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Berinde-Borcut tripled fixed point theorem in partially ordered (intuitionistic) fuzzy normed spaces
Journal of Inequalities and Applications volume 2014, Article number: 47 (2014)
Abstract
In this paper, we prove some tripled fixed point theorems in fuzzy normed spaces. Our results improve and restate the proof lines of the main results given in the paper (Abbas et al. in Fixed Point Theory Appl. 2012:187, 2012).
1 Introduction
Once the notion of coupled fixed point was given by Gnana Bhaskar and Lakshmikantham in [1], the theory of multidimensional fixed points has attracted much attention (see, for instance, [2–8]), specially in the tripled case (see [9–17]).
Recently, many authors have shown the existence of tripled fixed points and common tripled fixed points for some contractions in cone metric spaces, partially ordered metric spaces, fuzzy metric spaces, fuzzy normed spaces, intuitionistic fuzzy normed spaces and others. Especially in [18], Abbas et al. proved some tripled fixed point theorem for contractive mappings in partially complete intuitionistic fuzzy normed spaces. But the authors found some mistakes in the proof lines of their main result. In this paper we give a corrected version of the main theorem.
A t-norm (resp., a t-conorm) is a mapping (resp., ) that is associative, commutative, and non-decreasing in both arguments and has 1 (resp., 0) as identity.
For any , let the sequence be defined by and . Then a t-norm ∗ is said to be of H-type if the sequence is equicontinuous at .
Definition 2 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 ,
-
(F1)
;
-
(F2)
for all if and only if ;
-
(F3)
for all ;
-
(F4)
;
-
(F5)
is continuous;
-
(F6)
and .
Using the continuous t-norms and t-conorms, Saadati and Park [21] introduced the concept of an intuitionistic fuzzy normed space.
An intuitionistic fuzzy normed space (briefly, IFNS) is a 5-tuple where X is a vector space, ∗ is a continuous t-norm, ⋄ is a continuous t-conorm and are fuzzy sets such that, for all and ,
-
(IF1)
;
-
(IF2)
and ;
-
(IF3)
for all if and only if if and only if for all ;
-
(IF4)
and for all ;
-
(IF5)
and ;
-
(IF6)
are continuous;
-
(IF7)
and .
Obviously, if is a IFNS, then is a FNS. We refer to this space as its support.
Lemma 4 is a non-decreasing function on and is a non-increasing function on .
Some properties and examples of IFNS and the concepts of convergence and a Cauchy sequence in IFNS are given in [21].
Definition 5 Let be an IFNS.
-
(1)
A sequence is called a Cauchy sequence if, for any and , there exists such that and for all .
-
(2)
A sequence is said to be convergent to a point , denoted by or by , if, for any and , there exists such that and for all .
-
(3)
An IFNS in which every Cauchy sequence is convergent is said to be complete.
Definition 6 ([7])
Let and be two mappings.
-
We say that F and g are commuting if for all .
-
A point is called a tripled coincidence point of the mappings F and g if , and . If g is the identity, is called a tripled fixed point of F.
-
If is a partially ordered set, then F is said to have the mixed g-monotone property if it verifies the following properties:
If g is the identity mapping, then F is said to have the mixed monotone property.
-
If is a partially ordered set, then X is said to have the sequential g-monotone property if it verifies the following properties: If g is the identity mapping, then X is said to have the sequential monotone property.
-
(B1)
If is a non-decreasing sequence and , then for all .
-
(B2)
If is a non-increasing sequence and , then for all .
-
(B1)
If g is the identity mapping, then X is said to have the sequential monotone property..
Definition 7 Let X and Y be two IFNS. A function is said to be continuous at a point if, for any sequence in X converging to , the sequence in Y converges to . If f is continuous at each , then f is said to be continuous on X.
The following lemma proved by Haghi et al. [23] is useful for our main results:
Lemma 8 Let X be a nonempty set and be a mapping. Then there exists a subset such that and is one-to-one.
Definition 9 Let be an IFNS. The pair is said to satisfy the n-property on if and whenever , and .
In order to state our results, we recall the main result given in [18].
Theorem 10 (Abbas et al., Theorem 2.2)
Let be a partially ordered set and suppose that , for all . Let be a complete IFNS such that has the n-property. Let and be two mappings such that F has the mixed g-monotone property and
for which and and , where . Suppose either
-
(a)
F is continuous or
-
(b)
X has the sequential g-monotone property.
If there exist such that , and , then F and g have a tripled coincidence point.
2 Comments and revised tripled fixed point theorem
Firstly, we show that the conditions of Theorem 10 are inadequate and, further, the proof lines of Theorem 10 are not correct. We also would like to point out that the results in [18] can be corrected under the appropriate conditions on the t-norm and the FNS. The proof lines of Theorem 10 are not correct (see pp.7 and 8):
where such that . Hence the sequence is a Cauchy sequence. This is not correct since the same p would not be valid for all positive integers . For example, let be an ordinary normed space, define for any and and for all . Then is an IFNS. If and , we have
Now, by replacing in Theorem 10 the hypothesis that μ satisfies the n-property with the one that the t-norm is of H-type, we state and prove a tripled fixed point theorem as a modification.
Theorem 11 Let be a partially ordered set and be a complete FNS such that ∗ is of H-type and for all . Let be a number and be mapping such that F has the mixed monotone property and
for which , and . Suppose that either:
-
(a)
F is continuous or
-
(b)
X has the sequential monotone property.
If there exist such that , and , then F has a tripled fixed point. Furthermore, if and are comparable, then , that is, .
Proof As in [18] starting with such that , and , one can define inductively three sequences such that , and .
Define
Continuing as in [18], we have
Since for all , it follows that
This implies that
Since for all , we have for all .
Now, we claim that, for any and ,
In fact, it is obvious for by (2), and Lemma 4 since and is non-decreasing. Assume that (3) holds for some . By (2), we have
and so
Thus, from (1), (3) and , we have
Hence, by the monotonicity of the t-norm ∗, we have
Similarly, we have
Therefore, by induction, (3) holds for all . Suppose that and are given. By hypothesis, since ∗ is a t-norm of H-type, there exists such that for all and . Since , there exists such that for all . Hence, from (3), we get
Therefore, , and are Cauchy sequences. We can continue as in [18] to complete the proof. □
Theorem 12 Let be a partially ordered set and be a complete FNS such that ∗ is of H-type and for all . Let be a number and and be two mappings such that F has the mixed g-monotone property and
for which and and , where . Suppose either
-
(a)
F is continuous or
-
(b)
X has the sequential g-monotone property.
If there exist such that , and , then F and g have a tripled coincidence point.
Proof As in Theorem 2.2 in [18]. □
Of course, all the results are valid if X is intuitionistic.
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Acknowledgements
The first author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (Under NRU-CSEC Project No. NRU56000508). The last three authors have been partially supported by Junta de Andalucía, by projects FQM-268, FQM-178 and FQM-235 of the Andalusian CICYE.
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Kumam, P., Martínez-Moreno, J., Roldán-López-de-Hierro, AF. et al. Berinde-Borcut tripled fixed point theorem in partially ordered (intuitionistic) fuzzy normed spaces. J Inequal Appl 2014, 47 (2014). https://doi.org/10.1186/1029-242X-2014-47
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DOI: https://doi.org/10.1186/1029-242X-2014-47