Some fuzzy fixed point results for fuzzy mappings in complete b-metric spaces

Abstract

In this paper, we establish some fixed point results for fuzzy mapping in a complete b-metric space. Our results unify, extend and generalize several results in the existing literature. Example is also given to support our results.

Keywords:

• fixed point
• complete b-metric space
• fuzzy mapping
• Hausdorff metric space

AMS Subject Classifications:

• 46S40
• 47H10
• 54H25

Public Interest Statement

The concept of fuzziness is helpful in solving such real-world problems where uncertainty occurs and many authors solve such problems by mathematical modeling in terms of fuzzy differential equations. Fixed point theorems play a fundamental role in demonstrating the existence of solutions to a wide variety of problems arising in physics, mathematics, engineering, medicine and social sciences.

The derived results extend and generalize some results in the existing literature. To show the validity of the derived results, an appropriate example and applications are also discussed. Our results are also useful in geometric problems arising in high-energy physics. This is because events in this case are mostly fuzzy sets.

1. Introduction and preliminaries

Fixed point theory plays an important role in the various fields of mathematics. It provides very important tools for finding the existence and uniqueness of the solutions. The Banach contraction theorem has an important role in fixed point theory and became very popular due to iterations which can be easily implemented on the computers. The idea of fuzzy set was first laid down by Zadeh (1965). Later many researcher study many direction of fuzzy for extend in some research area for example in Nashine, Vetro, Kumam and Kumam (2014), Mursaleen, Srivastava and Sharma (2016), Phiangsungnoen, Sintunavarat and Kumam (2014), Xu, Tang, Yang and Srivastava (2016) and on Weiss (1975) and Butnariu (1982) give the idea of fuzzy mapping and obtained many fixed point results. Afterward, Heilpern (1981) initiated the idea of fuzzy contraction mappings and proved a fixed point theorem for fuzzy contraction mappings which is a fuzzy analogue of Nadler (1969) fixed point theorem for multivalued mappings. In 2005, Gupta et al. (2015) obtained some existence results of fixed points for contractive mappings in fuzzy metric spaces using control function. In 2015, Aghayan, Zireh and Ebadian (2017) studied some common best proximity points for non-self mappings between two subsets of complex valued b-metric spaces and generalized some well-known results that were proved in classic metric spaces on complex valued b-metric space by some new definitions. Also they presented a type of contractive condition and develop a common best proximity point theorem for non-self mappings in complex-valued b-metric spaces. Very recently, Shoaib, Kumam, Shahzad, Phiangsungnoen and Mahmood (2018) studied and established some fixed point results for fuzzy mappings in a complete dislocated b-metric space.

In this paper we extended and obtained an $\alpha$-fuzzy fixed point and an $\alpha$-fuzzy common fixed point for fuzzy mappings in a complete b-metric space. Example is also given which supports the proved results.

Definition 1.1

(Aydi, Bota, Karapnar, & Mitrovic, 2012) Let X be any nonempty set and $b\ge 1$ be any given real number. A function $d:X\times X\to {\mathbb{R}}^{+}$ is called a b-metric, if it satisfies the following conditions for all $x,y,z\in X$ :

(1)	$d(x,y)=0$ if and only if $x=y;$
(2)	$d(x,y)=d(y,x);$
(3)	$d(x,z)\le b[d(x,y)+d(y,z)].$

Then, the pair (X, d) is called as a b -metric space.

Definition 1.2

(Joseph, Roselin, & Marudai, 2016) Let (X, d) be a b-metric space and $\{x_n\}$ be a sequence in X. Then,

(1)	$\{x_n\}$ is called as convergent sequence if and only if there exists $x\in X$, such that for all $ϵ>0$ there exists $n(ϵ)\in \mathbb{N}$ such that for all $n\ge n(ϵ)$, we have $d(x_n,x)<ϵ$. So, we write $\underset{n\to \infty }{\mathit{lim}}{x}_n=x$;
(2)	$\{x_n\}$ is called as Cauchy sequence if and only if for all $ϵ>0$ there exists $n(ϵ)\in \mathbb{N}$ such that for each $m,n\ge n(ϵ)$, we have $d(x_n,x_m)<ϵ$.

Definition 1.3

(Nadler, 1969) Let (X, d) be a metric space. We define the Hausdorff metric on CB(X) induced by d. Then,$$\begin{array}{c}\hfill H(A,B)=max\{\underset{x\in A}{sup}d(x,B),\underset{y\in B}{sup}d(A,y)\},\end{array}$$

for all $A,B\in CB(X),$ where CB(X) denotes the family of closed and bounded subsets of X and$$\begin{array}{c}\hfill d(x,B)=inf\{d(x,a):a\in B\},\end{array}$$

for all $x\in X.$

A fuzzy set in X is a function with domain X and values in Azam (2011), $\mathcal{F}(X)$ is the collection of all fuzzy sets in X. If A is a fuzzy set and $x\in X$, then the function values A(x) is called as the grade of membership of x in A. The $\alpha$-level set of fuzzy set A,  is denoted by ${[A]}_{\alpha },$ and defined as:$$\begin{array}{cc}\hfill {[A]}_{\alpha }& =\{x:A(x)\ge \alpha \}\text{where}\alpha \in (0,1],\hfill \\ \hfill {[A]}_0& =\overline{\{x:A(x)>0\}}.\hfill \end{array}$$

Let X be any nonempty set and Y be a metric space. A mapping T is called as fuzzy mapping, if T is a mapping from X into $\mathcal{F}(Y)$. A fuzzy mapping T is a fuzzy subset on $X\times Y$ with membership function T(x)(y). The function T(x)(y) is the grade of membership of y in T(x) . For convenience, we denote the $\alpha$-level set of T(x) by ${[Tx]}_{\alpha }$ instead of ${[T(x)]}_{\alpha }$ (Azam, 2011).

Definition 1.4

(Azam, 2011) A point $x\in X$ is called an $\alpha$-fuzzy fixed point of a fuzzy mapping $T:X\to \mathcal{F}(X)$ if there exists $\alpha \in (0,1]$ such that $x\in {[Tx]}_{\alpha }$.

Definition 1.5

Let $S,T:X\to \mathcal{F}(X)$ be the two fuzzy mappings and for $x\in X,$ there exist ${\alpha }_S(x),{\alpha }_T(x)\in (0,1]$. A point x is said to be an $\alpha$-fuzzy common fixed point of S and T if $x\in {[Sx]}_{{\alpha }_S(x)}\cap {[Tx]}_{{\alpha }_T(x)}$.

Lemma 1.1

(Azam, 2011) Let A and B be nonempty closed and bounded subsets of a metric space (X, d). If $a\in A$, then$$\begin{array}{c}\hfill d(a,B)\le H(A,B).\end{array}$$

Lemma 1.2

(Azam, 2011) Let A and B be nonempty closed and bounded subsets of a metric space (X, d) and $0<\alpha \in R$. Then, for $a\in A$, there exists $b\in B$ such that$$\begin{array}{c}\hfill d(a,b)\le H(A,B)+\alpha .\end{array}$$

2. Main results

Now, we present our main results.

Theorem 2.1

Let (X, d) be a complete b-metric space with constant $b\ge 1$. Let $T:X\to \mathcal{F}(X)$ be a fuzzy mapping and for $x\in X,$ there exist $\alpha (x)\in (0,1]$ satisfying the following condition:(2.1) $$\begin{array}{cc}\hfill H({[Tx]}_{\alpha (x)},{[Ty]}_{\alpha (y)})& \le a_{1}d(x,{[Tx]}_{\alpha (x)})+a_{2}d(y,{[Ty]}_{\alpha (y)})+a_{3}d(x,{[Ty]}_{\alpha (y)})\hfill \\ \hfill & +a_{4}d(y,{[Tx]}_{\alpha (x)})+a_{5}d(x,y)+\hfill \\ \hfill & a_6\frac{d(x,{[Tx]}_{\alpha (x)})(1+d(x,{[Tx]}_{\alpha (x)}))}{1+d(x,y)},\hfill \end{array}$$(2.1)

for all $x,y\in X$. Also, $a_i\ge 0$, where $i=1,2,\dots 6$ with $a_1+a_2+2b{a}_3+a_4+a_5+a_6<1$ and $\underset{i=1}{\sum ^6}{a}_i<1.$ Then, T has an $\alpha$-fuzzy fixed point.

Proof

Let $x_0$ be any arbitrary point in X, such that $x_1\in {[T{x}_0]}_{\alpha (x_0)}.$ Then, by Lemma 1.2 there exists $x_2\in {[T{x}_1]}_{\alpha (x_1)}$, such that(2.2) $$\begin{array}{cc}\hfill d(x_1,x_2)& \le H({[T{x}_0]}_{\alpha (x_0)},{[T{x}_1]}_{\alpha (x_1)})+(a_1+b{a}_3+a_5+a_6)\hfill \\ \hfill & \le a_{1}d(x_0,{[T{x}_0]}_{\alpha (x_0)})+a_{2}d(x_1,{[T{x}_1]}_{\alpha (x_1)})+a_{3}d(x_0,{[T{x}_1]}_{\alpha (x_1)})\hfill \\ \hfill & +a_{4}d(x_1,{[T{x}_0]}_{\alpha (x_0)})+a_{5}d(x_0,x_1)+\hfill \\ \hfill & a_6\frac{d(x_0,{[T{x}_0]}_{\alpha (x_0)})(1+d(x_0,{[T{x}_0]}_{\alpha (x_0)}))}{1+d(x_0,x_1)}+(a_1+b{a}_3+a_5+a_6)\hfill \\ \hfill & \le a_{1}d(x_0,x_1)+a_{2}d(x_1,x_2)+a_{3}d(x_0,x_2)+a_{4}d(x_1,x_1)+a_{5}d(x_0,x_1)\hfill \\ \hfill & +a_6\frac{d(x_0,x_1)(1+d(x_0,x_1))}{1+d(x_0,x_1)}+(a_1+b{a}_3+a_5+a_6)\hfill \\ \hfill & \le a_{1}d(x_0,x_1)+a_{2}d(x_1,x_2)+b{a}_3[d(x_0,x_1)+d(x_1,x_2)]\hfill \\ \hfill & +a_{5}d(x_0,x_1)+a_{6}d(x_0,x_1)+(a_1+b{a}_3+a_5+a_6)\hfill \\ \hfill & \le \frac{a_1+b{a}_3+a_5+a_6}{1-(a_2+b{a}_3)}d(x_0,x_1)+\frac{(a_1+b{a}_3+a_5+a_6)}{1-(a_2+b{a}_3)}.\hfill \end{array}$$(2.2)

Let$$\begin{array}{c}\hfill \tau =\frac{(a_1+b{a}_3+a_5+a_6)}{1-(a_2+b{a}_3)}.\end{array}$$

then, by above 2.2, we have$$\begin{array}{c}\hfill d(x_1,x_2)\le \tau d(x_0,x_1)+\tau .\end{array}$$

Again by Lemma 1.2, $x_3\in {[T{x}_2]}_{\alpha (x_2)}$ such that(2.3) $$\begin{array}{cc}\hfill d(x_2,x_3)& \le H({[T{x}_1]}_{\alpha (x_1)},{[T{x}_2]}_{\alpha (x_2)})+\frac{{(a_1+b{a}_3+a_5+a_6)}^2}{1-(a_2+b{a}_3)}\hfill \\ \hfill & \le a_{1}d(x_1,{[T{x}_1]}_{\alpha (x_1)})+a_{2}d(x_2,{[T{x}_2]}_{\alpha (x_2)})+a_{3}d(x_1,{[T{x}_2]}_{\alpha (x_2)})\hfill \\ \hfill & +a_{4}d(x_2,{[T{x}_1]}_{\alpha (x_1)})+a_{5}d(x_1,x_2)+\hfill \\ \hfill & a_6\frac{d(x_1,{[T{x}_1]}_{\alpha (x_1)})(1+d(x_1,{[T{x}_1]}_{\alpha (x_1)}))}{1+d(x_1,x_2)}+\frac{{(a_1+b{a}_3+a_5+a_6)}^2}{1-(a_2+b{a}_3)}\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_2,x_3)+a_{3}d(x_1,x_3)+a_{4}d(x_2,x_2)+a_{5}d(x_1,x_2)\hfill \\ \hfill & +a_6\frac{d(x_1,x_2)(1+d(x_1,x_2))}{1+d(x_1,x_2)}+\frac{{(a_1+b{a}_3+a_5+a_6)}^2}{1-(a_2+b{a}_3)}\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_2,x_3)+b{a}_3[d(x_1,x_2)+d(x_2,x_3)]+a_{5}d(x_1,x_2)\hfill \\ \hfill & +a_{6}d(x_1,x_2)+\frac{{(a_1+b{a}_3+a_5+a_6)}^2}{1-(a_2+b{a}_3)}\hfill \\ \hfill & \le \frac{(a_1+b{a}_3+a_5+a_6)}{1-(a_2+b{a}_3)}d(x_1,x_2)+\frac{{(a_1+b{a}_3+a_5+a_6)}^2}{{(1-(a_2+b{a}_3))}^2}\hfill \\ \hfill & \le {\left(\frac{(a_1+b{a}_3+a_5+a_6)}{1-(a_2+b{a}_3)}\right)}^{2}d(x_0,x_1)+2{\left(\frac{(a_1+b{a}_3+a_5+a_6)}{1-(a_2+b{a}_3)}\right)}^2.\hfill \\ \hfill \end{array}$$(2.3)

By using 2.2, we get(2.4) $$\begin{array}{cc}\hfill d(x_2,x_3)& \le {\tau }^{2}d(x_0,x_1)+2{\tau }^2.\hfill \end{array}$$(2.4)

Continuing the same way by induction, we obtain a sequence $\{x_n\}$, such that $x_{n-1}\in {[T{x}_n]}_{\alpha (x_n)}$ and $x_n\in {[T{x}_{n+1}]}_{\alpha (x_{n+1})}$, we have(2.5) $$\begin{array}{c}\hfill d(x_n,x_{n+1})\le {\tau }^{n}d(x_0,x_1)+n{\tau }^n.\end{array}$$(2.5)

Now, for any positive integer m, n and $(n>m)$, we have$$\begin{array}{cc}\hfill d(x_m,x_n)& \le d(x_m,x_{m+1})+d(x_{m+1},x_{m+2})+\cdots +d(x_{n-1},x_n)\hfill \\ \hfill & \le {\tau }^{m}d(x_0,x_1)+m{\tau }^m+{\tau }^{m+1}d(x_0,x_1)+(m+1){\tau }^{m+1}+\cdots \hfill \\ \hfill & +{\tau }^{n-1}d(x_0,x_1)+(n-1){\tau }^{n-1}\hfill \\ \hfill & \le {\tau }^m(1+\tau +\cdots +{\tau }^{n-m-1})d(x_0,x_1)+\stackrel{n-1}{\underset{i=m}{\mathrm{\Sigma }}}i{\tau }^i\hfill \\ \hfill & \le \frac{{\tau }^m}{1-\tau }d(x_0,x_1)+\stackrel{n-1}{\underset{i=m}{\mathrm{\Sigma }}}i{\tau }^i.\hfill \end{array}$$

Since $\tau <1,$ it follows from Cauchy root test, $\mathrm{\Sigma }i{\tau }^i$ is convergent, hence $\{x_n\}$ is a Cauchy sequence in X. As, X is complete. So, there exists $z\in X$ such that $x_n\to z$ as $n\to \infty .$

Now, we consider$$\begin{array}{cc}\hfill d(z,{[Tz]}_{\alpha (z)})& \le b\left[d(z,x_{n+1})+d(x_{n+1},{[Tz]}_{\alpha (z)})\right]\hfill \\ \hfill & \le b\left[d(z,x_{n+1})+H({[T{x}_n]}_{\alpha (x_n)},{[Tz]}_{\alpha (z)})\right].\hfill \end{array}$$

Using 2.1, with $n\to \infty$ we get$$\begin{array}{c}\hfill (1-b(a_2+a_3))d(z,{[Tz]}_{\alpha (z)})\le 0.\end{array}$$

So, we get$$\begin{array}{c}\hfill z\in {[Tz]}_{\alpha (z)}.\end{array}$$

Hence, $z\in X$ is an $\alpha$-fuzzy fixed point.

$\square$

If we take $b=1$ in Theorem 2.1, then we have the following corollary which is a b-metric space extension of a metric space.

Corollary 2.1

Let (X, d) be a complete metric space. Let $T:X\to \mathcal{F}(X)$ be a fuzzy mapping and for $x\in X,$ there exist $\alpha (x)\in (0,1]$ satisfying the following condition:(2.6) $$\begin{array}{cc}\hfill H({[Tx]}_{\alpha (x)},{[Ty]}_{\alpha (y)})& \le a_{1}d(x,{[Tx]}_{\alpha (x)})+a_{2}d(y,{[Ty]}_{\alpha (y)})+a_{3}d(x,{[Ty]}_{\alpha (y)})\hfill \\ \hfill & +a_{4}d(y,{[Tx]}_{\alpha (x)})+a_{5}d(x,y)+\hfill \\ \hfill & a_6\frac{d(x,{[Tx]}_{\alpha (x)})(1+d(x,{[Tx]}_{\alpha (x)}))}{1+d(x,y)},\hfill \end{array}$$(2.6)

for all $x,y\in X$. Also, $a_i\ge 0$, where $i=1,2,\dots 6$ with $a_1+a_2+2b{a}_3+a_4+a_5+a_6<1$ and $\underset{i=1}{\sum ^6}{a}_i<1.$ Then, T has an $\alpha$-fuzzy fixed point.

Next, we replace the another condition in Theorem 2.1; we get the following result which is a b-metric space extension of fixed point theorem given by Heilpern (1981).

Corollary 2.2

Let (X, d) be a complete metric space. Let $T:X\to \mathcal{F}(X)$ be a fuzzy mapping and for $x\in X,$ there exist $\alpha (x)\in (0,1]$ satisfying the following condition:(2.7) $$\begin{array}{cc}\hfill H({[Tx]}_{\alpha (x)},{[Ty]}_{\alpha (y)})& \le k(d(x,y)),\hfill \end{array}$$(2.7)

for all $x,y\in X$, where $k\in (0,1)$. Then, T has an $\alpha$-fuzzy fixed point.

Theorem 2.2

Let (X, d) be a complete b-metric space with constant $b\ge 1$. Let $S,T:X\to \mathcal{F}(X)$ be the two fuzzy mappings and for $x\in X,$ there exist ${\alpha }_S(x),{\alpha }_T(x)\in (0,1]$ satisfying the following condition:(2.8) $$\begin{array}{cc}\hfill H({[Tx]}_{{\alpha }_T(x)},{[Sy]}_{{\alpha }_S(y)})& \le a_{1}d(x,{[Tx]}_{{\alpha }_T(x)})+a_{2}d(y,{[Sy]}_{{\alpha }_S(y)})+a_{3}d(x,{[Sy]}_{{\alpha }_S(y)})\hfill \\ \hfill & +a_{4}d(y,{[Tx]}_{{\alpha }_T(x)})+a_{5}d(x,y),\hfill \end{array}$$(2.8)

for all $x,y\in X$. Also $a_i\ge 0$, where $i=1,2,\dots 5$ with $(a_1+a_2)(b+1)+b(a_3+a_4)(b+1)+2b{a}_5<2$ and $\underset{i=1}{\sum ^5}{a}_i<1.$ Then, S and T have an $\alpha$-fuzzy common fixed point.

Proof

Let $x_0$ be any arbitrary point in X, such that $x_1\in {[T{x}_0]}_{\alpha (x_0)}.$ Then, by Lemma 1.2 there exists $x_2\in {[S{x}_1]}_{\alpha (x_1)}$, such that(2.9) $$\begin{array}{cc}\hfill d(x_1,x_2)& \le H({[T{x}_0]}_{\alpha (x_0)},{[S{x}_1]}_{\alpha (x_1)})+(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le a_{1}d(x_0,{[T{x}_0]}_{\alpha (x_0)})+a_{2}d(x_1,{[S{x}_1]}_{\alpha (x_1)})+a_{3}d(x_0,{[S{x}_1]}_{\alpha (x_1)})\hfill \\ \hfill & +a_{4}d(x_1,{[T{x}_0]}_{\alpha (x_0)})+a_{5}d(x_0,x_1)+(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le a_{1}d(x_0,x_1)+a_{2}d(x_1,x_2)+a_{3}d(x_0,x_2)+a_{4}d(x_1,x_1)+a_{5}d(x_0,x_1)\hfill \\ \hfill & +(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le a_{1}d(x_0,x_1)+a_{2}d(x_1,x_2)+b{a}_3[d(x_0,x_1)+d(x_1,x_2)]\hfill \\ \hfill & +a_{5}d(x_0,x_1)+(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le \frac{a_1+b{a}_3+a_5}{1-(a_2+b{a}_3)}d(x_0,x_1)+\frac{(a_1+b{a}_3+a_5)}{1-(a_2+b{a}_3)}.\hfill \end{array}$$(2.9)

Similarly, by symmetry, we have(2.10) $$\begin{array}{cc}\hfill d(x_2,x_1)& \le H({[S{x}_1]}_{\alpha (x_1)},{[T{x}_0]}_{\alpha (x_0)})+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,{[S{x}_1]}_{\alpha (x_1)})+a_{2}d(x_0,{[T{x}_0]}_{\alpha (x_0)})+a_{3}d(x_1,{[T{x}_0]}_{\alpha (x_0)})\hfill \\ \hfill & +a_{4}d(x_0,{[S{x}_1]}_{\alpha (x_1)})+a_{5}d(x_1,x_0)+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_0,x_1)+a_{3}d(x_1,x_1)+a_{4}d(x_0,x_2)+a_{5}d(x_1,x_0)\hfill \\ \hfill & +(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_0,x_1)+b{a}_4[d(x_0,x_1)+d(x_1,x_2)]\hfill \\ \hfill & +a_{5}d(x_1,x_0)+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le \frac{a_2+b{a}_4+a_5}{1-(a_1+b{a}_4)}d(x_0,x_1)+\frac{(a_2+b{a}_4+a_5)}{1-(a_1+b{a}_4)}.\hfill \end{array}$$(2.10)

Adding 2.10 and 2.11, we get(2.11) $$\begin{array}{c}\hfill d(x_1,x_2)\le \frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}d(x_0,x_1)+\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}.\end{array}$$(2.11)

Let,$$\begin{array}{c}\hfill \tau =\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}<\frac{1}{b},\end{array}$$

then, by above 2.11, we have(2.12) $$\begin{array}{c}\hfill d(x_1,x_2)\le \tau d(x_0,x_1)+\tau .\end{array}$$(2.12)

Again by Lemma 1.2, $x_3\in {[T{x}_2]}_{\alpha (x_2)}$ such that$$\begin{array}{cc}\hfill d(x_2,x_3)& \le H({[S{x}_1]}_{\alpha (x_1)},{[T{x}_2]}_{\alpha (x_2)})+\frac{{\left(a_1+a_2+b{a}_3+b{a}_4+2a_5\right)}^2}{2-(a_1+a_2+b{a}_3+b{a}_4)}\hfill \\ \hfill & \le {\tau }^{2}d(x_0,x_1)+2{\tau }^2.\hfill \end{array}$$

Continuing the same way, by induction, we have a sequence $\{x_n\}$ such that $x_{2n+1}\in {[T{x}_{2n}]}_{\alpha (x_{2n})}$ and $x_{2n+2}\in {[S{x}_{2n+1}]}_{\alpha (x_{2n+1})}$, with(2.13) $$\begin{array}{cc}\hfill d(x_{2n+1},x_{2n+2})& \le H({[T{x}_{2n}]}_{\alpha (x_{2n})},{[S{x}_{2n+1}]}_{\alpha (x_{2n+1})})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n}}\hfill \\ \hfill & \le a_{1}d(x_{2n},{[T{x}_{2n}]}_{\alpha (x_{2n})})+a_{2}d(x_{2n+1},{[S{x}_{2n+1}]}_{\alpha (x_{2n+1})})\hfill \\ \hfill & +a_{3}d(x_{2n},{[S{x}_{2n+1}]}_{\alpha (x_{2n+1})})+a_{4}d(x_{2n+1},{[T{x}_{2n}]}_{\alpha (x_{2n})})+a_{5}d(x_{2n},x_{2n+1})\hfill \\ \hfill & +\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n}}\hfill \\ \hfill & \le \frac{a_1+b{a}_3+a_5}{1-(a_2+b{a}_3)}d(x_{2n},x_{2n+1})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n+1}}.\hfill \end{array}$$(2.13)

Similarly,(2.14) $$\begin{array}{cc}\hfill d(x_{2n+2},x_{2n+1})& \le H({[S{x}_{2n+1}]}_{\alpha (x_{2n+1})},{[T{x}_{2n}]}_{\alpha (x_{2n})})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}\hfill \\ \hfill & \le a_{1}d(x_{2n+1},{[S{x}_{2n+1}]}_{\alpha (x_{2n+1})})+a_{2}d(x_{2n},{[T{x}_{2n}]}_{\alpha (x_{2n})})\hfill \\ \hfill & +a_{3}d(x_{2n+1},{[T{x}_{2n}]}_{\alpha (x_{2n})})+a_{4}d(x_{2n},{[S{x}_{2n+1}]}_{\alpha (x_{2n+1})})\hfill \\ \hfill & +a_{5}d(x_{2n+1},x_{2n})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}\hfill \\ \hfill & \le \frac{(a_2+b{a}_4+a_5)}{\left(1-(a_1+b{a}_4)\right)}d(x_{2n},x_{2n+1})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}.\hfill \end{array}$$(2.14)

Adding 2.13 and 2.14, we get$$\begin{array}{c}\hfill d(x_{2n+1},x_{2n+2})\le \tau d(x_{2n},x_{2n+1})+{\tau }^{2n+1}.\end{array}$$

Therefore,(2.15) $$\begin{array}{cc}\hfill d(x_n,x_{n+1})& \le \frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}d(x_{n-1},x_n)+{\left(\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}\right)}^n\hfill \\ \hfill & \le \tau d(x_{n-1},x_n)+{\tau }^n\hfill \\ \hfill & \le \tau \left[\tau d(x_{n-2},x_{n-1})+{\tau }^{n-1}\right]+{\tau }^n\hfill \\ \hfill & ={\tau }^{2}d(x_{n-2},x_{n-1})+2{\tau }^n\hfill \\ \hfill & \le \cdots \cdots \hfill \\ \hfill & \le {\tau }^{n}d(x_0,x_1)+n{\tau }^n.\hfill \end{array}$$(2.15)

Now, for any positive integer m, n and $(n>m)$, we have$$\begin{array}{cc}\hfill d(x_m,x_n)& \le d(x_m,x_{m+1})+d(x_{m+1},x_{m+2})+\cdots +d(x_{n-1},x_n)\hfill \\ \hfill & \le {\tau }^{m}d(x_0,x_1)+m{\tau }^m+{\tau }^{m+1}d(x_0,x_1)+(m+1){\tau }^{m+1}+\cdots \hfill \\ \hfill & +{\tau }^{n-1}d(x_0,x_1)+(n-1){\tau }^{n-1}\hfill \\ \hfill & \le {\tau }^m(1+\tau +\cdots +{\tau }^{n-m-1})d(x_0,x_1)+\stackrel{n-1}{\underset{i=m}{\mathrm{\Sigma }}}i{\tau }^i\hfill \\ \hfill & \le \frac{{\tau }^m}{1-\tau }d(x_0,x_1)+\stackrel{n-1}{\underset{i=m}{\mathrm{\Sigma }}}i{\tau }^i.\hfill \end{array}$$

Since $\tau <1,$ it follows from Cauchy root test, $\mathrm{\Sigma }i{\tau }^i$ is convergent, hence $\{x_n\}$ is a Cauchy sequence in X. As, X is complete. So, there exists $z\in X$ such that $x_n\to z$ as $n\to \infty .$

Now, we prove $z\in X$ be the $\alpha$-fuzzy common fixed point of S and T.$$\begin{array}{cc}\hfill d(z,{[Sz]}_{\alpha (z)})& \le b\left[d(z,x_{2n+1})+d(x_{2n+1},{[Sz]}_{\alpha (z)})\right]\hfill \\ \hfill & \le b\left[d(z,x_{2n+1})+H({[T{x}_{2n}]}_{\alpha (x_{2n})},{[Tz]}_{\alpha (z)})\right].\hfill \end{array}$$

Using 2.8, with $n\to \infty$ we get$$\begin{array}{c}\hfill (1-b(a_2+a_3))d(z,{[Sz]}_{\alpha (z)})\le 0.\end{array}$$

So, we get$$\begin{array}{c}\hfill z\in {[Sz]}_{\alpha (z)}.\end{array}$$

This implies that $z\in X$ is an $\alpha$-fuzzy fixed point for S. Similarly, we can show that $z\in {[Tz]}_{\alpha (z)}$. Hence, $z\in X,$ be an $\alpha$-fuzzy common fixed point.

If we take $b=1$ in Theorem 2.2, then we have the following corollary which is a b-metric space extension of a metric space.

Corollary 2.3

Let (X, d) be a complete metric space. Let $S,T:X\to \mathcal{F}(X)$ be the two fuzzy mappings and for $x\in X,$ there exist ${\alpha }_S(x),{\alpha }_T(x)\in (0,1]$ satisfying the following condition:(2.16) $$\begin{array}{cc}\hfill H({[Tx]}_{{\alpha }_T(x)},{[Sy]}_{{\alpha }_S(y)})& \le a_{1}d(x,{[Tx]}_{{\alpha }_T(x)})+a_{2}d(y,{[Sy]}_{{\alpha }_S(y)})+a_{3}d(x,{[Sy]}_{{\alpha }_S(y)})\hfill \\ \hfill & +a_{4}d(y,{[Tx]}_{{\alpha }_T(x)})+a_{5}d(x,y),\hfill \end{array}$$(2.16)

for all $x,y\in X$. Also $a_i\ge 0$, where $i=1,2,\dots 5$ with $(a_1+a_2)(2)+(a_3+a_4)(2)+2a_5<2$ and $\underset{i=1}{\sum ^5}{a}_i<1.$ Then, S and T have $\alpha$-fuzzy common fixed point.

In Theorem 2.2, if $T=S$ where $S,T:X\to \mathcal{F}(X)$ be the two fuzzy mappings and ${\alpha }_T(x)={\alpha }_S(x)=\alpha (x)$, then we have the following corollary.

Corollary 2.4

Let (X, d) be a complete metric space. Let $T:X\to \mathcal{F}(X)$ be the fuzzy mappings and for $x\in X,$ there exist ${\alpha }_{(}x)\in (0,1]$ satisfying the following condition:(2.17) $$\begin{array}{cc}\hfill H({[Tx]}_{{\alpha }_{(}x)},{[Ty]}_{\alpha (y)})& \le a_{1}d(x,{[Tx]}_{{\alpha }_{(}x)})+a_{2}d(y,{[Ty]}_{{\alpha }_{(}y)})+a_{3}d(x,{[Ty]}_{{\alpha }_{(}y)})\hfill \\ \hfill & +a_{4}d(y,{[Tx]}_{{\alpha }_{(}x)})+a_{5}d(x,y),\hfill \end{array}$$(2.17)

for all $x,y\in X$. Also $a_i\ge 0$, where $i=1,2,\dots 5$ with $(a_1+a_2)(2)+(a_3+a_4)(2)+2a_5<2$ and $\underset{i=1}{\sum ^5}{a}_i<1.$ Then, T has an $\alpha$-fuzzy fixed point.

Proof

Let $x_0$ be any arbitrary point in X, such that $x_1\in {[T{x}_0]}_{\alpha (x_0)}.$ Then, by Lemma 1.2 there exists $x_2\in {[T{x}_1]}_{\alpha (x_1)}$, such that(2.18) $$\begin{array}{cc}\hfill d(x_1,x_2)& \le H({[T{x}_0]}_{\alpha (x_0)},{[T{x}_1]}_{\alpha (x_1)})+(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le a_{1}d(x_0,{[T{x}_0]}_{\alpha (x_0)})+a_{2}d(x_1,{[T{x}_1]}_{\alpha (x_1)})+a_{3}d(x_0,{[T{x}_1]}_{\alpha (x_1)})\hfill \\ \hfill & +a_{4}d(x_1,{[T{x}_0]}_{\alpha (x_0)})+a_{5}d(x_0,x_1)+(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le a_{1}d(x_0,x_1)+a_{2}d(x_1,x_2)+a_{3}d(x_0,x_2)+a_{4}d(x_1,x_1)+a_{5}d(x_0,x_1)\hfill \\ \hfill & +(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le a_{1}d(x_0,x_1)+a_{2}d(x_1,x_2)+b{a}_3[d(x_0,x_1)+d(x_1,x_2)]\hfill \\ \hfill & +a_{5}d(x_0,x_1)+(a_1+b{a}_3+a_5)\hfill \\ \hfill & \le \frac{a_1+b{a}_3+a_5}{1-(a_2+b{a}_3)}d(x_0,x_1)+\frac{(a_1+b{a}_3+a_5)}{1-(a_2+b{a}_3)}.\hfill \end{array}$$(2.18)

Similarly, by symmetry, we have(2.19) $$\begin{array}{cc}\hfill d(x_2,x_1)& \le H({[T{x}_1]}_{\alpha (x_1)},{[T{x}_0]}_{\alpha (x_0)})+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,{[T{x}_1]}_{\alpha (x_1)})+a_{2}d(x_0,{[T{x}_0]}_{\alpha (x_0)})+a_{3}d(x_1,{[T{x}_0]}_{\alpha (x_0)})\hfill \\ \hfill & +a_{4}d(x_0,{[T{x}_1]}_{\alpha (x_1)})+a_{5}d(x_1,x_0)+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_0,x_1)+a_{3}d(x_1,x_1)+a_{4}d(x_0,x_2)+a_{5}d(x_1,x_0)\hfill \\ \hfill & +(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_0,x_1)+b{a}_4[d(x_0,x_1)+d(x_1,x_2)]\hfill \\ \hfill & +a_{5}d(x_1,x_0)+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le \frac{a_2+b{a}_4+a_5}{1-(a_1+b{a}_4)}d(x_0,x_1)+\frac{(a_2+b{a}_4+a_5)}{1-(a_1+b{a}_4)}.\hfill \end{array}$$(2.19)

Adding 2.19(2.19) $$\begin{array}{cc}\hfill d(x_2,x_1)& \le H({[T{x}_1]}_{\alpha (x_1)},{[T{x}_0]}_{\alpha (x_0)})+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,{[T{x}_1]}_{\alpha (x_1)})+a_{2}d(x_0,{[T{x}_0]}_{\alpha (x_0)})+a_{3}d(x_1,{[T{x}_0]}_{\alpha (x_0)})\hfill \\ \hfill & +a_{4}d(x_0,{[T{x}_1]}_{\alpha (x_1)})+a_{5}d(x_1,x_0)+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_0,x_1)+a_{3}d(x_1,x_1)+a_{4}d(x_0,x_2)+a_{5}d(x_1,x_0)\hfill \\ \hfill & +(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le a_{1}d(x_1,x_2)+a_{2}d(x_0,x_1)+b{a}_4[d(x_0,x_1)+d(x_1,x_2)]\hfill \\ \hfill & +a_{5}d(x_1,x_0)+(a_2+b{a}_4+a_5)\hfill \\ \hfill & \le \frac{a_2+b{a}_4+a_5}{1-(a_1+b{a}_4)}d(x_0,x_1)+\frac{(a_2+b{a}_4+a_5)}{1-(a_1+b{a}_4)}.\hfill \end{array}$$(2.19) and 2.20(2.20) $$\begin{array}{c}\hfill d(x_1,x_2)\le \frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}d(x_0,x_1)+\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}.\end{array}$$(2.20) , we get(2.20) $$\begin{array}{c}\hfill d(x_1,x_2)\le \frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}d(x_0,x_1)+\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}.\end{array}$$(2.20)

Let,$$\begin{array}{c}\hfill \tau =\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}<\frac{1}{b}.\end{array}$$

then, by above 2.20(2.20) $$\begin{array}{c}\hfill d(x_1,x_2)\le \frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}d(x_0,x_1)+\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}.\end{array}$$(2.20) , we have(2.21) $$\begin{array}{c}\hfill d(x_1,x_2)\le \tau d(x_0,x_1)+\tau .\end{array}$$(2.21)

Again by Lemma 1.2, $x_3\in {[T{x}_2]}_{\alpha (x_2)}$ such that$$\begin{array}{cc}\hfill d(x_2,x_3)& \le H({[T{x}_1]}_{\alpha (x_1)},{[T{x}_2]}_{\alpha (x_2)})+\frac{{\left(a_1+a_2+b{a}_3+b{a}_4+2a_5\right)}^2}{2-(a_1+a_2+b{a}_3+b{a}_4)}\hfill \\ \hfill & \le {\tau }^{2}d(x_0,x_1)+2{\tau }^2.\hfill \end{array}$$

Continuing the same way, by induction, we have a sequence $\{x_n\}$ such that $x_{2n+1}\in {[T{x}_{2n}]}_{\alpha (x_{2n})}$ and $x_{2n+2}\in {[T{x}_{2n+1}]}_{\alpha (x_{2n+1})}$, with(2.22) $$\begin{array}{cc}\hfill d(x_{2n+1},x_{2n+2})& \le H({[T{x}_{2n}]}_{\alpha (x_{2n})},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n}}\hfill \\ \hfill & \le a_{1}d(x_{2n},{[T{x}_{2n}]}_{\alpha (x_{2n})})+a_{2}d(x_{2n+1},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})\hfill \\ \hfill & +a_{3}d(x_{2n},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})+a_{4}d(x_{2n+1},{[T{x}_{2n}]}_{\alpha (x_{2n})})\hfill \\ \hfill & +a_{5}d(x_{2n},x_{2n+1})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n}}\hfill \\ \hfill & \le \frac{a_1+b{a}_3+a_5}{1-(a_2+b{a}_3)}d(x_{2n},x_{2n+1})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n+1}}.\hfill \end{array}$$(2.22)

Similarly,(2.23) $$\begin{array}{cc}\hfill d(x_{2n+2},x_{2n+1})& \le H({[T{x}_{2n+1}]}_{\alpha (x_{2n+1})},{[T{x}_{2n}]}_{\alpha (x_{2n})})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}\hfill \\ \hfill & \le a_{1}d(x_{2n+1},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})+a_{2}d(x_{2n},{[T{x}_{2n}]}_{\alpha (x_{2n})})\hfill \\ \hfill & +a_{3}d(x_{2n+1},{[T{x}_{2n}]}_{\alpha (x_{2n})})+a_{4}d(x_{2n},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})\hfill \\ \hfill & +a_{5}d(x_{2n+1},x_{2n})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}\hfill \\ \hfill & \le \frac{(a_2+b{a}_4+a_5)}{\left(1-(a_1+b{a}_4)\right)}d(x_{2n},x_{2n+1})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}.\hfill \end{array}$$(2.23)

Adding 2.22(2.22) $$\begin{array}{cc}\hfill d(x_{2n+1},x_{2n+2})& \le H({[T{x}_{2n}]}_{\alpha (x_{2n})},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n}}\hfill \\ \hfill & \le a_{1}d(x_{2n},{[T{x}_{2n}]}_{\alpha (x_{2n})})+a_{2}d(x_{2n+1},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})\hfill \\ \hfill & +a_{3}d(x_{2n},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})+a_{4}d(x_{2n+1},{[T{x}_{2n}]}_{\alpha (x_{2n})})\hfill \\ \hfill & +a_{5}d(x_{2n},x_{2n+1})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n}}\hfill \\ \hfill & \le \frac{a_1+b{a}_3+a_5}{1-(a_2+b{a}_3)}d(x_{2n},x_{2n+1})+\frac{{(a_1+b{a}_3+a_5)}^{2n+1}}{{\left(1-(a_2+b{a}_3)\right)}^{2n+1}}.\hfill \end{array}$$(2.22) and 2.23(2.23) $$\begin{array}{cc}\hfill d(x_{2n+2},x_{2n+1})& \le H({[T{x}_{2n+1}]}_{\alpha (x_{2n+1})},{[T{x}_{2n}]}_{\alpha (x_{2n})})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}\hfill \\ \hfill & \le a_{1}d(x_{2n+1},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})+a_{2}d(x_{2n},{[T{x}_{2n}]}_{\alpha (x_{2n})})\hfill \\ \hfill & +a_{3}d(x_{2n+1},{[T{x}_{2n}]}_{\alpha (x_{2n})})+a_{4}d(x_{2n},{[T{x}_{2n+1}]}_{\alpha (x_{2n+1})})\hfill \\ \hfill & +a_{5}d(x_{2n+1},x_{2n})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}\hfill \\ \hfill & \le \frac{(a_2+b{a}_4+a_5)}{\left(1-(a_1+b{a}_4)\right)}d(x_{2n},x_{2n+1})+\frac{{(a_2+b{a}_4+a_5)}^{2n+1}}{{\left(1-(a_1+b{a}_4)\right)}^{2n}}.\hfill \end{array}$$(2.23) , we get$$\begin{array}{c}\hfill d(x_{2n+1},x_{2n+2})\le \tau d(x_{2n},x_{2n+1})+{\tau }^{2n+1}.\end{array}$$

Therefore,(2.24) $$\begin{array}{cc}\hfill d(x_n,x_{n+1})& \le \frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}d(x_{n-1},x_n)+{\left(\frac{a_1+a_2+b{a}_3+b{a}_4+2a_5}{2-(a_1+a_2+b{a}_3+b{a}_4)}\right)}^n\hfill \\ \hfill & \le \tau d(x_{n-1},x_n)+{\tau }^n\hfill \\ \hfill & \le \tau \left[\tau d(x_{n-2},x_{n-1})+{\tau }^{n-1}\right]+{\tau }^n\hfill \\ \hfill & ={\tau }^{2}d(x_{n-2},x_{n-1})+2{\tau }^n\hfill \\ \hfill & \le \cdots \cdots \hfill \\ \hfill d(x_n,x_{n+1})& \le {\tau }^{n}d(x_0,x_1)+n{\tau }^n.\hfill \end{array}$$(2.24)

Now, for any positive integer m, n and $(n>m)$, we have$$\begin{array}{cc}\hfill d(x_m,x_n)& \le d(x_m,x_{m+1})+d(x_{m+1},x_{m+2})+\cdots +d(x_{n-1},x_n)\hfill \\ \hfill & \le {\tau }^{m}d(x_0,x_1)+m{\tau }^m+{\tau }^{m+1}d(x_0,x_1)+(m+1){\tau }^{m+1}+\cdots \hfill \\ \hfill & +{\tau }^{n-1}d(x_0,x_1)+(n-1){\tau }^{n-1}\hfill \\ \hfill & \le {\tau }^m(1+\tau +\cdots +{\tau }^{n-m-1})d(x_0,x_1)+\stackrel{n-1}{\underset{i=m}{\mathrm{\Sigma }}}i{\tau }^i\hfill \\ \hfill & \le \frac{{\tau }^m}{1-\tau }d(x_0,x_1)+\stackrel{n-1}{\underset{i=m}{\mathrm{\Sigma }}}i{\tau }^i.\hfill \end{array}$$

Since $\tau <1,$ it follows from Cauchy root test, $\mathrm{\Sigma }i{\tau }^i$ is convergent, hence $\{x_n\}$ is a Cauchy sequence in X. As, X is complete. So, there exists $z\in X$ such that $x_n\to z$ as $n\to \infty .$

Now, we prove $z\in X$ be the $\alpha$-fuzzy fixed point of T.$$\begin{array}{cc}\hfill d(z,{[Tz]}_{\alpha (z)})& \le b\left[d(z,x_{2n+1})+d(x_{2n+1},{[Tz]}_{\alpha (z)})\right]\hfill \\ \hfill & \le b\left[d(z,x_{2n+1})+H({[T{x}_{2n}]}_{\alpha (x_{2n})},{[Tz]}_{\alpha (z)})\right].\hfill \end{array}$$

Using 2.17(2.17) $$\begin{array}{cc}\hfill H({[Tx]}_{{\alpha }_{(}x)},{[Ty]}_{\alpha (y)})& \le a_{1}d(x,{[Tx]}_{{\alpha }_{(}x)})+a_{2}d(y,{[Ty]}_{{\alpha }_{(}y)})+a_{3}d(x,{[Ty]}_{{\alpha }_{(}y)})\hfill \\ \hfill & +a_{4}d(y,{[Tx]}_{{\alpha }_{(}x)})+a_{5}d(x,y),\hfill \end{array}$$(2.17) , with $n\to \infty$ we get$$\begin{array}{c}\hfill (1-b(a_2+a_3))d(z,{[Tz]}_{\alpha (z)})\le 0.\end{array}$$

So, we get$$\begin{array}{c}\hfill z\in {[Tz]}_{\alpha (z)}.\end{array}$$

This implies that $z\in X$ is an $\alpha$-fuzzy fixed point for T.

Example 2.1

Let X=(Azam, 2011) and $d(x,y)=\left|x-y\right|$, whenever $x,y\in X$, then (X, d) is a complete b-metric space. Define a fuzzy mapping $T:X\to F(X)$ by$$\begin{array}{c}\hfill T(x)(t)=\left\{\begin{array}{c}1,0\le t\le x/4;\hfill \\ 1/2,x/4<t\le x/3;\hfill \\ 1/4,x/3<t\le x/2;\hfill \\ 0,x/2<t\le 1.\hfill \end{array}\right.\end{array}$$

For all $x\in X$, there exists $\alpha (x)=1$, such that$$\begin{array}{c}\hfill {\left[Tx\right]}_{\alpha (x)}=\left[0,\frac{x}{4}\right]\text{.}\end{array}$$

Then,$$\begin{array}{cc}\hfill H({[Tx]}_{\alpha (x)},{[Ty]}_{\alpha (y)})& \le \frac{1}{5}\left|x-\frac{x}{4}\right|+\frac{1}{10}\left|y-\frac{y}{4}\right|+\frac{1}{15}\left|x-\frac{y}{4}\right|\hfill \\ \hfill & +\frac{1}{20}\left|y-\frac{x}{4}\right|+\frac{1}{25}\left|x-y\right|+\hfill \\ \hfill & \frac{1}{30}\left(\frac{\left|x-\frac{x}{4}\right|\left(1+\left|x-\frac{x}{4}\right|\right)}{1+\left|x-y\right|}\right).\hfill \end{array}$$

Since, all the conditions of Theorem 2.1 are satisfied. So, there exists $0\in X$ is an $\alpha$-fuzzy fixed point of T.

Authors Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

The authors acknowledge the financial support provided by King Mongkut’s University of Technology Thonburi through the KMUTT 55th Anniversary Commemorative Fund. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT. Furthermore, this work was financial supported by the Rajamangala University of Technology Thanyaburi (RMUTT).

Additional information

Funding

The authors acknowledge the financial support provided by King Mongkut’s University of Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. This project was supported by the Theoretical and Computational Science (TaCS) Center under ComputationaL and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT. Furthermore, Wiyada Kumam was financial supported by RMUTT annual government statement of expenditure in 2018.

Notes on contributors

Wiyada Kumam

Wiyada Kumam is currently a Assistant Professor at the Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT). Assistant Professor Dr. Wiyada Kumam, her field of interests are fuzzy optimization, fuzzy regression, least-squares method, minimization problem and fuzzy nonlinear mapping.