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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?

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Let me give you an example:
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

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