On fixed points and convergence results of sequences generated by uniformly convergent and point-wisely convergent sequences of operators in Menger probabilistic metric spaces

Manuel De la Sen; Asier Ibeas; Jorge Herrera

doi:10.1186/s40064-016-2057-0

On fixed points and convergence results of sequences generated by uniformly convergent and point-wisely convergent sequences of operators in Menger probabilistic metric spaces

Springerplus. 2016 May 4:5:557. doi: 10.1186/s40064-016-2057-0. eCollection 2016.

Authors

Manuel De la Sen¹, Asier Ibeas², Jorge Herrera³

Affiliations

¹ Institute of Research and Development of Processes IIDP, Faculty of Science and Technology, University of the Basque Country, PO Box 644, de Bilbao, Barrio Sarriena, 48940 Leioa, Bizkaia Spain.
² Department of Telecommunications and Systems Engineering, Faculty of Engineering, Universitat Autònoma de Barcelona (UAB), 08193 Bellaterra, Cerdanyola del Vallés, Barcelona Spain ; Departamento de Ingeniería, Facultad de Ciencias Naturales e Ingeniería, Universidad de Bogotá Jorge Tadeo Lozano, 22 Street, No. 4-96, Mod. 7A, Bogotá, D.C. 110311 Colombia.
³ Departamento de Ingeniería, Facultad de Ciencias Naturales e Ingeniería, Universidad de Bogotá Jorge Tadeo Lozano, 22 Street, No. 4-96, Mod. 7A, Bogotá, D.C. 110311 Colombia.

Abstract

In the framework of complete probabilistic Menger metric spaces, this paper investigates some relevant properties of convergence of sequences built through sequences of operators which are either uniformly convergent to a strict k-contractive operator, for some real constant k ∈ (0, 1), or which are strictly k-contractive and point-wisely convergent to a limit operator. Those properties are also reformulated for the case when either the sequence of operators or its limit are strict [Formula: see text]-contractions. The definitions of strict (k and [Formula: see text]) contractions are given in the context of probabilistic metric spaces, namely in particular, for the considered probability density function. A numerical illustrative example is discussed.

Keywords: Menger spaces; Probabilistic metric spaces; Strict [Formula: see text]-contractions; Strict contractions; Triangular norms.