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I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:

$$ |v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left( |v(x)+v(qx)|\right) dx. $$

Let $v(s)$ be a function defined on [0,1], $\epsilon$ and $\beta$ are positive constants, $\alpha \in [0, 1]$, $q \in [0, 1]$, and $s, x \in [0, 1]$. The inequality holds for all valid choices of $s$ and $x$.

I am now interested in determining a possible upper bound for the function $|v(s)|$ based on this inequality. Could someone provide guidance on how to do so? given these conditions and the inequality provided? I have looked for every extended Gronwall inequality and couldn't find any that could help me, especially that my integral limits are non-variables and the integral is singular.

I would appreciate any insights, approaches, or mathematical techniques that can be used to derive an upper bound for the function $|v(s)|$ in this context. Thank you in advance for your help!

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  • $\begingroup$ What specifically do you mean by "an upper bound for the function $|v(s)|$"? A pointwise bound? Or something else? $\endgroup$
    – Iosif Pinelis
    Commented Nov 3, 2023 at 14:28
  • $\begingroup$ just an upper bound in function of ϵ and β and s, something similar to what one gets from Gronwall inequality $\endgroup$
    – Takieddine Zeghida
    Commented Nov 4, 2023 at 0:00

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