Generalized contraction mapping principle and generalized best proximity point theorems in probabilistic metric spaces

Yongfu Su,Jen-Chih Yao,Wenbiao Gao

doi:10.1186/s13663-015-0323-4

Abstract

The purpose of this paper is to introduce some basic definitions about fixed point and best proximity point in two classes of probabilistic metric spaces and to prove contraction mapping principle and relevant best proximity point theorems. The first class is the so-called S-probabilistic metric spaces. In S-probabilistic metric spaces, the generalized contraction mapping principle and generalized best proximity point theorems have been proved by authors. These results improve and extend the recent results of Su and Zhang (Fixed Point Theory Appl. 2014:170, 2014). The second class is the so-called Menger probabilistic metric spaces. In Menger probabilistic metric spaces, the contraction mapping principle and relevant best proximity point theorems have been proved by authors. These results also improve and extend the results of many authors. In order to get the results of this paper, some new methods have been used. Meanwhile some error estimate inequalities have been established.

Highlights

Introduction and preliminariesProbabilistic metric spaces were introduced in by Menger [ ]
A mapping T is a probabilistic contraction if T is such that for some constant < k
The purpose of this paper is to introduce some basic definitions about fixed point and best proximity point in two classes of probabilistic metric spaces and to prove contraction mapping principle and relevant best proximity point theorems

Summary

It is easy to see that

In S-probabilistic metric spaces, the generalized contraction mapping principle and generalized best proximity point theorems have been proved by authors. In Menger probabilistic metric spaces, the contraction mapping principle and relevant best proximity point theorems have been proved by authors. These results improve and extend the results of many authors. Since ψ dF (xnk , xmk ) ≤ φ dF (xnk– , xmk– ) , by using condition ( ) we know ε = , this is a contradiction This shows that {xn} is a Cauchy sequence in the metric space (E, dF ). For any positive integer l, we have dF T lx, T ly ≤ hdF T l– x, T l– y ≤ hdF T l– x, T l– y ≤ · · · ≤ hldF (x, y)

For any given x

Finally we can get the error estimate formula