Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$.
$a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional unkown function such that
$\int_0^1 z(x,y)f(y,x)dy=a(x)$, $\forall x$ and
$\int_0^1 z(x,y)f(x,y)dx=b(y)$, $\forall y$
The question is there is any results that such system has a solution for $z(\cdot,\cdot)$?
Thanks.