Suppose we have the following two integral equations: $$ f_1(x) = \int K(x,t) \varphi(x,t) dt, \quad f_2(x) = \int K(h(x),t) \varphi(x,t) dt, $$ where $f_1,f_2,K,h$ are known functions, $h$ is strictly increasing, and $\varphi$ is unknown.
Both $x$ and $t$ lie in a compact interval on the real line. All the functions are smooth. $K$ is a conditional density with respect to $t$, i.e. $\int K(x,t) dt=1$ for all $x$, and $K >0$.
Is there anything we can say about $\varphi$? For example, is there going to be a unique solution $\varphi$ that satisfies the above two equations? Or any counterexample (non-unique solution)? Where can I possibly find related information?
Any help is greatly appreciated. Thanks in advance!