# Common Fixed Point for Generalized Bose-Mukherjee-Type Fuzzy Mappings

Authors: Ming-liang Song; Zhong-qian Wang
Journal: ISRN Applied Mathematics
 (2013)

---

## Abstract

We prove a common fixed point theorem for a pair of generalized Bose-Mukherjee-type fuzzy mappings in a complete metric space. An example is also provided to support the main result presented herein.


---

## Body

## 1. Introduction and Preliminaries

In many scientific and engineering applications, the fuzzy set concept plays an important role. The concept of fuzzy sets was introduced initially by Zadeh [1] in 1965. Since then, the study of fixed point theorems in fuzzy mathematics had been instigated by Weiss [2] and Butnariu [3]. Heilpern [4] introduced the concept of fuzzy contractive mappings and proved a fixed point theorem for these mappings in metric linear spaces. His result is a generalization of the fixed point theorem for point-to-set maps of Nadler [5]. Afterwards, several fixed point theorems for fuzzy contractive mappings have appeared in the literature (see [6–13]). Particularly, Vijayaraju and Marudai [6] studied a fixed point result for fuzzy (multivalued) mappings T:X→ℱ(X) in a metric space X. This result [6] is significant as it does not require the condition of approximate quantity for T(x) and linearity for X. However, Azam and Arshad [7] pointed out that its proof [6] is incorrect and incomplete and presented the right version of this result. In fact, although there exist mistakes in the proof of Theorem 3.1 in [6], its conclusion is correct.The aim of this work is to establish a common fixed point theorem for a pair of generalized Bose-Mukherjee-type fuzzy mappings in a complete metric space. Also, we give an example to show the validity of our result and by which indicate that our result improves and extends several known results in [6, 7, 14].LetX and Y be nonempty sets. A multivalued mapping T from X to Y, denoted by T:X→2Y, is defined to be a function that assigns to each element of X a nonempty subset of Y. Fixed points of the multivalued mapping T:X→2X will be the points x∈X such that x∈T(x).Let(X,d) be a metric space and let 𝒞ℬ(X) denote the set of all nonempty closed and bounded subsets of X. For A,B∈𝒞ℬ(X), define
(1)H(A,B)=max{supx∈Ad(x,B),supy∈Bd(A,y)},
where   d(x,A)=infy∈Ad(x,y).A fuzzy set inX is a function with domain X and values in [0,1]. If A is a fuzzy set and x∈X, then the function value A(x) is called the grade of membership of x in A. The α-level set of A is denoted by [A]α and is defined as follows:
(2)[A]α={x:A(x)≥α}ifα∈(0,1],[A]0={x:A(x)>0}¯.
Here, B¯ denotes the closure of the set B. Let ℱ(X) be the collection of all fuzzy sets in a metric space X. For A,B∈ℱ(X), A⊂B means A(x)≤B(x) for each x∈X.A mappingT from X to ℱ(Y) is called a fuzzy mapping if for each x∈X, T(x) (sometimes denoted by Tx) is a fuzzy set on Y and Tx(y) denotes the degree of membership of y in Tx. Let 𝒲(X) denote the set of all fuzzy sets on X such that each of its α-level is a nonempty closed bounded subset of X.The following lemmas are needed in the sequel.Lemma 1 (Nadler [5]).
Let(X,d) be a metric space and A,B∈𝒞ℬ(X); then  (1)
for eachx∈A,    d(x,B)≤H(A,B),(2)
for eachy∈X,    d(x,A)≤d(x,y)+d(y,A).Lemma 2 (Nadler [5]).
Let(X,d) be a metric space and A,B∈𝒞ℬ(X); then for each x∈A and ε>0 there exists an element y∈B such that  d(x,y)≤H(A,B)+ε.


## 2. Main Results

Lemma 3.
LetA1,A2,A3,A4, andA5 be five nonnegative real numbers with A1+A2+A3+A4+A5=1,A5>0, and either A1>A2,A3>A4 or A1<A2,A3<A4. Let a=(A1+A3+A5)/(A1+A4+A5),b=(A2+A4+A5)/(A2+A3+A5); then 0<ab<1.Proof.
IfA1>A2,A3>A4, then A1>0,A3>0. Note that A5>0; we have a=(A1+A3+A5)/(A1+A4+A5)>0,b=(A2+A4+A5)/(A2+A3+A5)>0; that is, ab>0. Moreover, it is evident that (A1-A2)(A3-A4)>0⇒A1A3+A2A4>A1A4+A2A3, which further implies that
(3)(A1+A4+A5)(A2+A3+A5)=A1A2+A1A3+A1A5+A4A2+A4A3+A4A5+A5A2+A5A3+A5A5>A1A2+A1A4+A1A5+A3A2+A3A4+A3A5+A5A2+A5A4+A5A5=(A1+A3+A5)(A2+A4+A5).
That is, ab=(A1+A3+A5)(A2+A4+A5)/(A1+A4+A5)(A2+A3+A5)<1.
Similarly, ifA1<A2,A3<A4, then 0<ab<1 holds.Theorem 4.
Let(X,d) be a complete metric space. Let S,T:X→ℱ(X) be two generalized Bose-Mukherjee-type fuzzy mappings. Suppose that, for each x∈X, there exists α(x)∈(0,1] such that [Sx]α(x)and[Tx]α(x) are nonempty closed bounded subsets of X and
(4)H([Sx]α(x),[Ty]α(y))≤A1d(x,[Sx]α(x))+A2d(y,[Ty]α(y))+A3d(x,[Ty]α(y))+A4d(y,[Sx]α(x))+A5d(x,y),
for all x,y∈X, where A1,A2,A3,A4, andA5 are five nonnegative real numbers with Σi=15Ai=1,A5>0 and either A1>A2,A3>A4 or A1<A2,A3<A4. Then there exists z∈X such that z∈[Sz]α(z)∩[Tz]α(z).Proof.
LetA1>A2,A3>A4 and a=(A1+A3+A5)/(A1+A4+A5),b=(A2+A4+A5)/(A2+A3+A5). By Lemma 3, we know that a>0,b>0 and 0<ab<1. Choosing x0∈X, by hypotheses, there exists α(x0)∈(0,1] such that [Sx0]α(x0) is nonempty closed bounded subset of X. For convenience, we denote α(x0) by α1. Let x1∈[Sx0]α1; for this x1 there exists α2∈(0,1] such that [Tx1]α2 is nonempty closed bounded subset of X. Since A1+A4+A5>0, by Lemma 2, there exists x2∈[Tx1]α2 such that
(5)d(x1,x2)≤H([Sx0]α1,[Tx1]α2)+(A1+A4+A5).
Since A2+A3+A5>0, by the same argument, we can find α3∈(0,1] and x3∈[Sx2]α3 such that
(6)d(x2,x3)≤H([Sx2]α3,[Tx1]α2)+ab(A2+A3+A5).
By induction, we produce a sequence {xn} of points of X,
(7)x2k+1∈[Sx2k]α2k+1,x2k+2∈[Tx2k+1]α2k+2,k=0,1,2,…,
such that
(8)d(x2k+1,x2k+2)≤H([Sx2k]α2k+1,[Tx2k+1]α2k+2)+(ab)k(A1+A4+A5),d(x2k+2,x2k+3)≤H([Sx2k+2]α2k+3,[Tx2k+1]α2k+2)+(ab)k+1(A2+A3+A5).
For k=0,1,2,…, applying (4), we obtain
(9)d(x2k+1,x2k+2)≤H([Sx2k]α2k+1,[Tx2k+1]α2k+2)+(ab)k(A1+A4+A5)≤A1d(x2k,[Sx2k]α2k+1)+A2d(x2k+1,[Tx2k+1]α2k+2)+A3d(x2k,[Tx2k+1]α2k+2)+A4d(x2k+1,[Sx2k]α2k+1)+A5d(x2k,x2k+1)+(ab)k(A1+A4+A5)≤A1d(x2k,x2k+1)+A2d(x2k+1,x2k+2)+A3d(x2k,x2k+2)+A4d(x2k+1,x2k+1)+A5d(x2k,x2k+1)+(ab)k(A1+A4+A5)≤A1d(x2k,x2k+1)+A2d(x2k+1,x2k+2)+A3d(x2k,x2k+1)+A3d(x2k+1,x2k+2)+A5d(x2k,x2k+1)+(ab)k(A1+A4+A5).
It implies that
(10)d(x2k+1,x2k+2)≤A1+A3+A51-A2-A3d(x2k,x2k+1)+(ab)kA1+A4+A51-A2-A3.
Note that Σi=15Ai=1; we have
(11)d(x2k+1,x2k+2)≤A1+A3+A5A1+A4+A5d(x2k,x2k+1)d(x2k+1,x2k+2)+(ab)kA1+A4+A5A1+A4+A5d(x2k+1,x2k+2)=ad(x2k,x2k+1)+(ab)k.
Similarly, we have
(12)d(x2k+2,x2k+3)≤bd(x2k+1,x2k+2)+(ab)k+1.
Using inductive method, for k=0,1,2,…, by (11) and (12), we can obtain
(13)d(x2k+1,x2k+2)≤ad(x2k,x2k+1)+(ab)k≤abd(x2k-1,x2k)+a(ab)k+(ab)k≤abad(x2k-2,x2k-1)+(ab)k+a(ab)k+(ab)k≤(ab)2d(x2k-3,x2k-2)+2(a+1)(ab)k≤⋯≤(ab)kd(x1,x2)+k(a+1)(ab)k≤(ab)kad(x0,x1)+(ab)k+k(a+1)(ab)k≤(ab)kd(x0,x1)+(k+1)(a+1)(ab)k,d(x2k+2,x2k+3)≤b((ab)kad(x0,x1)+(k+1)(a+1)(ab)k)+(ab)k+1=(ab)k+1d(x0,x1)+b(k+1)(a+1)(ab)k+(ab)k+1=(ab)k+1d(x0,x1)+(k+1)(a+1a)(ab)k+1+(ab)k+1≤(ab)k+1d(x0,x1)+(k+2)(a+1a)(ab)k+1.
Next, we prove that the sequence{xn} is a Cauchy sequence in X. For k<p, we have
(14)d(x2k+1,x2p+1)≤d(x2k+1,x2k+2)+⋯+d(x2p,x2p+1)≤(∑i=kp-1‍(ab)i+∑i=k+1p‍(ab)i)d(x0,x1)+((a+1)∑i=kp-1‍(i+1)(ab)i+(a+1a)∑i=k+1p‍(i+1)(ab)i)≤((ab)k1-ab+(ab)k+11-ab)d(x0,x1)+((a+1)∑i=k∞‍(i+1)(ab)i+(a+1a)∑i=k+1∞‍(i+1)(ab)i)≤2(ab)k1-abd(x0,x1)+2M∑i=k∞‍(i+1)(ab)i,
where M=max{a+1,(a+1)/a}. By the similar reasoning process, we can obtain
(15)d(x2k,x2p+1)≤2(ab)k1-abd(x0,x1)+2M∑i=k∞‍(i+1)(ab)i,d(x2k,x2p)≤2(ab)k1-abd(x0,x1)+2M∑i=k∞‍(i+1)(ab)i,d(x2k+1,x2p)≤2(ab)k1-abd(x0,x1)+2M∑i=k∞‍(i+1)(ab)i.
Then, there exists k with (n-1)/2≤k≤n/2, for any 0<n<m, such that
(16)d(xm,xn)≤2(ab)k1-abd(x0,x1)+2M∑i=k∞‍(i+1)(ab)i.
Since 0<ab<1, it follows from Cauchy’s root test that Σ(n+1)(ab)n is convergent and hence {xn} is a Cauchy sequence in X. Since X is a complete metric space, then there exists z∈X such that xn→z as n→∞. Then, by (4) and Lemma 1, we have
(17)d(z,[T(z)]α(z))≤d(z,x2n+1)+d(x2n+1,[T(z)]α(z))≤d(z,x2n+1)+H([Sx2n]α2n+1,[T(z)]α(z))≤d(z,x2n+1)+A1d(x2n,[Sx2n]α2n+1)+A2d(z,[T(z)]α(z))+A3d(x2n,[T(z)]α(z))+A4d(z,[Sx2n]α2n+1)+A5d(x2n,z)≤d(z,x2n+1)+A1d(x2n,x2n+1)+A2d(z,[T(z)]α(z))+A3d(x2n,z)+A3d(z,[T(z)]α(z))+A4d(z,x2n+1)+A5d(x2n,z).
Therefore,
(18)(1-A2-A3)d(z,[T(z)]α(z))≤d(z,x2n+1)+A1d(x2n,x2n+1)+A3d(x2n,z)+A4d(z,x2n+1)+A5d(x2n,z)
and hence d(z,[T(z)]α(z))→0 as n→∞. Thus, z∈[Tz]α(z).
Similarly, we can prove thatz∈[Sz]α(z). Hence, z∈[Sz]α(z)∩[Tz]α(z).
IfA1<A2,A3<A4, by the same argument, we can prove that the conclusion holds.Corollary 5 (Vijayaraju and Marudai [6]).
Let(X,d) be a complete metric space. Let S,T:X→ℱ(X) be two fuzzy mappings. Suppose that, for each x∈X, there exists α(x)∈(0,1] such that [Sx]α(x)and[Tx]α(x) are nonempty closed bounded subsets of X and
(19)H([Sx]α(x),[Ty]α(y))≤a1d(x,[Sx]α(x))+a2d(y,[Ty]α(y))+a3d(x,[Ty]α(y))+a4d(y,[Sx]α(x))+a5d(x,y)
for all x,y∈X, where a1,a2,a3,a4, anda5 are nonnegative real numbers with Σi=15ai<1 and either a1=a2 or a3=a4. Then, there exists z∈X such that z∈[Sz]α(z)∩[Tz]α(z).Proof.
Ifa1=a2,a3>a4, by Σi=15ai<1, we can take 2δ=1-(Σi=15ai)>0. And let A1=a1+δ,A2=a2,A3=a3,A4=a4, andA5=a5+δ; then we have Σi=15Ai=1,A1>A2,A3>A4,A5>0, and for all x,y∈X,
(20)H([Sx]α(x),[Ty]α(y))≤A1d(x,[Sx]α(x))+A2d(y,[Ty]α(y))+A3d(x,[Ty]α(y))+A4d(y,[Sx]α(x))+A5d(x,y),
which implies the conditions of Theorem 4 are satisfied. Similarly, we can prove that some cases of a1=a2,a3<a4 or a1>a2,a3=a4 or a1<a2,a3=a4, respectively. Therefore, by Theorem 4, the corollary is proved.Remark 6.
Corollary5 shows that, although there exist mistakes in the proof of Theorem 3.1 in [6], its conclusion is correct. Moreover, Corollary 5 also shows that Theorem 4 in [7] is not the right version of Theorem 3.1 in [6] but the special case of Theorem 3.1 in [6]. In addition, we give a correct proof of Theorem 3.1 in [6].Corollary 7 (Azam and Arshad [7]).
Let(X,d) be a complete metric space. Let S,T:X→ℱ(X) be two fuzzy mappings. Suppose that, for each x∈X, there exists α(x)∈(0,1] such that [Sx]α(x)and[Tx]α(x) are nonempty closed bounded subsets of X and
(21)H([Sx]α(x),[Ty]α(y))≤a1d(x,[Sx]α(x))+a2d(y,[Ty]α(y))+a3[d(x,[Ty]α(y))+d(y,[Sx]α(x))]+a4d(x,y)
for all x,y∈X, where a1,a2,a3, anda4 are nonnegative real numbers with a1+a2+2a3+a4<1. Then, there exists z∈X such that z∈[Sz]α(z)∩[Tz]α(z).Theorem 8.
Let(X,d) be a complete metric space. Let S,T:X→𝒞ℬ(X) be two generalized Bose-Mukherjee-type multivalued mappings. Suppose that, for all x,y∈X,
(22)H(Sx,Ty)≤A1d(x,Sx)+A2d(y,Ty)+A3d(x,Ty)+A4d(y,Sx)+A5d(x,y),
where A1,A2,A3,A4, andA5 are five nonnegative real numbers with Σi=15Ai=1,A5>0 and either A1>A2,A3>A4 or A1<A2,A3<A4. Then, there exists z∈X such that z∈Sz∩Tz.Proof.
Let the fuzzy mappingsS,T:X→ℱ(X) be defined as S(x)=χS(x) and T(x)=χT(x), where χA is the characteristic function on any subset A of X. Using the facts [Sx]α(x)=S(x)and[Tx]α(x)=T(x) for any α(x)∈(0,1], it is evident that S and T satisfy the condition of Theorem 4.Similarly, as the proof of Corollary5, from Theorem 8, we can obtain that common fixed point theorem for Bose-Mukherjee-type multivalued mappings in [14].Corollary 9 (Bose and Mukherjee [14]).
Let(X,d) be a complete metric space. Let S,T:X→𝒞ℬ(X) be two multivalued mappings. Suppose that, for all x,y∈X,
(23)H(Sx,Ty)≤a1d(x,Sx)+a2d(y,Ty)+a3d(x,Ty)+a4d(y,Sx)+a5d(x,y),
where a1,a2,a3,a4, anda5 are nonnegative real numbers with Σi=15ai<1 and either a1=a2 or a3=a4. Then, there exists z∈X such that z∈Sz∩Tz.Example 10.
LetX={0,1,2}; d is an ordinary metric; then (X,d) is a complete metric space. Define two fuzzy mappings S,T:X→ℱ(X) as follows:
(24)(Sx)(z)={1,ifz=0,0,ifz=1or2,∀x∈X;(T2)(z)={1,ifz=1,0,ifz=0or2,andfory∈X∖{2},(Ty)(z)={1,ifz=0,0,ifz=1or2.
Then, we have
(25)[Sx]1=[Sx]α={0}∀x∈X,α∈(0,1],[Ty]1=[Ty]α={{1},ify=2,{0},ify=0or1,∀α∈(0,1].
Now we take A1=2/45,A2=22/45,A3=1/9,A4=23/90,A5=1/10; then we have Σi=15Ai=1,A5>0 and A1<A2,A3<A4.
Moreover, ifx∈X and y=0or1, then, for all   α∈(0,1],
(26)H([Sx]α,[Ty]α)=0H([Sx]α,[Ty]α)≤245d(x,[Sx]α)+2245d(y,[Ty]α)H([Sx]α,[Ty]α)+19d(x,[Ty]α)+2390d(y,[Sx]α)H([Sx]α,[Ty]α)+110d(x,y).
If x=0 and y=2, then, for all α∈(0,1],
(27)H([S0]α,[T2]α)=1H([S0]α,[T2]α)<245d(0,{0})+2245d(2,{1})H([S0]α,[T2]α)+19d(0,{1})+2390d(2,{0})+110d(0,2).
If x=1 and y=2, then, for all α∈(0,1],
(28)H([S1]α,[T2]α)=1H([S1]α,[T2]α)<245d(1,{0})+2245d(2,{1})H([S1]α,[T2]α)+19d(1,{1})+2390d(2,{0})+110d(1,2).
If x=2 and y=2, then, for all α∈(0,1],
(29)H([S2]α,[T2]α)=1H([S2]α,[T2]α)<245d(2,{0})+2245d(2,{1})H([S2]α,[T2]α)+19d(2,{1})+2390d(2,{0})+110d(2,2).
Hence, the conditions of Theorem 4 are satisfied, and there exists 0∈X such that 0∈{0}=[S0]α∩[T0]α. But, for any nonnegative real numbers a1,a2,a3,a4, anda5 with a1+a2+2a3+a5<1, we have
(30)H([S1]α,[T2]α)=1H([S1]α,[T2]α)>a1d(1,{0})+a2d(2,{1})H([S1]α,[T2]α)+a3[d(1,{1})+d(2,{0})]+a5d(1,2)
for all α∈(0,1]. Thus, S,T cannot satisfy the general contractive condition a1+a2+2a3+a5<1.


## 3. Conclusion

The aim of this work is to establish a common fixed point theorem for a pair of generalized Bose-Mukherjee-type fuzzy mappings in a complete metric space. Also, we give an example to show the validity of our result and by which indicate that our result improves and extends several known results in [6, 7, 14]. Moreover, we give a correct proof of Theorem 3.1 in [6] and point out the conclusion of Theorem 4.1 in [7] is the special case of Theorem 3.1 in [6]. Finally, we hope that this theory would provide a mathematical background to the ongoing work in the problems of scientific and engineering applications.



---

*Source: 935386-2013-07-28.xml*