$f:[0,1]^2\to R_+$ is a continuous conditional density function.
For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq 0,\phantom{0}\phantom{0} \frac{\partial h}{\partial x}\geq 0\phantom{0} on \phantom{0} \{(x,y)\in [0,1]^2|x\geq y\} $$ $$ \int_{x}^1 \frac{\partial g}{\partial x} (x,y) f(y|x) dy+\int_{0}^x \frac{\partial h}{\partial x} (x,y) f(y|x) dy= [ g(x,x)-h(x,x)]f(x|x), \forall x $$ $$ \int_{x}^1 g(x,y)f(y|x)dy + \int_{0}^x h(x,y)f(y|x)dy=K-x, \forall x $$
Is it possible to show that for all $f\in F$ (or on the subsets), the system has a nontrivial solution such that $(g, h)$ are not constants everywhere?
Thanks.