Kernel of an integral equation

Summary

The kernel of a linear integral equation is the function $K(x, s)$ of two variables that multiplies the unknown function inside the integral. In $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$ it is $K$. The properties of the kernel, such as continuity, symmetry or the form of its singularity, decide which theory applies to the equation.