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Workshop on Proof Theory, Modal Logic and Reflection Principles
October 17, 2017 12:15, Moscow, Steklov Mathematical Institute
 


The reverse mathematics of Ekeland's variational principle

P. Shafer
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MP4 913.8 Mb
MP4 913.8 Mb

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P. Shafer


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Abstract: (Joint with David Fernández-Duque, Henry Towsner, and Keita Yokoyama.)
Let $X$ be a complete metric space, and let $V$ be a lower semi-continuous function from $X$ to the non-negative reals. Ekeland's variational principle states that $V$ has a ‘critical point’, which is a point $x^*$ such that $d(x^*, y)> V(x^*) - V(y)$ whenever y is not $x^*$. This theorem has a variety of applications in analysis. For example, it implies that certain optimization problems have approximate solutions, and it implies a number of interesting fixed point theorems, including Caristi's fixed point theorem.
We analyze the proof-theoretic strength of Ekeland's variational principle in the context of second-order arithmetic. We show that the full theorem is equivalent to $\Pi^1_1-CA_0$. We also show that a few natural special cases, such as when $V$ is assumed to be continuous and/or X is assumed to be compact, are equivalent to the much weaker systems $ACA_0$ and $WKL_0$.

Language: English

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