Minimax theorems in fuzzy metric spaces
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Minimax theorems in fuzzy metric spaces M. H. M. Rashid1
Received: 7 September 2016 / Revised: 12 January 2017 / Accepted: 16 January 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
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. Keywords Probabilistic metric space · Chainable subset · L-space · L-convex set · L-KKM mapping · Variational inequality · KKM theorem · Matching theorem · Minimax inequality Mathematics Subject Classification 35J87 · 49J35 · 49J40 · 47J20
1 Introduction The concept of fuzzy sets was introduced initially by Zadeh (1965). Since then, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and applications. The strong points about fuzzy mathematics are its fruitful applications,
Communicated by Rosana Sueli da.
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M. H. M. Rashid [email protected] Department of Mathematics and Statistics, Faculty of Science, Mu’tah University, P. O. Box 7, Al-Karak, Jordan
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M. H. M. Rashid
especially outside mathematics, such as in quantum particle physics studies by ElNaschie (2004). To use this concept in topology and analysis, Kramosil and Michalek (1975) have introduced the concept of fuzzy metric space using the concept of continuous triangular norm defined by Schweizer and Sklar (1960). Recently, Gregori et al. (2011) utilized the concept of fuzzy metric spaces to color images processing and also studied several interesting examples of fuzzy metrics in the sense of George and Veeramani (1994). Most Recently, a number of fixed point theorems for single-valued and multi-valued mappings in fuzzy metric spaces have been proved by many authors (Cho 1997; Deng 1982; George and Veeramani 1994; Grabiec 1998; Kramosil and Michalek 1975). Since every metric space is a fuzzy metric space, we can use many results in fuzzy metric spaces to prove some fixed point theorems in metric spaces. For notations and properties of fuzzy metric spaces, refer to Cho (1997), Deng (1982), and Ereeg (1979). The classical KKM theore
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