I asked the following question 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.
I'm currently reading this paper (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.
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}
Best Answer
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 $\delta>0$.