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A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application
Journal of Inequalities and Applications volume 2014, Article number: 122 (2014)
Abstract
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.
1 Introduction
In 1975, Matkowski [[1], Theorem 1.2] proved the following distinguished generalization of Banach’s contraction principle.
Theorem 1 [1]
Let be a complete fuzzy metric space and a self-map such that
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 .
Matkowski’s theorem has been generalized or extended in several directions (see e.g. [2–6]). In particular, Jachymski obtained in [[4], Theorem 1] the following nice fuzzy version of it.
Theorem 2 [4]
Let be a complete fuzzy metric space, with ∗ a continuous t-norm of Hadžić type, and let be a self-map such that
for all and , where is a function satisfying and for all . Then f has a unique fixed point.
Remark 2 See [7] or [4] for the notion of a t-norm of Hadžić type (or of h-type).
Recently, Ricarte and Romaguera [[8], 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. [9], and that generalizes C-contractions as defined by Hicks in [10].
Theorem 3 [8]
Let be a complete fuzzy metric space and a self-map such that
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).
2 Background
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 [11] 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.
If d is a quasi-metric on X, then the relation on X given by
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 :
(KM1) ;
(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 [16]).
A simple but useful fact (see e.g. [12, 13]) is that for each KM-fuzzy quasi-metric on a set X and each , the function is nondecreasing.
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.
Remark 3 If is a fuzzy quasi-metric on X, then the relation on X given by
is an order on X, called the specialization order of .
It is interesting to note that, however, the relation given by
is a preorder on X (see e.g. [[17], 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.
Example 2 [12, 13] (see also [[18], Example 2.9])
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 for all and
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 .
3 The fixed point theorem and some of its consequences
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. [19]).
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 .
Theorem 4 Let be a preordered ⪯-complete fuzzy quasi-metric space and a ⪯-continuous nondecreasing self-map such that
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.
Proof Take . Let with . Since , it follows that
Since f is nondecreasing, we deduce that for all , so, by (1) and (2), we immediately deduce that
for all with and .
Now let such that . Put for all . Since f is nondecreasing it follows that is a nondecreasing sequence for ⪯.
Similarly to the proof of [[8], Theorem 2.2], we show that is a left K-Cauchy sequence in . Choose and . Then there exists such that for all . Let . Then for some , so, by (3)
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 .
Finally suppose that for some . From our hypothesis it follows, exactly as in the first part of the proof, that
for all . Since and we deduce, for , and with , that
Since ε is arbitrary, we conclude that for all . Similarly for all , so . This completes the proof. □
Corollary 1 Let be a preordered ⪯-complete fuzzy quasi-metric space and a ⪯-nondecreasing self-map such that
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.
Corollary 2 Let be a complete fuzzy metric space and a self-map such that
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.
Proof Let be any function satisfying and for all . Then, for each such that , we have , and thus
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.
Example 3 Let and let M be the fuzzy set in defined as
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.
Now let defined as
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.)
Let and . Then and , so
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.
Example 4 Let and let be the fuzzy metric on X defined as for all , for all , for all , and otherwise. Now define a preorder ⪯ on X as follows: for all , , and . Obviously ⪯ is not an order on X. Clearly is ⪯-complete. Let be such that , and . Then f is a ⪯-nondecreasing self-map with (also ). Let be any function such that and for all . Since for all , we deduce that the conditions of Corollary 1, and hence of Theorem 4, for with , are satisfied. However, we cannot apply Corollary 2 to this example. Indeed, suppose that there exists a function satisfying and for all , and such that
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 .
4 An application
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 [20] 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.
The complexity (quasi-metric) space (see [20]) consists of the pair , where
and is the quasi-metric on given by
for all . (We adopt the convention that .) The elements of are called complexity functions and is said to be the complexity quasi-metric. Observe that
and thus condition , can be computationally interpreted as f to be ‘more efficient’ than g on all inputs (see [[20], Section 6]).
In our context we shall work on the subset of defined as
and we shall use the function introduced in [26] and defined, for each and , as
where , .
The following well-known facts will be useful.
Remark 5 [[26], Remark 1]
for all and .
Remark 6 whenever and .
Remark 7 [[27], 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 .
Now we construct a fuzzy set in as
Note that for all if and only if .
Then we have:
Lemma 1 is a -complete fuzzy quasi-metric space.
Proof We first see that is a fuzzy quasi-metric on (recall that for all ). Indeed, conditions (KM1), (KM2) and (KM4) of Definition 1 are almost trivially satisfied, while condition (KM3) follows immediately from the fact showed in [[26], Lemma 1] that
for all and . Hence is a fuzzy quasi-metric space.
Now let be a nondecreasing left K-Cauchy sequence in . Then for all , so, by Remark 7,
Since for all (note that ), we deduce that
for all and . On the other hand, and assuming that for all , we deduce from Remark 5 and (5) that
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.
Example 5 Consider the recurrence equation T given by , and
for . (This recurrence was obtained by Kruse [[28], Section 4.8.4] in discussing the average case analysis of Quicksort algorithms.)
In this case, we define a functional as
for all , where (see for instance [29, 30]).
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.
Example 6 It is well known that Divide and Conquer algorithms solve a problem by recursively splitting it into subproblems each of which is solved separately by the same algorithm, after which the results are combined into a solution of the original problem (see e.g. [20, 31]). Thus, the complexity of a Divide and Conquer algorithm typically is the solution of the recurrence equation given by
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.
References
Matkowski J: Integrable solutions of functional equations. Diss. Math. 1975, 127: 1-68.
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109-116. 10.1080/00036810701556151
Jachymski J: Equivalent conditions and the Meir-Keeler type theorems. J. Math. Anal. Appl. 1995, 194: 293-303. 10.1006/jmaa.1995.1299
Jachymski J: On probabilistic φ -contractions on Menger spaces. Nonlinear Anal. TMA 2010, 73: 2199-2203. 10.1016/j.na.2010.05.046
Karapinar E, Romaguera S, Tas K: Fixed points for cyclic orbital generalized contractions on complete metric spaces. Cent. Eur. J. Math. 2013, 11: 552-560. 10.2478/s11533-012-0145-0
Romaguera S: Matkowski’s type theorems for generalized contractions on (ordered) partial metric spaces. Appl. Gen. Topol. 2011, 12: 213-220.
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht; 2001.
Ricarte LA, Romaguera S: On φ -contractions in fuzzy metric spaces with application to the intuitionistic setting. Iran. J. Fuzzy Syst. 2013,10(6):63-72.
Huang X, Zhu C, Wen X:On -contraction in intuitionistic fuzzy metric spaces. Math. Commun. 2010, 15: 425-435.
Hicks TL: Fixed point theory in probabilistic metric spaces. Zb. Rad. Prir.-Mat. Fak. (Novi Sad) 1983, 13: 63-72.
Künzi HPA: Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology. 3. In Handbook of the History of General Topology. Edited by: Aull CE, Lowen R. Kluwer Academic, Dordrecht; 2001:853-968.
Cho YJ, Grabiec M, Radu V: On Nonsymmetric Topological and Probabilistic Structures. Nova Science Publishers, New York; 2006.
Gregori V, Romaguera S: Fuzzy quasi-metric spaces. Appl. Gen. Topol. 2004, 5: 129-136.
Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 314-334.
Schweizer B, Sklar A: Probabilistic Metric Spaces. North-Holland, New York; 1983.
Kramosil I, Michalek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975, 11: 326-334.
Rodríguez-López J, Romaguera S, Sánchez-Álvarez JM: The Hausdorff fuzzy quasi-metric. Fuzzy Sets Syst. 2010, 161: 1078-1096. 10.1016/j.fss.2009.09.019
George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64: 395-399. 10.1016/0165-0114(94)90162-7
George A, Veeramani P: On some results of analysis of fuzzy metric spaces. Fuzzy Sets Syst. 1997, 90: 365-368. 10.1016/S0165-0114(96)00207-2
Schellekens M: The Smyth completion: a common foundation for denotational semantics and complexity analysis. Electronic Notes Theoret. Comput. Sci. 1. Proc. MFPS 11 1995, 535-556.
García-Raffi LM, Romaguera S, Sánchez-Pérez EA: Sequence spaces and asymmetric norms in the theory of computational complexity. Math. Comput. Model. 2002, 36: 1-11. 10.1016/S0895-7177(02)00100-0
García-Raffi LM, Romaguera S, Schellekens M: Applications of the complexity space to the general probabilistic Divide and Conquer algorithms. J. Math. Anal. Appl. 2008, 348: 346-355. 10.1016/j.jmaa.2008.07.026
Romaguera S, Schellekens M: Quasi-metric properties of complexity spaces. Topol. Appl. 1999, 98: 311-322. 10.1016/S0166-8641(98)00102-3
Romaguera S, Schellekens M: Partial metric monoids and semivaluation spaces. Topol. Appl. 2005, 153: 948-962. 10.1016/j.topol.2005.01.023
Romaguera S, Valero O: On the structure of the complexity space of partial functions. Int. J. Comput. Math. 2008, 85: 631-640. 10.1080/00207160701210117
Romaguera S, Tirado P: The complexity probabilistic quasi-metric space. J. Math. Anal. Appl. 2011, 376: 732-740. 10.1016/j.jmaa.2010.11.056
Romaguera S, Schellekens MP, Valero O: Complexity spaces as quantitative domains of computation. Topol. Appl. 2011, 158: 853-860. 10.1016/j.topol.2011.01.005
Kruse R: Data Structures and Program Design. Prentice Hall, New York; 1984.
Romaguera S, Sapena A, Tirado P: The Banach fixed point theorem in fuzzy quasi-metric spaces with application to the domain of words. Topol. Appl. 2007, 154: 2196-2203. 10.1016/j.topol.2006.09.018
Saadati R, Vaezpour SM, Cho YJ: Quicksort algorithm: application of fixed point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words. J. Comput. Appl. Math. 2009, 228: 219-225. 10.1016/j.cam.2008.09.013
Aho V, Hopcroft J, Ullman J: Data Structures and Algorithms. Addison-Wesley, Reading; 1987.
Acknowledgements
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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Castro-Company, F., Romaguera, S. & Tirado, P. A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application. J Inequal Appl 2014, 122 (2014). https://doi.org/10.1186/1029-242X-2014-122
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DOI: https://doi.org/10.1186/1029-242X-2014-122