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Tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces
Fixed Point Theory and Applications volume 2012, Article number: 187 (2012)
Abstract
The aim of this paper is to prove the existence of tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces (IFNS). Our results generalize and extend recent coupled fixed point theorems in IFNS.
MSC:47H09, 47H10, 54H25.
1 Introduction and preliminaries
The evolution of fuzzy mathematics commenced with an introduction of the notion of fuzzy sets by Zadeh [1] in 1965 as a new way to represent vagueness in every day life. The idea of intuitionistic fuzzy sets (IFS) was introduced by Atanassov [2]. Saadati and Park [3, 4] introduced intuitionistic fuzzy normed spaces (IFNS). For the detailed survey on fixed point results in fuzzy metric spaces, fuzzy normed spaces and IFNS, we refer the reader to [5–8]. Recently coupled fixed point theorems have been proved in IFNS; for details of these we refer to Gordji [9] and Sintunavarat et al. [10]. More recently, tripled fixed point theorems have been introduced in partially ordered metric spaces by Berinde [11]. In this paper, we have proved tripled fixed point and tripled coincidence point theorems in IFNS. Now we give some definitions, examples and lemmas for our main results.
For the sake of completeness, we recall some definitions and known results in a fuzzy metric space.
Definition 1.1 ([1])
Let X be any set. A fuzzy set A in X is a function with domain X and values in .
Definition 1.2 ([12])
A binary operation is called a continuous t-norm if
-
(1)
∗ is associative and commutative;
-
(2)
∗ is continuous;
-
(3)
for all ;
-
(4)
whenever and .
Example 1.3 Three typical examples of continuous t-norms are (minimum t-norm), (product t-norm), and (Lukasiewicz t-norm).
Definition 1.4 ([12])
A binary operation is called a continuous t-conorm if
-
(1)
⋄ is associative and commutative;
-
(2)
⋄ is continuous;
-
(3)
for all ;
-
(4)
whenever and .
Example 1.5 Two typical examples of continuous t-conorms are and .
Using the continuous t-norm and continuous t-conorm, Saadati and Park [3] introduced the concept of intuitionistic fuzzy normed spaces.
Definition 1.6 ([3])
The 5-tuple is called an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space, ∗ and ⋄ are continuous t-norm and continuous t-conorm respectively and μ, υ are fuzzy sets on satisfying the following conditions: for all and ,
(IF1) ;
(IF2) ;
(IF3) if and only if ;
(IF4) for all ;
(IF5) ;
(IF6) is continuous;
(IF7) μ is a non-decreasing function on ,
(IF8) ;
(IF9) if and only if ;
(IF10) for all ;
(IF11) ;
(IF12) is continuous;
(IF13) υ is a non-increasing function on ,
In this case is called an intuitionistic fuzzy norm.
Definition 1.7 ([3])
Let be an IFNS. A sequence in X is said to be:
-
(1)
convergent to a point with respect to an intuitionistic fuzzy norm if for any and , there exists such that
In this case, we write .
-
(2)
Cauchy sequence with respect to an intuitionistic fuzzy norm if for any and , there exists such that
Definition 1.8 ([3])
An IFNS is said to be complete if every Cauchy sequence in is convergent.
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 a point , the sequence in Y converges to . If g is continuous at each , then is said to be continuous on X.
Examples 1.10 Let be an ordinary normed space and ϕ be an increasing and continuous function from into such that . Four typical examples of these functions are as follows:
Let ∗ and ⋄ be a continuous t-norm and a continuous t-conorm such that
For any , we define
then is an IFNS.
For further details regarding IFNS, we refer to [3].
Definition 1.11 ([9])
Let be an IFNS. is said to satisfy the n-property on if
where , , and .
Throughout this paper, we assume that satisfies the n-property on .
Definition 1.12 ([11])
Let X be a non-empty set. An element is called a tripled fixed point of if
Definition 1.13 Let X be a non-empty set. An element is called a tripled coincidence point of mappings and if
Definition 1.14 ([11])
Let be a partially ordered set. A mapping is said to have the mixed monotone property if F is monotone non-decreasing in its first and third argument and is monotone non-increasing in its second argument; that is, for any
and
Definition 1.15 Let be a partially ordered set, and . A mapping is said to have the mixed g-monotone property if F is monotone g-non-decreasing in its first and third argument and is monotone g-non-increasing in its second argument; that is, for any ,
and
Lemma 1.16 ([15])
Let X be a non-empty set and be a mapping. Then there exists a subset such that and is one-to-one.
2 Main results
Theorem 2.1 Let be a complete IFNS, ⪯ be a partial order on X and suppose that
for all . Suppose that has the mixed monotone property and
for all those x, y, z, u, v, w in X for which , , , where . If either
-
(a)
F is continuous or
-
(b)
X has the following property:
(bi) if is a non-decreasing sequence and , then for all ,
(bii) if is a non-decreasing sequence and , then for all ,
(biii) if is a non-decreasing sequence and , then for all ,
then F has a tripled fixed point provided that there exist such that
Proof Let be such that
As , so we can construct sequences , and in X such that
Now we show that
Since
(2.4) holds for . Suppose that (2.4) holds for any . That is,
As F has the mixed monotone property so by (2.5) we obtain
which on replacing y by and z by in (i) implies that ; replacing x by and z by in (ii), we obtain ; replacing y by and x by in (iii), we get . Thus, we have , that is, . Similarly, we have
which on replacing y by and x by in (iv) implies that ; replacing x by and y by in (v), we obtain ; replacing y by in (vi), we get . Thus, we have , that is, . Similarly, we have
which on replacing y by and x by in (vii) implies that ; replacing x by and z by in (viii), we obtain ; replacing y by and z by in (xi), we get . Thus, we have , that is, . So, by induction, we conclude that (2.5) holds for all , that is,
Define
Consider
Also,
Now,
Using the properties of a t-norm, (2.9)-(2.12) and (2.1), we obtain
which implies that
Now, repetition of the above process gives
Hence,
It is obvious to note that
Consider
where such that . Since has the n-property on , therefore
Hence,
Next, we show that
Define
Note that
and
Using the properties of a t-conorm, (2.15)-(2.18) and (2.1), we obtain
that is,
Now, repetition of the above process gives
which further implies that
Using the properties of a t-conorm, we get
where such that . Since has the n-property on , we have
So,
Now, (2.14) and (2.20) imply that , and are Cauchy sequences in X. Since X is complete, there exist x, y and z such that , and . If the assumption (a) does hold, then we have
and
Suppose that the assumption (b) holds then
which, on taking limit as , gives , . Also,
which, on taking limit as , implies , . Finally, we have
which, on taking limit as , gives , . □
Theorem 2.2 Let be an IFNS, ⪯ be a partial order on X, and suppose that
for all . Let and be mappings such that F has the mixed g-monotone property and
for all those x, y, z, and u, v, w for which , , , where . Assume that is complete, and g is continuous. If either
-
(a)
F is continuous or
-
(b)
X has the following property:
(bi) if is a non-decreasing sequence and , then for all ,
(bii) if is a non-decreasing sequence and , then for all , and
(biii) if is a non-decreasing sequence and , then for all .
Then F has a tripled coincidence point provided that there exist such that
Proof By Lemma 1.16, there exists such that is one-to-one and . Now, define a mapping by
Since g is one-to-one, so is well defined. Now, (2.22) and (2.23) imply that
for all for which , , . Since F has the mixed g-monotone property for all , so we have
Now, from (2.23) and (2.25), we have
Hence, has the mixed monotone property. Suppose that the assumption (a) holds. Since F is continuous, is also continuous. By using Theorem 2.1, has a tripled fixed point . If the assumption (b) holds, then using the definition of , following similar arguments to those given in Theorem 2.1, has a tripled fixed point . Finally, we show that F and g have a tripled coincidence point. Since has a tripled fixed point , we get
Hence, there exist such that , , and . Now, it follows from (2.27) that
Thus, is a tripled coincidence point of F and g. □
Example 2.3 Let be a usual normed, and be defined by
It is easy to see that ∗ is a continuous t-norm and ⋄ is a continuous t-conorm satisfy
Let be defined by for all . Now we have is an IFNS, where
such that satisfies the n-property on .
If X is endowed with usual order as , then is a partially ordered set. Define mappings and by
Obviously, F and g both are onto maps so . Also, F and g are continuous and F has the mixed g-monotone property. Indeed,
Similarly, we can prove that
and
If , , , then
So, there exist such that
Now, for all , for which , , , we have
for . Hence, there exists such that
for all , for which , , .
Now, for all , for which , , , we have
for . Hence, there exists such that
for all , for which , , .
Therefore, all the conditions of Theorem 2.2 are satisfied. So, F and g have a tripled coincidence point and here is a tripled coincidence point of F and g.
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Acknowledgements
The third author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the fourth author would like to thank the National Research University Project of Thailand’s Office of the Higher Education Commission for financial support (under the CSEC Project No. 55000613).
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Abbas, M., Ali, B., Sintunavarat, W. et al. Tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl 2012, 187 (2012). https://doi.org/10.1186/1687-1812-2012-187
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DOI: https://doi.org/10.1186/1687-1812-2012-187