Common fixed point theorems for weakly compatible hybrid mappings

Definition 2.1 Sklar and Schweizer (1960)

A mapping $*:[0\text{,}1]\times [0\text{,}1]\to [0\text{,}1]$ is a continuous $t$ -norm if it satisfies the following conditions:

• $*$ is associative and commutative,

• $*$ is continuous,

• $a*1=a$ for every $a\in [0\text{,}1]$ ,

• $a*b⩽c*d$ whenever $a⩽c$ and $b⩽d$ for each $a\text{,}b\text{,}c\text{,}d\in [0\text{,}1]$ .

Definition 2.2 Kramosil and Michalek (1975)

A triple $(X\text{,}M\text{,}*)$ is a fuzzy metric space if $X$ is an arbitrary set, $*$ is a continuous $t$ norm and $M$ is a fuzzy set on $X\times X\times [0\text{,}\infty )\to [0\text{,}1]$ satisfying, $\forall x\text{,}y\in X$ ,the following conditions:

• $M(x\text{,}y\text{,}0)=0$ ,

• $M(x\text{,}y\text{,}t)=1\text{,}\forall t>0$ iff $x=y$ ,

• $M(x\text{,}y\text{,}t)=M(y\text{,}x\text{,}t)$ ,

• $M(x\text{,}y\text{,}t)*M(y\text{,}z\text{,}s)⩽M(x\text{,}z\text{,}s+t)\text{,}s\text{,}t\in [0\text{,}1)$ ,

• $M(x\text{,}y\text{,}·):[0\text{,}\infty )\to [0\text{,}1]$ is left continuous.

Note that $M(y\text{,}x\text{,}t)$ can be thought of as the degree of nearness between $x$ and $y$ with respect to $t$ .

Definition 2.3 Grebiec (1988)

A sequence $\{x_n\}$ in a fuzzy metric space $(X\text{,}M\text{,}*)$ is said to be convergent to a point $x\in X$ if ${\lim }_{n\to \infty }M(x_n\text{,}x\text{,}t)=1\text{,}\forall t>0$ .

A sequence $\{x_n\}$ in a fuzzy metric space $(X\text{,}M\text{,}*)$ is Cauchy sequence if ${\lim }_{n\to \infty }M(x_{n+p}\text{,}{x}_n\text{,}t)=1\text{,}\forall t\text{,}p>0$ .

A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

Definition 2.4 Abu-donia et al. (2000) and Sharma (2002)

A mapping $*:[0\text{,}1]\times [0\text{,}1]\times [0\text{,}1]\to [0\text{,}1]$ is a continuous $t$ -norm if it satisfies the following conditions:

• $*$ is associative and commutative,

• $*$ is continuous,

• $a*1=a$ for every $a\in [0\text{,}1]$ ,

• $a_1*b_1*c_1⩽a_2*b_2*c_2$ whenever $a_1⩽a_2\text{,}{b}_1⩽b_2$ and $c_1⩽c_2$ for each $a_1\text{,}{b}_1\text{,}{c}_1\text{,}{a}_2\text{,}{b}_2\text{,}{c}_2\in [0\text{,}1]$ .

Definition 2.5 Abu-donia et al. (2000) and Sharma (2002)

A triple $(X\text{,}M\text{,}*)$ is a fuzzy 2-metric space if $X$ is an arbitrary set, $*$ is a continuous $t$ norm and $M$ is a fuzzy set on $X\times X\times X\times [0\text{,}\infty )$ satisfying, $\forall x\text{,}y\text{,}z\in X$ , the following conditions:

• $M(x\text{,}y\text{,}z\text{,}0)=0$ ,

• $M(x\text{,}y\text{,}z\text{,}t)=1\text{,}\forall t>0$ when at least two of the three point are equal,

• $M(x\text{,}y\text{,}z\text{,}t)=M(x\text{,}z\text{,}y\text{,}t)=M(y\text{,}z\text{,}x\text{,}t)=\cdots$ Symmetry about three variables,

• $M(x\text{,}y\text{,}u\text{,}{t}_1)*M(x\text{,}u\text{,}z\text{,}{t}_1)*M(u\text{,}y\text{,}z\text{,}{t}_3)⩽M(x\text{,}y\text{,}z\text{,}{t}_1+t_2+t_3)\text{,}{t}_1\text{,}{t}_2\text{,}{t}_3\in [0\text{,}1)$ ,

• $M(x\text{,}y\text{,}z\text{,}·):[0\text{,}\infty )\to [0\text{,}1]$ is left continuous.

Definition 2.6 Abu-donia et al. (2000) and Sharma (2002)

A sequence $\{x_n\}$ in a fuzzy 2-metric space $(X\text{,}M\text{,}*)$ is said to be convergent to a point $x\in X$ if ${\lim }_{n\to \infty }M(x_n\text{,}x\text{,}z\text{,}t)=1\text{,}\forall t>0\text{,}z\in X$ .

A sequence $\{x_n\}$ in a fuzzy 2-metric space $(X\text{,}M\text{,}*)$ is Cauchy sequence if ${\lim }_{n\to \infty }M(x_{n+p}\text{,}{x}_n\text{,}z\text{,}t)=1\text{,}\forall z\in X\text{and}t\text{,}p>0$ .

Example 2.1 George and Veeramani (1994)

Let $(X\text{,}d)$ be a metric space. Define $a*b=\mathit{ab}$ and for all $x\text{,}y\in X\text{,}t>0$ , $$M(x\text{,}y\text{,}t)=\frac{t}{t+d(x\text{,}y)}\text{.}$$

We call $M$ is a fuzzy metric on $X$ induced by metric $d$ .

Definition 2.7 Jungck and Rhoades (1998)

The mappings $I:X\to X$ and $F:X\to B(X)$ are weakly compatible if they commute at coincidence points, i.e., for each point $u\in X$ such that $\mathit{Fu}=\{\mathit{Iu}\}$ , we have $\mathit{FIu}=\mathit{IFu}$ . Not that the equation $\mathit{Fu}=\{\mathit{Iu}\}$ implies that $\mathit{Fu}$ is singleton.

Definition 2.8 Vasuki (1999)

The mappings $I:X\to X$ and $F:X\to B(X)$ are $R$ -weakly commuting if, for all $R\text{,}t>0\text{,}M(\mathit{FIx}\text{,}\mathit{IFx}\text{,}t)⩾M(\mathit{Fx}\text{,}\mathit{Ix}\text{,}t/R)$ , such that $x\in X\text{,}\mathit{IFx}\in B(X)$ .

$R$ -weakly commuting is weakly compatible but the converse is not true (Pathak and Singh, 2007).

Theorem 2.1 Som (1985)

Let $S$ and $T$ be two continuous self mappings of a complete fuzzy metric space $(X\text{,}M\text{,}*)$ . Let $A$ and $B$ be two self mappings of $X$ satisfying following conditions:

• $A(X)\bigcup B(X)\subset S(X)\bigcap T(X)$ ,

• $\{A\text{,}T\}$ and $\{B\text{,}S\}$ are $R$ -weakly commuting pairs,

• $\mathit{aM}(\mathit{Tx}\text{,}\mathit{Sy}\text{,}t)+\mathit{bM}(\mathit{Tx}\text{,}\mathit{Ax}\text{,}t)+\mathit{cM}(\mathit{Sy}\text{,}\mathit{By}\text{,}t)+\max \{M(\mathit{Ax}\text{,}\mathit{Sy}\text{,}t)\text{,}M(\mathit{By}\text{,}\mathit{Tx}\text{,}t)\}⩽\mathit{qM}(\mathit{Ax}\text{,}\mathit{By}\text{,}t)$ ,

Theorem 2.2 Pathak and Singh (2007)

Let $S$ and $T$ be two continuous self mappings of a complete fuzzy metric space $(X\text{,}M\text{,}*)$ . Let $A$ and $B$ be two self mappings of $X$ satisfying following conditions:

• $A(X)\bigcup B(X)\subset S(X)\bigcap T(X)$ ,

• $\{A\text{,}T\}$ and $\{B\text{,}S\}$ are weakly compatible pairs,

• $\mathit{aM}(\mathit{Tx}\text{,}\mathit{Sy}\text{,}t)+\mathit{bM}(\mathit{Tx}\text{,}\mathit{Ax}\text{,}t)+\mathit{cM}(\mathit{Sy}\text{,}\mathit{By}\text{,}t)+\max \{M(\mathit{Ax}\text{,}\mathit{Sy}\text{,}t)\text{,}M(\mathit{By}\text{,}\mathit{Tx}\text{,}t)\}⩽\mathit{qM}(\mathit{Ax}\text{,}\mathit{By}\text{,}t)$ ,

Let $S$ and $T$ be two self mappings of a fuzzy metric space $(X\text{,}M\text{,}*)$ and $A\text{,}B:X\to \mathit{CB}(X)$ set-valued mappings satisfying following conditions:

• $\bigcup A(X)\subseteq S(X)$ and $\bigcup B(X)\subseteq T(X)$ ,

• $\{A\text{,}T\}$ and $\{B\text{,}S\}$ are weakly compatible pairs,

• $\mathit{aM}(\mathit{Tx}\text{,}\mathit{Sy}\text{,}t)+\mathit{bM}(\mathit{Tx}\text{,}\mathit{Ax}\text{,}t)+\mathit{cM}(\mathit{Sy}\text{,}\mathit{By}\text{,}t)+\max \{M(\mathit{Ax}\text{,}\mathit{Sy}\text{,}t)\text{,}M(\mathit{By}\text{,}\mathit{Tx}\text{,}t)\}⩽\mathit{qM}(\mathit{Ax}\text{,}\mathit{By}\text{,}t)$ ,

Let $x_0$ be an arbitrary point in $X$ . From the condition (1), we chose a point $x_1$ in $X$ such that ${\mathit{Sx}}_1\in {\mathit{Ax}}_0$ . For this point $x_1$ there exist a point $x_2$ in $X$ such that ${\mathit{Tx}}_2\in {\mathit{Bx}}_1$ and so on. Inductively, we can define a sequence $\{Z_n\}$ in $X$ such that $${\mathit{Sx}}_{2n+1}\in {\mathit{Ax}}_{2n}=Z_{2n}\text{,}{\mathit{Tx}}_{2n+2}\in {\mathit{Bx}}_{2n+1}=Z_{2n+1}\text{,}\forall n=0\text{,}1\text{,}2\text{,}\dots \text{.}$$

We will prove that $\{Z_n\}$ is Cauchy sequence.

Using inequality (3), we obtain $$\mathit{qM}(Z_{2n}\text{,}{Z}_{2n+1}\text{,}t)=\mathit{qM}({\mathit{Ax}}_{2n}\text{,}{\mathit{Bx}}_{2n+1}\text{,}t)⩾\mathit{aM}({\mathit{Tx}}_{2n}\text{,}{\mathit{Sx}}_{2n+1}\text{,}t)+\mathit{bM}({\mathit{Tx}}_{2n}\text{,}{\mathit{Ax}}_{2n}\text{,}t)+\mathit{cM}({\mathit{Sx}}_{2n+1}\text{,}{\mathit{Bx}}_{2n+1}\text{,}t)+\max \{M({\mathit{Ax}}_{2n}\text{,}{\mathit{Sx}}_{2n+1}\text{,}t)\text{,}M({\mathit{Bx}}_{2n+1}\text{,}{\mathit{Tx}}_{2n}\text{,}t)\}⩾\mathit{aM}(Z_{2n-1}\text{,}{Z}_{2n}\text{,}t)+\mathit{bM}(Z_{2n-1}\text{,}{Z}_{2n}\text{,}t)+\mathit{cM}(Z_{2n}\text{,}{Z}_{2n+1}\text{,}t)+\max \{M(Z_{2n}\text{,}{Z}_{2n}\text{,}t)\text{,}M(Z_{2n+1}\text{,}{Z}_{2n-1}\text{,}t)\}\text{.}$$

Then $M(Z_{2n}\text{,}{Z}_{2n+1}\text{,}t)⩾\beta M(Z_{2n-1}\text{,}{Z}_{2n}\text{,}t)$ , where $\beta =\frac{a+b+1}{q-c}>1$ .

Since $\beta >1$ , we obtain $$M(Z_{2n+1}\text{,}{Z}_{2n}\text{,}t)>M(Z_{2n}\text{,}{Z}_{2n-1}\text{,}t)\text{.}$$

Similarly $$M(Z_{2n+2}\text{,}{Z}_{2n+1}\text{,}t)>M(Z_{2n+1}\text{,}{Z}_{2n}\text{,}t)\text{.}$$

Now for any positive integer $p$ , $$M\left(Z_n\text{,}{Z}_{n+p}\text{,}t\right)⩾M\left(Z_n\text{,}{Z}_{n+1}\text{,}\frac{t}{p}\right)*M\left(Z_{n+1}\text{,}{Z}_{n+2}\text{,}\frac{t}{p}\right)*\cdots *M\left(Z_{n+p-1}\text{,}{Z}_{n+p}\text{,}\frac{t}{p}\right)\text{.}$$

As $n\to \infty$ , we get $M(Z_n\text{,}{Z}_{n+p}\text{,}t)\to 1*1*\cdots *1\to 1$ .

Hence $Z_n$ is a Cauchy sequence. Suppose that $\mathit{SX}$ is complete, therefore by the above, $\{{\mathit{Sx}}_{2n+1}\}$ is a Cauchy sequence and hence ${\mathit{Sx}}_{2n+1}\to z=\mathit{Sv}$ for some $v\in X$ . Hence, $Z_n\to z$ and the subsequences ${\mathit{Tx}}_{2n+2}\text{,}{\mathit{Ax}}_{2n}$ and ${\mathit{Bx}}_{2n+1}$ converge to $z$ .

We shall prove that $z=\mathit{Sv}\in \mathit{Bv}$ , by (3), we have $$\mathit{qM}({\mathit{Ax}}_{2n}\text{,}\mathit{Bv}\text{,}t)⩾\mathit{aM}({\mathit{Tx}}_{2n}\text{,}\mathit{Sv}\text{,}t)+\mathit{bM}({\mathit{Tx}}_{2n}\text{,}{\mathit{Ax}}_{2n}\text{,}t)+\mathit{cM}(\mathit{Sv}\text{,}\mathit{Bv}\text{,}t)+\max \{M({\mathit{Ax}}_{2n}\text{,}\mathit{Sv}\text{,}t)\text{,}M(\mathit{Bv}\text{,}{\mathit{Tx}}_{2n}\text{,}t)\}\text{.}$$

As $n\to \infty$ , we obtain $$\begin{array}{cc}\hfill & \mathit{qM}(z\text{,}\mathit{Bv}\text{,}t)⩾\mathit{aM}(z\text{,}z\text{,}t)+\mathit{bM}(z\text{,}z\text{,}t)+\mathit{cM}(z\text{,}\mathit{Bv}\text{,}t)\hfill \\ \hfill & +\max \{M(z\text{,}z\text{,}t)\text{,}M(\mathit{Bv}\text{,}z\text{,}t)\}\text{,}\hfill \\ \hfill & M(z\text{,}\mathit{Bv}\text{,}t)⩾\left(\frac{a+b+1}{q-c}\right)>1\text{,}\hfill \end{array}$$ which yields $\{z\}=\{\mathit{Sv}\}=\mathit{Bv}$ .

Since $\bigcup B(X)\subseteq T(X)$ , thus, there exist $u\in X$ such that $\{\mathit{Tu}\}=\mathit{Bv}=\{z\}=\{\mathit{Sv}\}$ .

Now if $\mathit{Au}\ne \mathit{Bv}$ ,we get $$\mathit{qM}(\mathit{Au}\text{,}\mathit{Bv}\text{,}t)⩾\mathit{aM}(\mathit{Tu}\text{,}\mathit{Sv}\text{,}t)+\mathit{bM}(\mathit{Tu}\text{,}\mathit{Au}\text{,}t)+\mathit{cM}(\mathit{Sv}\text{,}\mathit{Bv}\text{,}t)+\max \{M(\mathit{Au}\text{,}\mathit{Sv}\text{,}t)\text{,}M(\mathit{Bv}\text{,}\mathit{Tu}\text{,}t)\}\text{,}$$ $$\mathit{qM}(\mathit{Au}\text{,}z\text{,}t)⩾\mathit{aM}(z\text{,}z\text{,}t)+\mathit{bM}(z\text{,}\mathit{Au}\text{,}t)+\mathit{cM}(z\text{,}z\text{,}t)+\max \{M(\mathit{Au}\text{,}z\text{,}t)\text{,}M(z\text{,}z\text{,}t)\}\text{,}$$ $$M(\mathit{Au}\text{,}z\text{,}t)⩾\left(\frac{a+c+1}{q-b}\right)>1\text{,}$$ which yields $\mathit{Au}=\{z\}=\{\mathit{Tu}\}=\{\mathit{Sv}\}=\mathit{Bv}$ .

Since $\mathit{Au}=\{\mathit{Tu}\}$ and $\{A\text{,}T\}$ is weakly compatible, $\mathit{ATv}=\mathit{TAv}$ gives $\mathit{Az}=\{\mathit{Tz}\}$ .

On using (3), we obtain $$\mathit{qM}(\mathit{Az}\text{,}\mathit{Bv}\text{,}t)⩾\mathit{aM}(\mathit{Tz}\text{,}\mathit{Sv}\text{,}t)+\mathit{bM}(\mathit{Tz}\text{,}\mathit{Az}\text{,}t)+\mathit{cM}(\mathit{Sv}\text{,}\mathit{Bv}\text{,}t)+\max \{M(\mathit{Az}\text{,}\mathit{Sv}\text{,}t)\text{,}M(\mathit{Bv}\text{,}\mathit{Tz}\text{,}t)\}\text{,}$$ $$\mathit{qM}(\mathit{Az}\text{,}z\text{,}t)⩾\mathit{aM}(\mathit{Tz}\text{,}z\text{,}t)+\mathit{bM}(z\text{,}\mathit{Az}\text{,}t)+\mathit{cM}(z\text{,}z\text{,}t)+\max \{M(\mathit{Az}\text{,}z\text{,}t)\text{,}M(z\text{,}z\text{,}t)\}\text{.}$$

Hence, $\mathit{Az}=\{z\}=\{\mathit{Tz}\}$ . Similarly, $\mathit{Bz}=\{z\}=\{\mathit{Sz}\}$ where $\{B\text{,}S\}$ is weakly compatible. Then, $\mathit{Az}=\{\mathit{Tz}\}=\{z\}=\{\mathit{Sz}\}=\mathit{Bz}$ , i.e., $z$ is the common fixed point of $A\text{,}B\text{,}S$ and $T$ have a unique.

To see $z$ is unique, suppose that $p\ne z$ such that $\mathit{Ap}=\{\mathit{Tp}\}=\{p\}=\{\mathit{Sp}\}=\mathit{Bp}$ .

On using (3), we get $$\mathit{qM}(\mathit{Az}\text{,}\mathit{Bp}\text{,}t)⩾\mathit{aM}(\mathit{Tz}\text{,}\mathit{Sp}\text{,}t)+\mathit{bM}(\mathit{Tz}\text{,}\mathit{Az}\text{,}t)+\mathit{cM}(\mathit{Sp}\text{,}\mathit{Bp}\text{,}t)+\max \{M(\mathit{Az}\text{,}\mathit{Sp}\text{,}t)\text{,}M(\mathit{Bp}\text{,}\mathit{Tz}\text{,}t)\}\text{,}$$ $$M(z\text{,}p\text{,}t)⩾\left(\frac{b+c}{q-a-1}\right)\text{,}$$ which is impossible, $z=p$ . Then $A\text{,}B\text{,}S$ and $T$ have a unique common fixed point. □

Theorem 3.1 is a generalization, extension and improvement for results of Pathak and Singh (2007) in fuzzy metric space.

Theorem 3.1 is a generalization, extension and improvement for results of Sharma and Tiwari (2005) in fuzzy metric space.

Let $X=[0\text{,}\infty ]$ endowed with the Euclidean metric $d$ and $M(\mathit{Ax}\text{,}\mathit{By}\text{,}t)=\frac{t}{t+\delta (\mathit{Ax}\text{,}\mathit{By})}\text{,}\delta (A\text{,}B)=\max \{d(a\text{,}b):a\in A\text{,}b\in B\}$ . Define $$\begin{array}{cc}\hfill & \mathit{Ax}=\left[0\text{,}\frac{x^6}{6}\right]\text{,}\mathit{Bx}=\left[0\text{,}\frac{x^3}{6}\right]\text{,}\hfill \\ \hfill & \mathit{Sx}=\frac{x^{12}}{2}+x^6+\frac{x^3}{2}\text{,}\mathit{Tx}=x^6+6x^3\text{,}\hfill \end{array}$$ for all $x\in X$ . We have $\bigcup A(X)=T(X)=\bigcup B(X)=S(X)=X$ .

From the above, we have that $M(A(0)\text{,}0\text{,}t)=1\text{,}M(B(0)\text{,}0\text{,}t)=1\text{,}M(S(0)\text{,}0\text{,}t)=1$ and $M(T(0)\text{,}0\text{,}t)=1$ .

Thus 0 is a common fixed point of $A\text{,}B\text{,}S$ and $T$ . Also, $\{A\text{,}T\}$ and $\{B\text{,}S\}$ are weakly compatible pairs, where, $M(\mathit{AT}(0)\text{,}\mathit{TA}(0)\text{,}t)=1$ and $M(\mathit{BS}(0)\text{,}\mathit{SB}(0)\text{,}t)=1$ .

For any $x\text{,}y\in X\text{,}x\ne y$ $$\delta (\mathit{Ax}\text{,}\mathit{By})=\max \left\{\frac{x^6}{6}\text{,}\frac{y^3}{6}\right\}=\max \left\{\frac{1}{3}\frac{x^6}{2}\text{,}\frac{1}{3}\frac{y^3}{2}\right\}⩽\frac{1}{3}\max \left\{x^6+6x^3\text{,}\frac{y^{12}}{2}+y^6+\frac{y^3}{2}\right\}⩽\frac{1}{3}\max \{\delta (\mathit{By}\text{,}\mathit{Tx})\text{,}\delta (\mathit{Ax}\text{,}\mathit{Sy})\}\text{.}$$

Thus $$\begin{array}{cc}\hfill & \frac{1}{\delta (\mathit{Ax}\text{,}\mathit{By})}⩾3\max \left\{\frac{1}{\delta (\mathit{By}\text{,}\mathit{Tx})}\text{,}\frac{1}{\delta (\mathit{Ax}\text{,}\mathit{Sy})}\right\}\text{,}\hfill \\ \hfill & \frac{t}{t+\delta (\mathit{Ax}\text{,}\mathit{By})}⩾3\max \left\{\frac{t}{t+\delta (\mathit{By}\text{,}\mathit{Tx})}\text{,}\frac{t}{t+\delta (\mathit{Ax}\text{,}\mathit{Sy})}\right\}\text{,}\hfill \\ \hfill & M(\mathit{Ax}\text{,}\mathit{By}\text{,}t)⩾3\max \{M(\mathit{Ax}\text{,}\mathit{Sy}\text{,}t)\text{,}M(\mathit{By}\text{,}\mathit{Tx}\text{,}t)\}\text{.}\hfill \end{array}$$

Then $$\frac{2}{3}M(\mathit{Ax}\text{,}\mathit{By}\text{,}t)⩾\frac{1}{4}M(\mathit{Tx}\text{,}\mathit{Sy}\text{,}t)+\frac{1}{5}M(\mathit{Tx}\text{,}\mathit{Ax}\text{,}t)+\frac{1}{3}M(\mathit{Sy}\text{,}\mathit{By}\text{,}t)+\max \{M(\mathit{Ax}\text{,}\mathit{Sy}\text{,}t)\text{,}M(\mathit{By}\text{,}\mathit{Tx}\text{,}t)\}\text{,}$$ where $q=\frac{2}{3}\text{,}a=\frac{1}{4}\text{,}b=\frac{1}{5}$ and $c=\frac{1}{3}$ .

Let $S$ and $T$ be two self mappings of a fuzzy 2-metric space $(X\text{,}M\text{,}*)$ and $A\text{,}B:X\to \mathit{CB}(X)$ set-valued mappings satisfying following conditions:

• $\bigcup A(X)\subseteq S(X)$ and $\bigcup B(X)\subseteq T(X)$ ,

• $\{A\text{,}T\}$ and $\{B\text{,}S\}$ are weakly compatible pairs,

• $\mathit{aM}(\mathit{Tx}\text{,}\mathit{Sy}\text{,}w\text{,}t)+\mathit{bM}(\mathit{Tx}\text{,}\mathit{Ax}\text{,}w\text{,}t)+\mathit{cM}(\mathit{Sy}\text{,}\mathit{By}\text{,}w\text{,}t)+\max \{M(\mathit{Ax}\text{,}\mathit{Sy}\text{,}w\text{,}t)\text{,}M(\mathit{By}\text{,}\mathit{Tx}\text{,}w\text{,}t)\}⩽\mathit{qM}(\mathit{Ax}\text{,}\mathit{By}\text{,}w\text{,}t)$ ,

We can define a sequence $\{Z_n\}$ in $X$ such that $${\mathit{Sx}}_{2n+1}\in {\mathit{Ax}}_{2n}=Z_{2n}\text{,}{\mathit{Tx}}_{2n+2}\in {\mathit{Bx}}_{2n+1}=Z_{2n+1}\text{,}\forall n=0\text{,}1\text{,}2\text{,}\dots \text{.}$$

We will prove that $\{Z_n\}$ is Cauchy sequence.

Using inequality (3), we obtain $$\mathit{qM}(Z_{2n}\text{,}{Z}_{2n+1}\text{,}w\text{,}t)=\mathit{qM}({\mathit{Ax}}_{2n}\text{,}{\mathit{Bx}}_{2n+1}\text{,}w\text{,}t)⩾\mathit{aM}({\mathit{Tx}}_{2n}\text{,}{\mathit{Sx}}_{2n+1}\text{,}w\text{,}t)+\mathit{bM}({\mathit{Tx}}_{2n}\text{,}{\mathit{Ax}}_{2n}\text{,}w\text{,}t)+\mathit{cM}({\mathit{Sx}}_{2n+1}\text{,}{\mathit{Bx}}_{2n+1}\text{,}w\text{,}t)+\max \{M({\mathit{Ax}}_{2n}\text{,}{\mathit{Sx}}_{2n+1}\text{,}w\text{,}t)\text{,}M({\mathit{Bx}}_{2n+1}\text{,}{\mathit{Tx}}_{2n}\text{,}w\text{,}t)\}⩾\mathit{aM}(Z_{2n-1}\text{,}{Z}_{2n}\text{,}w\text{,}t)+\mathit{bM}(Z_{2n-1}\text{,}{Z}_{2n}\text{,}w\text{,}t)+\mathit{cM}(Z_{2n}\text{,}{Z}_{2n+1}\text{,}w\text{,}t)+\max \{M(Z_{2n}\text{,}{Z}_{2n}\text{,}w\text{,}t)\text{,}M(Z_{2n+1}\text{,}{Z}_{2n-1}\text{,}w\text{,}t)\}⩾(a+b)M(Z_{2n-1}\text{,}{Z}_{2n}\text{,}w\text{,}t)+\mathit{cM}(Z_{2n}\text{,}{Z}_{2n+1}\text{,}w\text{,}t)+\max \{1\text{,}M(Z_{2n}\text{,}{Z}_{2n-1}\text{,}w\text{,}t)*M(Z_{2n+1}\text{,}{Z}_{2n}\text{,}w\text{,}t)*M(Z_{2n+1}\text{,}{Z}_{2n-1}\text{,}{Z}_{2n}\text{,}t)\}\text{,}$$ $$M(Z_{2n}\text{,}{Z}_{2n+1}\text{,}w\text{,}t)⩾\beta M(Z_{2n-1}\text{,}{Z}_{2n}\text{,}w\text{,}t)\text{,}\text{where}\beta =\frac{a+b+1}{q-c}>1\text{.}$$

Since $\beta >1$ , we obtain $$M(Z_{2n+1}\text{,}{Z}_{2n}\text{,}w\text{,}t)>M(Z_{2n}\text{,}{Z}_{2n-1}\text{,}w\text{,}t)\text{.}$$

Similarly $$M(Z_{2n+2}\text{,}{Z}_{2n+1}\text{,}w\text{,}t)>M(Z_{2n+1}\text{,}{Z}_{2n}\text{,}w\text{,}t)\text{.}$$

Now for any positive integer $p$ , $$M(Z_n\text{,}{Z}_{n+p}\text{,}w\text{,}t)⩾M\left(Z_n\text{,}{Z}_{n+1}\text{,}w\text{,}\frac{t}{p}\right)*M\left(Z_{n+1}\text{,}{Z}_{n+2}\text{,}{Z}_{n+1}\text{,}\frac{t}{p}\right)*\cdots *M\left(Z_{n+p-1}\text{,}{Z}_{n+p}\text{,}w\text{,}\frac{t}{p}\right)\text{.}$$

As $n\to \infty$ , we get $M(Z_n\text{,}{Z}_{n+p}\text{,}w\text{,}t)\to 1$ .

Hence $Z_n$ is a Cauchy sequence. Suppose that $\mathit{SX}$ is complete, therefore by the above, $\{{\mathit{Sx}}_{2n+1}\}$ is a Cauchy sequence and hence ${\mathit{Sx}}_{2n+1}\to z=\mathit{Sv}$ for some $v\in X$ . Hence, $Z_n\to z$ and the subsequences ${\mathit{Tx}}_{2n+2}\text{,}{\mathit{Ax}}_{2n}$ and ${\mathit{Bx}}_{2n+1}$ converge to $z$ .

We shall prove that $z=\mathit{Sv}\in \mathit{Bv}$ , by (3), we have $$\mathit{qM}({\mathit{Ax}}_{2n}\text{,}\mathit{Bv}\text{,}w\text{,}t)⩾\mathit{aM}({\mathit{Tx}}_{2n}\text{,}\mathit{Sv}\text{,}w\text{,}t)+\mathit{bM}({\mathit{Tx}}_{2n}\text{,}{\mathit{Ax}}_{2n}\text{,}w\text{,}t)+\mathit{cM}(\mathit{Sv}\text{,}\mathit{Bv}\text{,}w\text{,}t)+\max \{M({\mathit{Ax}}_{2n}\text{,}\mathit{Sv}\text{,}w\text{,}t)\text{,}M(\mathit{Bv}\text{,}{\mathit{Tx}}_{2n}\text{,}w\text{,}t)\}\text{.}$$

As $n\to \infty$ , we obtain $$\mathit{qM}(z\text{,}\mathit{Bv}\text{,}w\text{,}t)⩾\mathit{aM}(z\text{,}z\text{,}w\text{,}t)+\mathit{bM}(z\text{,}z\text{,}w\text{,}t)+\mathit{cM}(z\text{,}\mathit{Bv}\text{,}w\text{,}t)+\max \{M(z\text{,}z\text{,}w\text{,}t)\text{,}M(\mathit{Bv}\text{,}z\text{,}w\text{,}t)\}\text{,}$$ $$M(z\text{,}\mathit{Bv}\text{,}w\text{,}t)⩾\left(\frac{a+b+1}{q-c}\right)>1\text{,}$$ which yields $\{z\}=\{\mathit{Sv}\}=\mathit{Bv}$ .

Since $\bigcup B(X)\subseteq T(X)$ , thus, there exist $u\in X$ such that $\{\mathit{Tu}\}=\mathit{Bv}=\{z\}=\{\mathit{Sv}\}$ . Now if $\mathit{Au}\ne \mathit{Bv}$ , we get $$\mathit{qM}(\mathit{Au}\text{,}\mathit{Bv}\text{,}w\text{,}t)⩾\mathit{aM}(\mathit{Tu}\text{,}\mathit{Sv}\text{,}w\text{,}t)+\mathit{bM}(\mathit{Tu}\text{,}\mathit{Au}\text{,}w\text{,}t)+\mathit{cM}(\mathit{Sv}\text{,}\mathit{Bv}\text{,}w\text{,}t)+\max \{M(\mathit{Au}\text{,}\mathit{Sv}\text{,}w\text{,}t)\text{,}M(\mathit{Bv}\text{,}\mathit{Tu}\text{,}w\text{,}t)\}\text{,}$$ $$\mathit{qM}(\mathit{Au}\text{,}z\text{,}w\text{,}t)⩾\mathit{aM}(z\text{,}z\text{,}w\text{,}t)+\mathit{bM}(z\text{,}\mathit{Au}\text{,}w\text{,}t)+\mathit{cM}(z\text{,}z\text{,}w\text{,}t)+\max \{M(\mathit{Au}\text{,}z\text{,}w\text{,}t)\text{,}1\}\text{,}$$ $$M(\mathit{Au}\text{,}z\text{,}t)⩾\left(\frac{a+c+1}{q-b}\right)>1\text{,}$$ which yields $\mathit{Au}=\{z\}=\{\mathit{Tu}\}=\{\mathit{Sv}\}=\mathit{Bv}$ .

Since $\mathit{Au}=\{\mathit{Tu}\}$ and $\{A\text{,}T\}$ is weakly compatible, $\mathit{ATv}=\mathit{TAv}$ gives $\mathit{Az}=\{\mathit{Tz}\}$ .

On using (3), we obtain $$\mathit{qM}(\mathit{Az}\text{,}\mathit{Bv}\text{,}w\text{,}t)⩾\mathit{aM}(\mathit{Tz}\text{,}\mathit{Sv}\text{,}w\text{,}t)+\mathit{bM}(\mathit{Tz}\text{,}\mathit{Az}\text{,}w\text{,}t)+\mathit{cM}(\mathit{Sv}\text{,}\mathit{Bv}\text{,}w\text{,}t)+\max \{M(\mathit{Az}\text{,}\mathit{Sv}\text{,}w\text{,}t)\text{,}M(\mathit{Bv}\text{,}\mathit{Tz}\text{,}w\text{,}t)\}\text{,}$$ $$\mathit{qM}(\mathit{Az}\text{,}z\text{,}w\text{,}t)⩾\mathit{aM}(\mathit{Tz}\text{,}z\text{,}w\text{,}t)+\mathit{bM}(z\text{,}\mathit{Az}\text{,}w\text{,}t)+\mathit{cM}(z\text{,}z\text{,}w\text{,}t)+\max \{M(\mathit{Az}\text{,}z\text{,}w\text{,}t)\text{,}1\}\text{.}$$

Hence, $\mathit{Az}=\{z\}=\{\mathit{Tz}\}$ . Similarly, $\mathit{Bz}=\{z\}=\{\mathit{Sz}\}$ where $\{B\text{,}S\}$ is weakly compatible. Then, $\mathit{Az}=\{\mathit{Tz}\}=\{z\}=\{\mathit{Sz}\}=\mathit{Bz}$ , i.e., $z$ is the common fixed point of $A\text{,}B\text{,}S$ and $T$ .

To see $z$ is unique, suppose that $p\ne z$ such that $\mathit{Ap}=\{\mathit{Tp}\}=\{p\}=\{\mathit{Sp}\}=\mathit{Bp}$ .

By (3), we get $$\mathit{qM}(\mathit{Az}\text{,}\mathit{Bp}\text{,}w\text{,}t)⩾\mathit{aM}(\mathit{Tz}\text{,}\mathit{Sp}\text{,}w\text{,}t)+\mathit{bM}(\mathit{Tz}\text{,}\mathit{Az}\text{,}w\text{,}t)+\mathit{cM}(\mathit{Sp}\text{,}\mathit{Bp}\text{,}w\text{,}t)+\max \{M(\mathit{Az}\text{,}\mathit{Sp}\text{,}w\text{,}t)\text{,}M(\mathit{Bp}\text{,}\mathit{Tz}\text{,}w\text{,}t)\}\text{,}$$ $$M(z\text{,}p\text{,}w\text{,}t)⩾\left(\frac{b+c}{q-a-1}\right)\text{,}$$ which yields $z=p$ . Then $A\text{,}B\text{,}S$ and $T$ have a unique common fixed point.

In Theorem 3.2, we replaced a complete fuzzy metric space by one mapping is complete and three self mappings into four mappings, two self mappings and two set-valued mappings. □

Theorem 3.2 is a generalization, extension and improvement for results of Sharma and Tiwari (2005) in fuzzy 2-metric space.