Lyapunov–Schmidt reduction

Definition

Let $X$, $Y$ and $\Lambda$ be Banach spaces, let $U \subseteq X \times \Lambda$ be an open neighbourhood of $(0, 0)$, and let $F \colon U \to Y$ be continuously differentiable with $F(0, 0) = 0$. Let $L = D_x F(0, 0) \colon X \to Y$, and assume that $L$ is a Fredholm operator:

1. the kernel $N = \ker L$ has finite dimension;
2. the range $R = L(X)$ is closed and has a complement of finite dimension in $Y$.

Choose closed subspaces $X_1$ and $Y_1$ with $X = N \oplus X_1$ and $Y = R \oplus Y_1$, and let $Q \colon Y \to R$ be the continuous projection onto $R$ along $Y_1$. The restriction of $QL$ to $X_1$ is an isomorphism onto $R$. Write $x = v + w$ with $v \in N$ and $w \in X_1$. By the implicit function theorem, the equation $Q F(v + w, \lambda) = 0$ has a unique solution $w = W(v, \lambda)$ near $0$. Here $(v, \lambda)$ is near $(0, 0)$, and $W(0, 0) = 0$. The Lyapunov–Schmidt reduction of $F = 0$ at $(0, 0)$ is the bifurcation equation $$(I - Q)\, F\big(v + W(v, \lambda), \lambda\big) = 0,$$ for $v \in N$ and $\lambda$ near $0$. Its values lie in $Y_1$.