16 - Triangular norms in probabilistic metric spaces and fixed point theory

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This chapter discusses probabilistic metric spaces and some constructions methods of triangle functions. One of the basic abstract tools in mathematics for solving nonlinear equations is the well-known classical Banach fixed point theorem, which has many generalizations and applications. The chapter introduces some necessary tools and presents a definition of probabilistic metric spaces. The first simple construction of triangle functions goes through point wise extension of triangular norms (t-norms). An important method of construction of triangle functions is also based on t-norms. It is important to obtain a rich source of different triangle functions to enable the construction of new probabilistic metric spaces. Starting from a metric space and a distribution function, it is possible to generate an important class of probabilistic metric spaces. New probabilistic metric spaces can be obtained making their direct products. Using the notion of geometrically convergent t-norms, some fixed point results are obtained in this chapter. Special semi groups on the unit interval triangular norms play an important role in the theory of probabilistic metric spaces.