Positive solutions of integral equations

Summary

A positive solution of an integral equation is a solution that is nonnegative and not identically zero, or in a stronger sense strictly positive. In models of densities, concentrations, populations or probabilities only such solutions have a meaning, so existence of a positive solution is often the real question. The main tools are order methods: the cone of nonnegative functions, positive operators that map it into itself, and monotone iteration. Fixed point and eigenvalue theorems on cones, such as the Krein–Rutman theorem, complete the toolbox. For linear equations with nonnegative kernels, positivity is closely tied to the spectral radius of the integral operator.