# Minimax theorems in fuzzy metric spaces

## 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.

- Authors:

- Mu’tah University, Department of Mathematics and Statistics, Faculty of Science (Jordan)

- Publication Date:

- OSTI Identifier:
- 22769334

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CONTROL THEORY; DIFFERENTIAL EQUATIONS; FUZZY LOGIC; GAME THEORY; MATHEMATICAL SOLUTIONS; METRICS; OPTIMIZATION; PROBABILISTIC ESTIMATION; VARIATIONAL METHODS

### Citation Formats

```
Rashid, M. H. M.,.
```*Minimax theorems in fuzzy metric spaces*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0417-1.

```
Rashid, M. H. M.,.
```*Minimax theorems in fuzzy metric spaces*. United States. doi:10.1007/S40314-017-0417-1.

```
Rashid, M. H. M.,. Tue .
"Minimax theorems in fuzzy metric spaces". United States. doi:10.1007/S40314-017-0417-1.
```

```
@article{osti_22769334,
```

title = {Minimax theorems in fuzzy metric spaces},

author = {Rashid, M. H. M.,},

abstractNote = {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.},

doi = {10.1007/S40314-017-0417-1},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}