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Some fixed point results in semi-metric spaces as well as in symmetric spaces are proved. Applications of our results to probabilistic spaces are also presented.

There have been a number of generalizations of metric space. Two of them are the notions of symmetric spaces and semi-metric spaces introduced and studied by Wilson [

In this paper we prove some fixed point results in semi-metric spaces and symmetric spaces. We also present applications of our results to probabilistic spaces. Our results generalize earlier results obtained by Aranđelović and Kečkić [

A symmetric space is a pair

Let

Many properties and notions in symmetric spaces are similar to those in metric spaces (but not all, because of the absence of the triangle inequality). For example, a sequence

In every symmetric space

The following conditions can be used as partial replacements for the triangle inequality's absence in the symmetric space

there exists

The properties (W3) and (W4) were induced by Wilson [

Next statement gives the characterization of symmetric space which satisfies the property (JMS).

Let

The convergence of a sequence

The following two propositions have been well known for a long time, but for the convenience of the reader we will state them without proofs, which can also be found in [

If

A topological space

It is worth mentioning that this basis need not consist of open sets. Moreover, in [

Let

Let

A symmetric space

Next statement was proved in [

Let

Let

Let

The function

If

If

where

Let

if

Let

The implication (W)

Now let

A semi-metric space in which all balls

Let

Suppose that

In this section, we obtain generalizations of fixed point results of Browder [

Let

Let

Each element

We want to prove that

Now, let

Therefore (

Now, by (

Hence,

Since

If, in addition to the hypothesis of Theorem

Since

Let

By Theorem

Next, assume that

Thus, it is possible to choose two sequences

Since there exists

So,

In both cases, one can conclude that there exists

If

On the other hand, if

Let

In this section, we extend results attributed to Maiti et al. [

Let

Define

Let

Let

Let

Let

We will prove that, for all

If

From

The next example of [

Let

Let

But

We now present applications of our results to probabilistic spaces. We begin with some essential definitions.

Let

If

If

The topology

The space

(1) The condition (W) is equivalent to

The following lemma was proved in [

Let

Let

if

Let

Let

Define

Let

Define

Let

Does Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the reviewers for their valuable suggestions. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first and third authors acknowledge with thanks DSR for the financial support. The second author was supported by the Ministry of Education, Science and Technological Development of Serbia, Grant no. 174002.