I asked the following [question](https://math.stackexchange.com/questions/4223194/a-uniform-continuity-type-condition-on-an-integral) on MathStackExchange, but I have not received the answer that I'm looking for. Although it may not be a research-level question, I thought I could ask it here.




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I'm currently reading this [paper](https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03154-1/S0002-9939-96-03154-1.pdf) (and working on a similar one). Specifically, I'm trying to improve the two hypotheses (2.5) and (2.6) of [O'Regan's paper](https://link.springer.com/article/10.1007/s000130050243).


The main goal is to study the Hammerstein integral equation (in $\mathcal{C}(I,E))$:


$$x(t) = \int\_{0}^{t} K(t,s)f\big(s,x(s)\big)ds,\quad t\in I;$$


where $I=[0,1]$, $K $ is a scalar kernel, $E$ a Banach space and $f:I \times E \rightarrow E$ is a given function.


**Suppose that $s \mapsto K(t, s)$ is integrable and $t \mapsto K(t, s)$ is continuous.**


**My goal is to see if the following is true:**


\begin{array}{l}
\text { For each } \epsilon>0 \text { , there exists } \delta>0 \text { such that, for any } t\_{1}, t\_{2} \in I, \text { if }\left|t\_{1}-t\_{2}\right|<\delta \text { then }\\
\int\_{t\_1}^{t\_2} K(t\_2,s) d s< \epsilon
\end{array}




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A "uniform continuity" type condition on a Hammerstein integral equation
This does certainly not follow from your other hypotheses, as what you want to conclude is not much weaker than the equi-integrability of $\{K(t,\cdot):t\in I\}$ (sometimes also called absolute continuity in the $L\_1$-norm), but what you assume is only a continuity in one variable which implies nothing about the $L\_1$-norm, not even the boundedness.


A counterexample might look as follows: Around the line $t=s/2$ with $t\in(0,1)$, say, for $s\in[t/3,t/2]$, the function $K(t,s)$ assumes huge values (for instance, $1/t^2$). Then you extend the function to a nonnegative function preserving your condition such that $K(t,s)=0$ for $t\in\{0,1\}$ or $s\ge t$.


Then you have even $\sup\_{\lvert t\_1-t\_2\rvert\le\delta}\int\_{t\_1}^{t\_2}K(t\_2,s)ds=\infty$ for every $\delta>0$.


