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Extensions of minimization theorems and fixed point theorems on a quasimetric space. (English) Zbl 1157.49022

Summary: We introduce the new concepts of \(e\)-distance, \(e\)-type mapping with respect to some \(e\)-distance and \(S\)-complete quasimetric space, and prove minimization theorems, fixed point theorems, and variational principles on an \(S\)-complete quasimetric space. We also give some examples of quasimetrics, \(e\)-distances, and \(e\)-type mapping with respect to some \(e\)-distance. Our results extend, improve, and unify many known results due to Caristi, Ekeland, Ćirić, Kada-Suzuki-Takahashi, Ume, and others.

MSC:

49J52 Nonsmooth analysis
47H10 Fixed-point theorems
49J27 Existence theories for problems in abstract spaces
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References:

[1] doi:10.2307/1999724 · Zbl 0305.47029 · doi:10.2307/1999724
[2] doi:10.1090/S0273-0979-1979-14595-6 · Zbl 0441.49011 · doi:10.1090/S0273-0979-1979-14595-6
[6] doi:10.2307/2040075 · Zbl 0291.54056 · doi:10.2307/2040075
[7] doi:10.1006/jmaa.1998.6030 · Zbl 0917.54047 · doi:10.1006/jmaa.1998.6030
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