Integral equation of the first kind

Remarks

Example: $\int_0^1 (x + s) \varphi(s) \, ds = x$ on $[0, 1]$ is of the first kind. With $\alpha = \int_0^1 \varphi$ and $\beta = \int_0^1 s \varphi(s) \, ds$ it says $\alpha x + \beta = x$, so $\alpha = 1$ and $\beta = 0$. The function $\varphi(s) = 4 - 6s$ satisfies both, since $\int_0^1 (4 - 6s) \, ds = 1$ and $\int_0^1 (4s - 6s^2) \, ds = 0$; so do infinitely many others.

Non-example: $\varphi(x) = x + \int_0^1 (x + s) \varphi(s) \, ds$ is not of the first kind, because $\varphi$ also appears outside the integral.

With $I_x = [a, b]$ it is a Fredholm equation of the first kind; with $I_x = [a, x]$ it is a Volterra equation of the first kind. The example shows that the solution need not be unique. The right side $f(x) = x^2$ shows that it need not exist, since the left side is always affine in $x$.