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On maximal element theorems, variants of Ekeland’s variational principle and their applications. (English) Zbl 1133.58006

Summary: We establish several different versions of the generalized Ekeland’s variational principle and maximal element theorem for \(\tau \)-functions in \(\lesssim\) complete metric spaces. The equivalence relations between maximal element theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.

MSC:

58E30 Variational principles in infinite-dimensional spaces
49J35 Existence of solutions for minimax problems
54H25 Fixed-point and coincidence theorems (topological aspects)
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