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,Jorge Herrera,Asier Ibeas

doi:10.1186/s40064-016-2057-0

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 varphi-contractions. The definitions of strict (k and varphi) 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.

Highlights

Fixed point theory is an important tool to investigate the convergence of sequences to limits and unique limits in metric spaces and normed spaces
Menger probabilistic metric spaces are a special class of the wide class of probabilistic metric spaces which are endowed with a triangular norm, (Pap et al 1996; Sehgal and Bharucha-Reid 1972; Choudhury et al 2011; De la Sen and Karapinar 2015a, b; Choudhury and Das 2008; Gopal et al 2014) and which are very useful in the context of fixed point theory since the triangular norm plays a close role to that of the norm in normed spaces
This paper investigates some properties of convergence of sequences being 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

Summary

Introduction

Fixed point theory is an important tool to investigate the convergence of sequences to limits and unique limits in metric spaces and normed spaces. The subsequent result is concerned with the probabilistic convergence properties of the sequences {xn} ⊂ X generated by the iterated scheme xn+1 = Tnxn; ∀n ∈ Z+ for any given x1 ∈ X: Theorem 2 Let (X, F , ∆M) be a complete Menger space and let {Tn} be a sequence of operators Tn : X → X, such that FTn =

Results

Conclusion