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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:
 [1] Dugundji, Fixed point theory 1 (1982) [2] Danes, Boll. Un. Mat. Ital. 6 pp 369– (1972) [3] DOI: 10.1007/BFb0081133 [4] Brøndsted, Pacific J. Math. 55 pp 335– (1974) · Zbl 0248.46009 [5] DOI: 10.1016/0022-247X(74)90025-0 · Zbl 0286.49015 [6] Oettli, Quantitative Wirtschafts-forschung pp 535– (1977) [7] DOI: 10.1016/0362-546X(86)90069-6 · Zbl 0612.49011 [8] DOI: 10.1016/0022-247X(88)90187-4 · Zbl 0647.49009 [9] de Figueiredo, The Ekeland variational principle with applications and detours (1989) · Zbl 0688.49011 [10] Takahashi, Fixed point theory and applications pp 397– (1991)
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