Abstract

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.

1. Introduction

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 [1]. For historical remarks about these spaces see [2]. Fixed point theory of various classes of maps in a metric space and its generalizations has been studied by a number of authors; see, for example, [39] and the references cited therein. In 1976, Cicchese proved the first fixed point theorem for contractions in semi-metric spaces. Further fixed point results for this class of spaces were obtained by Jachymski et al. [10], Hicks and Rhoades [11], Aamri and El Moutawakil [12], Aamri et al. [13], Zhu et al. [14], Miheţ [15], Imdad et al. [16], Aliouche [17], and Radenović and Kadelburg [18]. For more information on fixed point theory in symmetric spaces and semi-metric spaces, we refer the reader to [2].

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 [2], Browder [19], Walter [20], and Maiti et al. [21].

2. Preliminary Notes

A symmetric space is a pair consisting of a nonempty set and a function such that for all in the following conditions hold:(W1) if and only if ;(W2).

Let be symmetric space. The open ball with center and radius is defined by Also if is a subset of , then denotes the diameter of .

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 is said to be -Cauchy sequence if given there is such that , for all .

In every symmetric space one may introduce the topology by defining the family of closed sets as follows: a set is closed if and only if for each , implies , where

The following conditions can be used as partial replacements for the triangle inequality's absence in the symmetric space (W3) and imply ;(W4) and imply ;(W) and imply ;(JMS) and imply ;(CC) implies ;(SC) implies ;(MT)there exists such that for any

The properties (W3) and (W4) were induced by Wilson [1], (W) by Miheţ [15], (JMS) by Jachymski et al. [10], (CC) by Cho et al. [22] and earlier by Borges [23] (as 1-continuity property), (MT) by Czerwik [24] (see also [25]), and (SC) by Aranđelović and Kečkić [2].

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

Proposition 1 (Jachymski et al. [10]). Let be a symmetric space. Then the following conditions are equivalent.(i) satisfies property (JMS).(ii)There exists such that for any , (iii)There exists such that

The convergence of a sequence in the topology need not imply , although the converse is true (see Proposition 2).

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 [2].

Proposition 2. If is a symmetric space, then the family forms a local basis at . Also, if , then (or ) in the topology .

Definition 3. A topological space is semimetrizable if there is a symmetric function such that and that the mapping is the closure operator in . In terms of it can be expressed by saying that the operator is idempotent. In this case we say that is semi-metric space; is said to be semi-metric function on (or admissible semi-metric for ).

It is worth mentioning that this basis need not consist of open sets. Moreover, in [26], a semimetrizable space was constructed with the property that, for any that generates , there exist and such that is not open.

Proposition 4. Let be a symmetric space. Then is a semi-metric space if and only if the following conditions hold.(1) is first countable.(2)For any sequence , is equivalent to    in the topology .

Example 5. Let . Define by Then is a -Cauchy complete semi-metric.

A symmetric space is said to be -Cauchy complete if every -Cauchy sequence converges to some in the topology , and it is said to be -weakly complete if every decreasing sequence of nonempty closed subsets, such that there exists a sequence , with has a nonempty intersection.

Proposition 6 (Galvin and Shore [27]). Let be a semi-metric space. Then the following are equivalent:(1) is -weakly complete;(2)every -Cauchy sequence in has a convergent subsequence;(3)every decreasing sequence of nonempty closed subsets of such that for each has a nonempty intersection.

Let be a nonempty set and . Then is called a fixed point of if . Let . The sequence defined by is called the sequence of Picard iterates of at point . This sequence is also called the orbit of at point . We will denote it by and use to denote set .

Let denote the set of all functions satisfying the following properties:(a) is monotone nondecreasing;(b) for any .

The function is known as the comparison function (see [29]). As a consequence of the above properties, we have the following (see [29]).

Lemma 7. If then for all and .

Definition 8. If is a metric space and , then is called a(1)contraction if there exists real number such that (2)-contraction if there exists a function such that for any (3)generalized  -contraction if there exists a function such that for any where

Lemma 9 (Aranđelović and Kečkić [2]). Let be a nonempty set, let , and let be a fixed positive integer such that the iterate has a unique fixed point . Then(1) is a unique fixed point of ;(2)if is a topological space and any sequence of Picard iterates defined by converges to , then the sequence of Picard iterates defined by always converges to .

3. Some Topological Results

Proposition 10. Let be a symmetric space satisfying (W). Then it satisfies (W4) and (JMS).

Proof. The implication (W)(W4) is straightforward (see [15]).
Now let , and be sequences in such that From (W) it follows that So, satisfies (JMS).

A semi-metric space in which all balls , and , are open will be called a semi-metric space with open balls.

Proposition 11. Let be a compact semi-metric space with open balls and a nonempty compact set. Then is bounded.

Proof. is first countable [2, Proposition  3] and -space [2, page 5161]. Also, satisfies the property (SC) because all are open sets [2, Theorem  1]. is countably compact because it is compact [30, Theorem  11.9]. It is sequentially compact, as a first countable countably compact set [30, Problem  10.7].
Suppose that is not bounded. Let . For each positive integer there exists such that . Then there exists and an increasing sequence of positive integers such that, in the topology , because is sequentially compact. So, we get that which is a contradiction because

4. Bounded Semi-Metric Spaces: Fixed Point Results

In this section, we obtain generalizations of fixed point results of Browder [19] and Walter [20] (see also [7]).

Theorem 12. Let be a bounded semi-metric and -Cauchy complete space satisfying (W4). Suppose satisfies that, for , there exists such that, for any and , with . Then there exists such that in the topology (or equivalently as ), .

Proof. Let and .
Each element has one of forms or , with . Now let and consider the case with and ; then Other cases will lead to the same inequality: Now, define the sequences and by , , and , .
We want to prove that By (20), we get that (21) is valid for .
Now, let be arbitrary and set , and ; then But and . Thus .
Therefore (21) holds for all .
Now, by (21) and the monotonicity of , we get that as that is, , which is equivalent to , . But which implies Similarly
Hence, and are -Cauchy sequences, and by the -completeness of , there exists such that and in the topology .
Since and , (W4) implies that . But in the topology and so . Since (W4) implies (W3), we have . Since is arbitrary in , in the topology , .

Corollary 13. If, in addition to the hypothesis of Theorem 12, one assumes that is -continuous (i.e, in the topology implies in the topology ) then has a fixed point.

Proof. Since in the topology , by the -continuity of , in the topology . Therefore, since (W4) implies (W3), . Hence, is a fixed point.

Theorem 14. Let be a bounded semi-metric and -Cauchy complete space satisfying (W4), (CC) and (JMS). Suppose that is a self-map on , and for . Then has a unique fixed point and in the topology (or equivalently as ), .

Proof. By Theorem 12, there exists such that   for all .
Next, assume that ; that is, .
Thus, it is possible to choose two sequences such that So one can pick with a corresponding , such that .
Since there exists such that therefore , by Proposition 1(ii)
So, for infinitely many , with ; thus there exists a sequence such that . So, either for infinitely many (i.e., or there exists a sequence with as which implies .
In both cases, one can conclude that there exists such that .
If , since and by (CC) of , we get which is a contradiction with , for .
On the other hand, if , by (26), which is also a contradiction. Hence ; that is, .

Corollary 15. Let be a bounded semi-metric and -Cauchy complete space satisfying (W) and (CC). Suppose that is a self-map on , and for Then has a unique fixed point and in the topology (or equivalently as ), for all .

5. Symmetric Spaces: Fixed Point Results

In this section, we extend results attributed to Maiti et al. [21, Theorem  4] and Aranđelović and Kečkić [2, Theorem  3].

Theorem 16. Let be a -Cauchy complete symmetric space satisfying (W3) and (JMS). Let be a -continuous map such that for all , and . Then has a unique fixed point and for each , the sequence of Picard iterates defined by at converges to in the topology .

Proof. Define as follows: for and otherwise. Then the space is a symmetric space. Also, we have for any . So, if is an arbitrary -Cauchy sequence in , then is a -Cauchy sequence in .
Let . From it follows that So is a -contraction on .
Let be defined as in (ii) of Proposition 1. Then there exists the least positive integer such that .
Let . We have that is continuous (in ). Then So is a -contraction on .
Let and . Then and So which implies that Then there exists such that
We will prove that, for all , By definition of , we get that (41) is valid for . Now, assume that (41) is satisfied for some . From it follows that which by Proposition 1 implies that So, by induction we get that (41) is satisfied for any . Thus Hence is a -Cauchy sequence in , which implies that is a -Cauchy sequence in . It follows that there exists such that (in the topology ) because is -Cauchy complete. Then (in the topology ) because is -continuous. Now we get that because satisfies (W3).
If is another fixed point of , then for all we have So is a unique fixed point of . By Lemma 9 we get that is a unique fixed point of .
From it follows that for each the sequence of Picard iterates defined by at converges, in the topology , to , which implies their convergence in the topology . So, by Lemma 9, we obtain that for each the sequence of Picard iterates defined by at converges, in the topology , to .

Remark 17. The next example of [10] illustrates that the continuity of in Theorem 16 can not be omitted.

Example 18. Let and let be defined as follows:; for ;;  otherwise .
Let given by Then is a bounded -Cauchy complete semi-metric space and for all , (see [10, Example  3]). satisfies (W3) and (JMS).
But does not have a fixed point in . Note that is not continuous.

6. Applications

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

Definition 19. Let be a set and a mapping of into a collection of all distribution functions (a distribution function is a nondecreasing and left continuous mapping of reals into with and ). Consider the following conditions:(I) for all , where denotes the value of at .(II) if and only if , where denotes the distribution function defined by if and if .(III).(IV)If and , then .

If satisfies (I) and (II), then it is called a PPM-structure on and the pair is called a PPM-space. satisfying (III) is said to be symmetric. A symmetric PPM-space satisfying (IV) is a probabilistic metric space (or briefly PM-space).

The topology in is generated by the family where the set is called -neighborhood of . A sequence is said to be a Cauchy sequence if, for every given , there exists a positive integer such that for all . A topology on is defined as follows: if, for any , there exists such that . If , then is said to be topological.

The space is called -complete if for every Cauchy sequence there exists such that for all .

Remark 20. (1) The condition (W) is equivalent to (2) The condition (W4) is equivalent to

The following lemma was proved in [11].

Lemma 21 (Hicks and Rhoades [11]). Let be a symmetric PPM-space. Set Then is a bounded compatible symmetric for .

Lemma 22 (Hicks and Rhoades [11]). Let be a symmetric PPM-space. Define as in (52). Then(1) if and only if ;(2) is compatible symmetric for ;(3) is complete if and only if is -Cauchy complete symmetric space;(4)if is topological, is semi-metric.

is -continuous if for all implies . This is equivalent to the continuity of , where is as in Lemma 21.

Let denote the set of all functions satisfying for all .

Theorem 23. Let be a complete symmetric PPM-space that satisfies , where is a topological. Suppose is -continuous and satisfies that for there exists such that for any and for every with . Then has a fixed point.

Proof. Define as in (52). According to Lemmas 21 and 22, is a bounded -Cauchy complete semi-metric space satisfying (W4). Now assume that (54) is satisfied. Let be given and let . Then gives and so and so . Since was arbitrary, we have that By Corollary 13, has a fixed point.

Theorem 24. Let be a complete symmetric PPM-space that satisfies . Let be -continuous such that for all , and . Then has a unique fixed point.

Proof. Define as in (52). According to Lemma 21, is a bounded compatible symmetric for and is -Cauchy complete symmetric space satisfying (W3) and (JMS). Suppose (56) is satisfied and let be given. Let . Then and so , which implies . This further implies that and so Since was arbitrary, we have that Now Theorem 16 guarantees that has a unique fixed point .

7. Some Open Problems

Problem 1 (see [2]). Let be a symmetric space which satisfies the property (MT). Is it a semi-metric space (not necessarily with open balls)?

Problem 2. Does Theorem 16 hold if is replaced with

Conflict of Interests

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

Acknowledgments

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.