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Fixed point theorems in probabilistic metric spaces
Chaos, Solitons and Fractals 41 (2009) 1014–1019
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Fixed point theorems in probabilistic metric spaces Dorel Mihetß * West University of Timisßoara, Faculty of Mathematics and Computer Science, Bv. V. Parvan 4, 300223 Timisßoara, Romania
a r t i c l e
i n f o
Article history: Accepted 23 April 2008
a b s t r a c t We extend some results in [Ghaemi MB, Razani A. Fixed and periodic points in the probabilistic normed and metric spaces. Chaos, Solitons & Fractals 2006;28:1181–7] and answer in the affirmative an open question raised in the above quoted paper. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries The advances in the art of measurement in the 19th century stimulated a corresponded concern with the accompanying errors. Motivated by the idea that in practice and – as, e.g., quantum mechanics implies – even in theory, some measurements are necessarily inexact, in his note Statistical Metrics [9] Karl Menger proposed transferring the probabilistic notions of quantum mechanics from the physics to the underlying geometry and showed how one could replace a numerical distance between points p and q by a distribution function F pq . Subsequently, numerous authors studied such spaces, as did e.g., the excellent books ‘‘Probabilistic Metric Spaces” [14] by Berthold Schweizer and Abe Sklar and ‘‘Fixed Point Theory in Probabilistic Metric Spaces” [6] by Olga Hadzˇic´ and Endre Pap. The contribution of Menger to resolving the interpretative issue of quantum mechanics turned out to be of fundamental importance in probabilistic functional analysis and nonlinear analysis; see e.g. [6]. As it has been described in [5], probabilistic metric spaces are also useful in modelling some phenomena connected with both string and E-infinity theory [2,3]. For instance, the process in the analysis of the probability involved in the two-slit experiment can be modelled by means of a probabilistic metric. In the following we recall some well-known definitions and results in the theory of probabilistic metric spaces used later on in the paper. For more details we refer the reader to [6, Chapters 1 and 2] and [14, Chapters 8 and 12]. A triangular norm (shorter t-norm) is a binary operation T on [0,1], which is associative, commutative, non-decreasing at both places and has 1 as the unit element. Basic examples are the t-norms T L (Łukasiewicz t-norm), T P and T M (Min norm), defined by T L ða; bÞ ¼ maxfa þ b 1; 0g; T P ða; bÞ ¼ ab and T M ða; bÞ ¼ minfa; bg. As regard to pointwise ordering, T M is the strongest t-norm, that is T 6 T M for every t-norm T. A distance distribution function is any mapping F : ½0; 1Þ ! ½0; 1 which is non-decreasing and left continuous on ð0; 1Þ and Fð0Þ ¼ 0. The class of all distribution functions is denoted by Dþ . Dþ is the subset of Dþ containing all functions F which also satisfy limt!1 FðtÞ ¼ 1. A special element of Dþ is the function e0 , defined by 0; if t ¼ 0 : e0 ðtÞ ¼ 1; if t > 0 Definition 1.1 [14]. A probabilistic semi-metric space (shortly PSM space) is a pair ðS; FÞ where S is a nonempty set and F is a mapping from S S to Dþ satisfying the following properties (as usually, Fðx; yÞ is denoted by F xy ):
* Fax: +40 256 592316. E-mail addresses: [email protected], [email protected] 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.030
D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019
ðPM0Þ :
F pq ¼ e0 () p ¼ q:
ðPM1Þ :
F pq ¼ F qp 8p; q 2 S:
1015
A PSM space satisfying a certain kind of ‘‘triangle inequality” is called a probabilistic metric space. Definition 1.2 [14]. A probabilistic metric space in the sense of Schweizer and Sklar (or a generalized Menger space) is a triple ðS; F; TÞ, where ðS; FÞ is a PSM space, T is a t-norm and the following triangle inequality holds: F pr ðx þ yÞ P TðF pq ðxÞ; F qr ðyÞÞ
8p; q; r 2 S;
8x; y P 0:
A Menger PM space is a generalized Menger space such that RangeðFÞ Dþ . Definition 1.3 [8]. A PSM space ðS; FÞ satisfying the triangle inequality: 8e > 09d > 0 : F pq ðdÞ > 1 d;
F qr ðdÞ > 1 d ) F pr ðeÞ > 1 e
is called an H-space. Every generalized Menger space ðS; F; TÞ with T satisfying supa<1 Tða; aÞ ¼ 1 is an H-space. If ðX; FÞ is an H-space, then ([14]) the family fU e;k ge>0;k2ð0;1Þ where U e;k ¼ fðx; yÞ 2 X XjF xy ðeÞ > 1 kg is a base for a metrizable uniformity on X, called the F-uniformity and denoted by UF . The F-uniformity is also generated by the family fV d gd>0 , where V d :¼ U d;d . UF naturally determines a topology TF on X, named the F-topology or the strong topology. It is a basic fact that in a metric space ðS; dÞ the distance function d is a continuous mapping from S S to ½0; 1Þ. As ð½0; 1; TÞ is only a semigroup, the corresponding result does not hold in Menger spaces. However, a well-known theorem of Schweizer and Sklar asserts that the continuity of T implies the continuity of the probabilistic metric, in the following sense: Theorem 1.1. ([14], Theorem 12.2.3). Let ðS; F; TÞ be a generalized Menger space with T continuous. If ðpn Þ and ðqn Þ are sequences in S such that pn ! p 2 S and qn ! q 2 S, then F pn qn ðtÞ ! F pq ðtÞ for every continuity point t of F pq . Fixed point theory in probabilistic metric spaces is a part of a very dynamic area of mathematical research, Probabilistic Analysis. It has been developed starting with the paper of Sehgal and Bharucha-Reid [16], where the notion of probabilistic B-contraction was introduced and a generalization of the classical Banach fixed point principle to complete Menger PM spaces was given. Another class of probabilistic contractions (named Hicks C-contractions or C-contractions) has been introduced by Hicks in [7]. Some results in fixed point theory in probabilistic metric spaces have applications to control theory, system theory and optimization problems; in a recent paper [13] a fixed point theorem in Menger quasi-metric spaces was applied to prove the existence of solution for some recurrence equations associated to the asymptotic complexity analysis of Quicksort algorithms and Divide & Conquer algorithms. Definition 1.4. Let ðS; FÞ is a PSM space. According to [15], a mapping A : S ! S is called (a) a probabilistic B-contraction if, for some k 2 ð0; 1Þ, F AðpÞAðqÞ ðktÞ P F pq ðtÞ for all p; q 2 S; t > 0; (b) a probabilistic C-contraction if the following implication holds for every p; q 2 S and t > 0: F pq ðtÞ > 1 t ) F AðpÞAðqÞ ðktÞ > 1 kt
ðHÞ
(k 2 ð0; 1Þ is given). The present paper is related to some fixed point theorems in [4]. We extend the results of Ghaemi and Razani concerning Hicks C-contractions in Menger PM spaces to a larger class of contractions in H-spaces. Then we answer in the affirmative an open question about periodic points of a contractive-type mapping in Menger PM spaces, raised in this paper.
2. Hicks-type probabilistic contractions in H-spaces The following general result about Hicks C-probabilistic contractions has been proved by Radu: Theorem 2.1 [11]. Let ðS; F; TÞ be a complete Menger PM space such that supa<1 Tða; aÞ ¼ 1. Then every C-contraction A on S has a unique fixed point which is the limit of the sequence ðAn ðpÞÞn2N for every p 2 S. In [4] the existence of a fixed point for a C-contraction was obtained from the convergence of a subsequence of the sequence of successive approximations.
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D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019
Theorem 2.2 [4], Theorem 2.3. Let ðS; F; TÞ be a Menger PM space. If A is a C-contraction on S and, for some u in S, ðAni ðuÞÞ is a convergent subsequence of ðAn ðuÞÞ, then f ¼ limi!1 Ani ðuÞ is the unique fixed point of A. We are going to show that in the above theorem instead Menger PM spaces one can consider even H-spaces (which are generalized Menger spaces). Then, we obtain a similar result by weakening the contractive condition. The next theorem is an extension of Theorem 2.2 to Hicks C-contractions on H-spaces. Its proof is based on the remark that every Cauchy sequence in a uniform space containing a convergent subsequence is convergent and the following: Lemma 2.1. (see [11]). Let ðS; FÞ be an H-space and A be a C-contraction on S. Then, for every p 2 S, the sequence ðAn ðpÞÞ is a Cauchy sequence. The proof of the lemma immediately follows from the implication: k > 1 ) F pq ðkÞ > 1 k: Theorem 2.3. Let ðS; FÞ be an H-space and A be a C-contraction on S. Then A has a unique fixed point iff there is p 2 S such that ðAn ðpÞÞ has a convergent subsequence. Proof. From the above lemma it follows that the sequence ðAn ðpÞÞ converges to a fixed point of A, for every Hicks C-contraction is (uniformly) continuous. The uniqueness can be proved in the usual way. In the following we prove a similar result for a larger class of probabilistic contractions, recently introduced in [10]. h Definition 2.1 [10]. Let ðS; FÞ be a PSM space. A self-mapping A of S is called a weak–Hicks contraction (shortly w–H contraction) if ðHÞ holds for all p; q 2 S and t 2 ð0; 1Þ, that is, the following implication takes place for every p; q 2 ð0; 1Þ and t 2 ð0; 1Þ: F pq ðtÞ > 1 t ) F AðpÞAðqÞ ðktÞ > 1 kt (k 2 ð0; 1Þ is given). The difference between this type of contractions and Hicks C-contractions has nicely been illustrated by Radu in the following example: Example 2.1 [12]. Let X be a set containing at least two elements. Consider the discrete Menger space under T M determined (for x–y) by the probabilistic metric 0; if t 6 1 : F xy ðtÞ ¼ 1; if t > 1 Then any mapping A : X ! X is a weak-Hicks contraction, while the only Hicks C-contractions on ðX; F; T M Þ are the constant mappings. Because in the case of w-H contractions the relation ðHÞ takes place only for t 2 ð0; 1Þ, the implication justifying Lemma 2.1 cannot be used. However, as one can see in the next theorem, the existence of a fixed point for a weak-Hicks contraction A is guaranteed by the weak condition F xAðxÞ ð1Þ > 0. Theorem 2.4. Let ðX; FÞ be an H space and A be a weak-Hicks contraction on X. If there exists x 2 X such that F xAðxÞ ð1Þ > 0 and the sequence ðAn ðxÞÞn2N has a convergent subsequence, then ðAn ðxÞÞ converges to a fixed point of A. Proof. As F xAðxÞ ð1Þ > 0 and F xAðxÞ is left continuous, there is d 2 ð0; 1Þ such that F xAðxÞ ðdÞ > 1 d. Using (H), it immediately follows by induction that n
n
F An ðxÞAnþ1 ðxÞ ðk dÞ > 1 k d
8n 2 N: n
If e > 0 is given and n0 2 N is such that k 0 d < e, by the above inequality one obtains F An ðxÞAnþ1 ðxÞ ðeÞ > 1 e
8n P n0 :
Let xn ¼ An ðxÞ and ðxni Þi2N be a subsequence of ðxn Þ converging to y. We show that limi!1 xni þ1 ¼ y. Indeed, we know that for every e > 0 there is d > 0 such that F pq ðdÞ > 1 d;
F qr ðdÞ > 1 d ) F pr ðeÞ > 1 e:
Choosing i0 for which F xni xn þ1 ðdÞ > 1 d8i P i0 and F xni y ðdÞ > 1 d8i P i0 , it follows F xn þ1 y ðeÞ > 1 e8i P i0 , as claimed. i i On the other hand, as any weak-Hicks contraction on an H-space is uniformly continuous, limi!1 xni þ1 ¼ AðyÞ. Therefore, we have Ay ¼ y. Let us now prove that ðxn Þn2N converges to y. We know that for every e > 0 and d 2 ð0; 1Þ there is l 2 N such that F xni y ðeÞ > 1 d8i P l. Let e > 0; d 2 ð0; 1Þ be given and m be an arbitrary natural number P nl . If m ¼ nl þ s, then from
D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019 s
1017
s
F Anl ðxÞy ðeÞ ¼ F xnl y ðeÞ > 1 d it follows F xm y ðk eÞ ¼ F Anl þs ðxÞAs ðyÞ ðk eÞ > 1 d, hence F xm y ðeÞ > 1 d8m P nl , which conclude the proof. h We note that a weak-Hicks contraction may have more than one fixed point (see [10, Example 2.1]). 3. Periodic points This section is devoted to a question settled in [4], concerning the existence of periodic points for an e-contractive mapping in Menger PM spaces ðS; F; TÞ with T–T M . The next definition is a slight modification of Definition 3.2 in [4]. Definition 3.1. Let ðX; FÞ be a PSM space and e 2 ð0; 1Þ. A mapping f : X ! X is called probabilistic e-contractive if F f ðxÞf ðyÞ ðtÞ > F xy ðtÞ whenever F xy ðtÞ > 1 e and x–y. It is easy to see that every probabilistic e-contractive mapping in a generalized Menger space ðS; F; TÞ with T such that supa<1 Tða; aÞ ¼ 1 is continuous. Indeed, let k 2 ð0; 1Þ be given. Choose d 2 ð0; minfe; kgÞ and let x; y and t be such that F xy ðtÞ > 1 d. If x–y, then F f ðxÞf ðyÞ ðtÞ > F xy ðtÞ > 1 k, while if x ¼ y then f ðxÞ ¼ f ðyÞ and again F f ðxÞf ðyÞ ðtÞ > 1 k. Definition 3.2 [4]. A point n 2 X is called a periodic point for f if there is a positive integer k such that f k ðnÞ ¼ n. Theorem 3.1 [4], Theorem 3.3. Let ðX; F; T M Þ be a generalized Menger space. Suppose that f is a probabilistic e-contractive selfmapping of X such that there exists a point x 2 X whose sequence of iterates ðf n ðxÞÞ contains a convergent subsequence ðf ni ðxÞÞ. Then n ¼ limi!1 f ni ðxÞ is a periodic point for f. In the next theorem we prove that Theorem 3.1 remains true even if T M is replaced by an arbitrary continuous t-norm. Thus, we answer into affirmative Question 3.7 in [4]. The following continuity lemma is an improvement of Theorem 1.1. Lemma 3.1 [1]. Let ðS; F; TÞ be a generalized Menger space with T satisfying the condition ðcÞ : lim Tðt; sÞ ¼ s t!1
for all t; s 2 ½0; 1. If ðpn Þ and ðqn Þ are sequences in S such that pn ! p 2 S and qn ! q 2 S, then F pn qn ðtÞ ! F pq ðtÞ for every continuity point t of F pq . Theorem 3.2. Let (X; F; TÞ be a generalized Menger space with the t-norm T satisfying limt!1 Tðt; sÞ ¼ s for all t; s 2 ½0; 1. Then for every probabilistic e-contractive mapping f on X with the property that there exists a point x 2 X whose sequence of iterates ðf n ðxÞÞn2N contains a convergent subsequence, the point n ¼ limi!1 f ni ðxÞ is a periodic point. Proof. Since T is continuous at (1,1), there is d 2 ð0; eÞ such that Tð1 d; 1 dÞ > 1 e. Also, there is a positive integer N 1 such that i P N 1 implies F f ni ðxÞn ðt=2Þ > 1 d; 8t > 0. Fix a k P N 1 and denote nkþ1 nk by s. As f is probabilistic e-contractive and F f nk ðxÞn ðt=2Þ > 1 e, we have F f nk þ1 ðxÞf ðnÞ ðt=2Þ PF f nk ðxÞn ðt=2Þ >1 d > 1 e and, after nkþ1 nk iterations, F f nkþ1 ðxÞf s ðnÞ ðt=2Þ > 1 d. Therefore, F n;f s ðnÞ ðtÞ P TðF f nkþ1 ðxÞn ðt=2Þ; F f nkþ1 ðxÞf s ðnÞ ðt=2ÞÞ P Tð1 d; 1 dÞ > 1 e ni
Since f is continuous, limi!1 f ðxÞ ¼ n implies limi!1 f lim F f ni ðxÞf ni þs ðxÞ ðtÞ ¼ F nf s ðnÞ ðtÞ i!1
ni þs
8t > 0:
s
ðxÞ ¼ f ðnÞ, therefore, by Theorem 1.1,
8t > 0:
The sequence of real numbers ðzn ÞnPnk ; zn :¼ F f n ðxÞf nþs ðxÞ ðtÞnPnk is convergent for every t > 0 (being non-decreasing and bounded), therefore lim F f n ðxÞf nþs ðxÞ ðtÞ ¼ F nf s ðnÞ ðtÞ
n!1
8t > 0:
On the other hand, from f ni ðf ðxÞÞ ¼ f ðf ni ðxÞÞ!i!1 f ðnÞ and f ni ðf sþ1 ðxÞÞ ¼ f sþ1 ðf ni ðxÞÞ ! f sþ1 ðnÞ it follows that lim F f ni þ1 ðxÞf ni þ1þs ðxÞ ðtÞ ¼ F f ðnÞf sþ1 ðnÞ ðtÞ i!1
8t > 0
that is, F nf s ðnÞ ðtÞ ¼ F f ðnÞf sþ1 ðnÞ ðtÞ
8t > 0:
We claim that f s ðnÞ ¼ n. Indeed, if we had f s ðnÞ–n, then 1 e < F nf s ðnÞ ðtÞ for all t > 0 would imply F nf s ðnÞ ðt0 Þ < F f ðnÞf sþ1 ðnÞ ðt 0 Þ, which is a contradiction. Therefore, f s ðnÞ ¼ n, concluding the proof. h
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D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019
Next we show that if ðX; F; TÞ is a Menger PM space, then any periodic point is actually a fixed point. Theorem 3.3. Let (X; F; TÞ be a Menger PM space with the t-norm satisfying the condition (c) from Lemma 3.1 and f : X ! X be a probabilistic e-contractive mapping. Suppose that, for some x 2 X, the sequence ðf n ðxÞÞn2N contains a convergent subsequence and let x 2 X be its limit. Then x is the unique fixed point of f. Proof. The proof is similar to that of Theorem 3.2. Let xn ¼ f n ðxÞ and ðxnk Þk2N a subsequence converging to x . Since f is continuous, the sequence (f ðxnk ÞÞk2N converges to f ðx Þ and ðf ðf ðxnk ÞÞÞk2N converges to f ðf ðx ÞÞ. Therefore, by Lemma 3.1, F xnk f ðxnk Þ ðtÞ ! F x f ðx Þ ðtÞ and F f ðxn
k
Þf 2 ðxnk Þ ðtÞ
! F f ðx Þf 2 ðx Þ ðtÞ
for every continuity point t of F x f ðx Þ and F f ðx Þf 2 ðx Þ . Choose any t 0 > 0 such that F x f ðx Þ ðt 0 Þ > 1 e; F f ðx Þf 2 ðx Þ ðt0 Þ > 1 e and F x f ðx Þ ; F f ðx Þf 2 ðx Þ are continuous in t0 (this is possible, because F x f ðx Þ and F f ðx Þf 2 ðx Þ are in Dþ hence they have only a countable number of discontinuity points). Since F xf ðxÞ ðt 0 Þ > 1 e, the sequence ðzn Þn2N ; zn :¼ F xn f ðxn Þ ðt0 Þ is a non-decreasing sequence of numbers in ½0; 1, therefore it is convergent. Since its subsequence ðznk Þk2N converges to F x f ðx Þ ðt0 Þ, we have limn!1 zn ¼ F x f ðx Þ ðt 0 Þ. As limn!1 znþ1 ¼ limn!1 F f ðxn Þf 2 ðxn Þ ðt 0 Þ ¼ F f ðx Þf 2 ðx Þ ðt0 Þ, the equality F f ðx Þf 2 ðx Þ ðt0 Þ ¼ F x f ðx Þ ðt0 Þ holds. If we had x –f ðx Þ, then as F x f ðx Þ ðt0 Þ > 1 e and f is probabilistic e-contractive, we would also have F f ðx Þf 2 ðx Þ ðt0 Þ > F x f ðx Þ ðt0 Þ, contradicting the above equality. For the uniqueness, let u; v be fixed points of f. As F uv 2 Dþ , there is t > 0 such that F uv ðtÞ > 1 e. If u–v, then F uv ðtÞ ¼ F f ðuÞf ðvÞ ðtÞ > F uv ðtÞ, which is impossible. Therefore, it must be the case that u ¼ v. It is worth noting that the condition RangeðFÞ Dþ among the hypotheses of the preceding theorem is essential. A probabilistic e-contractive mapping in a generalized Menger space may have more than one fixed point, as well as we can find probabilistic e-contractive mappings for which the limit of ðxnk Þ is not a fixed point of f. h Example 3.1. It is easy to see that the triple ðN ; F; T M Þ where, for x–y; F xy ðtÞ ¼ 12 8t > 0 is a (generalized) Menger space. The mapping f : N ! N 1; if x is even f ðxÞ ¼ 2; if x is odd is probabilistic 12-contractive, for the implication x–y; 1 e < F xy ðtÞ ) F f ðxÞf ðyÞ ðtÞ > F xy ðtÞ takes place in absence. The sequence of the successive approximations of 1 is: 2,1,2,1,2,1. . ., and its subsequence 1,1,. . ., converges to 1, which is not a fixed point, but a periodic point for f. Also, the probabilistic 12-contractive mapping g : X ¼ N , 1; if x is odd gðxÞ ¼ 2; if x is even has two fixed points, x ¼ 1 and x ¼ 2. 4. Conclusions In this work we have extended a fixed point theorem by Ghaemi and Razani [4] to a larger class of PM spaces and have shown that the result can be yet improved by weakening the contractive condition. In the final part of the paper we have answered in the affirmative an open question raised in [4]. References [1] Chang SS, Lee BS, Cho YJ, Chen YQ, Kang SM, Jung JS. Generalized contraction mapping principle and differential equations in probabilistic metric spaces. Proc AMS 1996;124:2367–76. [2] El Naschie MS. On a fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int J Nonlin Sci Numer Simul 2005;6:517–29. [3] El Naschie MS. The two-slit experiment as the foundation of E-infinity of high energy physics. Chaos, Solitons & Fractals 2005;25:509–14. [4] Ghaemi MB, Razani A. Fixed and periodic points in the probabilistic normed and metric spaces. Chaos, Solitons & Fractals 2006;28:1181–7. [5] Gregori V, Romaguera S, Veeramani P. A note on intutionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2006;28:902–5. [6] Hadzˇic´ O, Pap E. Fixed point theory in probabilistic metric spaces. Dordrecht: Kluwer Academic Publishers; 2001. [7] Hicks TL. Fixed point theory in probabilistic metric spaces. Zb Rad Prir Mat Fak Ser Mat 1983;13:63–72. [8] Hicks TL, Sharma PL. Probabilistic metric structures: Topological classification. Zb Rad Prirod -Mat Fak Ser Mat Univ u Novom Sadu 1984;14:43–50. [9] Menger K. Statistical metrics. Proc Nat Acad Sci USA 1942;28:535–7. [10] Mihetß D. Weak-Hicks contractions. Fixed Point Theory 2005;6(1):71–8. [11] Radu V. Some fixed point theorems in probabilistic metric spaces. Lect Notes Math 1987;12333:125–33.
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[12] Radu V. Some suitable metrics on fuzzy metric spaces. Fixed Point Theory 2004;5(2):323–47. [13] 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;154:2196–203. [14] Schweizer B, Sklar A. Probabilistic metric spaces. North-Holland, New York: Elsevier; 1983. [15] Schweizer B, Sherwood H, Tardiff R. Contractions on PM spaces: examples and counter examples. Stochastica 1988;12(1):5–17. [16] Sehgal VM, Bharucha-Reid AT. Fixed points of contraction mappings on PM-spaces. Math Syst Theory 1972;6:97–100.
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Fixed point theorems in probabilistic metric spaces Dorel Mihetß * West University of Timisßoara, Faculty of Mathematics and Computer Science, Bv. V. Parvan 4, 300223 Timisßoara, Romania
a r t i c l e
i n f o
Article history: Accepted 23 April 2008
a b s t r a c t We extend some results in [Ghaemi MB, Razani A. Fixed and periodic points in the probabilistic normed and metric spaces. Chaos, Solitons & Fractals 2006;28:1181–7] and answer in the affirmative an open question raised in the above quoted paper. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries The advances in the art of measurement in the 19th century stimulated a corresponded concern with the accompanying errors. Motivated by the idea that in practice and – as, e.g., quantum mechanics implies – even in theory, some measurements are necessarily inexact, in his note Statistical Metrics [9] Karl Menger proposed transferring the probabilistic notions of quantum mechanics from the physics to the underlying geometry and showed how one could replace a numerical distance between points p and q by a distribution function F pq . Subsequently, numerous authors studied such spaces, as did e.g., the excellent books ‘‘Probabilistic Metric Spaces” [14] by Berthold Schweizer and Abe Sklar and ‘‘Fixed Point Theory in Probabilistic Metric Spaces” [6] by Olga Hadzˇic´ and Endre Pap. The contribution of Menger to resolving the interpretative issue of quantum mechanics turned out to be of fundamental importance in probabilistic functional analysis and nonlinear analysis; see e.g. [6]. As it has been described in [5], probabilistic metric spaces are also useful in modelling some phenomena connected with both string and E-infinity theory [2,3]. For instance, the process in the analysis of the probability involved in the two-slit experiment can be modelled by means of a probabilistic metric. In the following we recall some well-known definitions and results in the theory of probabilistic metric spaces used later on in the paper. For more details we refer the reader to [6, Chapters 1 and 2] and [14, Chapters 8 and 12]. A triangular norm (shorter t-norm) is a binary operation T on [0,1], which is associative, commutative, non-decreasing at both places and has 1 as the unit element. Basic examples are the t-norms T L (Łukasiewicz t-norm), T P and T M (Min norm), defined by T L ða; bÞ ¼ maxfa þ b 1; 0g; T P ða; bÞ ¼ ab and T M ða; bÞ ¼ minfa; bg. As regard to pointwise ordering, T M is the strongest t-norm, that is T 6 T M for every t-norm T. A distance distribution function is any mapping F : ½0; 1Þ ! ½0; 1 which is non-decreasing and left continuous on ð0; 1Þ and Fð0Þ ¼ 0. The class of all distribution functions is denoted by Dþ . Dþ is the subset of Dþ containing all functions F which also satisfy limt!1 FðtÞ ¼ 1. A special element of Dþ is the function e0 , defined by 0; if t ¼ 0 : e0 ðtÞ ¼ 1; if t > 0 Definition 1.1 [14]. A probabilistic semi-metric space (shortly PSM space) is a pair ðS; FÞ where S is a nonempty set and F is a mapping from S S to Dþ satisfying the following properties (as usually, Fðx; yÞ is denoted by F xy ):
* Fax: +40 256 592316. E-mail addresses: [email protected], [email protected] 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.030
D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019
ðPM0Þ :
F pq ¼ e0 () p ¼ q:
ðPM1Þ :
F pq ¼ F qp 8p; q 2 S:
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A PSM space satisfying a certain kind of ‘‘triangle inequality” is called a probabilistic metric space. Definition 1.2 [14]. A probabilistic metric space in the sense of Schweizer and Sklar (or a generalized Menger space) is a triple ðS; F; TÞ, where ðS; FÞ is a PSM space, T is a t-norm and the following triangle inequality holds: F pr ðx þ yÞ P TðF pq ðxÞ; F qr ðyÞÞ
8p; q; r 2 S;
8x; y P 0:
A Menger PM space is a generalized Menger space such that RangeðFÞ Dþ . Definition 1.3 [8]. A PSM space ðS; FÞ satisfying the triangle inequality: 8e > 09d > 0 : F pq ðdÞ > 1 d;
F qr ðdÞ > 1 d ) F pr ðeÞ > 1 e
is called an H-space. Every generalized Menger space ðS; F; TÞ with T satisfying supa<1 Tða; aÞ ¼ 1 is an H-space. If ðX; FÞ is an H-space, then ([14]) the family fU e;k ge>0;k2ð0;1Þ where U e;k ¼ fðx; yÞ 2 X XjF xy ðeÞ > 1 kg is a base for a metrizable uniformity on X, called the F-uniformity and denoted by UF . The F-uniformity is also generated by the family fV d gd>0 , where V d :¼ U d;d . UF naturally determines a topology TF on X, named the F-topology or the strong topology. It is a basic fact that in a metric space ðS; dÞ the distance function d is a continuous mapping from S S to ½0; 1Þ. As ð½0; 1; TÞ is only a semigroup, the corresponding result does not hold in Menger spaces. However, a well-known theorem of Schweizer and Sklar asserts that the continuity of T implies the continuity of the probabilistic metric, in the following sense: Theorem 1.1. ([14], Theorem 12.2.3). Let ðS; F; TÞ be a generalized Menger space with T continuous. If ðpn Þ and ðqn Þ are sequences in S such that pn ! p 2 S and qn ! q 2 S, then F pn qn ðtÞ ! F pq ðtÞ for every continuity point t of F pq . Fixed point theory in probabilistic metric spaces is a part of a very dynamic area of mathematical research, Probabilistic Analysis. It has been developed starting with the paper of Sehgal and Bharucha-Reid [16], where the notion of probabilistic B-contraction was introduced and a generalization of the classical Banach fixed point principle to complete Menger PM spaces was given. Another class of probabilistic contractions (named Hicks C-contractions or C-contractions) has been introduced by Hicks in [7]. Some results in fixed point theory in probabilistic metric spaces have applications to control theory, system theory and optimization problems; in a recent paper [13] a fixed point theorem in Menger quasi-metric spaces was applied to prove the existence of solution for some recurrence equations associated to the asymptotic complexity analysis of Quicksort algorithms and Divide & Conquer algorithms. Definition 1.4. Let ðS; FÞ is a PSM space. According to [15], a mapping A : S ! S is called (a) a probabilistic B-contraction if, for some k 2 ð0; 1Þ, F AðpÞAðqÞ ðktÞ P F pq ðtÞ for all p; q 2 S; t > 0; (b) a probabilistic C-contraction if the following implication holds for every p; q 2 S and t > 0: F pq ðtÞ > 1 t ) F AðpÞAðqÞ ðktÞ > 1 kt
ðHÞ
(k 2 ð0; 1Þ is given). The present paper is related to some fixed point theorems in [4]. We extend the results of Ghaemi and Razani concerning Hicks C-contractions in Menger PM spaces to a larger class of contractions in H-spaces. Then we answer in the affirmative an open question about periodic points of a contractive-type mapping in Menger PM spaces, raised in this paper.
2. Hicks-type probabilistic contractions in H-spaces The following general result about Hicks C-probabilistic contractions has been proved by Radu: Theorem 2.1 [11]. Let ðS; F; TÞ be a complete Menger PM space such that supa<1 Tða; aÞ ¼ 1. Then every C-contraction A on S has a unique fixed point which is the limit of the sequence ðAn ðpÞÞn2N for every p 2 S. In [4] the existence of a fixed point for a C-contraction was obtained from the convergence of a subsequence of the sequence of successive approximations.
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D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019
Theorem 2.2 [4], Theorem 2.3. Let ðS; F; TÞ be a Menger PM space. If A is a C-contraction on S and, for some u in S, ðAni ðuÞÞ is a convergent subsequence of ðAn ðuÞÞ, then f ¼ limi!1 Ani ðuÞ is the unique fixed point of A. We are going to show that in the above theorem instead Menger PM spaces one can consider even H-spaces (which are generalized Menger spaces). Then, we obtain a similar result by weakening the contractive condition. The next theorem is an extension of Theorem 2.2 to Hicks C-contractions on H-spaces. Its proof is based on the remark that every Cauchy sequence in a uniform space containing a convergent subsequence is convergent and the following: Lemma 2.1. (see [11]). Let ðS; FÞ be an H-space and A be a C-contraction on S. Then, for every p 2 S, the sequence ðAn ðpÞÞ is a Cauchy sequence. The proof of the lemma immediately follows from the implication: k > 1 ) F pq ðkÞ > 1 k: Theorem 2.3. Let ðS; FÞ be an H-space and A be a C-contraction on S. Then A has a unique fixed point iff there is p 2 S such that ðAn ðpÞÞ has a convergent subsequence. Proof. From the above lemma it follows that the sequence ðAn ðpÞÞ converges to a fixed point of A, for every Hicks C-contraction is (uniformly) continuous. The uniqueness can be proved in the usual way. In the following we prove a similar result for a larger class of probabilistic contractions, recently introduced in [10]. h Definition 2.1 [10]. Let ðS; FÞ be a PSM space. A self-mapping A of S is called a weak–Hicks contraction (shortly w–H contraction) if ðHÞ holds for all p; q 2 S and t 2 ð0; 1Þ, that is, the following implication takes place for every p; q 2 ð0; 1Þ and t 2 ð0; 1Þ: F pq ðtÞ > 1 t ) F AðpÞAðqÞ ðktÞ > 1 kt (k 2 ð0; 1Þ is given). The difference between this type of contractions and Hicks C-contractions has nicely been illustrated by Radu in the following example: Example 2.1 [12]. Let X be a set containing at least two elements. Consider the discrete Menger space under T M determined (for x–y) by the probabilistic metric 0; if t 6 1 : F xy ðtÞ ¼ 1; if t > 1 Then any mapping A : X ! X is a weak-Hicks contraction, while the only Hicks C-contractions on ðX; F; T M Þ are the constant mappings. Because in the case of w-H contractions the relation ðHÞ takes place only for t 2 ð0; 1Þ, the implication justifying Lemma 2.1 cannot be used. However, as one can see in the next theorem, the existence of a fixed point for a weak-Hicks contraction A is guaranteed by the weak condition F xAðxÞ ð1Þ > 0. Theorem 2.4. Let ðX; FÞ be an H space and A be a weak-Hicks contraction on X. If there exists x 2 X such that F xAðxÞ ð1Þ > 0 and the sequence ðAn ðxÞÞn2N has a convergent subsequence, then ðAn ðxÞÞ converges to a fixed point of A. Proof. As F xAðxÞ ð1Þ > 0 and F xAðxÞ is left continuous, there is d 2 ð0; 1Þ such that F xAðxÞ ðdÞ > 1 d. Using (H), it immediately follows by induction that n
n
F An ðxÞAnþ1 ðxÞ ðk dÞ > 1 k d
8n 2 N: n
If e > 0 is given and n0 2 N is such that k 0 d < e, by the above inequality one obtains F An ðxÞAnþ1 ðxÞ ðeÞ > 1 e
8n P n0 :
Let xn ¼ An ðxÞ and ðxni Þi2N be a subsequence of ðxn Þ converging to y. We show that limi!1 xni þ1 ¼ y. Indeed, we know that for every e > 0 there is d > 0 such that F pq ðdÞ > 1 d;
F qr ðdÞ > 1 d ) F pr ðeÞ > 1 e:
Choosing i0 for which F xni xn þ1 ðdÞ > 1 d8i P i0 and F xni y ðdÞ > 1 d8i P i0 , it follows F xn þ1 y ðeÞ > 1 e8i P i0 , as claimed. i i On the other hand, as any weak-Hicks contraction on an H-space is uniformly continuous, limi!1 xni þ1 ¼ AðyÞ. Therefore, we have Ay ¼ y. Let us now prove that ðxn Þn2N converges to y. We know that for every e > 0 and d 2 ð0; 1Þ there is l 2 N such that F xni y ðeÞ > 1 d8i P l. Let e > 0; d 2 ð0; 1Þ be given and m be an arbitrary natural number P nl . If m ¼ nl þ s, then from
D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019 s
1017
s
F Anl ðxÞy ðeÞ ¼ F xnl y ðeÞ > 1 d it follows F xm y ðk eÞ ¼ F Anl þs ðxÞAs ðyÞ ðk eÞ > 1 d, hence F xm y ðeÞ > 1 d8m P nl , which conclude the proof. h We note that a weak-Hicks contraction may have more than one fixed point (see [10, Example 2.1]). 3. Periodic points This section is devoted to a question settled in [4], concerning the existence of periodic points for an e-contractive mapping in Menger PM spaces ðS; F; TÞ with T–T M . The next definition is a slight modification of Definition 3.2 in [4]. Definition 3.1. Let ðX; FÞ be a PSM space and e 2 ð0; 1Þ. A mapping f : X ! X is called probabilistic e-contractive if F f ðxÞf ðyÞ ðtÞ > F xy ðtÞ whenever F xy ðtÞ > 1 e and x–y. It is easy to see that every probabilistic e-contractive mapping in a generalized Menger space ðS; F; TÞ with T such that supa<1 Tða; aÞ ¼ 1 is continuous. Indeed, let k 2 ð0; 1Þ be given. Choose d 2 ð0; minfe; kgÞ and let x; y and t be such that F xy ðtÞ > 1 d. If x–y, then F f ðxÞf ðyÞ ðtÞ > F xy ðtÞ > 1 k, while if x ¼ y then f ðxÞ ¼ f ðyÞ and again F f ðxÞf ðyÞ ðtÞ > 1 k. Definition 3.2 [4]. A point n 2 X is called a periodic point for f if there is a positive integer k such that f k ðnÞ ¼ n. Theorem 3.1 [4], Theorem 3.3. Let ðX; F; T M Þ be a generalized Menger space. Suppose that f is a probabilistic e-contractive selfmapping of X such that there exists a point x 2 X whose sequence of iterates ðf n ðxÞÞ contains a convergent subsequence ðf ni ðxÞÞ. Then n ¼ limi!1 f ni ðxÞ is a periodic point for f. In the next theorem we prove that Theorem 3.1 remains true even if T M is replaced by an arbitrary continuous t-norm. Thus, we answer into affirmative Question 3.7 in [4]. The following continuity lemma is an improvement of Theorem 1.1. Lemma 3.1 [1]. Let ðS; F; TÞ be a generalized Menger space with T satisfying the condition ðcÞ : lim Tðt; sÞ ¼ s t!1
for all t; s 2 ½0; 1. If ðpn Þ and ðqn Þ are sequences in S such that pn ! p 2 S and qn ! q 2 S, then F pn qn ðtÞ ! F pq ðtÞ for every continuity point t of F pq . Theorem 3.2. Let (X; F; TÞ be a generalized Menger space with the t-norm T satisfying limt!1 Tðt; sÞ ¼ s for all t; s 2 ½0; 1. Then for every probabilistic e-contractive mapping f on X with the property that there exists a point x 2 X whose sequence of iterates ðf n ðxÞÞn2N contains a convergent subsequence, the point n ¼ limi!1 f ni ðxÞ is a periodic point. Proof. Since T is continuous at (1,1), there is d 2 ð0; eÞ such that Tð1 d; 1 dÞ > 1 e. Also, there is a positive integer N 1 such that i P N 1 implies F f ni ðxÞn ðt=2Þ > 1 d; 8t > 0. Fix a k P N 1 and denote nkþ1 nk by s. As f is probabilistic e-contractive and F f nk ðxÞn ðt=2Þ > 1 e, we have F f nk þ1 ðxÞf ðnÞ ðt=2Þ PF f nk ðxÞn ðt=2Þ >1 d > 1 e and, after nkþ1 nk iterations, F f nkþ1 ðxÞf s ðnÞ ðt=2Þ > 1 d. Therefore, F n;f s ðnÞ ðtÞ P TðF f nkþ1 ðxÞn ðt=2Þ; F f nkþ1 ðxÞf s ðnÞ ðt=2ÞÞ P Tð1 d; 1 dÞ > 1 e ni
Since f is continuous, limi!1 f ðxÞ ¼ n implies limi!1 f lim F f ni ðxÞf ni þs ðxÞ ðtÞ ¼ F nf s ðnÞ ðtÞ i!1
ni þs
8t > 0:
s
ðxÞ ¼ f ðnÞ, therefore, by Theorem 1.1,
8t > 0:
The sequence of real numbers ðzn ÞnPnk ; zn :¼ F f n ðxÞf nþs ðxÞ ðtÞnPnk is convergent for every t > 0 (being non-decreasing and bounded), therefore lim F f n ðxÞf nþs ðxÞ ðtÞ ¼ F nf s ðnÞ ðtÞ
n!1
8t > 0:
On the other hand, from f ni ðf ðxÞÞ ¼ f ðf ni ðxÞÞ!i!1 f ðnÞ and f ni ðf sþ1 ðxÞÞ ¼ f sþ1 ðf ni ðxÞÞ ! f sþ1 ðnÞ it follows that lim F f ni þ1 ðxÞf ni þ1þs ðxÞ ðtÞ ¼ F f ðnÞf sþ1 ðnÞ ðtÞ i!1
8t > 0
that is, F nf s ðnÞ ðtÞ ¼ F f ðnÞf sþ1 ðnÞ ðtÞ
8t > 0:
We claim that f s ðnÞ ¼ n. Indeed, if we had f s ðnÞ–n, then 1 e < F nf s ðnÞ ðtÞ for all t > 0 would imply F nf s ðnÞ ðt0 Þ < F f ðnÞf sþ1 ðnÞ ðt 0 Þ, which is a contradiction. Therefore, f s ðnÞ ¼ n, concluding the proof. h
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D. Mihetß / Chaos, Solitons and Fractals 41 (2009) 1014–1019
Next we show that if ðX; F; TÞ is a Menger PM space, then any periodic point is actually a fixed point. Theorem 3.3. Let (X; F; TÞ be a Menger PM space with the t-norm satisfying the condition (c) from Lemma 3.1 and f : X ! X be a probabilistic e-contractive mapping. Suppose that, for some x 2 X, the sequence ðf n ðxÞÞn2N contains a convergent subsequence and let x 2 X be its limit. Then x is the unique fixed point of f. Proof. The proof is similar to that of Theorem 3.2. Let xn ¼ f n ðxÞ and ðxnk Þk2N a subsequence converging to x . Since f is continuous, the sequence (f ðxnk ÞÞk2N converges to f ðx Þ and ðf ðf ðxnk ÞÞÞk2N converges to f ðf ðx ÞÞ. Therefore, by Lemma 3.1, F xnk f ðxnk Þ ðtÞ ! F x f ðx Þ ðtÞ and F f ðxn
k
Þf 2 ðxnk Þ ðtÞ
! F f ðx Þf 2 ðx Þ ðtÞ
for every continuity point t of F x f ðx Þ and F f ðx Þf 2 ðx Þ . Choose any t 0 > 0 such that F x f ðx Þ ðt 0 Þ > 1 e; F f ðx Þf 2 ðx Þ ðt0 Þ > 1 e and F x f ðx Þ ; F f ðx Þf 2 ðx Þ are continuous in t0 (this is possible, because F x f ðx Þ and F f ðx Þf 2 ðx Þ are in Dþ hence they have only a countable number of discontinuity points). Since F xf ðxÞ ðt 0 Þ > 1 e, the sequence ðzn Þn2N ; zn :¼ F xn f ðxn Þ ðt0 Þ is a non-decreasing sequence of numbers in ½0; 1, therefore it is convergent. Since its subsequence ðznk Þk2N converges to F x f ðx Þ ðt0 Þ, we have limn!1 zn ¼ F x f ðx Þ ðt 0 Þ. As limn!1 znþ1 ¼ limn!1 F f ðxn Þf 2 ðxn Þ ðt 0 Þ ¼ F f ðx Þf 2 ðx Þ ðt0 Þ, the equality F f ðx Þf 2 ðx Þ ðt0 Þ ¼ F x f ðx Þ ðt0 Þ holds. If we had x –f ðx Þ, then as F x f ðx Þ ðt0 Þ > 1 e and f is probabilistic e-contractive, we would also have F f ðx Þf 2 ðx Þ ðt0 Þ > F x f ðx Þ ðt0 Þ, contradicting the above equality. For the uniqueness, let u; v be fixed points of f. As F uv 2 Dþ , there is t > 0 such that F uv ðtÞ > 1 e. If u–v, then F uv ðtÞ ¼ F f ðuÞf ðvÞ ðtÞ > F uv ðtÞ, which is impossible. Therefore, it must be the case that u ¼ v. It is worth noting that the condition RangeðFÞ Dþ among the hypotheses of the preceding theorem is essential. A probabilistic e-contractive mapping in a generalized Menger space may have more than one fixed point, as well as we can find probabilistic e-contractive mappings for which the limit of ðxnk Þ is not a fixed point of f. h Example 3.1. It is easy to see that the triple ðN ; F; T M Þ where, for x–y; F xy ðtÞ ¼ 12 8t > 0 is a (generalized) Menger space. The mapping f : N ! N 1; if x is even f ðxÞ ¼ 2; if x is odd is probabilistic 12-contractive, for the implication x–y; 1 e < F xy ðtÞ ) F f ðxÞf ðyÞ ðtÞ > F xy ðtÞ takes place in absence. The sequence of the successive approximations of 1 is: 2,1,2,1,2,1. . ., and its subsequence 1,1,. . ., converges to 1, which is not a fixed point, but a periodic point for f. Also, the probabilistic 12-contractive mapping g : X ¼ N , 1; if x is odd gðxÞ ¼ 2; if x is even has two fixed points, x ¼ 1 and x ¼ 2. 4. Conclusions In this work we have extended a fixed point theorem by Ghaemi and Razani [4] to a larger class of PM spaces and have shown that the result can be yet improved by weakening the contractive condition. In the final part of the paper we have answered in the affirmative an open question raised in [4]. References [1] Chang SS, Lee BS, Cho YJ, Chen YQ, Kang SM, Jung JS. Generalized contraction mapping principle and differential equations in probabilistic metric spaces. Proc AMS 1996;124:2367–76. [2] El Naschie MS. On a fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int J Nonlin Sci Numer Simul 2005;6:517–29. [3] El Naschie MS. The two-slit experiment as the foundation of E-infinity of high energy physics. Chaos, Solitons & Fractals 2005;25:509–14. [4] Ghaemi MB, Razani A. Fixed and periodic points in the probabilistic normed and metric spaces. Chaos, Solitons & Fractals 2006;28:1181–7. [5] Gregori V, Romaguera S, Veeramani P. A note on intutionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2006;28:902–5. [6] Hadzˇic´ O, Pap E. Fixed point theory in probabilistic metric spaces. Dordrecht: Kluwer Academic Publishers; 2001. [7] Hicks TL. Fixed point theory in probabilistic metric spaces. Zb Rad Prir Mat Fak Ser Mat 1983;13:63–72. [8] Hicks TL, Sharma PL. Probabilistic metric structures: Topological classification. Zb Rad Prirod -Mat Fak Ser Mat Univ u Novom Sadu 1984;14:43–50. [9] Menger K. Statistical metrics. Proc Nat Acad Sci USA 1942;28:535–7. [10] Mihetß D. Weak-Hicks contractions. Fixed Point Theory 2005;6(1):71–8. [11] Radu V. Some fixed point theorems in probabilistic metric spaces. Lect Notes Math 1987;12333:125–33.
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