Homogeneous integral equation

Summary

A homogeneous integral equation is a linear integral equation whose free term $f$ is zero, such as $\varphi(x) = \lambda \int_a^b K(x, s) \varphi(s) \, ds$. It always has the zero solution, and its solutions form a vector space. The values of $\lambda$ for which it has a nonzero solution are the characteristic values of the kernel. By the Fredholm alternative, these values decide whether the inhomogeneous equation is uniquely solvable.