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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 wellknown 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 \ast :[0,1]\times [0,1]\to [0,1] is a continuous tnorm if ∗ satisfies the following conditions:

(i)
∗ is commutative and associative;

(ii)
∗ is continuous;

(iii)
a\ast 1=a, \mathrm{\forall}a\in [0,1];

(iv)
a\ast b\le c\ast d, whenever a\le c and b\le d for all a,b,c,d\in [0,1].
A tnorm ∗ is said to be of Htype if the sequence of functions {\{{\ast}^{n}a\}}_{n=1}^{\mathrm{\infty}} is equicontinuous at a=1.
The tnorm {\ast}_{m} defined by a{\ast}_{m}b=min\{a,b\} is an example of an Htype tnorm ∗.
Definition 1.3 ([13])
A binary operation \star :[0,1]\times [0,1]\to [0,1] is a continuous tconorm if ⋆ satisfies the following conditions:

(i)
⋆ is commutative and associative;

(ii)
⋆ is continuous;

(iii)
a\star 0=a, \mathrm{\forall}a\in [0,1];

(iv)
a\star b\le c\star d, whenever a\le c and b\le d for all a,b,c,d\in [0,1].
Definition 1.4 ([23])
A fuzzy normed space (briefly, FNS) is a triple (X,\mu ,\ast ), where X is a vector space, ∗ is a continuous tnorm and \mu :X\times (0,\mathrm{\infty})\to [0,1] is a fuzzy set such that, for all x,y\in X and t,s>0,

(i)
\mu (x,t)>0;

(ii)
\mu (x,t)=1 if and only if x=\theta;

(iii)
\mu (cx,t)=\mu (x,\frac{t}{c}) for all c\ne 0;

(iv)
\mu (x,s)\ast \mu (y,t)\le \mu (x+y,s+t);

(v)
\mu (x,\cdot ) is a continuous function of {\mathbb{R}}^{+} and
\underset{t\to \mathrm{\infty}}{lim}\mu (x,t)=1,\phantom{\rule{2em}{0ex}}\underset{t\to 0}{lim}\mu (x,t)=0.
By the results in George and Veeramani [19], we can know that every fuzzy norm (\mu ,\ast ) on X generates a Hausdorff first countable topology {\tau}_{\mu} on X which has as a base the family of open sets of the form
where B(x,r,t)=\{y\in X:\mu (x,y,t)>1r\} for all x\in X, r\in (0,1) and t>0.
The 5tuple (X,\mu ,\nu ,\ast ,\star ) is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a linear space, ∗ is a continuous tnorm, ⋆ is a continuous tconorm and μ, ν are fuzzy sets on X\times (0,\mathrm{\infty}) satisfying the following conditions:

(i)
\mu (x,t)+\nu (x,t)\le 1, \mathrm{\forall}(x,t)\in X\times (0,\mathrm{\infty});

(ii)
\mu (x,t)>0;

(iii)
\mu (x,t)=1 if and only if x=\theta;

(iv)
\mu (cx,t)=\mu (x,\frac{t}{c}) for all c\ne 0;

(v)
\mu (x,s)\ast \mu (y,t)\le \mu (x+y,s+t);

(vi)
\mu (x,\cdot ) is a continuous function of {\mathbb{R}}^{+} and
\underset{t\to \mathrm{\infty}}{lim}\mu (x,t)=1,\phantom{\rule{2em}{0ex}}\underset{t\to 0}{lim}\mu (x,t)=0; 
(vii)
\nu (x,t)<1;

(viii)
\nu (x,t)=0 if and only if x=\theta;

(ix)
\nu (cx,t)=\nu (x,\frac{t}{c}) for all c\ne 0;

(x)
\nu (x,s)\star \nu (y,t)\ge \nu (x+y,s+t);

(xi)
\nu (x,\cdot ) is a continuous function of {\mathbb{R}}^{+} and
\underset{t\to \mathrm{\infty}}{lim}\nu (x,t)=0,\phantom{\rule{2em}{0ex}}\underset{t\to 0}{lim}\nu (x,t)=1.
Park proved in [22], among other results, that each intuitionistic fuzzy norm (\mu ,\nu ) on X generates a Hausdorff first countable topology {\tau}_{(\mu ,\nu )} on X which has as a base the family of open sets of the form \{B(x,r,t):x\in X,r\in (0,1),t>0\}, where B(x,r,t)=\{y\in X:\mu (x,y,t)>1r,\upsilon (x,y,t)<r\} for all x\in X, r\in (0,1) and t>0. According to this topology, Park [22] gave the following definitions.
Definition 1.6 A sequence \{{x}_{n}\} in an intuitionistic fuzzy normed linear space (X,\mu ,\nu ,\ast ,\star ) is said to converge to x\in X with respect to the intuitionistic fuzzy norm (\mu ,\nu ) if, for any \epsilon >0, t>0, 0<\epsilon <1, there exists an integer {n}_{0}\in \mathbb{N} such that
Definition 1.7 A sequence \{{x}_{n}\} in an intuitionistic fuzzy normed linear space (X,\mu ,\nu ,\ast ,\star ) is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm (\mu ,\nu ) if, for any \epsilon >0, t>0, 0<\epsilon <1, there exists an integer {n}_{0}\in \mathbb{N} such that
Definition 1.8 Let (X,\mu ,\nu ,\ast ,\star ) be an IFNS. Then (X,\mu ,\nu ,\ast ,\star ) is said to be complete if every Cauchy sequence in (X,\mu ,\nu ,\ast ,\star ) is convergent.
Definition 1.9 Let X and Y be two intuitionistic fuzzy normed spaces. A mapping f:X\to Y is said to be continuous at {x}_{0}\in X if, for any sequence \{{x}_{n}\} in X converging to {x}_{0}, the sequence \{f({x}_{n})\} in Y converges to f({x}_{0})\in Y. If f:X\to Y is continuous at each x\in X, then f is said to be continuous on X.
For the topology {\tau}_{(\mu ,\nu )}, Gregori et al. [20] proved the following result.
Lemma 1.1 Let (X,\mu ,\nu ,\ast ,\star ) be an intuitionistic fuzzy metric space. Then the topologies {\tau}_{(\mu ,\nu )} and {\tau}_{\mu} coincide on X.
The following lemma was proved by Haghi et al. [15].
Lemma 1.2 Let X be a nonempty set, and let g:X\to X be a mapping. Then there exists a subset E\subset X such that g(E)=g(X) and g:E\to X is onetoone.
Definition 1.10 ([16])
A point (x,y)\in X\times X is called a coupled coincidence point of the mappings F:X\times X\to X and g:X\to X if
Definition 1.11 ([16])
Let (X,\u2291) be a partially ordered set, and let F:X\times X\to X, g:X\to X be two mappings. Then F is said to have the mixed gmonotone property if F is monotone gnondecreasing in the first argument and is monotone gnonincreasing in the second argument, that is, for any x,y\in X,
and
If g:X\to X is an identity mapping, we say that F has the mixed monotone property.
2 Main results
Theorem 2.1 Let (X,\u2291) be a partially ordered set, and let (X,\mu ,\ast ) be a complete FNS such that the tnorm ∗ is of Htype. Let F:X\to X be a mapping such that F is nondecreasing and
for which x\u2291u and t>0, where 0<k<1. Suppose either

(a)
F is continuous, or

(b)
if \{{x}_{n}\} is a nondecreasing sequence and {lim}_{n\to \mathrm{\infty}}{x}_{n}=x, then {x}_{n}\u2291x for all n\in \mathbb{N}.
If there exists {x}_{0}\in X such that
then F has a fixed point in X.
Proof Let {x}_{0}\in X such that {x}_{0}\u2291F({x}_{0}), and let {x}_{n}=F({x}_{n1}), n=1,2,\dots , then we have that
Now, put
Then, by using (2.1), we have
Thus, it follows that {\delta}_{n}(kt)\ge {\ast}^{2}{\delta}_{n1}(t), 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 tnorm ∗ is of Htype; for all \epsilon \in (0,1), there exists \eta >0 such that
for all s\in (1\eta ,1] and for all p. Note that
for all t>0 and 0<k<1, we have that there exists {n}_{0} such that
for all m>n>{n}_{0}. Thus, \{{x}_{n}\} is a Cauchy sequence. Since X is complete, there exists x\in X 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 {x}_{n}\u2291x for all n\in \mathbb{N}. It follows from (2.1) that
Thus, \mu (xF(x),kt)=1, that is, x=F(x). Therefore, x is a fixed point of F. The proof is completed. □
Theorem 2.2 Let (X,\u2291) be a partially ordered set, and let (X,\mu ,\ast ) be a complete FNS such that the tnorm ∗ is of Htype and a\ast b>0 for any a,b\in (0,1]. Let F:X\times X\to X be a mapping such that F has the mixed monotone property and
for which x\u2291u,y\u2292v and t>0, where 0<k<1. Suppose either

(a)
F is continuous, or

(b)
X has the following property:

(i)
if \{{x}_{n}\} is a nondecreasing sequence and {lim}_{n\to \mathrm{\infty}}{x}_{n}=x, then {x}_{n}\u2291x for all n\in \mathbb{N},

(ii)
if \{{y}_{n}\} is a nonincreasing sequence and {lim}_{n\to \mathrm{\infty}}{y}_{n}=y, then {y}_{n}\u2292y for all n\in \mathbb{N}.
If there exist {x}_{0},{y}_{0}\in X such that
then F has a coupled fixed point x,y\in X, that is,
Proof First, we define a partial order ≼ on X\times X as follows: (x,y)\preccurlyeq (u,v) if and only if x\u2291u and y\u2292v. Second, we define a fuzzy set on X\times X as follows: \tilde{\mu}((x,y),t)=\mu (x,t)\ast \mu (y,t) for any (x,y)\in X\times X and any t>0. Since (X,\mu ,\ast ) is a complete FNS, we can easily prove that (X\times X,\tilde{\mu},\ast ) is a complete FNS. Lastly, we define a mapping \tilde{F}:X\times X\to X\times X by
Since F has the mixed monotone property, if (x,y)\preccurlyeq (u,v), we have that
that is, \tilde{F}(x,y)\preccurlyeq \tilde{F}(u,v). Therefore, \tilde{F}:X\times X\to X\times X is a nondecreasing mapping. Since {x}_{0},{y}_{0}\in X and
we have that ({x}_{0},{y}_{0})\preccurlyeq (F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}))=\tilde{F}({x}_{0},{y}_{0}). If (x,y)\preccurlyeq (u,v), by (2.5) we have that
Thus, all the assumptions of Theorem 2.1 hold for \tilde{F} and (X\times X,\tilde{\mu},\ast ). By Theorem 2.1 we get that \tilde{F} has a fixed point (x,y)\in X\times X, that is, (x,y)=(F(x,y),F(y,x)). This implies that x=F(x,y), y=F(y,x), that is, (x,y) 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 (X,\u2291) be a partially ordered set, and let (X,\mu ,\nu ,\ast ,\star ) be a complete IFNS such that the tnorm ∗ is of Htype and a\ast b>0 for any a,b\in (0,1]. Let F:X\times X\to X be a mapping such that F has the mixed monotone property and
for which x\u2291u,y\u2292v and t>0, where 0<k<1. Suppose either

(a)
F is continuous, or

(b)
X has the following property:

(i)
if \{{x}_{n}\} is a nondecreasing sequence and {lim}_{n\to \mathrm{\infty}}{x}_{n}=x, then {x}_{n}\u2291x for all n\in \mathbb{N},

(ii)
if \{{y}_{n}\} is a nonincreasing sequence and {lim}_{n\to \mathrm{\infty}}{y}_{n}=y, then {y}_{n}\u2292y for all n\in \mathbb{N}.
If there exist {x}_{0},{y}_{0}\in X such that
then F has a coupled fixed point x,y\in X, that is,
Proof Assume that \{{x}_{n}\} is a sequence in (X,\mu ,\nu ,\ast ,\star ). Let t>0, 0<\epsilon <1. If \mu ({x}_{m}{x}_{n},t)>1\epsilon, then by Definition 1.5(i) we can deduce that \nu ({x}_{m}{x}_{n},t)<\epsilon. Thus, a sequence \{{x}_{n}\} in (X,\mu ,\nu ,\ast ,\star ) is Cauchy if and only if \{{x}_{n}\} is Cauchy in (X,\mu ,\ast ). By Lemma 1.1, we know that the topology of (X,\mu ,\nu ,\ast ,\star ) is the same as the topology of (X,\mu ,\ast ). This implies that (X,\mu ,\nu ,\ast ,\star ) is a complete IFNS if and only if (X,\mu ,\ast ) is a complete FNS. Therefore, by using Theorem 2.2 to (X,\mu ,\ast ) and F, we get that F has a coupled fixed point x,y\in X. The proof is completed. □
Theorem 2.4 Let (X,\u2291) be a partially ordered set, and let (X,\mu ,\nu ,\ast ,\star ) be a complete IFNS such that the tnorm ∗ is of Htype and a\ast b>0 for any a,b\in (0,1]. Let F:X\times X\to X, g:X\to X be two mappings such that F has the mixed g monotone property and
for which g(x)\u2291g(u) , g(y)\u2292g(v) and t>0, where 0<k<1, F(X\times X)\subseteq g(X) and g is continuous. Suppose either

(a)
F is continuous, or

(b)
X has the following property:

(i)
if \{{x}_{n}\} is a nondecreasing sequence and {lim}_{n\to \mathrm{\infty}}{x}_{n}=x, then g{x}_{n}\u2291gx for all n\in \mathbb{N},

(ii)
if \{{y}_{n}\} is a nonincreasing sequence and {lim}_{n\to \mathrm{\infty}}{y}_{n}=y, then g{y}_{n}\u2292gy for all n\in \mathbb{N}.
If there exist {x}_{0},{y}_{0}\in X such that
then there exist x,y\in X 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 2.2Theorem 2.4 are equivalent.
Remark 2.2 In [13] and [17], the condition a\star b\le ab for all a,b\in [0,1] 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 1=1\star 0\le 1\cdot 0=0, 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 X=\mathbb{R}, \mu (x,t)=\frac{t}{t+x}, \nu (x,t)=1\frac{t}{t+x} for every x\in X, and let t>0, a\ast b=min\{a,b\}, a\star b=max\{a,b\} for all a,b\in [0,1]. Then (X,\mu ,\nu ,\ast ,\star ) is a complete intuitionistic fuzzy normed linear space, and the tnorm ∗ and tconorm ⋆ are of Htype. If X is endowed with the usual order x\u2291y\iff xy\le 0, then (X,\u2291) is a partially ordered set. Let 0<k<1, and define F(x,y)=\frac{xy}{4}, gx=\frac{x}{k} for any x,y\in X. Then F:X\times X\to X is a mixed gmonotone mapping, F(X,X)\subseteq g(X), and g is continuous. Let {x}_{0}=1 and {y}_{0}=1, then
For any x,y,u,v\in X, with gx\u2291gu,gv\u2291gy, we have
Thus, all the conditions of Theorem 2.4 are satisfied. By Theorem 2.4, there is ({x}^{\ast},{y}^{\ast})\in X\times X such that F({x}^{\ast},{y}^{\ast})=g({x}^{\ast}) and F({y}^{\ast},{x}^{\ast})=g({y}^{\ast}). But υ does not satisfy the contractive conditions in [13] and [17]. In fact, for u=x+1, y=v+1,
This shows that υ does not satisfy the contractive conditions in [13] and [17].
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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/10.1186/16871812201379
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DOI: https://doi.org/10.1186/16871812201379