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Abstract
In this paper, the concept of a Q-function defined on a quasi-metric space is introduced. This concept generalizes the notions of a τ-function and an ω-distance. We then present Ekeland-type variational principles within the context of quasi-metric spaces with a Q-function, followed by an equilibrium version of the Ekeland-type variational principle for complete quasi-metric spaces with a Q-function. The equivalence of these Ekeland-type variational principles with Caristi–Kirk type fixed point theorems for multivalued maps, the Takahashi minimization theorem, and some other related results in the setting of complete quasi-metric spaces with a Q-function are also stated and proven.
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References
- S. Ansari, Q.H. Al-Homidan, J. C, Yao, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. (2008), pp. 69, 126–139.
- J.-P. Aubin, H. Frankowska, Set-Valued Analysis, Birkh¨auser, Boston, Basel, Berlin,(1990).
- M. Bianchi, G. Kassay, R. Pini, Existence of equilibria via Ekeland’s principle, J. Math. Anal. Appl. 305, (2005), PP. 502–512.
- Y. Chen, Y.J. Cho, L. Yang, Note on the results with lower semi-continuity, Bull. Korean Math. Soc. 39 (4) (2002), PP. 535–541.
- Ekeland, On the variational principle, J. Math. Anal. Appl. 47, (1974) PP. 324–353.
- D.G. De Figueiredo, The Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research, Bombay, (1989).
- O. Kada, T. Suzuki, W.Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japonica 44 (2), (1996), PP. 381–391.
- L.-J. Lin, W.-S. Du, Ekelend’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces, J. Math. Anal. Appl. 323, (2006), PP. 360–370.
- J.-P. Penot, The drop theorem, the petal theorem and Ekeland’s variational principle, Nonlinear Anal. TMA 10, (1986), PP. 813–822.
- T. Suzuki: Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl. 253, (2001), PP. 440–458.
References
S. Ansari, Q.H. Al-Homidan, J. C, Yao, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. (2008), pp. 69, 126–139.
J.-P. Aubin, H. Frankowska, Set-Valued Analysis, Birkh¨auser, Boston, Basel, Berlin,(1990).
M. Bianchi, G. Kassay, R. Pini, Existence of equilibria via Ekeland’s principle, J. Math. Anal. Appl. 305, (2005), PP. 502–512.
Y. Chen, Y.J. Cho, L. Yang, Note on the results with lower semi-continuity, Bull. Korean Math. Soc. 39 (4) (2002), PP. 535–541.
Ekeland, On the variational principle, J. Math. Anal. Appl. 47, (1974) PP. 324–353.
D.G. De Figueiredo, The Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research, Bombay, (1989).
O. Kada, T. Suzuki, W.Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japonica 44 (2), (1996), PP. 381–391.
L.-J. Lin, W.-S. Du, Ekelend’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces, J. Math. Anal. Appl. 323, (2006), PP. 360–370.
J.-P. Penot, The drop theorem, the petal theorem and Ekeland’s variational principle, Nonlinear Anal. TMA 10, (1986), PP. 813–822.
T. Suzuki: Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl. 253, (2001), PP. 440–458.