Compatibility of some systems of inequalities
DOI:
https://doi.org/10.33993/jnaat321-729Keywords:
quasi-convex inequalities, concave-convex-like function, minimax inequalityAbstract
In this paper, necessary and sufficient conditions for the compatibility of some systems of quasi-convex, or convex inequalities are established. Finally a new proof for a theorem of Shioji and Takahashi (1988) [10] is given.Downloads
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Balaj M., Finite families of convex sets with convex union, Seminar on Mathematical Analysis, Preprint 93-7, Univ. "Babeş--Bolyai" Cluj-Napoca, pp. 69-74, 1993.
Barbu, V. and Precupanu, T., Convexity and Optimization in Banach Spaces, Ed. Academiei RSR, Bucureşti, 1975 (in Romanian).
Berge, M. C., Sur une proprieté combinatoire des ensembles convexes, C. R. Acad. Sci. Paris, 248, pp. 2698-2699, 1959.
Fan, K., Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations, Math. Z., 68, pp. 205-217, 1957, https://doi.org/10.1007/BF01160340. DOI: https://doi.org/10.1007/BF01160340
Granas, A. and Liu, F. C., Remark on a theorem of Ky Fan concerning systems of inequalities, Bull. Inst. Math. Acad. Sinica., 11, pp. 639-643, 1983.
Itoh, S., Takahashi, W. and Yanagi, K., Variational inequalities and complementary problems, J. Math. Soc. Japan, 30, pp. 23-28, 1978, https://doi.org/10.2969/jmsj/03010023. DOI: https://doi.org/10.2969/jmsj/03010023
Kassay, G. and Kolumban, J., On a generalized sup-inf problem, J. Optim. Theory Appl., 62, pp. 127-138, 1989, https://doi.org/10.1007/BF02190126. DOI: https://doi.org/10.1007/BF02190126
Lin, F. C., A note on the von Neumann--Sion minimax principle, Bull. Inst. Math. Acad. Sinica, 6, pp. 517--524, 1978.
Mawhin, J. and Willem, M., Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New-York, 1989. DOI: https://doi.org/10.1007/978-1-4757-2061-7
Shioji, N. and Takahashi, W., Fan's theorem concerning systems of convex inequalities and its applications, J. Math. Anal. Appl., 135, pp. 383-398, 1988, https://doi.org/10.1016/0022-247X(88)90162-X. DOI: https://doi.org/10.1016/0022-247X(88)90162-X
Shioji, N., A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, Proc. Amer. Math. Soc., 111, pp. 187-195, 1991, https://doi.org/10.1090/S0002-9939-1991-1045601-X. DOI: https://doi.org/10.1090/S0002-9939-1991-1045601-X
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