By using the fixed point method, we obtain a version of a stability result of Baker in probabilistic metric and quasimetric spaces under triangular norms of Hadžić type. As an application, we prove a theorem regarding the stability of the additive Cauchy functional equation in random normed spaces.

The use of the fixed point theory in the study of Ulam-Hyers stability was initiated by Baker in the paper [

Suppose

Starting with the papers [

In this paper we study the stability of (

In this section, we study the stability of the equation

We assume that the reader is familiar with the basic concepts of the theory of probabilistic metric spaces. As usual,

We recall some facts from the fixed point theory in probabilistic metric spaces.

A

A trivial example of a

The theorem below provides a characterization of continuous

Let

Let

The following lemma completes Theorem

Let

This lemma can be extended to the case of probabilistic metric spaces under a continuous

Let

We have to prove only the last part of the theorem. We show by induction on

The case

Let

In order to state our first stability result, we define an appropriate concept of approximate solution for the functional equation (

A probabilistic uniform approximate solution of (

Let

Let

If

Denote by

The assumptions on the space

Moreover, the relation

Now we can apply Lemma

Next, let

The result of Baker [

From Theorem

Recall (see [

A probabilistic uniform approximate solution of (

Let

If

We apply Theorem

It is easy to see that

It remains to show that

The defining feature of quasimetric structures is the absence of symmetry. This allows one to consider different notions of convergence and completeness. We state the terminology and notations, following [

A probabilistic quasimetric space is a triple

We note that if

If

Let

right

Let

The probabilistic quasimetric space

The following lemma is a quasimetric analogue of Lemma

Let

Suppose

We proceed in the classical manner to show that the sequence of iterates

Let

The probabilistic quasimetric version of Baker’s theorem can be stated as follows.

Let

If

We only sketch the proof, as it is very similar to that of Theorem

As in the mentioned proof, denote by

The assumptions on the space

We can now apply Lemma

The estimation (

The work of D. Miheţ was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, no. PN-II-ID-PCE-2011-3-0087. The work of C. Zaharia was supported by the strategic Grant POSDRU/CPP107/DMI1.5/S/78421, Project ID 78421 (2010), cofinanced by the European Social Fund—Investing in People, within the Sectoral Operational Programme Human Resources Development 2007–2013.