Abstract
Recently, Gordji et al. [Math. Comput. Model. 54, 18971906 (2011)] prove the coupled coincidence point theorems for nonlinear contraction mappings satisfying commutative condition in intuitionistic fuzzy normed spaces. The aim of this article is to extend and improve some coupled coincidence point theorems of Gordji et al. Also, we give an example of a nonlinear contraction mapping which is not applied by the results of Gordji et al., but can be applied to our results.
2000 MSC: primary 47H10; secondary 54H25; 34B15.
Keywords:
intuitionistic fuzzy normed space; coupled fixed point; coupled coincidence point; partially ordered set; commutative condition1. Introduction
The classical Banach's contraction mapping principle first appear in [1]. Since this principle is a powerful tool in nonlinear analysis, many mathematicians have much contributed to the improvement and generalization of this principle in many ways (see [210] and others).
One of the most interesting is study to other spaces such as probabilistic metric spaces (see [1115]). The fuzzy theory was introduced simultaneously by Zadeh [16]. The idea of intuitionistic fuzzy set was first published by Atanassov [17]. Since then, Saadati and Park [18] introduced the concept of intuitionistic fuzzy normed spaces (IFNSs). In [19], Saadati et al. have modified the notion of IFNSs of Saadati and Park [18].
Several researchers have applied fuzzy theory to the wellknown results in many fields, for example, quantum physics [20], nonlinear dynamical systems [21], population dynamics [22], computer programming [23], fixed point theorem [2427], fuzzy stability problems [2830], statistical convergence [3134], functional equation [35], approximation theory [36], nonlinear equation [37,38] and many others.
In the other hand, coupled fixed points and their applications for binary mappings in partially ordered metric spaces were introduced by Bhaskar and Lakshmikantham [39]. They applied coupled fixed point theorems to show the existence and uniqueness of a solution for a periodic boundary value problem. After that, Lakshmikantham and Ćirić [40] proved some more generalizations of coupled fixed point theorems in partially ordered sets.
Recently, Gordji et al. [41] proved some coupled coincidence point theorems for contractive mappings satisfying commutative condition in partially complete IFNSs as follows:
Theorem 1.1 (Gordji et al. [41]). Let (X, ≼) be a partially ordered set and (X, μ, ν, *, ◊) a complete IFNS such that (μ, ν) has nproperty and
Let F: X × X → X and g : X → X be two mappings such that F has the mixed gmonotone property and
for which g(x) ≼ g(u) and g(y) ≽ z g(v), where 0 <k < 1, F(X × X) ⊆ g(X), g is continuous and g commuting with F. Suppose that either
(1) F is continuous or
(2) X has the following properties:
(a) if {x_{n}} is a nondecreasing sequence with {x_{n}} → x, then gx_{n }≼ gx for all n ∈ ℕ,
(b) if {y_{n}} is a nonincreasing sequence with {y_{n}} → y, then gy ≼ gy_{n }for all n ∈ ℕ.
If there exist x_{0}, y_{0 }∈ X such that
then F and g have a coupled coincidence point in X × X.
In this article, we improve the result given by Gordji et al. [41] without using the commutative condition and also give an example to validate the main results in this article. Our results improve and extend some couple fixed point theorems due to Gordji et al. [41] and other couple fixed point theorems.
2. Preliminaries
Now, we give some definitions, examples and lemmas for our main results in this article.
Definition 2.1 ([42]). A binary operation *: [0,1]^{2 }→ [0,1] is called a continuous tnorm if ([0,1], *) is an abelian topological monoid, i.e.,
(1) * is associative and commutative;
(2) * is continuous;
(3) a * 1 = a for all a ∈ [0,1];
(4) a * b ≤ c * d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0,1].
Definition 2.2 ([42]). A binary operation ◊: [0,1]^{2 }→ [0,1] is called a continuous tconorm if ([0,1],◊) is an abelian topological monoid, i.e.,
(1) ◊ is associative and commutative;
(2) ◊ is continuous;
(3) a ◊ 0 = a for all a ∈ [0,1];
(4) a ◊ b ≤ c ◊ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0,1].
Using the continuous tnorm and tconorm, Saadati and Park [18] introduced the concept of IFNSs.
Definition 2.3 ([18]). The 5tuple (X, μ, ν, *,◊) is called an IFNS if X is a vector space, * is a continuous tnorm, ◊ is a continuous tconorm and μ, ν are fuzzy sets on X × (0, ∞) satisfying the following conditions: for all x, y ∈ X and s, t > 0,
(IF_{1}) μ(x, t) + ν(x, t) ≤ 1;
(IF_{2}) μ(x, t) > 0;
(IF_{3}) μ(x, t) = 1 if and only if x = 0;
(IF_{4})
(IF_{5}) μ(x, t) * μ(y, s) ≤ μ(x + y, t + s);
(IF_{6}) μ(x,.): (0, ∞) → [0,1] is continuous;
(IF_{7}) μ is a nondecreasing function on ℝ^{+},
(IF_{8}) ν(x, t) < 1;
(IF_{9}) ν(x, t) = 0 if and only if x = 0;
(IF_{10})
(IF_{11}) ν(x, t) ◊ ν(y, s) ≥ ν(x + y, t + s);
(IF_{12}) ν(x,·): (0, ∞) → [0,1] is continuous;
(IF_{13}) ν is a nonincreasing function on ℝ^{+},
In this case, (μ, ν) is called an intuitionistic fuzzy norm.
Definition 2.4 ([18]). Let (X, μ, ν, *,◊) be an IFNS.
(1) A sequence {x_{n}} in X is said to be convergent to a point x ∈ X with respect to the intuitionistic fuzzy norm (μ, ν) if, for any ε > 0 and t > 0, there exists k ∈ ℕ such that
In this case, we write lim_{n}_{→∞ }x_{n }= x. In fact that lim_{n}_{→∞ }x_{n }= x if μ(x_{n } x, t) → 1 and ν(x_{n } x, t) → 0 as n → ∞ for every t > 0.
(2) A sequence {x_{n}} in X is called a Cauchy sequence with respect to the intuitionistic fuzzy norm (μ, ν) if, for any ε > 0 and t > 0, there exists k ∈ ℕ such that
This implies {x_{n}} is Cauchy if μ(x_{n } x_{m}, t) → 1 and ν(x_{n } x_{m}, t) → 0 as n, m → ∞ for every t > 0.
(3) An IFNS (X, μ, ν, *, ◊) is said to be complete if every Cauchy sequence in (X, μ, ν, *, ◊) is convergent.
Definition 2.5 ([43,44]). Let X and Y be two IFNS. A function g : X → Y is said to be continuous at a point x_{0 }∈ X if, for any sequence {x_{n}} in X converging to a point x_{0 }∈ X, the sequence {g(x_{n})} in Y converges to a point g(x_{0}) ∈ Y. If g : X → Y is continuous at each x ∈ X, then g : X → Y is said to be continuous on X.
Example 2.6 ([41]). Let (X,  · ) be an ordinary normed space and θ an increasing and continuous function from ℝ^{+ }into (0,1) such that lim_{t→∞ }θ(t) = 1. Four typical examples of these functions are as follows:
Let a * b = ab and a ◊ b ≥ ab for all a, b ∈ [0,1]. If, for any t ∈ (0, ∞), we define
then (X, μ, ν, *, ◊) is an IFNS.
The other basic properties and examples of IFNSs are given in [18].
Definition 2.7 ([41]). Let (X, μ, ν, *,◊) be an IFNS. (μ, ν) is said to satisfy the nproperty on X × (0, ∞) if
whenever x ∈ X, k > 1 and p > 0.
For examples for nproperty see in [41]. Next, we give some notion in coupled fixed point theory.
Definition 2.8 ([39]). Let X be a nonempty set. An element (x, y) ∈ X × X is call a coupled fixed point of the mapping F : X × X → X if
Definition 2.9 ([40]). Let X be a nonempty set. An element (x, y) ∈ X × X is call a coupled coincidence point of the mappings F : X × X → X and g : X → X if
Definition 2.10 ([39]). Let (X, ≼) be a partially ordered set and F : X × X → X be a mapping. The mapping F is said to has the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument, that is, for any x, y ∈ X
and
Definition 2.11 ([40]). Let (X, ≼) be a partially ordered set and F : X × X → X, g : X → X be mappings. The mapping F is said to has the mixed gmonotone property if F is monotone gnondecreasing in its first argument and is monotone gnonincreasing in its second argument, that is, for any x, y ∈ X,
and
Definition 2.12 ([40]). Let X be a nonempty set and F : X × X → X, g : X → X be mappings. The mappings F and g are said to be commutative if
The following lemma proved by Haghi et al. [45] is useful for our main results:
Lemma 2.13 ([45]). Let X be a nonempty set and g : X → X be a mapping. Then, there exists a subset E ⊆ X such that g(E) = g(X) and g : E → X is onetoone.
3. Main Results
First, we prove a coupled fixed point theorem for a mapping F : X × X → X which is an essential tool in the partial order IFNSs to show the existence of coupled fixed point. Although the proof in Theorem 3.1 is not difficult to modify, it is an important theorem which is helpful in proving some coupled coincidence point theorems without commutative condition.
Theorem 3.1. Let (X, ≼) be a partially ordered set and (X, μ, ν, *, ◊) a complete IFNS such that (μ, ν) has nproperty and
Let F : X × X → X be mapping such that F has the mixed monotone property and
for which x ≼ u and y ≽ v, where 0 <k < 1. Suppose that either
(1) F is continuous or
(2) X has the following properties:
(a) if {x_{n}} is a nondecreasing sequence with {x_{n}} → x, then x_{n }≼ x for all n ∈ ℕ,
(b) if {y_{n}} is a nonincreasing sequence with {y_{n}} → y, then y ≼ y_{n }for all n ∈ ℕ.
If there exist x_{0}, y_{0 }∈ X such that
then F has a coupled fixed point in X × X.
Proof. Let x_{0}, y_{0 }∈ X be such that
Since F(X × X) ⊆ X, we can construct the sequences {x_{n}} and {y_{n}} in X such that
Now, we show that
In fact, by induction, we prove this. For n = 0, since x_{0 }≼ F(x_{0}, y_{0}) = x_{1 }and y_{0 }= F(y_{0}, x_{0}) ≽ y_{1}, we show that (3.4) holds for n = 0. Suppose that (3.4) holds for any n ≥ 0. Then, we have
Since F has the mixed monotone property, it follows from (3.5) and (2.1) that
and also it follows from (3.5) and (2.2) that
If we take y = y_{n }and x = x_{n }in (3.6), then we get
If we take y = y_{n+1 }and x = x_{n+1 }in (3.7), then we get
Hence, it follows from (3.8) and (3.9) that
Therefore, by induction, we conclude that (3.4) holds for all n ≥ 0, that is,
and
Define α_{n}(t): = μ(x_{n } x_{n+1}, t) * μ(y_{n } y_{n+1}, t). Then, using (3.2) and (3.3), we have
and
From the tnorm property, (3.13) and (3.14), it follows that
From (3.1), we have
By (3.15) and (3.16), we get α_{n}(kt) ≥ [α_{n1}(t)]^{2 }for all n ≥ 1. Repeating this process, we have
which implies that
On the other hand, we have
By property of tnorm, we get
where p > 0 such that m <n^{p}. Sine (μ, ν) has the nproperty, we have
and so
Next, we claim that
Define β_{n}(t) := ν(x_{n } x_{n+1}, t) ◊ ν(y_{n } y_{n+1}, t). It follows from (3.2) and (3.3) that
and
Thus, it follows from the notion of tconorm, (3.21) and (3.22) that
From (3.1), we have
Thus, by (3.23) and (3.24), we get β_{n}(kt) ≤ [β_{n1}(t)]^{2 }for all n ≥ 1. Repeating this process again, we have
that is,
Since we have
by the tconorm property, we get
where p > 0 such that m <n^{p}. Sine (μ, ν) has the nproperty, we have
and so
From (3.20) and (3.28), we know that the sequences {x_{n}} and {y_{n}} are Cauchy sequences in X. Since X complete, there exist x, y ∈ X such that
Next, we show that x = F(x, y) and y = F(y, x). If the assumption (1) holds, then we have
and
Therefore, x = F(x, y) and y = F(y, x), that is, F has a coupled fixed point.
Suppose that the assumption (2) holds. Since {x_{n}} is nondecreasing and x_{n }→ x, it follows from (a) that x_{n }≼ x for all n ∈ ℕ. Similarly, we can conclude that y_{n }≽ y for all n ∈ ℕ. Then, by (3.2), we get
Taking the limit as n → ∞, we have μ(x  F(x, y), kt) = 1 and so x = F(x, y). Using (3.2) again, we have
Taking the limit as n → ∞ in both sides of (3.33), we have ν(y  F(y, x), kt) = 0 and then y = F(y, x). Therefore, F has a coupled fixed point at (x, y). This completes the proof. □
Next, we prove the existence of coupled coincidence point theorem, where we do not require that F and g are commuting.
Theorem 3.2. Let (X, ≼) be a partially ordered set and (X, μ, ν, *,◊) a IFNS such that (μ, ν) has nproperty and
Let F : X × X → X and g : X → X be two mappings such that F has the mixed gmonotone property and
for which gx ≼ gu and gy ≽ gv, where 0 <k < 1, F(X × X) ⊆ g(X) and g is continuous, g(X) is complete. Suppose that either
(1) F is continuous or
(2) X has the following property:
(a) if {x_{n}} is a nondecreasing sequence with {x_{n}} → x, then x_{n }≼ x for all n ∈ ℕ,
(b) if {y_{n}} is a nonincreasing sequence with {y_{n}} → y, then y ≼ y_{n }for all n ∈ ℕ.
If there exist x_{0}, y_{0 }∈ X such that
then F and g have a coupled coincidence point in X × X.
Proof. Using Lemma 2.13, there exists E ⊆ X such that g(E) = g(X) and g : E → X is onetoone. We define a mapping
As g is one to one on g(E), so A is welldefined. Thus, it follows from (3.35) and (3.36) that
and
for all gx, gy, gu, gv ∈ g(E) with gx ≼ gy and gy ≽ gv. Since F has the mixed gmonotone property, for all x, y ∈ X, we have
and
Thus, it follows from (3.36), (3.39) and (3.40) that, for all gx, gy ∈ g(E),
and
which implies that
Suppose that the assumption (1) holds. Since F is continuous,
Suppose that the assumption (2) holds. We can conclude similarly in the proof of Theorem
3.1 that the mapping
Finally, we prove that F and g have a coupled coincidence point in X. Since (u, v) is a coupled fixed point of
Since (u, v) ∈ g(X) × g(X), there exists a point
Thus, it follows from (3.43) and (3.44) that
Also, from (3.36) and (3.45), we get
Therefore,
Next, we give example to validate Theorem 3.2.
Example 3.3. Let X = ℝ, a * b = ab ≥ a ◊ b for all a, b ∈ [0,1] and
that (μ, ν) satisfies the nproperty on X × (0, ∞). If X is endowed with the usual order as x ≼ y ⇔ y  x ∈ [0, ∞), then (X, ≼) is a partially ordered set. Define mappings F : X × X → X and g : X → X by
and
Since
for all x, y ∈ X, the mappings F and g do not satisfy the commutative condition. Hence, Theorem 2.5 of Gordji et al. [41] cannot be applied to this example. But, by simple calculation, we see that F(X × X) ⊆ g(X), g and F are continuous and F has the mixed gmonotone property. Moreover, there exist x_{0 }= 1 and y_{0 }= 3 with
and
Now, for any x, y, u, v ∈ X with gx ≼ gu and gy ≽ gv, we get
and
where 0 <k < 1. Therefore, all the conditions of Theorem 3.2 hold and so F and g have a coupled coincidence point in X × X. In fact, a point (2,2) is a coupled coincidence point of F and g.
Remark 3.4. Although Theorem 2.5 of Gordji et al. [41] is essential tool in the partially ordered fuzzy normed spaces to claim the existence of coupled coincidence points of two mappings. However, some mappings do not have the commutative property as in the above example. Therefore, it is very interesting to use Theorem 3.2 as another auxiliary tool to claim the existence of a coupled coincidence point.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the third author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under the Project NRUCSEC No. 54000267) for financial support during the preparation of this manuscript. This project was partially completed while the first and third authors visit Department of Mathematics Education, Gyeongsang National University, Korea. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 20110021821).
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