Abstract
This paper studies the convergence properties of an iterative process which involves sequences of convergent self-mappings in probabilistic Menger spaces which are used to generate the sequences of interest. The convergent self-mappings under consideration satisfy conditions of either uniform or point-wise convergence, in a probabilistic sense, to a self-mapping on the same abstract space of the considered probabilistic metric space. Furthermore, the self-mappings of the considered sequence satisfy a probabilistic ϕ-contractive condition which is based on the use of a control ϕ-function. Some illustrative examples are also discussed.
Highlights
Fixed point theory [ – ] is receiving important research attention in the framework of probabilistic metric spaces
Menger probabilistic metric spaces are a special case of the wider class of probabilistic metric spaces which are endowed with a triangular norm, [, ]
Under an intuition-based point of view, the deterministic notion of distance is considered to be probabilistic in the sense that, given any two points x and y of a metric space, a measure of the distance between them is a probabilistic metric Fx,y(t), rather than the deterministic distance d(x, y), which is interpreted as the probability of the distance between x and y being less than t (t > ) [ ]
Summary
Fixed point theory [ – ] is receiving important research attention in the framework of probabilistic metric spaces. Fixed point theorems in complete Menger probabilistic metric spaces for probabilistic concepts of B and C-contractions can be found in [ ] together with a new notion of contraction, referred to as ( , C)-contraction. Such a contraction was proved to be useful for multivalued mappings while it generalizes the previous concept of C-contraction. Some interesting general fixed point theorems have been very recently obtained in [ ] for two new classes of contractive mappings in Menger probabilistic metric spaces. It is assumed that the limit self-mapping is φ-contractive, while in the second one it is assumed that the members of the sequence of self-mappings of the iterative scheme are φ-contractive
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