Simple example of Hammerstein integral equation

Question

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 a 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 $a(.)\in L^{\infty}(I)$ and $b:[0,+\infty) \rightarrow(0,+\infty)$ a nondecreasing function such that $\|f(s, x)\| \leq a(s) b\big(\|x\|\big)$ for a.e. $s \in I$ and $x\in E$,

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

But I'm wondering if those are reasonable assumptions. I'm looking for an example of the Hammerstein integral equation with the hypothesis $(H_1)$ and $(H_2)$, take $\color{blue}{E=\mathbb{R}}$ for simplification.

Answer 1

It is a question what is reasonable and whether you can conclude something interesting from it: If, for instance, the integral operator with kernel $\lvert K\rvert$ acts in $L_\infty$ and $f$ grows sublinear in its second argument (with a bound independent of its first argument), the hypothesis will be satisfied with a sufficiently large constant $r$.

It is unlikely to find a real-world problem in which the former fails but your hypotheses are satisfied (though it is of course easily possible to construct artifical such examples - for instance, just modify $f$ arbitrarily for very large arguments after you calculated a possible constant $r$).

As mentioned, the main question is whether you can conclude something interesting from the hypotheses: If it is just another existence result, it is unlikely that your result is not already covered by the hundreds or thousands of existence results for Hammerstein equations with very similar hypotheses which are all essentially only such small generalizations of the sublinear growth condition.