Bifurcation from a characteristic value

Summary

Consider a nonlinear equation $\varphi = \lambda\bigl(K\varphi + R(\varphi)\bigr)$ with a compact linear operator $K$ and a term $R$ that is small compared with $\varphi$ near $0$. Then $\varphi = 0$ is a solution for every $\lambda$. A value $\lambda_0$ is a bifurcation point if nonzero solutions branch off the trivial solution at $\lambda = \lambda_0$. Such a point must be a characteristic value of $K$, that is, a value where the linearized equation $\varphi = \lambda K\varphi$ has a nonzero solution. This is the classical setting of bifurcation theory for nonlinear integral equations, with results such as Krasnosel'skii's theorem on characteristic values of odd multiplicity.