Solution of an integral equation Consider a simple Wiener-Hopf integral equation of the first kind with unknown function $\phi(x)$ for $x\geq 0$:
$$f(x)=\int_0^\infty \phi(y)\min\{x,y\}\,\mathrm{d}y$$
where $f(x)=x-a$ and $a \geq 0$.
Although it arose from a well stated physical problem, I suggest there are no solutions of any of these equations in $\phi(x)$. Or are they?
Note: Would there be a solution if we choose $f(x)=
\begin{cases}
0 &;x<a\\
x-a &;x\geq a
\end{cases}
$ instead?
 A: $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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I'll assume the equation is
  $\ds{\,\mathrm{f}\pars{x} =
\int_{0}^{\infty}\phi\pars{y}\min\braces{x,y}\,\dd y}$.

\begin{align}
\,\mathrm{f}\pars{x} & =
\int_{0}^{x}\phi\pars{y}y\,\dd y + x\int_{x}^{\infty}\phi\pars{y}\,\dd y
\\[3mm] \imp\
\,\mathrm{f}'\pars{x} & =
\phi\pars{x}x + \int_{x}^{\infty}\phi\pars{y}\,\dd y -
x\phi\pars{x}
\\[3mm] \imp\
\,\mathrm{f}''\pars{x} & =
-\phi\pars{x}\quad\imp\quad\color{#f00}{\phi\pars{x}} =
\color{#f00}{-\,\mathrm{f}''\pars{x} = -\delta\pars{x - a}}
\end{align}

$$\mbox{because}\
\mathrm{f}\pars{x} = \Theta\pars{x - a}\pars{x - a}\ \imp\
\mathrm{f}'\pars{x} = \Theta\pars{x - a}\ \imp\
\mathrm{f}''\pars{x} = \delta\pars{x - a}
$$


Checking ?:
  $$
\int_{0}^{\infty}\bracks{-\delta\pars{y - a}}\min\braces{x,y}\,\dd y =
-\min\pars{x,a} =
\left\lbrace\begin{array}{rcl}
\ds{-x} & \mbox{if} & \ds{x < a}
\\
\ds{-a} & \mbox{if} & \ds{x > a}
\end{array}\right.
$$

Because the solution is expresed in terms of the second derivative
$\,\mathrm{f}''\pars{x}$, it's still a solution of $\ds{\,\mathrm{f}\pars{x} + cx + d}$ for some arbitrary constants $\ds{c}$ and $\ds{d}$. It means the 'above checking' becomes
$$
\left\lbrace\begin{array}{rcl}
\ds{-x + cx + d} & \mbox{if} & \ds{x < a}
\\
\ds{-a + cx + d} & \mbox{if} & \ds{x > a}
\end{array}\right.
$$
The original $\ds{\,\mathrm{f}\pars{x}}$ is recovered with
$\color{#f00}{\ds{c = 1}}$ and $\color{#f00}{\ds{d = 0}}$.

In general, it means there should be additional conditions besides the integral equation.

