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Xuan Duc Ha Truong(txdhamath.ac.vn) Abstract: The wellknown Ekeland variational principle (in brief, EVP) concerns a function $f:X\to\R\cup\{+\infty\}$, which is lower semicontinuous (in brief, lsc) and bounded from below on a complete metric space $X$. In this note, we answer the question: What happens when the assumption ``$f$ is lsc on $X$" is relaxed to ``$f$ is lsc on its domain"? We show that, in such a situation, the conclusion of EVP holds for $\epsilon$ varying in some interval $]0,\gamma \inf_Xf[$. Here, $\gamma$ is the supremum of scalars $t$ with $t>\inf_Xf$ such that lower level sets $[f\leq t]$ are closed and it also is calculated based on information about the behavior of $f$ at the boundary of its domain. When $f$ is lsc on the whole space $X$, our version of EVP reduces to the classical one with $\gamma =+\infty$. Keywords: Ekeland variational principle, lower semicontinuity, Gateaux differentiability Category 1: Nonlinear Optimization (Other ) Citation: Download: [PDF] Entry Submitted: 05/31/2015 Modify/Update this entry  
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