I'm currently reading this paper (and working on a similar one). 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 an scalar kernel, $E$ a Banach space and $f:I \times E \rightarrow E$ is a given function.

Let's say that in my hypothesis, I need:

$H_1$. There exist $a:I \rightarrow (0,+\infty)$ with $\left |K(t,.)\right |a(.)\in L^1(I)$ for each $t\in I$ and $b:[0,+\infty) \rightarrow(0,+\infty)$ a nondecreasing continuous function such that $\|f(s, x)\| \leq a(s) b\big(\|x\|\big)$ for a.e. $s \in I$ and $x, y\in E$,

$H_2$. there exists at least one solution $r \in \mathcal{C}\big(I,(0, \infty)\big)$ to the inequality $$ b\left(\|r\|_{\infty}\right) \int_{0}^{t} K(t, s) a(s) d s \leq r(t), \quad t \in I$$ where $\|.\|_{\infty}$ is the sup norm in $\mathcal{C}\big(I,(0, \infty)\big)$.

But I'm wondering if this hypothesis is plausible. I'm looking for an example of Hammerstein integral equation, with $\color{blue}{E=\mathbb{R}}$.