Consider $t:[0,1]^2\to R$ that is differentiable a.e. and satisfies conditions (i)-(ii):
(i)
$$ \int_0^1 \frac{\partial t}{\partial t_1}(x,y)f(y\mid x)\,dy=0, \quad \forall x\in[0,1] \\ \int_0^1 \frac{\partial t}{\partial t_2}(x,y)f(x\mid y)\,dx=0, \quad \forall y\in[0,1] $$
where $f$ is a continuously differentiable density function $f:[0,1]^2\to R_+$.
(ii)
$$ \frac{\partial t}{\partial t_1}\leq 0, \quad \text{if } x\geq y; \qquad \frac{\partial t}{\partial t_1}\geq 0, \quad \text{if } x< y; \\ \frac{\partial t}{\partial t_2}\geq 0, \quad \text{if } x\geq y; \qquad \frac{\partial t}{\partial t_2}\leq 0, \quad \text{if } x< y. $$
The question is that without constructing a solution, how could I show that the system has a nontrivial solution such that $t$ is not a constant ?
Thanks.