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Equivalents of Ekeland’s principle. (English) Zbl 0793.54025
Let \((V,d)\) be a complete metric space; \(f: V \times V \mapsto (- \infty,+\infty]\) a function which is lower semicontinuous in the second argument and such that: (1) \(f(v,v) \equiv 0\); (2) \(f(u,v) \leq f(u,w) + f(w,v)\) for \(u,v,w \in V\); (3) there exists \(v_ 0 \in V\) such that \(\inf\{f(v_ 0,v): v\in V\} > -\infty\).
The authors prove the following theorem and demonstrate its equivalence to the Ekeland variational principle, the Caristi-Kirk fixed point theorem and the Takahashi minimization principle. Let \(S_ 0 = \{v \in V: f(v_ 0,v) + d(v_ 0,v) \leq 0\}\) and \(\Psi \subset V\) be such that for every \(\overline{v} \in S_ 0 \setminus \Psi\) there exists \(v\in V\) such that \(v\neq \overline{v}\) and \(f(\overline{v},v) + d(\overline{v},v) \leq 0\). Then there exists \(v^* \in S_ 0 \cap \Psi\).

MSC:
54E50 Complete metric spaces
49J40 Variational inequalities
47H10 Fixed-point theorems
49J27 Existence theories for problems in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
54C60 Set-valued maps in general topology
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References:
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