- Open Access
A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application
© Castro-Company et al.; licensee Springer. 2014
- Received: 18 November 2013
- Accepted: 19 February 2014
- Published: 27 March 2014
We obtain a fixed point theorem for a type of generalized contractions on preordered complete fuzzy quasi-metric spaces which is applied to deduce, among other results, a procedure to show in a direct and easy fashion the existence of solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively.
MSC:47H10, 54H25, 06A06, 68Q25.
- fixed point
- preordered fuzzy quasi-metric space
- recurrence equation
In 1975, Matkowski [, Theorem 1.2] proved the following distinguished generalization of Banach’s contraction principle.
Theorem 1 
for all , where is a nondecreasing function satisfying for all . Then f has a unique fixed point.
Remark 1 It is well known and easy to check that if is a nondecreasing function such that for all , then for all .
Theorem 2 
for all and , where is a function satisfying and for all . Then f has a unique fixed point.
Recently, Ricarte and Romaguera [, Theorem 2.2] established the following new fuzzy version of Matowski’s theorem by using a type of contraction introduced in the fuzzy intuitionistic context by Huang et al. , and that generalizes C-contractions as defined by Hicks in .
Theorem 3 
for all and , where is a nondecreasing function satisfying for all . Then f has a unique fixed point.
In this paper we obtain a generalization of Theorem 3 to preordered fuzzy quasi-metric spaces which is applied to deduce, among other results, a procedure to show in a direct and easy way the existence of solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively. The key for this application is the nice fact that, for the specialization order of a fuzzy quasi-metric space, the contraction condition of Theorem 3 is automatically satisfied whenever the self-map f is nondecreasing for the specialization order and is any function verifying for all (see Theorem 5 in Section 3).
In this section we recall several notions and properties which will be useful in the rest of the paper. Our basic reference for quasi-metric spaces is  and for fuzzy (quasi-)metric spaces they are [12, 13].
The letters ℝ, ℕ and ω will denote the set of real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively.
A preorder on a nonempty set X is a reflexive and transitive binary relation ⪯ on X.
The preorder ⪯ is called a partial order, or simply an order, if it is antisymmetric (i.e., condition and , implies ).
Note that for any nonempty set X, the binary relation defined by if and only if , is obviously a preorder on X, the so-called trivial preorder on X.
A quasi-metric on a set X is a function such that for all : (i) if and only if , and (ii) .
A quasi-metric space is a pair such that X is a nonempty set and d is a quasi-metric on X.
Given a quasi-metric d on X the function defined by for all , is a metric on X.
is an order on X, called the specialization order of d.
According to [14, 15], a binary operation is a continuous t-norm if ∗ satisfies the following conditions: (i) ∗ is associative and commutative; (ii) ∗ is continuous; (iii) for every ; (iv) whenever and , with .
Two interesting examples are ∧, and , where, for all , , and is the well-known Lukasiewicz t-norm defined by .
It seems appropriate to point out that ∗ ≤ ∧ for any continuous t-norm ∗.
A KM-fuzzy quasi-metric on a set X is a pair such that ∗ is a continuous t-norm and M is a fuzzy set in (i.e., a function from into ) such that for all :
(KM2) if and only if for all ;
(KM3) for all ;
(KM4) is left continuous.
A KM-fuzzy quasi-metric on X such that for each :
(KM5) for all
is a fuzzy metric on X (in the sense of Kramosil and Michalek ).
In the following, KM-fuzzy quasi-metrics will be simply called fuzzy quasi-metrics.
If is a fuzzy quasi-metric on a set X, then the pair is a fuzzy metric on X where is the fuzzy set in defined by for all and .
As in the fuzzy metric case, each fuzzy quasi-metric on a set X induces a topology on X which has as a base the family of open balls , where .
It immediately follows that a sequence in a fuzzy quasi-metric space converges to a point with respect to if and only if for all .
Definition 2 A fuzzy (quasi-)metric space is a triple such that X is a set and is a fuzzy (quasi-)metric on X.
Definition 3 A preordered fuzzy (quasi-)metric space is 4-tuple such that is a fuzzy (quasi-)metric space and ⪯ is a preorder on X.
The notion of an ordered fuzzy (quasi-)metric space is defined in the obvious manner.
is an order on X, called the specialization order of .
is a preorder on X (see e.g. [, Remark 3]).
We conclude this section with two typical examples of fuzzy quasi-metrics induced by a given quasi-metric space.
Example 1 Let be a quasi-metric space. Then the pair is a fuzzy quasi-metric on X where ∗ is any continuous t-norm and is the fuzzy set on given by if and , and otherwise.
for all and .
Remark 4 Let be a quasi-metric space. It is clear that the specialization orders , and coincide, while the preorder coincides with the trivial preorder on X, and the preorder coincides with the specialization order .
We start this section with the notions of fuzzy quasi-metric completeness and continuity of self-maps which will be used in our main result (Theorem 4 below).
A left K-Cauchy sequence in a fuzzy quasi-metric space is a sequence in X such that for each and each there is such that whenever .
A preordered fuzzy quasi-metric space will be called ⪯-complete if for each nondecreasing left K-Cauchy sequence there is such that converges to x with respect to , and for all .
Observe that if is a preordered fuzzy metric space which is -complete, then is a complete fuzzy metric space in the usual sense (see e.g. ).
Let be a preordered fuzzy quasi-metric space. A self-map is said to be ⪯-nondecreasing if condition implies for all , and it will be called ⪯-continuous if whenever is a nondecreasing sequence for ⪯, which converges with respect to to some such that for all , then the sequence converges to fx with respect to .
for all with , and , where satisfies for all . If there is such that , then .
If, in addition, condition (1) is satisfied for each , then f has a unique fixed point.
for all with and .
Now let such that . Put for all . Since f is nondecreasing it follows that is a nondecreasing sequence for ⪯.
Therefore is a nondecreasing left K-Cauchy sequence in . Since is ⪯-complete there exists such that for all , and for all . So for all , by ⪯-continuity of f. Hence .
Since ε is arbitrary, we conclude that for all . Similarly for all , so . This completes the proof. □
for all with , and , where satisfies and for all . If there is such that , then .
If, in addition, condition (4) is satisfied for each , then f has a unique fixed point.
Proof We shall show that f is ⪯-continuous in . Let be a nondecreasing sequence for ⪯, convergent with respect to to some such that for all . Given there is such that . By condition (4) and the fact that it follows that for all . Therefore f is ⪯-continuous. The conclusions follow from Theorem 4. □
As an immediate consequence of Corollary 1 we obtain the following improvement of Theorem 3.
for all and , where satisfies and for all . Then f has a unique fixed point.
Proof It is clear that is a preordered -complete fuzzy metric space. Now the result follows from Corollary 1. □
The contraction (4) is almost trivially satisfied in the case that the preorder ⪯ is the specialization order and f is nondecreasing for . This situation, which will be crucial in our application in Section 4, is described in the following result.
Theorem 5 If the ordered fuzzy quasi-metric space is -complete and is a -nondecreasing self-map such that there is satisfying , then f has a fixed point.
for all . This shows that condition (4) is satisfied, and thus f has a fixed point by Corollary 1. □
We conclude this section with two examples illustrating the obtained results.
It is routine to check that is a fuzzy quasi-metric. Note that the specialization order coincides with the usual order ≤ on X. Moreover, a sequence in X is left K-Cauchy in if and only it is eventually constant, so is an ordered ≤-complete fuzzy quasi-metric space.
Clearly for all . We show that if is any ≤-nondecreasing self-map, the contraction condition (1) is satisfied for with and . (Note that f is automatically continuous and hence ≤-continuous, because is the discrete topology on X.)
so condition (1) is trivially satisfied. If, in addition, there exists such that , then f has a fixed point by Theorem 4 or Theorem 5. Observe that in this example we cannot apply Corollary 1 because for ; in fact, it is not nondecreasing.
for all . Since condition holds for , we deduce that, for , , so because . Repeating this argument, we deduce that for all , which contradicts the fact that .
In this section we apply Theorem 5 to obtain a general procedure from which we can deduce in a fast and easy fashion the existence the solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively.
Let us recall that Schellekens introduced in  the so-called complexity quasi-metric space in order to construct a topological foundation for the complexity analysis of programs and algorithms. In that paper, he also applied his theory to show that the existence and uniqueness of solution for the recurrence associated to Divide and Conquer algorithms. Further contributions to the study of these spaces and of other related ones may be found in [12, 21–25], etc.
and thus condition , can be computationally interpreted as f to be ‘more efficient’ than g on all inputs (see [, Section 6]).
where , .
The following well-known facts will be useful.
Remark 5 [, Remark 1]
for all and .
Remark 6 whenever and .
Remark 7 [, Lemma 2]
Let be a sequence in such that for all , and let defined as for all . Then .
In the following, for any , by we mean that for all .
Note that for all if and only if .
Then we have:
Lemma 1 is a -complete fuzzy quasi-metric space.
for all and . Hence is a fuzzy quasi-metric space.
for all , and thus for all . By (6) and (7) we deduce that converges to F with respect to . We have shown that is a -complete fuzzy quasi-metric space. □
Taking into account the preceding constructions and results, we immediately obtain the following consequence of Theorem 5.
Theorem 6 If is a -nondecreasing map and there is such that , then Φ has a fixed point.
We finish the paper by applying Theorem 6 to show the existence of solution for the recurrence equations associated to Quicksort and Divide and Conquer algorithm, respectively.
for . (This recurrence was obtained by Kruse [, Section 4.8.4] in discussing the average case analysis of Quicksort algorithms.)
It is clear that if then , so by the definition of , Φ is nondecreasing for . Moreover, the complexity function defined as for all , satisfies , i.e., . Therefore, we can apply Theorem 6 and thus Φ has a fixed point . Consequently, the function given by and for all is solution of the recurrence equation T.
where with , n ranges over the set and if .
This recurrence equation induces, in a natural way the associated functional defined by , , if and if .
As in Example 5, Φ is -nondecreasing. Since the complexity function defined as for all , satisfies , we can apply Theorem 6 and thus Φ has a fixed point which is solution of the recurrence equation T.
The second and third named authors thank the supports of the Universitat Politècnica de València, grant PAID-06-12-SP20120471, and the Ministry of Economy and Competitiveness of Spain, grant MTM2012-37894-C02-01.
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