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Fixed point theorems for contractions in fuzzy normed spaces and intuitionistic fuzzy normed spaces
Fixed Point Theory and Applications volume 2013, Article number: 79 (2013)
Abstract
In this paper, we prove that some coupled fixed point theorems and coupled coincidence point theorems for contractions in fuzzy normed spaces and intuitionistic fuzzy normed spaces can be directly deduced from fixed point theorems for contractions in fuzzy normed spaces. We also prove that these results are equivalent.
MSC:47H10, 54A40, 54E50, 54H25.
1 Introduction
The well-known Banach contraction mapping principle [1] is a powerful tool in nonlinear analysis; many mathematicians have much contributed to the improvement and generalization of this principle in many ways. Especially, some recent meaningful results have been obtained in [2–18].
In this paper, we first prove a simple fixed point theorem for an increasing mapping defined on fuzzy normed spaces, and by using this result, we can easily prove some coupled fixed point theorems and coupled coincidence point theorems in fuzzy normed spaces and intuitionistic fuzzy normed spaces. Also, we prove that these results are essentially equivalent. Finally, we give an example to show that our contractive conditions is a real improvement over the contractive conditions used in [13] and [17]. Our results are also an improvement over the results in [13] and [17].
For the reader’s convenience, we restate some definitions and results that will be used in this paper.
Definition 1.1 ([13])
A binary operation is a continuous t-norm if ∗ satisfies the following conditions:
-
(i)
∗ is commutative and associative;
-
(ii)
∗ is continuous;
-
(iii)
, ;
-
(iv)
, whenever and for all .
A t-norm ∗ is said to be of H-type if the sequence of functions is equicontinuous at .
The t-norm defined by is an example of an H-type t-norm ∗.
Definition 1.3 ([13])
A binary operation is a continuous t-conorm if ⋆ satisfies the following conditions:
-
(i)
⋆ is commutative and associative;
-
(ii)
⋆ is continuous;
-
(iii)
, ;
-
(iv)
, whenever and for all .
Definition 1.4 ([23])
A fuzzy normed space (briefly, FNS) is a triple , where X is a vector space, ∗ is a continuous t-norm and is a fuzzy set such that, for all and ,
-
(i)
;
-
(ii)
if and only if ;
-
(iii)
for all ;
-
(iv)
;
-
(v)
is a continuous function of and
By the results in George and Veeramani [19], we can know that every fuzzy norm on X generates a Hausdorff first countable topology on X which has as a base the family of open sets of the form
where for all , and .
The 5-tuple is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a linear space, ∗ is a continuous t-norm, ⋆ is a continuous t-conorm and μ, ν are fuzzy sets on satisfying the following conditions:
-
(i)
, ;
-
(ii)
;
-
(iii)
if and only if ;
-
(iv)
for all ;
-
(v)
;
-
(vi)
is a continuous function of and
-
(vii)
;
-
(viii)
if and only if ;
-
(ix)
for all ;
-
(x)
;
-
(xi)
is a continuous function of and
Park proved in [22], among other results, that each intuitionistic fuzzy norm on X generates a Hausdorff first countable topology on X which has as a base the family of open sets of the form , where for all , and . According to this topology, Park [22] gave the following definitions.
Definition 1.6 A sequence in an intuitionistic fuzzy normed linear space is said to converge to with respect to the intuitionistic fuzzy norm if, for any , , , there exists an integer such that
Definition 1.7 A sequence in an intuitionistic fuzzy normed linear space is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm if, for any , , , there exists an integer such that
Definition 1.8 Let be an IFNS. Then is said to be complete if every Cauchy sequence in is convergent.
Definition 1.9 Let X and Y be two intuitionistic fuzzy normed spaces. A mapping is said to be continuous at if, for any sequence in X converging to , the sequence in Y converges to . If is continuous at each , then f is said to be continuous on X.
For the topology , Gregori et al. [20] proved the following result.
Lemma 1.1 Let be an intuitionistic fuzzy metric space. Then the topologies and coincide on X.
The following lemma was proved by Haghi et al. [15].
Lemma 1.2 Let X be a nonempty set, and let be a mapping. Then there exists a subset such that and is one-to-one.
Definition 1.10 ([16])
A point is called a coupled coincidence point of the mappings and if
Definition 1.11 ([16])
Let be a partially ordered set, and let , be two mappings. Then F is said to have the mixed g-monotone property if F is monotone g-non-decreasing in the first argument and is monotone g-non-increasing in the second argument, that is, for any ,
and
If is an identity mapping, we say that F has the mixed monotone property.
2 Main results
Theorem 2.1 Let be a partially ordered set, and let be a complete FNS such that the t-norm ∗ is of H-type. Let be a mapping such that F is non-decreasing and
for which and , where . Suppose either
-
(a)
F is continuous, or
-
(b)
if is a non-decreasing sequence and , then for all .
If there exists such that
then F has a fixed point in X.
Proof Let such that , and let , , then we have that
Now, put
Then, by using (2.1), we have
Thus, it follows that , and so
On the other hand, we have
By Definition 1.4, we get that
It follows from (2.2) and (2.3) that
By the hypothesis, the t-norm ∗ is of H-type; for all , there exists such that
for all and for all p. Note that
for all and , we have that there exists such that
for all . Thus, is a Cauchy sequence. Since X is complete, there exists such that
If the assumption (a) holds, then by the continuity of F, we get that
If the assumption (b) holds, then we have that for all . It follows from (2.1) that
Thus, , that is, . Therefore, x is a fixed point of F. The proof is completed. □
Theorem 2.2 Let be a partially ordered set, and let be a complete FNS such that the t-norm ∗ is of H-type and for any . Let be a mapping such that F has the mixed monotone property and
for which and , where . Suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if is a non-decreasing sequence and , then for all ,
-
(ii)
if is a non-increasing sequence and , then for all .
If there exist such that
then F has a coupled fixed point , that is,
Proof First, we define a partial order ≼ on as follows: if and only if and . Second, we define a fuzzy set on as follows: for any and any . Since is a complete FNS, we can easily prove that is a complete FNS. Lastly, we define a mapping by
Since F has the mixed monotone property, if , we have that

that is, . Therefore, is a non-decreasing mapping. Since and
we have that . If , by (2.5) we have that
Thus, all the assumptions of Theorem 2.1 hold for and . By Theorem 2.1 we get that has a fixed point , that is, . This implies that , , that is, is a coupled fixed point of F. The proof is completed. □
By using Theorem 2.2, we can prove the following coupled fixed point theorem in intuitionistic fuzzy normed spaces.
Theorem 2.3 Let be a partially ordered set, and let be a complete IFNS such that the t-norm ∗ is of H-type and for any . Let be a mapping such that F has the mixed monotone property and
for which and , where . Suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if is a non-decreasing sequence and , then for all ,
-
(ii)
if is a non-increasing sequence and , then for all .
If there exist such that
then F has a coupled fixed point , that is,
Proof Assume that is a sequence in . Let , . If , then by Definition 1.5(i) we can deduce that . Thus, a sequence in is Cauchy if and only if is Cauchy in . By Lemma 1.1, we know that the topology of is the same as the topology of . This implies that is a complete IFNS if and only if is a complete FNS. Therefore, by using Theorem 2.2 to and F, we get that F has a coupled fixed point . The proof is completed. □
Theorem 2.4 Let be a partially ordered set, and let be a complete IFNS such that the t-norm ∗ is of H-type and for any . Let , be two mappings such that F has the mixed g- monotone property and
for which , and , where , and g is continuous. Suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if is a non-decreasing sequence and , then for all ,
-
(ii)
if is a non-increasing sequence and , then for all .
If there exist such that
then there exist such that
that is, F and g have a coupled coincidence point in X.
Proof The conclusion of Theorem 2.4 can be proved by using Lemma 1.2 and Theorem 2.3. Since the proof is similar to the proof of Theorem 3.2 in [17], we delete the details of the proof. The proof is completed. □
Remark 2.1 It follows from the proof of the above theorems that the following implications hold: Theorem 2.1 ⟹ Theorem 2.2 ⟹Theorem 2.3 ⟹ Theorem 2.4. Conversely, it is clear that the following implications hold: Theorem 2.4 ⟹ Theorem 2.3 ⟹ Theorem 2.2. Thus, we have the following conclusion.
Theorem 2.5 Theorem 2.2-Theorem 2.4 are equivalent.
Remark 2.2 In [13] and [17], the condition for all is used. But this condition cannot hold in intuitionistic fuzzy normed spaces. In fact, if this condition holds, by using (iii) and (iv) in the definition of IFNS, we can get , which yields a contradiction. Furthermore, the proofs of the results in [13] and [17] have the same errors as noted in [18]. Therefore, our results improve and correct results in [13] and [17].
In the following, we give an example to show that our contractive conditions are a real improvement over the contractive conditions used in [13] and [17].
Example 2.1 Let , , for every , and let , , for all . Then is a complete intuitionistic fuzzy normed linear space, and the t-norm ∗ and t-conorm ⋆ are of H-type. If X is endowed with the usual order , then is a partially ordered set. Let , and define , for any . Then is a mixed g-monotone mapping, , and g is continuous. Let and , then
For any , with , we have
Thus, all the conditions of Theorem 2.4 are satisfied. By Theorem 2.4, there is such that and . But υ does not satisfy the contractive conditions in [13] and [17]. In fact, for , ,
This shows that υ does not satisfy the contractive conditions in [13] and [17].
References
Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux éuations intérales. Fundam. Math. 1922, 3: 133–181.
Ćirić LB, Agarwal R, Samet B: Mixed monotone generalized contractions in partially ordered probabilistic metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 56
Gordji ME, Savadkouhi MB: Stability of a mixed type additive, quadratic and cubic functional equation in random normed spaces. Filomat 2012, 25(3):43–54.
Ćirić LB, Abbas M, Damjanovic B, Saadati R: Common fuzzy fixed point theorems in ordered metric spaces. Math. Comput. Model. 2011, 53: 1737–1741. 10.1016/j.mcm.2010.12.050
Shakeri S, Ćirić LB, Saadati R: Common fixed point theorem in partially ordered L -fuzzy metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 125082
Ćirić LB: Solving Banach fixed point principle for nonlinear contractions in probabilistic metric spaces. Nonlinear Anal. 2010, 72: 2009–2018. 10.1016/j.na.2009.10.001
Ćirić LB, Mihet D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topol. Appl. 2009, 156: 2838–2844. 10.1016/j.topol.2009.08.029
Ćirić LB, Lakshmikantham V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stoch. Anal. Appl. 2009, 27(6):1246–1259. 10.1080/07362990903259967
Shakeri S, Jalili M, Saadati R, Vaezpour SM, Ćirić LB: Quicksort algorithms: application of fixed point theorem in probabilistic quasi-metric spaces at domain of words. J. Appl. Sci. 2009, 9: 397–400. 10.3923/jas.2009.397.400
Ćirić LB: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos Solitons Fractals 2009, 42: 146–154. 10.1016/j.chaos.2008.11.010
Ćirić LB, Jesic SN, Ume JS: The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2008, 37: 781–791. 10.1016/j.chaos.2006.09.093
Abbas M, Babu GVR, Alemayehu GN: On common fixed points of weakly compatible mappings satisfying generalized condition. Filomat 2011, 25(2):9–19. 10.2298/FIL1102009A
Gordji ME, Baghani H, Cho YJ: Coupled fixed point theorem for contractions in intuitionistic fuzzy normed spaces. Math. Comput. Model. 2011, 54: 1897–1906. 10.1016/j.mcm.2011.04.014
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht; 2001.
Haghi RH, Rezapour SH, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052
Lakshmikantham V, Ćirić LB: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011, 2011: 81. 10.1186/1687-1812-2011-81
Zhu XH, Xiao JZ: Note on ‘Coupled fixed point theorems for contractions in fuzzy metric spaces’. Nonlinear Anal. 2011, 74: 5475–5479. 10.1016/j.na.2011.05.034
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
Gregori V, Romaguera S, Veeramani P: A note on intuitionstic fuzzy spaces. Chaos Solitons Fractals 2006, 28: 902–905. 10.1016/j.chaos.2005.08.113
Mursaleen M, Mohiuddine SA: On stability of a cubic functional equation in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 42: 2997–3005. 10.1016/j.chaos.2009.04.041
Park JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051
Saadati R, Vaezpour S: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 2005, 17: 475–484. 10.1007/BF02936069
Acknowledgements
First, the authors are very grateful to the referees for their careful reading of the manuscript, valuable comments and suggestions. Next, this research was supported by the National Natural Science Foundation of China (11171286).
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Zhu, J., Wang, Y. & Zhu, CC. Fixed point theorems for contractions in fuzzy normed spaces and intuitionistic fuzzy normed spaces. Fixed Point Theory Appl 2013, 79 (2013). https://doi.org/10.1186/1687-1812-2013-79
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DOI: https://doi.org/10.1186/1687-1812-2013-79