Abstract
In this paper, we provide coincidence point and fixed point theorems satisfying an implicit relation, which extends and generalizes the result of Gregori and Sapena, for set-valued mappings in complete partially ordered fuzzy metric spaces. Also we prove a fixed point theorem for set-valued mappings on complete partially ordered fuzzy metric spaces which generalizes results of Mihet and Tirado.
MSC:54E40, 54E35, 54H25.
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1 Preliminaries
The concept of fuzzy metric space was introduced by Kramosil and Michalek [1] and the modified concept by George and Veeramani [2] (for other modifications see [3, 4]). Furthermore, the fixed point theory in this kind of spaces has been intensively studied (see [5–14]).
The applications of fixed point theorems are remarkable in different disciplines of mathematics, engineering, and economics in dealing with problems in approximation theory, game theory, and many others (see [15] and references therein).
In 2004 Rodríguez-López and Romaguera [16] introduced the Hausdorff fuzzy metric of a given fuzzy metric space in the sense of George and Veeramani on the set of non-empty compact subsets.
Some fixed point results for set-valued mappings on fuzzy metric space can be found in [17, 18] and references therein.
The aim of this paper is to prove a coincidence point and fixed point theorem on a partially ordered fuzzy metric space satisfying an implicit relation and another fixed point theorem. Our result substantially generalizes and extends the result of Gregori and Sapena [8] and results of Miheţ [19] and Tirado [20] and also the result of Latif and Beg [21] for set-valued mappings in complete partially ordered fuzzy metric spaces. Implicit relations have been considered by several authors in connection with solving nonlinear functional equations (see [22–25]).
For the sake of completeness, we briefly recall some basic concepts used in the following.
Definition 1.1 [26]
A binary operation is called a continuous t-norm if it satisfies the following conditions:
-
(1)
∗ is associative and commutative,
-
(2)
∗ is continuous,
-
(3)
for all ,
-
(4)
whenever and , for each .
The three basic continuous t-norms are: (i) The minimum t-norm is defined by . (ii) The product t-norm is defined by . (iii) The Łukasiewicz t-norm is defined by .
-
(i)
A t-norm ∗ is said to be Hadžić-type t-norm, if the family of its iterates defined for each by , , for all , are equi-continuous at , that is, given , there exists such that for all
The t-norm ∗, defined by is a trivial example of the t-norm of Hadžić-type, but there are other t-norms of Hadžić-type (see [27]).
-
(ii)
If ∗ be a t-norm and is a sequence of numbers in , one defines recurrently by and , . is defined as and as .
If is given, we say that the t-norm is geometrically convergent (g-convergent) if .
The Łukasiewicz t-norm and t-norms of Hadžić-type are examples of g-convergent t-norms. Other examples be found in [28]. Also note that if the t-norm ∗ is g-convergent, then .
Proposition 1.3 [28]
-
(i)
For the following implication holds:
-
(ii)
If ∗ is of Hadžić-type, then , for every sequence in such that .
Definition 1.4 [2]
A 3-tuple is called a fuzzy metric space if X is an arbitrary non-empty set, ∗ is a continuous t-norm and M is a fuzzy set on , satisfying the following conditions for each and :
(FM-1) ,
(FM-2) for all if and only if ,
(FM-3) ,
(FM-4) ,
(FM-5) is continuous.
Definition 1.5 [2]
Let be a fuzzy metric space. A sequence in X is called a Cauchy sequence, if, for each and , there exists such that for all . A sequence in a fuzzy metric space is said to be convergent to if for all . A 3-tuple is complete if every Cauchy sequence is convergent in X.
Lemma 1.6 [7]
Let be a fuzzy metric space. Then is non-decreasing with respect to t for all .
Definition 1.7 [16]
Let be a fuzzy metric space. M is said to be continuous on , if
whenever a sequence in converges to a point , that is,
Lemma 1.8 [16]
Let be a fuzzy metric space. Then M is continuous function on .
Definition 1.9 [6]
Let be a fuzzy metric space. The fuzzy metric M is triangular if it satisfies the condition
for every and every .
Example 1.10 [2]
Let be a metric space. Define (or ) and for all and ,
Then is a fuzzy metric space. We call the fuzzy metric induced by the metric d the standard fuzzy metric. Note that every standard fuzzy metric is triangular.
Definition 1.11 Let is a fuzzy metric space and . (i): A subset is said to be closed if for each convergent sequence with and as , we have .
(ii): is said to be compact if each sequence in A has a convergent subsequence.
Throughout the article, let , , and denote the set of all non-empty subsets, the set of all non-empty closed subsets, and the set of all non-empty compact subsets of X, respectively.
Definition 1.12 Let X be a non-empty set. A point is called a coincidence point of the mappings and if . Point is called a fixed point of the mappings if .
Theorem 1.13 [16]
Let be a fuzzy metric space. For each and define
where . Then the 3-tuple is a fuzzy metric space.
The fuzzy metric will be called the Hausdorff fuzzy metric of on .
Lemma 1.14 [16]
Let be a fuzzy metric space. Then, for each , and , there is such that
2 Main results
Throughout this section, ∗ denotes a continuous t-norm and the set of all continuous real-valued mappings satisfying the following properties:
: is non-increasing in .
: If there exists such that for each , we have
where are non-decreasing functions with , then .
: For each and some , the condition
implies .
Now we give our main result.
Theorem 2.1 Let be a complete fuzzy metric space with Hadžić-type t-norm ∗ such that as , for all . Let ⪯ be a partial order defined on X. Let be a set-valued mapping with non-empty compact values and a mapping such that is closed and for some and all comparable elements , and , we have
Also suppose that the following conditions are satisfied:
-
(i)
,
-
(ii)
implies ,
-
(iii)
if is a sequence such that , then for all n.
Then F and f have a coincidence point, that is, there exists such that .
Proof Let be fixed and . By using (i) and (ii), there exists such that and . Now from (i), (ii), and by Lemma 1.14, for there is such that and with
thus
On the other hand by and in (2.1), we have
Now since , , also
and , and by using , we get
This means that
where , , , then from , we have
hence by (2.2), we obtain
Again by (i), (ii), and by Lemma 1.14, there exists such that with that satisfying in
Since thus by replacing and in (2.1) and from , we obtain
Now by , , , the property implies
thus from (2.3), we get
Repeatedly, there exists with such that and , and
therefore
Continuing the process, we can have a sequence in X with such that, for , , and
and
From (2.4), we conclude that, for each ,
Next, we prove that the sequence is Cauchy. Suppose that and are given. Then, by Lemma 1.6 and (FM-4), for all ,
On the other hand, putting in (2.6), for all , , we get
Then by replacing the above inequality in (2.7), we obtain, for all ,
By hypothesis, ∗ is a t-norm of Hadžić-type, and there exists such that for all ,
By as , there exists such that, for all ,
From (2.5) and the above inequality, we have
therefore, (2.8) and (2.9) imply that, for all and each ,
This shows that is a Cauchy sequence. Since X is complete, there exists some such that
Now, since is closed, there exists such that . Also (ii) implies that for any n. Thus from (2.1), we have
By taking the limit as , by the continuity of T, and from Lemma 1.8, we get
Now by using , we have
on the other hand , so
It follows that for each . Now since is closed (note that is compact), we get , thus is a coincidence point of F and f. The proof is complete. □
Remark 2.2 In Theorem 2.1, we proved that the sequence is Cauchy; one can replace the condition ‘∗ is Hadžić-type t-norm and as , for all ’ with the following: ‘ for each ’. To see this, choose some and such that and . Then from (FM-4) and (2.5), for every , we have
Thus, is a Cauchy sequence. Then we have the following theorem.
Theorem 2.3 Let be a complete fuzzy metric space and suppose for each , . Let ⪯ be a partial order defined on X. Let be a set-valued mapping with non-empty compact values and a mapping such that is closed and for some and all comparable elements , and , we have
Also suppose that the following conditions are satisfied:
-
(i)
,
-
(ii)
implies ,
-
(iii)
if is a sequence such that , then for all n.
Then F and f have a coincidence point, that is, there exists such that .
If in Theorem 2.1 and 2.3 we put , where , then we have the following corollaries.
Corollary 2.4 Let be a complete fuzzy metric space with Hadžić-type t-norm ∗ such that as , for all . Let ⪯ be a partial order defined on X. Let be a set-valued mapping with non-empty compact values and a mapping such that be closed and for all comparable elements , and , we have
Also suppose that the following conditions are satisfied:
-
(i)
,
-
(ii)
implies ,
-
(iii)
if is a sequence such that , then for all n.
Then there exists such that .
Corollary 2.5 Let be a complete fuzzy metric space and suppose for each , . Let ⪯ be a partial order defined on X. Let be a set-valued mapping with non-empty compact values and a mapping such that be closed and for all comparable elements , and , we have
Also suppose that the following conditions are satisfied:
-
(i)
,
-
(ii)
implies ,
-
(iii)
if is a sequence such that , then for all n.
Then there exists such that .
Putting (the identity mapping) in Corollary 2.4 and 2.5, we get the following corollaries.
Corollary 2.6 Let be a complete fuzzy metric space with Hadžić-type t-norm ∗ such that as , for some and . Let ⪯ be a partial order defined on X. Let be a set-valued mapping with non-empty compact values for all comparable elements , and , we have
Also suppose that the following conditions are satisfied:
-
(i)
implies ,
-
(ii)
if is a sequence such that , then for all n.
Then F has a fixed point.
Corollary 2.7 Let be a complete fuzzy metric space and suppose for each , for some and . Let ⪯ be a partial order defined on X. Let be a set-valued mapping with non-empty compact values for all comparable elements , and , we have
Also suppose that the following conditions are satisfied:
-
(i)
implies ,
-
(ii)
if is a sequence such that , then for all n.
Then F has a fixed point.
Remark 2.8 Note that we assumed the implicit relation (2.1) only for the comparable elements of the partially ordered fuzzy metric space.
Remark 2.9 Corollary 2.7 improves and generalizes the mentioned result of Gregori and Sapena (see Theorem 4.8 of [8]) for set-valued mappings in complete partially ordered fuzzy metric spaces.
In continuation, in the spirit of Miheţ [19], we introduce the notion of a set-valued fuzzy order ψ-contraction of -type mappings and give a fixed point theorem in partially ordered fuzzy metric spaces.
Definition 2.10 Let be a fuzzy metric space and . A mapping is a set-valued fuzzy order ψ-contraction of -type if the following implication holds:
for every , and all comparable elements .
If for some , then F will be called a set-valued fuzzy order α-contraction of -type.
Also note that if for all , then every set-valued fuzzy order ψ-contraction of -type satisfies the relation
for all comparable elements and . Indeed, if for some comparable and there exists such that for all , we have ; then there is such that , that is, and , which is a contradiction.
Example 2.11 Let be a fuzzy metric space. Let be a set-valued mapping, where . If there is such that
for all comparable elements and , then F is a set-valued fuzzy order α-contraction of -type. Indeed, if , then for every comparable elements and some , we have
thus .
Now we state our main theorem.
Theorem 2.12 Let be a complete fuzzy metric space with . and be a set-valued fuzzy order ψ-contraction of -type, where for all . Let ‘⪯’ be a partial order defined on X, and for all . Suppose that there exist and such that and the following two conditions hold:
-
(i)
implies ,
-
(ii)
if is a sequence with and , then for all n.
Then F has a fixed point.
Proof Since there exist and such that , we have with . We may suppose that . For, if we assume the contrary, then for all , that is, and we have finished the proof. Therefore, for some and every , , we have
Since F is a set-valued fuzzy order ψ-contraction of -type mapping, there exists with such that . Repeating this argument, we get a sequence in Y such that with and such that
Suppose that and are given. Since for all , there exists such that for all and all , and we have
Now by using (FM-4) and from (2.11)-(2.12), for all , we get
This shows that is a Cauchy sequence. Since X is complete, converges to some , that is, . Now we prove that . But ; then it is enough to show that for every and there exists such that .
Let and be arbitrary. From , it follows that there exists such that
Also for there are such that
Now put . We prove that there exists such that . For, if for every , then for every and every , therefore for all , which means that , and this is a contradiction.
Since for all , there exists such that for all , and we have ; thus, since and by using (2.10), there exists such that
On the other hand for every . Therefore (2.11) implies the existence of the element such that for all , we have
Also since , there exists such that for all ,
Now if , then by (2.13)-(2.17), we get
Hence , consequently is a fixed point of F. The theorem is proved. □
Corollary 2.13 Let be a complete fuzzy metric space with Lukasiewicz t-norm and ‘⪯’ be a partial order defined on X. Let and be a set-valued mapping with the property that there is such that
for all comparable elements and , and the following conditions hold:
-
(i)
implies ,
-
(ii)
if is a sequence with and , then for all n.
Then F has a fixed point.
Proof By using Definition 1.2, . Also, from Example 2.11 it follows that F is a set-valued fuzzy order ψ-contraction of -type with . Since, for all , , from Proposition 1.3, we have . Next, since
for all comparable elements and , there exist and such that . Consequently, by the preceding theorem, F has a fixed point. □
Corollary 2.14 Let be a complete fuzzy metric space with a continuous g-convergent t-norm and ‘⪯’ be a partial order defined on X. Let and be a set-valued fuzzy order α-contraction of -type. If there exist and such that and the following two conditions hold:
-
(i)
implies ,
-
(ii)
if is a sequence with and , then for all n.
Then F has a fixed point.
Theorem 2.12 and Corollary 2.13 are, respectively, generalizations of the theorems of Mihet [19] and Tirado [20] to the set-valued case in partial ordered fuzzy metric spaces.
Now we introduce a definition and, by using it, we shall state fixed and common fixed point theorems in the partially ordered fuzzy metric space. Our results generalize and extend Theorems 4.1 and 4.2 of [21] to set-valued mappings in complete partially ordered fuzzy metric spaces.
Definition 2.15 Let Y be a non-empty subset of fuzzy metric space . Mapping is called fuzzy order K-set-valued mapping, if for all , , there exists with such that
for every and with and some .
Theorem 2.16 Let be a complete fuzzy metric space, with M triangular, and ‘⪯’ a partial order on X. Let and be a fuzzy order K-set-valued mapping. Also let there for some exist with , and the following condition is satisfied:
If is a sequence in Y whose consecutive terms are comparable, then , for all n.
Then F has a fixed point in X.
Proof By the hypothesis, for there exists such that . Now because F is a fuzzy order K-set-valued mapping, there exists such that and
thus
Then it follows by induction that
where is a sequence whose consecutive terms are comparable, that is, . Now we prove that is a Cauchy sequence. By putting , and by (2.19), and since M is triangular, we have for all
For each and each , we can choose a sufficiently large such that
Thus from (2.20) and (2.21), , for all and . This shows that the sequence is Cauchy, and, since X is complete, it converges to a point . But Y is closed, thus and also by using the hypothesis . Now we show that . From , and for all n, since F is a fuzzy order K-set-valued mapping, there exists such that , and
Now since M is triangular, by using (2.22), we get
and so, letting , . Consequently, since Fx is closed, we have . Then F has a fixed point. □
From the above theorem we can immediately obtain the following generalization for getting a common fixed point.
Theorem 2.17 Let be a complete fuzzy metric space, with M triangular, and ‘⪯’ a partial order on X. Let and, for every , be a sequence of mappings such that, for every two mappings , and for all , , there exists with such that
for every and with and some . Also let there exist, for some , with , and the following condition be satisfied:
If is a sequence in Y whose consecutive terms are comparable, then , for all n.
Then there exists such that , that is, has a common fixed point.
Proof We can find such that and that
Also for there exists with and such that
By continuing this process, we get
where is a sequence with . Now similar to the proof of the preceding theorem, we can prove that is a Cauchy sequence and by the completeness of X it follows that converges to some . Furthermore, and . Now suppose that is any arbitrary member of . Since , for all n, and by the hypothesis, there exists such that , and
thus
Next by the letting , we get , and then . As is an arbitrary member of , , and x is a common fixed point of . The theorem is proved. □
Example 2.18 Let with t-norm defined for all and , for all and . Then is a complete fuzzy metric space. Let the natural ordering ≤ of the numbers as the partial ordering ⪯. Define and as for each , and for each , where is an arbitrary. If such that and , then there exists such that and (2.18) is satisfied. Thus F is a fuzzy order K-set-valued mapping. But if , then three cases arise.
Case (i). If , then for every
Case (ii). If , then for every
Case (iii). If , then for every
Hence F is a fuzzy order K-set-valued mapping with . Moreover, there exists (or ) with () such that . Thus all the hypotheses of Theorem 2.16 are satisfied and (or ) is the fixed point of F.
References
Kramosil I, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetika 1975, 11: 336–344.
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
Deng Z: Fuzzy pseudometric spaces. J. Math. Anal. Appl. 1982, 86: 74–95. 10.1016/0022-247X(82)90255-4
Kaleva O, Seikkala S: On fuzzy metric spaces. Fuzzy Sets Syst. 1984, 12: 215–229. 10.1016/0165-0114(84)90069-1
Ćirić L: Some new results for Banach contractions and Edelestein contractive mappings on fuzzy metric spaces. Chaos Solitons Fractals 2009, 42: 146–154. 10.1016/j.chaos.2008.11.010
Di Bari C, Vetro C: Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space. J. Fuzzy Math. 2005, 13: 973–982.
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4
Gregori V, Sapena A: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 125: 245–252. 10.1016/S0165-0114(00)00088-9
Miheţ D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 2004, 144: 431–439. 10.1016/S0165-0114(03)00305-1
Miheţ D: Multivalued generalizations of probabilistic contractions. J. Math. Anal. Appl. 2005, 304: 464–472. 10.1016/j.jmaa.2004.09.034
Miheţ D: On the existence and the uniqueness of fixed points of Sehgal contractions. Fuzzy Sets Syst. 2005, 156: 135–141. 10.1016/j.fss.2005.05.024
Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos Solitons Fractals 2006,27(2):331–344. 10.1016/j.chaos.2005.03.019
Saadati R, Razani A, Adibi H: A common fixed point theorem in L -fuzzy metric spaces. Chaos Solitons Fractals 2007,33(2):358–363. 10.1016/j.chaos.2006.01.023
Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Anal., Theory Methods Appl. 2010,72(3–4):1298–1304. 10.1016/j.na.2009.08.018
Pathak HK, Hussain N: Common fixed points for Banach pairs with applications. Nonlinear Anal. 2008, 69: 2788–2802. 10.1016/j.na.2007.08.051
Rodríguez-López J, Romaguera S: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets Syst. 2004, 147: 273–283. 10.1016/j.fss.2003.09.007
Hadžić O, Pap E: Fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 127: 333–344. 10.1016/S0165-0114(01)00144-0
Kiani F, Amini-Harandi A: Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 94
Miheţ D: A class of contractions in fuzzy metric spaces. Fuzzy Sets Syst. 2010, 161: 1131–1137. 10.1016/j.fss.2009.09.018
Tirado P:Contraction mappings in fuzzy quasi-metric spaces and -fuzzy posets. VII Iberoamerican Cont. on Topology and Its Applications Valencia, Spain 2008, 25–28.
Latif A, Beg I: Geometric fixed points for single and multivalued mappings. Demonstr. Math. 1997,30(4):791–800.
Altun I, Turkoglu D: Some fixed point theorems on fuzzy metric spaces with implicit relations. Commun. Korean Math. Soc. 2008,23(1):111–124. 10.4134/CKMS.2008.23.1.111
Beg I, Butt AR: Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal. 2009, 71: 3699–3704. 10.1016/j.na.2009.02.027
Popa V: A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation. Demonstr. Math. 2000, 33: 159–164.
Sedghi S, Rao KPR, Shobe N: A general common fixed point theorem for multimaps satisfying an implicit relation on fuzzy metric spaces. Filomat 2008,22(1):1–11. 10.2298/FIL0801001S
Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 314–334.
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Space. Kluwer Academic, Dordrecht; 2001.
Hadžić O, Pap E, Budinčević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetika 2002,38(3):363–381.
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All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Sadeghi, Z., Vaezpour, S.M., Park, C. et al. Set-valued mappings in partially ordered fuzzy metric spaces. J Inequal Appl 2014, 157 (2014). https://doi.org/10.1186/1029-242X-2014-157
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DOI: https://doi.org/10.1186/1029-242X-2014-157