be a distraction.