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Generalized convexity and applications to fixed points and equilibria

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Abstract

In this paper, we give a uniform approach for generalized convexity by using the concept of L-convexity defined by Ben El-Mechaiekh et al. (J Math Anal Appl 222:138–150, 1998). We prove that the generalized notion of L-space contains well-known generalized convex spaces defined in the literature in topological vector spaces as well as several generalized convexity structures defined on metric spaces. In this context, we give a generalized version of the Fan–Knaster–Kuratowski–Mazurkiewicz Principle (FKKM Principle) in L-spaces and a Browder-Fan type theorem about the existence of fixed points for open lower section set-valued maps defined in an L-space. As an application, we prove the existence of equilibria for an abstract economy with an infinite number of agents.

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Notes

  1. A subset X of a topological space E is said to be \(C^\infty \) if for each integer n, any continuous function \(f:\partial \triangle _n \rightarrow X\) can be continuously extended to a continuous function \(g:\triangle _n\rightarrow X\). A contractible subset of a topological space is an example of a \(C^\infty \) set.

  2. For the definition of \(\triangle _\mathrm{geo}\) and \(\triangle _{\mu }\) see [14].

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Authors and Affiliations

  1. Department of Mathematics, College of Science, King Saud University, Box 2455, Riyadh, 11451, Saudi Arabia

    Najla Altwaijry & Souhail Chebbi

  2. Paris School of Economics, University of Paris 1, Panthéon Sorbonne, Paris, France

    Senda Ounaies

  3. Department of Mathematics, College of Science, University El-Manar, Tunis, Tunisia

    Senda Ounaies

Authors
  1. Najla Altwaijry
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  2. Senda Ounaies
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  3. Souhail Chebbi
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Corresponding author

Correspondence to Souhail Chebbi.

Additional information

This paper is dedicated to the Memory of Monique Florenzano.

This project was funded by the National Plan for Science Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, award number (12-MAT2703-02).

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Altwaijry, N., Ounaies, S. & Chebbi, S. Generalized convexity and applications to fixed points and equilibria. J. Fixed Point Theory Appl. 20, 31 (2018). https://doi.org/10.1007/s11784-018-0517-6

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  • DOI: https://doi.org/10.1007/s11784-018-0517-6

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