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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.

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    That $|f(x)$ is bounded in $A$.2011-04-18

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I guess that you are aiming for a result similar to Gronwall's inequality, that is, a bound on $f$ depending only on $g$ and $\phi$. I am afraid that no suuch bound is possible in general. Consider for rxample the case $A=[0,2]$, $A^*=[0,1]$, $g(x)\equiv1$ and $\phi(x,y)\equiv1$. Then you have continuous functions $f\colon[0,2]\to\mathbb{R}$ such that $ |f(x)|\le1+\int_0^1|f(y)|dy\quad\forall x\in[0,2] $ with $\sup_{x\in A}|f(x)|$ as large as you want.

On the other hand, if $\alpha=\sup_{x\in A}\int_{A^*}\phi(x,y)dy<1$ and $g$ is bounded, then $ \sup_{x\in A}|f(x)|\le\frac{1}{1-\alpha}\sup_{x\in A}|g(x)|. $