Fredholm equation of the second kind

Remarks

Example: on $[0, 1]$, take $K(x, y) = xy$, $\lambda = 1$ and $f(x) = x$. Then $\varphi(x) = 3x/2$ is a solution, since $\int_0^1 x y \cdot \tfrac{3y}{2} \, dy = \tfrac{3x}{2} \cdot \tfrac{1}{3} = \tfrac{x}{2}$ and $\tfrac{3x}{2} - \tfrac{x}{2} = x$.

Non-example: $\int_a^b K(x, y) \varphi(y) \, dy = f(x)$ is not of the second kind, because $\varphi$ does not appear outside the integral; it is a Fredholm equation of the first kind.

Some books write the equation as $\varphi - \lambda T_K \varphi = f$, where $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$. Others put $\lambda$ on the other side, as $\lambda \varphi - T_K \varphi = f$; for $\lambda \ne 0$ the two forms are equivalent after dividing by $\lambda$.