Resumen de Generalized fuzzy metric spaces defined by means of t-norms

David Miravet Fortuño

• In 1965, L. Zadeh introduced the concept of fuzzy set, and thus established a new topic of research, known as fuzzy mathematics. Since then, several authors have been investigating the approach of a consistent fuzzy metric space theory. In 1994, George and Veeramani introduced and studied a concept of fuzzy metric space which was a proper modification of the concept given by Kramosil and Michalek. These notions have been studied and developed in several ways during the last 25 years. With the purpose of contributing to the development of the study of the fuzzy theory, in this thesis we have introduced and studied the following items:

  1. We have introduced the concept of extended fuzzy metric M0 which is an appropriate extension of a GV -fuzzy metric M where the parameter t can take the value 0. Furthermore, we have studied convergence and Cauchyness concepts in this context, as well as contractivity and fixed point theorems.

  2. We have proved the existence of contractive sequences in the sense of D. Mihet in a GV -fuzzy metric space which are not Cauchy. Then we have given and studied an appropriate concept of strictly contractive sequence and we have corrected Lemma 3.2 of [V. Gregori and J.J. Miñana, On fuzzy psi-contractive sequences and fixed point theorems, Fuzzy Sets and Systems 300 (2016), 93-101].

  3. We have introduced and studied a concept of (GV -)fuzzy partial metric space (X,P,*) without any extra conditions on the continuous t-norm *. Then we have defined a topology T_P on X deduced from P and we have proved that (X, T_P) is a T0 space.

  4. We have related the aforementioned notion of GV -fuzzy partial metric space with the concept of GV -fuzzy quasi-metric space given by Gregori and Romaguera in [V. Gregori and S. Romaguera, Fuzzy quasi-metric spaces, Applied General Topology 5 (2004), 129-136]. A duality is studied by mimicking the techniques used in [S.G.Matthews, Partial metric topology, Annals of the New York Academy of Sciences 728 (1994), 183-197] by Matthews.