Consider a compact $A\subset \mathbb{R}$ and a compact $A^*\subset A$. For a positive functions $g(x)$ and $\phi(x,y)$ the following inequality holds on $f$ $ |f(x)|\leq g(x)+\int\limits_{A^*}|f(y)|\phi(x,y)\,dy. $ How to derive bounds on $|f(x)|$ for all $x\in A$?
Here $g(x),\phi(x,y)$ are Lipschitz continuous functions.