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Working through the proof of the mountain-pass theorem given in Applied Nonlinear Analysis by Aubin & Ekeland, at what seems to be a critical point of the proof (the top of page 274) they refer to "the inf sup theorem 5.2.7;" yet the actual theorem 5.2.7 (theorem 7 of section 2 of chapter 5, page 253) is about Cesarò means. I could not find the so-called inf sup theorem anywhere except maybe in chapter 6 on game theory, but there are so many inf sup theorems there that I cannot figure out which one (if any) is being used.

The inf sup theorem seems to involve the interchange of infimum and supremum when one of the two sets involved is compact. The theorem is used to justify the following proof of the mountain-pass theorem, which I reproduce for convenience. Perhaps the use of the theorem here can be bypassed. Suppose we have:

  • $X$ a Banach space;
  • $U\colon X \to \mathbb R$ a $C^1$ function (Fréchet-differentiable with continuous derivative $U'\colon X \to X^*$);
  • for some $\alpha>0$ and some $z \in X$ with $\lVert z\rVert > \alpha$, $\inf_{\lVert x\rVert = \alpha} U(x) > U(0),U(z)$.

Then $\inf_{x\in X} \lVert U'(x) \rVert = 0$.

Briefly the proof is as follows. The set $\mathcal C$ of continuous paths $[0,1] \to X$ from $0$ to $z$ is a a complete metric space for the distance $d(c_1,c_2) = \max_t \lVert c_2(t) - c_1(t) \rVert$. The function $I\colon \mathcal C \to \mathbb R$ defined by $I(c) = \max_t U(c(t))$ is lower semicontinuous, and bounded from below. Thus we can apply Ekeland's variational principle to obtain for every $\varepsilon > 0$ a path $c_\varepsilon \in \mathcal C$ with, for every $c \in \mathcal C$, $$ I(c) \ge I(c_\varepsilon) - \varepsilon\, d(c_\varepsilon,c) . $$ There then follows a rather sophisticated argument to show that this implies there exists a $t_\varepsilon \in [0,1]$ at which $\lVert U'(c_\varepsilon(t_\varepsilon)) \rVert \le \varepsilon$. It is within this argument that the "inf sup theorem" is used. What is the exact statement of the theorem used? Or else, would there be a more elementary justification?

Only the problematic part of the proof is visible on the Google Books page. I can reproduce more detailed context if needed.

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If you assume the contrary, that is $U^\prime(c_\epsilon(t))$ is big in norm for all $t$, it would be possible to define, using the linearisation, a direction of descend. Considering the perturbed path will provide a contradiction.

Cf. our work based on an idea of Ekeland and Sere:

Milen Ivanov and Nadia Zlateva, "Hadamard Inverse Function Theorem Proved by Variational Analysis". Preprint, arXiv:2202.09327 [math.FA], to appear in Serdica, (2022), Zbl arXiv:2202.09327.

Note that this apparently works only for Frechet differentiability. At any rate, I was not able to make it work for Gateaux.

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