I'm writing down a cumulative distribution function $F(x)$, with pdf $f(x)$. Below is a simplified version of the expression for $F(x)$ when $x \in [0,1]$.
$F(x) = h(x) + \int_0^x \int_x^1 g(y,z) \cdot f(y) ~dy ~dz$
where $h$ and $g$ are some known functions.
This looks like a differential equation to me, so I'm wondering (1) whether a CDF can be defined implicitly or as a differential equation and (2) if so, how to show that $F$ is a valid CDF? I can show that $F(0)=0$ and $F(1)=1$, so letting
$F(x) = 0 ~\forall ~x <0$
and
$F(x) = 1 ~\forall ~x >1$
takes care of the limit conditions and right-continuity. How can I show that it's non-decreasing?
It's not a differential equation, it's a double integral. Assuming it satisfies the usual requirements for a CDF, it's a perfectly valid definition.
As for being non-decreasing, you might note that using the Fundamental Theorem of Calculus, if $h$ is differentiable
$$ \dfrac{dF}{dx} = h'(x) + \int_x^1 g(y,x) f(y)\; dy - \int_0^x g(x,z) f(z)\; dz$$
You'll need this to be always nonnegative.