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Minimax theorems in fuzzy metric spaces

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Abstract

A minimax theorem is a theorem providing conditions which guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann’s minimax theorem, which was considered the starting point of game theory. Since then, several alternative generalizations of von Neumann’s original theorem have appeared in the literature. Variational inequality and minimax problems are of fundamental importance in modern non-linear analysis. They are widely applied in mechanics, differential equations, control theory, mathematical economics, game theory, and optimization. The purpose of this paper is first to establish a minimax theorem for mixed lower–upper semi-continuous functions in fuzzy metric spaces which extends the minimax theorems of many von Neumann types. As applications, we utilize this result to study the existence problems of solutions for abstract variational inequalities and quasi-variational inequalities in fuzzy metric spaces and to study the coincidence problems and saddle problems in fuzzy metric spaces.

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Acknowledgements

The author wishes to express his sincere thanks to the referee for his constructive comments which resulted in an improvement of the first draft of the paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Mu’tah University, P. O. Box 7, Al-Karak, Jordan

    M. H. M. Rashid

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  1. M. H. M. Rashid
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Correspondence to M. H. M. Rashid.

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Communicated by Rosana Sueli da.

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Rashid, M.H.M. Minimax theorems in fuzzy metric spaces. Comp. Appl. Math. 37, 1703–1720 (2018). https://doi.org/10.1007/s40314-017-0417-1

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