# Connection between eigenvalue problems and Lagrange multiplier rule?

In Zeidler's Nonlinear Functional Analysis Part III, Chapter 43.3 contains the following statement:

$F,G$ are both functionals on a Banach space $X$. We consider the minimum problem with a side condition $$min_{u \in N_a} F(u) = F(u_a)$$ where $N_a = \{ u\in X: G(u) = a \}$, and the eigenvalue problem which corresponds to the Lagrange multiplier rule: $$F'(u_a) = \lambda_a G'(u_a)$$ With $\lambda_a \neq 0, u_a \neq 0$.

How is that second equation an eigenvalue problem? My understanding of an eigenvalue problem is that if you have an operator $A: X \rightarrow X$, then the eigenvalue problem is to find a $\lambda$ for which there exists a $v\in X$ such that $$Av = \lambda v$$

What is the connection between the eigenvalue problem written in Zeidler and this more elementary statement of the eigenvalue problem here?

The Euler-Lagrange equation can become an eigenvalue equation. For example, consider the problem in an approriate space $X$: $$\int_0^1(v'(x))^2dx \to min$$ subject to $$\int_0^1 (v(x))^2dx=1 \\ v(0)=v(1)=0$$ The Euler-Langrange equation with multiplier $\lambda$ becomes $$-u-\lambda u''=0$$ which is indeed an eigenvalue equation. So you can formulate some eigenvalue problems as variational problems with constraints