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Equivalents of Ekeland’s principle. (English) Zbl 0793.54025

Let (V,d) be a complete metric space; f:V×V(-,+] a function which is lower semicontinuous in the second argument and such that: (1) f(v,v)0; (2) f(u,v)f(u,w)+f(w,v) for u,v,wV; (3) there exists v 0 V such that inf{f(v 0 ,v):vV}>-.

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 ={vV:f(v 0 ,v)+d(v 0 ,v)0} and ΨV be such that for every v ¯S 0 Ψ there exists vV such that vv ¯ and f(v ¯,v)+d(v ¯,v)0. Then there exists v * S 0 Ψ.


MSC:
54E50Complete metric spaces
49J40Variational methods including variational inequalities
47H10Fixed point theorems for nonlinear operators on topological linear spaces
49J27Optimal control problems in abstract spaces (existence)
49J45Optimal control problems involving semicontinuity and convergence; relaxation
54C60Set-valued maps (general topology)