Lipschitz perturbations and Hammerstein integral equations

[S.G. Janssens]

Recently I was a witness and a minor contributor to this thread, which more or less derailed, in spite of the efforts by @Samy_A. This is a pity and it angered me a bit, because the topic touches upon some interesting questions in elementary functional analysis. Here I would like to briefly discuss these questions.

 

[S.G. Janssens]

The response has been so massive that it is maybe good to point out that Hammerstein integral equations usually arise rather indirectly, as equivalent formulations of nonlinear boundary value problems (BVPs). When the domain is one-dimensional (as above, where we work on the interval $[a,b]$) these are BVPs for ordinary differential equations, whereas if the domain is multidimensional, these are BVPs for nonlinear elliptic partial differential equations. For a very simple example, consider (with $n = 1$, $a = 0$ and $b = 1$)
$$
\left\{
\begin{aligned}
&u''(x) + \alpha(u(x)) = \beta(x) \qquad (0 \le x \le 1)\\
& u(0) = 0 = u(1)
\end{aligned}
\qquad \text{(BVP)}
\right.
$$
where $\alpha \in C(\mathbb{R})$ and $\beta \in C([0,1])$. If we now set
$$
k(x,y) :=
\begin{cases}
x(1 - y) &\text{if } 0 \le x \le y \le 1\\
y(1 - x) &\text{if } 0 \le y \le x \le 1
\end{cases}
$$
then $k$ is continuous on $[a,b] \times [a,b]$. Let us also set
$$
f(x,z) := \alpha(z), \quad v(x) := -\int_0^1{k(x,y)\beta(y)\,dy} \qquad \forall\,x \in [0,1],\,\forall\,z \in \mathbb{R}
$$
With these choices for $k$, $f$ and $v$ a little bit of calculation shows that (BVP) and ($*$) are equivalent: If $u \in C([0,1])$ is a solution of ($*$), then $u \in C^2([0,1])$ and $u$ solves (BVP) and vice versa. This is a very useful result, because integral operators are often easier to deal with than differential operators, both from a purely theoretical as well as a numerical point of view. In particular, note that
$$
\|k\|_{\infty} = \sup_{0 \le x, y \le 1}{|k(x,y)|} = \frac{1}{4}
$$
so in view of post #1 we conclude that (BVP) has a unique solution provided that $\alpha$ is Lipschitz with $\text{Lip}(\alpha) < 4$. (In fact, we can take $\text{Lip}(\alpha) < 8$ because for this particular choice of $k$ the estimate in post #1 is easily refined.)

In these two posts, I wrote about some of the things that crossed my mind when I read through the thread I mentioned at the top of post #1. I simply didn't want these simple and classical (but beautiful, in my opinion) applications to nonlinear integral and differential equations to go unnoticed. Also, I welcome examples of problems of the type (BVP) (or of the type ($*$), for that matter) from physics. I know BVPs of this sort arise in continuum mechanics, but at present I'm not yet sufficiently well introduced to the physical aspects of that field.