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Workshop on Proof Theory, Modal Logic and Reflection Principles
October 17, 2017 12:15–12:50, Moscow, Steklov Mathematical Institute  The reverse mathematics of Ekeland's variational principle

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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