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Abstract convexity in KKM theorems via set-valued maps

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Abstract

Different aspects of the KKM Lemma besides its vast applications in nonlinear analysis have been investigated over the years. Some researchers tried to present the KKM results such as generalized L-KKM type theorems in the absence of usual convexity. Here, first we present a counterexample to show that the main results in some generalized KKM type theorems and their consequences are not valid. Afterward, the uniform mapconvex space is introduced and a nonempty intersection result in this space is proved.

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Acknowledgements

The authors would like to express their best gratitude and deep respects to Professor Charles Horvath for his valuable remarks and also his illuminating researches in the KKM theory. The authors also thank the referees for their thoughtful comments and their help in improving this article.

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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran

    M. Fakhar

  2. Department of Mathematics, Faculty of Science, Fasa University, Fars, 74616-86131, Iran

    M. Lotfipour

  3. Sheikhbahaee University, Isfahan, Iran

    J. Zafarani

  4. University of Isfahan, Isfahan, Iran

    J. Zafarani

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  1. M. Fakhar
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  2. M. Lotfipour
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  3. J. Zafarani
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Correspondence to M. Lotfipour.

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Fakhar, M., Lotfipour, M. & Zafarani, J. Abstract convexity in KKM theorems via set-valued maps. Optim Lett 15, 923–931 (2021). https://doi.org/10.1007/s11590-020-01536-9

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  • DOI: https://doi.org/10.1007/s11590-020-01536-9

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