arXiv subjects
Publications and source records attributed to Bernardo Lafuerza-Guillen.
In this paper, we present the generalization of B-contraction and C-contraction due to Sehgal and Hicks respectively. We also study some properties of C-contraction in probabilistic metric space.
We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.
In this paper we consider probabilistic normed spaces as defined by Alsina, Sklar, and Schweizer, but equipped with non necessarily continuous triangle functions. Such spaces endow a generalized topology that is Fréchet-separable, translation-invariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable.
The motivation of this paper is a suggestion by Höle of comparing the notions of $\D$-boundedness and boundedness in Probabilistic Normed spaces (briefly PN spaces), with non necessarily continuous triangle functions. Such spaces are here called ``pre-PN spaces''. Some results on Šerstnev spaces due to B. Lafuerza, J. A. Rodriguez, and C. Sempi, are here extended to generalized Šerstnev spaces (these are pre-PN spaces satisfying a more general Šerstnev condition). We also prove some facts on PN spaces (with continuous triangle functions). First, a connection between fuzzy normed spaces defined by Felbin and certain Šerstnev PN spaces is established. We further observe that topological vector PN spaces are $F$-normable and paranormable, and also that locally convex topological vector PN spaces are bornological. This last fact allows to describe continuous linear operators between certain generalized Šerstnev spaces in terms of bounded subsets.
In this work we first give for PN spaces results parallel to those obtained by Egbert for the product of PM spaces, and generalize results by Alsina and Schweizer in order to study non-trivial products and the product of $m$-- transforms of several PN spaces. In addition we present a detailed study of $α$--simple product PN spaces and, finally, the product topologies in PN spaces which are products of countable families of PN spaces.