

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 
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MP4 
913.8 Mb 
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Abstract:
(Joint with David FernándezDuque, Henry Towsner, and Keita Yokoyama.)
Let $X$ be a complete metric space, and let $V$ be a lower semicontinuous function from $X$ to the nonnegative 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 prooftheoretic strength of Ekeland's variational principle in the context of secondorder arithmetic. We show that the full theorem is equivalent to $\Pi^1_1CA_0$. 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

