# Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces

- Xin-Qi Hu
^{1}Email author, - Ming-Xue Zheng
^{1}, - Boško Damjanović
^{2}and - Xiao-Feng Shao
^{3}

**2013**:220

https://doi.org/10.1186/1687-1812-2013-220

© Hu et al.; licensee Springer. 2013

**Received: **1 March 2013

**Accepted: **1 August 2013

**Published: **19 August 2013

## Abstract

In this paper, we prove a common fixed point theorem for weakly compatible mappings under *ϕ*-contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716). We also indicate a minor mistake in Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716).

## Keywords

## 1 Introduction

In 1965, Zadeh [1] introduced the concept of fuzzy sets. Then many authors gave the important contribution to development of the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of a fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space, which have very important applications in quantum particle physics, particularly, in connection with both string and E-infinity theory.

Bhaskar and Lakshmikantham [4], Lakshmikantham and Ćirić [5] discussed the mixed monotone mappings and gave some coupled fixed point theorems, which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi *et al.* [6] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Jin-xuan Fang [7] gave some common fixed point theorems for compatible and weakly compatible *ϕ*-contractions mappings in Menger probabilistic metric spaces. Xin-Qi Hu [8] proved a common fixed point theorem for mappings under *φ*-contractive conditions in fuzzy metric spaces. Many authors [9–26] proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.

In this paper, we give a new coupled fixed point theorem under weaker conditions than in [8] and give an example, which shows that the result is a genuine generalization of the corresponding result in [8].

## 2 Preliminaries

First, we give some definitions.

**Definition 2.1** (see [2])

*t*-norm if ∗ satisfies the following conditions:

- (1)
∗ is commutative and associative,

- (2)
∗ is continuous,

- (3)
$a\ast 1=a$ for all $a\in [0,1]$,

- (4)
$a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$.

**Definition 2.2** (see [27])

*t*-norm Δ is said to be of H-type if the family of functions ${\{{\mathrm{\Delta}}^{m}(t)\}}_{m=1}^{\mathrm{\infty}}$ is equicontinuous at $t=1$, where

The *t*-norm ${\mathrm{\Delta}}_{M}=min$ is an example of *t*-norm of H-type, but there are some other *t*-norms Δ of H-type [27].

Obviously, Δ is a *t*-norm of H-type if and only if for any $\lambda \in (0,1)$, there exists $\delta (\lambda )\in (0,1)$ such that ${\mathrm{\Delta}}^{m}(t)>1-\lambda $ for all $m\in \mathbb{N}$, when $t>1-\delta $.

**Definition 2.3** (see [2])

A 3-tuple $(X,M,\ast )$ is said to be a fuzzy metric space if *X* is an arbitrary nonempty set, ∗ is a continuous *t*-norm and *M* is a fuzzy set on ${X}^{2}\times (0,+\mathrm{\infty})$ satisfying the following conditions for each $x,y,z\in X$ and $t,s>0$,

(FM-1) $M(x,y,t)>0$,

(FM-2) $M(x,y,t)=1$ if and only if $x=y$,

(FM-3) $M(x,y,t)=M(y,x,t)$,

(FM-4) $M(x,y,t)\ast M(y,z,s)\le M(x,z,t+s)$,

(FM-5) $M(x,y,\cdot ):(0,\mathrm{\infty})\to [0,1]$ is continuous.

A subset $A\subset X$ is called open if for each $x\in A$, there exist $t>0$ and $0<r<1$ such that $B(x,r,t)\subset A$. Let *τ* denote the family of all open subsets of *X*. Then *τ* is called the topology on *X*, induced by the fuzzy metric *M*. This topology is Hausdorff and first countable.

**Example 2.4** Let $(X,d)$ be a metric space. Define *t*-norm $a\ast b=ab$ or $a\ast b=min\{a,b\}$ and for all $x,y\in X$ and $t>0$, $M(x,y,t)=\frac{t}{t+d(x,y)}$. Then $(X,M,\ast )$ is a fuzzy metric space.

**Definition 2.5** (see [2])

- (1)a sequence $\{{x}_{n}\}$ in
*X*is said to be convergent to*x*(denoted by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$) if$\underset{n\to \mathrm{\infty}}{lim}M({x}_{n},x,t)=1$for all $t>0$.

- (2)A sequence $\{{x}_{n}\}$ in
*X*is said to be a Cauchy sequence if for any $\epsilon >0$, there exists ${n}_{0}\in \mathbb{N}$, such that$M({x}_{n},{x}_{m},t)>1-\epsilon $for all $t>0$ and $n,m\ge {n}_{0}$.

- (3)
A fuzzy metric space $(X,M,\ast )$ is said to be complete if and only if every Cauchy sequence in

*X*is convergent.

**Remark 2.6** (see [9])

- (1)
for all $x,y\in X$, $M(x,y,\cdot )$ is non-decreasing;

- (2)if ${x}_{n}\to x$, ${y}_{n}\to y$, ${t}_{n}\to t$, then$\underset{n\to \mathrm{\infty}}{lim}M({x}_{n},{y}_{n},{t}_{n})=M(x,y,t);$
- (3)
if $M(x,y,t)>1-r$ for

*x*,*y*in*X*, $t>0$, $0<r<1$, then we can find a ${t}_{0}$, $0<{t}_{0}<t$ such that $M(x,y,{t}_{0})>1-r$; - (4)
for any ${r}_{1}>{r}_{2}$, we can find a ${r}_{3}$ such that ${r}_{1}\ast {r}_{3}\ge {r}_{2}$, and for any ${r}_{4}$, we can find a ${r}_{5}$ such that ${r}_{5}\ast {r}_{5}\ge {r}_{4}$ (${r}_{1},{r}_{2},{r}_{3},{r}_{4},{r}_{5}\in (0,1)$).

Define $\mathrm{\Phi}=\{\varphi :{R}^{+}\to {R}^{+}\}$, where ${R}^{+}=[0,+\mathrm{\infty})$ and each $\varphi \in \mathrm{\Phi}$ satisfies the following conditions:

(*ϕ*-1) *ϕ* is non-decreasing,

(*ϕ*-2) *ϕ* is upper semi-continuous from the right,

(*ϕ*-3) ${\sum}_{n=0}^{\mathrm{\infty}}{\varphi}^{n}(t)<+\mathrm{\infty}$ for all $t>0$, where ${\varphi}^{n+1}(t)=\varphi ({\varphi}^{n}(t))$, $n\in \mathbb{N}$.

It is easy to prove that if $\varphi \in \mathrm{\Phi}$, then $\varphi (t)<t$ for all $t>0$.

**Lemma 2.7** (see [7])

*Let*$(X,M,\ast )$

*be a fuzzy metric space*,

*where*∗

*is a continuous*

*t*-

*norm of H*-

*type*.

*If there exists*$\varphi \in \mathrm{\Phi}$

*such that*

*for all* $t>0$, *then* $x=y$.

**Definition 2.8** (see [4])

**Definition 2.9** (see [5])

**Definition 2.10** (see [5])

**Definition 2.11** (see [5])

**Definition 2.12** (see [8])

*X*, such that

for all $x,y\in X$ are satisfied.

**Definition 2.13** (see [20])

The mappings $F:X\times X\to X$ and $g:X\to X$ are called weakly compatible mappings if $F(x,y)=g(x)$, $F(y,x)=g(y)$ implies that $gF(x,y)=F(gx,gy)$, $gF(y,x)=F(gy,gx)$ for all $x,y\in X$.

**Remark 2.14** It is easy to prove that if *F* and *g* are compatible, then they are weakly compatible, but the converse need not be true. See the example in the next section.

## 3 Main results

for all $n\in \mathbb{N}$.

Xin-Qi Hu [8] proved the following result.

**Theorem 3.1** (see [8])

*Let*$(X,M,\ast )$

*be a complete FM*-

*space*,

*where*∗

*is a continuous*

*t*-

*norm of H*-

*type satisfying*(2.2).

*Let*$F:X\times X\to X$

*and*$g:X\to X$

*be two mappings*,

*and there exists*$\varphi \in \mathrm{\Phi}$

*such that*

*for all* $x,y,u,v\in X$, $t>0$.

*Suppose that* $F(X\times X)\subseteq g(X)$, *g* *is continuous*, *F* *and* *g* *are compatible*. *Then there exist* $x,y\in X$ *such that* $x=g(x)=F(x,x)$; *that is*, *F* *and* *g* *have a unique common fixed point in X*.

Now we give our main result.

**Theorem 3.2** *Let* $(X,M,\ast )$ *be a FM*-*space*, *where* ∗ *is a continuous* *t*-*norm of H*-*type satisfying* (2.2). *Let* $F:X\times X\to X$ *and* $g:X\to X$ *be two weakly compatible mappings*, *and there exists* $\varphi \in \mathrm{\Phi}$ *satisfying* (3.1).

*Suppose that* $F(X\times X)\subseteq g(X)$ *and* $F(X\times X)$ *or* $g(X)$ *is complete*. *Then* *F* *and* *g* *have a unique common fixed point in* *X*.

*Proof*Let ${x}_{0},{y}_{0}\in X$ be two arbitrary points in

*X*. Since $F(X\times X)\subseteq g(X)$, we can choose ${x}_{1},{y}_{1}\in X$ such that $g({x}_{1})=F({x}_{0},{y}_{0})$ and $g({y}_{1})=F({y}_{0},{x}_{0})$. Continuing this process, we can construct two sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in

*X*such that

The proof is divided into 4 steps.

Step 1: We shall prove that $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ are Cauchy sequences.

for all $k\in \mathbb{N}$.

*ϕ*-3), we have ${\sum}_{n=1}^{\mathrm{\infty}}{\varphi}^{n}({t}_{0})<\mathrm{\infty}$. Then for any $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that

for all $m,n\in \mathbb{N}$ with $m>n\ge {n}_{0}$ and $t>0$. So $\{g({x}_{n})\}$ is a Cauchy sequence.

Similarly, we can prove that $\{g({y}_{n})\}$ is also a Cauchy sequence.

Step 2: Now, we prove that *g* and *F* have a coupled coincidence point.

*M*is continuous, taking limit as $n\to \mathrm{\infty}$, we have

which implies that $F(a,b)=g(a)=x$.

Similarly, we can show that $F(b,a)=g(b)=y$.

Since *F* and *g* are weakly compatible, we get that $gF(a,b)=F(g(a),g(b))$ and $gF(b,a)=F(g(b),g(a))$, which implies that $g(x)=F(x,y)$ and $g(y)=F(y,x)$.

Step 3: We prove that $g(x)=y$ and $g(y)=x$.

for all $k\in \mathbb{N}$.

Since $M(x,y,\cdot )$ is continuous and ${lim}_{t\to +\mathrm{\infty}}M(x,y,t)=1$ for all $x,y\in X$, there exists ${t}_{0}>0$ such that $M(gx,y,{t}_{0})\ge 1-\mu $ and $M(gy,x,{t}_{0})\ge 1-\mu $.

*ϕ*-3), we have ${\sum}_{n=1}^{\mathrm{\infty}}{\varphi}^{n}({t}_{0})<\mathrm{\infty}$. Thus, for any $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that $t>{\sum}_{k={n}_{0}}^{\mathrm{\infty}}{\varphi}^{k}({t}_{0})$. Since

for all $t>0$. Hence conclude that $gx=y$ and $gy=x$.

Step 4: Now, we prove that $x=y$.

for all $k\in \mathbb{N}$.

Since $M(x,y,\cdot )$ is continuous and ${lim}_{t\to +\mathrm{\infty}}M(x,y,t)=1$, there exists ${t}_{0}>0$ such that $M(x,y,{t}_{0})\ge 1-\mu $.

On the other hand, since $\varphi \in \mathrm{\Phi}$, by condition (*ϕ*-3), we have ${\sum}_{n=1}^{\mathrm{\infty}}{\varphi}^{n}({t}_{0})<\mathrm{\infty}$. Then, for any $t>0$, there exists ${n}_{0}\in \mathbb{N}$ such that $t>{\sum}_{k={n}_{0}}^{\mathrm{\infty}}{\varphi}^{k}({t}_{0})$.

which implies that $x=y$.

Thus, we proved that *F* and *g* have a common fixed point in *X*.

The uniqueness of the fixed point can be easily proved in the same way as above. This completes the proof of Theorem 3.2. □

Taking $g=I$ (the identity mapping) in Theorem 3.2, we get the following consequence.

**Corollary 3.3**

*Let*$(X,M,\ast )$

*be a FM*-

*space*,

*where*∗

*is a continuous*

*t*-

*norm of H*-

*type satisfying*(2.2).

*Let*$F:X\times X\to X$,

*and there exists*$\varphi \in \mathrm{\Phi}$

*such that*

*for all* $x,y,u,v\in X$, $t>0$. $F(X)$ *is complete*.

*Then there exist* $x\in X$ *such that* $x=F(x,x)$; *that is*, *F* *admits a unique fixed point in* *X*.

**Remark 3.4**Comparing Theorem 3.2 with Theorem 3.1 in [8], we can see that Theorem 3.2 is a genuine generalization of Theorem 3.2.

- (1)
We only need the completeness of $g(X)$ or $F(X\times X)$.

- (2)
The continuity of

*g*is relaxed. - (3)
The concept of compatible has been replaced by weakly compatible.

**Remark 3.5** The Example 3 in [8] is wrong, since the *t*-norm $a\ast b=ab$ is not the *t*-norm of H-type.

Next, we give an example to support Theorem 3.2.

**Example 3.6** Let $X=\{0,1,\frac{1}{2},\frac{1}{3},\dots ,\frac{1}{n},\dots \}$, $\ast =min$, $M(x,y,t)=\frac{t}{|x-y|+t}$, for all $x,y\in X$, $t>0$. Then $(X,M,\ast )$ is a fuzzy metric space.

so *g* and *F* are not compatible. From $F(x,y)=g(x)$, $F(y,x)=g(y)$, we can get $(x,y)=(0,0)$, and we have $gF(0,0)=F(g0,g0)$, which implies that *F* and *g* are weakly compatible.

*M*and

*ϕ*and the result above, we can get that inequality (3.1)

Now, we verify inequality (3.11). Let $A=\{\frac{1}{2n},n\in \mathbb{N}\}$, $B=X-A$. By the symmetry and without loss of generality, $(x,y)$, $(u,v)$ have 6 possibilities.

Case 1: $(x,y)\in B\times B$, $(u,v)\in B\times B$. It is obvious that (3.11) holds.

Case 2: $(x,y)\in B\times B$, $(u,v)\in B\times A$. It is obvious that (3.11) holds.

which implies that (3.11) holds.

Case 4: $(x,y)\in B\times A$, $(u,v)\in B\times A$. It is obvious that (3.11) holds.

(3.11) holds.

Case 6: $(x,y)\in A\times A$, $(u,v)\in A\times A$.

If $x\ne y$, $u\ne v$, (3.11) holds.

(3.11) holds.

(3.11) holds.

Then all the conditions in Theorem 3.2 are satisfied, and 0 is the unique common fixed point of *g* and *F*.

## Declarations

### Acknowledgements

This work of Xin-qi Hu was supported by the National Natural Science Foundation of China (under grant No. 71171150). The research of B. Damjanović was supported by Grant No. 174025 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.

## Authors’ Affiliations

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