TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 853032 10.1155/2014/853032 853032 Research Article L -Fuzzy Fixed Points Theorems for L-Fuzzy Mappings via βL-Admissible Pair Rashid Maliha 1 Azam Akbar 2 http://orcid.org/0000-0003-3661-5712 Mehmood Nayyar 2 Bonanno G. Wang J. 1 Department of Mathematics International Islamic University Sector H-10 Islamabad 44000 Pakistan iiu.edu.pk 2 Department of Mathematics COMSATS Institute of Information Technology Chack Shahzad Islamabad 44000 Pakistan ciit.edu.pk 2014 522014 2014 30 08 2013 10 11 2013 5 2 2014 2014 Copyright © 2014 Maliha Rashid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We define the concept of βL-admissible for a pair of L-fuzzy mappings and establish the existence of common L-fuzzy fixed point theorem. Our result generalizes some useful results in the literature. We provide an example to support our result.

1. Introduction

A large variety of the problems of analysis and applied mathematics relate to finding solutions of nonlinear functional equations which can be formulated in terms of finding the fixed points of nonlinear mappings. Heilpern  first introduced the concept of fuzzy mappings and established a fixed point theorem for fuzzy contraction mappings in complete metric linear spaces, which is a fuzzy extension of Banach contraction principle and Nadler’s  fixed point theorem. Subsequently several other authors  generalized this result and studied the existence of fixed points and common fixed points of fuzzy mappings satisfying a contractive type condition.

Zadeh published his important paper “Fuzzy sets” , after that Goguen published the paper “L-Fuzzy sets” . The concept of L-fuzzy sets is a generalization of the concept of fuzzy sets. Fuzzy set is a special case of L-fuzzy set when L=[0,1]. There are basically two understandings of the meaning of L, one is when L is a complete lattice equipped with a multiplication * operator satisfying certain conditions as shown in the initial paper  and the second understanding of the meaning of L is that L is a completely distributive complete lattice with an order-reversing involution (see, e.g., , etc.).

In 2012, Samet et al.  introduced the concept of β-admissible mapping and established fixed point theorems via β-admissible and also showed that these results can be utilized to derive fixed point theorems in partially ordered spaces and coupled fixed point theorems. Moreover, they applied the main results to ordinary differential equations. Afterwards, Asl et al.  extended the concept of β-admissible for single valued mappings to multivalued mappings. Recently, Mohammadi et al.  introduced the concept of β-admissible for multivalued mappings which is different from the notion of β*-admissible which has been provided in  and Azam and Beg  obtained a common α-fuzzy fixed point of a pair of fuzzy mappings on a complete metric space under a generalized contractive condition for α-level sets via Hausdorff metric for fuzzy sets.

In this paper we introduce the concept of βL-admissible for a pair of L-fuzzy mappings and establish the existence of common L-fuzzy fixed point theorem. We also have given an example to support our main theorem.

2. Preliminaries

Let (X,d) be a metric space, and denote

CB(X)={A:A is nonempty closed and bounded subset of X},

C(X)={A:A is nonempty compact subset of X}.

For ϵ>0 and the sets A,BCB(X) define (1)d(x,A)=infyAd(x,y),d(A,B)=infxA,yBd(x,y),N(ϵ,A)={xX:d(x,a)<ϵ,forsomeaA},EA,B={ϵ>0:AN(ϵ,B),BN(ϵ,A)}.

Then the Hausdorff metric dH on CB(X) induced by d is defined as (2)dH(A,B)=infEA,B.

Lemma 1 (see [<xref ref-type="bibr" rid="B15">2</xref>]).

Let (X,d) be a metric space and A,BCB(X); then for each aA(3)d(a,B)H(A,B).

Lemma 2 (see [<xref ref-type="bibr" rid="B15">2</xref>]).

Let (X,d) be a metric space and A,BCB(X); then for each aA, ϵ>0, there exists an element bB such that (4)d(a,b)H(A,B)+ϵ.

Definition 3 (see [<xref ref-type="bibr" rid="B10">19</xref>]).

A partially ordered set (L,L) is called

a lattice, if abL,  abL for any a,bL;

a complete lattice, if AL,  AL for any AL;

distributive if a(bc)=(ab)(ac),  a(bc)=(ab)(ac) for any a,b,cL.

Definition 4 (see [<xref ref-type="bibr" rid="B10">19</xref>]).

Let L be a lattice with top element 1L and bottom element 0L and let a,bL. Then b is called a complement of a, if ab=1L, and ab=0L. If aL has a complement element, then it is unique. It is denoted by a´.

Definition 5 (see [<xref ref-type="bibr" rid="B10">19</xref>]).

An L-fuzzy set A on a nonempty set X is a function A:XL, where L is complete distributive lattice with 1L and 0L.

Remark 6.

The class of L-fuzzy sets is larger than the class of fuzzy sets as an L-fuzzy set is a fuzzy set if L=[0,1].

The αL-level set of L-fuzzy set A is denoted by AαL and is defined as follows: (5)AαL={x:αLLA(x)}ifαLL{0L},A0L=cl({x:0LLA(x)}).

Here cl(B) denotes the closure of the set B.

We denote and define the characteristic function χLA of an L-fuzzy set A as follows: (6)χLA:={0LifxA,1LifxA.

Definition 7.

Let X be an arbitrary set and Y a metric space. A mapping T is called L-fuzzy mapping if T is a mapping from X into L(Y). An L-fuzzy mapping T is an L-fuzzy subset on X×Y with membership function T(x)(y). The function T(x)(y) is the grade of membership of y in T(x).

Definition 8.

Let (X,d) be a metric space and S,TL-fuzzy mappings from X into L(X). A point zX is called an L-fuzzy fixed point of T if z[Tz]αL, where αLL{0L}. The point zX is called a common L-fuzzy fixed point of S and T if z[Sz]αL[Tz]αL.

Definition 9 (see [<xref ref-type="bibr" rid="B19">23</xref>]).

Let X be a nonempty set, T:XX, and β:X×X[0,). We say that T is β-admissible if for all x,yX we have (7)β(x,y)1β(Tx,Ty)1.

Definition 10 (see [<xref ref-type="bibr" rid="B1">24</xref>]).

Let X be a nonempty set, T:X2X, where 2X is a collection of subset of X, β:X×X[0,) and β*:2X×2X[0,). We say that T is β*-admissible if for all x,yX we have (8)β(x,y)1β*(Tx,Ty)1.

Definition 11 (see [<xref ref-type="bibr" rid="B14">25</xref>]).

Let X be a nonempty set, T:X2X, where 2X is a collection of subset of X and β:X×X[0,). We say that T is β-admissible whenever for each xX and yTx with β(x,y)1 we have β(y,z)1 for all zTy.

3. Main Result

In this section, we introduce a new concept of βL-admissible for a pair of L-fuzzy mappings and establish the existence of common L-fuzzy fixed point theorem.

Definition 12.

Let (X,d) be a metric space, β:X×X[0,), and S,TL-fuzzy mappings from X into L(X). The order pair (S,T) is said to be βL-admissible if it satisfies the following conditions:

for each xX and y[Sx]αL(x), where αL(x)L{0L}, with β(x,y)1, we have β(y,z)1 for all z[Ty]αL(y)ϕ, where αL(y)L{0L};

for each xX and y[Tx]αL(x), where αL(x)L{0L}, with β(x,y)1, we have β(y,z)1 for all z[Sy]αL(y)ϕ, where αL(y)L{0L}.

If S=T then T is called βL-admissible.

Remark 13.

It is easy to see that if (S,T) is βL-admissible, then (T,S) is also βL-admissible.

Next, we give a common L-fuzzy fixed point theorem for βF-admissible pair.

Theorem 14.

Let (X,d) be a complete metric space, β:X×X[0,), and S,TL-fuzzy mappings from X into L(X) satisfying the following conditions.

For each xX, there exists αL(x)L{0L} such that [Sx]αL(x), [Tx]αL(x) are nonempty closed bounded subsets of X and for x0X, there exists x1[Sx0]αL(x0) with β(x0,x1)1.

For all x,yX, we have (9)max{β(x,y),β(y,x)}[H([Sx]αL(x),[Ty]αL(y))]a1(dx,[Sx]αL(x))+a2(dy,[Ty]αL(y))+a3(dx,[Ty]αL(y))+a4(dy,[Sx]αL(x))+a5d(x,y),

where a1, a2, a3, a4, and a5 are nonnegative real numbers and i=15ai<1 and either a1=a2 or a3=a4.

(S,T) is βL-admissible pair.

If {xn}X, such that β(xn,xn+1)1 and xnx, then β(xn,x)1.

Then there exists zX such that z[Sz]αL(z)[Tz]αL(z).

Proof.

We will prove the above result by considering the following three cases:

a1+a3+a5=0,

a2+a4+a5=0,

a1+a3+a50 and a2+a4+a50.

Case 1. For x0X in condition (a), there exist αL(x0)L{0L} and x1[Sx0]αL(x0) such that β(x0,x1)1 and also there exists αL(x1)L{0L} such that [Sx0]αL(x0) and [Tx1]αL(x1) are nonempty closed bounded subsets of X. From Lemma 1, we obtain that (10)d(x1,[Tx1]αL(x1))H([Sx0]αL(x0),[Tx1]αL(x1))β(x0,x1)[H([Sx0]αL(x0),[Tx1]αL(x1))]max{β(x0,x1),β(x1,x0)}×[H([Sx0]αL(x0),[Tx1]αL(x1))].

Now, inequality (9) implies that (11)d(x1,[Tx1]αL(x1))a1d(x0,[Sx0]αL(x0))+a2d(x1,[Tx1]αL(x1))+a3d(x0,[Tx1]αL(x1))+a4d(x1,[Sx0]αL(x0))+a5d(x0,x1).

Using a1+a3+a5=0 together with the fact that d(x1,[Sx0]αL(x0))=0, we get (12)d(x1,[Tx1]αL(x1))a2d(x1,[Tx1]αL(x1)).

It follows that x1[Tx1]αL(x1), which further implies that (13)d(x1,[Sx1]αL(x1))H([Tx1]αL(x1),[Sx1]αL(x1)).

By condition (c), for x0X and x1[Sx0]αL(x0) such that β(x0,x1)1, we have β(x1,z)1 for all z[Tx1]αL(x1). Since x1[Tx1]αL(x1), therefore β(x1,x1)1 and hence (14)d(x1,[Sx1]αL(x1))β(x1,x1)[H([Sx1]αL(x1),[Tx1]αL(x1))].

Again, inequality (9) implies that (15)d(x1,[Sx1]αL(x1))a1d(x1,[Sx1]αL(x1))+a2d(x1,[Tx1]αL(x1))+a3d(x1,[Tx1]αL(x1))+a4d(x1,[Sx1]αL(x1))+a5d(x1,x1).

Since a1+a3+a5=0 and d(x1,[Tx1]αL(x1))=0, we get (16)d(x1,[Sx1]αL(x1))a4d(x1,[Sx1]αL(x1)), which implies that x1[Sx1]αL(x1) and hence (17)x1[Sx1]αL(x1)[Tx1]αL(x1).

Case 2. For x0X in condition (a), there exist αL(x0)L{0L} and x1[Sx0]αL(x0) such that β(x0,x1)1 and also there exists αL(x1)L{0L} such that [Sx0]αL(x0) and [Tx1]αL(x1) are nonempty closed bounded subsets of X. By condition (c), we have β(x1,x2)1 for all x2[Tx1]αL(x1). From Lemma 1, we obtain that (18)d(x2,[Sx2]αL(x2))H([Tx1]αL(x1),[Sx2]αL(x2))β(x1,x2)[H([Sx2]αL(x2),[Tx1]αL(x1))]max{β(x1,x2),β(x2,x1)}×[H([Sx2]αL(x2),[Tx1]αL(x1))]a1d(x2,[Sx2]αL(x2))+a2d(x1,[Tx1]αL(x1))+a3d(x2,[Tx1]αL(x1))+a4d(x1,[Sx2]αL(x2))+a5d(x2,x1).

Using a2+a4+a5=0 together with the fact that d(x2,[Tx1]αL(x1))=0, we get (19)d(x2,[Sx2]αL(x2))a1d(x2,[Sx2]αL(x2)).

It follows that x2[Sx2]αL(x2), which further implies that (20)d(x2,[Tx2]αL(x2))H([Sx2]αL(x2),[Tx2]αL(x2)).

By condition (c), we have β(x2,x2)1, and hence (21)d(x2,[Tx2]αL(x2))β(x2,x2)[H([Sx2]αL(x2),[Tx2]αL(x2))].

Again, inequality (9) implies that (22)d(x2,[Tx2]αL(x2))a1d(x2,[Sx2]αL(x2))+a2d(x2,[Tx2]αL(x2))+a3d(x2,[Tx2]αL(x2))+a4d(x2,[Sx2]αL(x2))+a5d(x2,x2).

Since a2+a4+a5=0 and d(x2,[Sx2]αL(x2))=0, we get (23)d(x2,[Tx2]αL(x2))a3d(x2,[Tx2]αL(x2)), which implies that x2[Tx2]αL(x2) and hence (24)x2[Sx2]αL(x2)[Tx2]αL(x2).

Case 3. Let λ=((a1+a3+a5)/(1-a2-a3)) and μ=((a2+a4+a5)/(1-a1-a4)). Next, we show that if a1=a2 or a3=a4, then 0<λμ<1.

If a3=a4, then λ,μ<1 and so 0<λμ<1. Now if a1=a2, then (25)0<λμ=(a1+a3+a51-a2-a3)(a2+a4+a51-a1-a4)=(a1+a3+a51-a1-a3)(a1+a4+a51-a1-a4)=(a1+a3+a51-a1-a4)(a1+a4+a51-a1-a3)<1. By condition (a), for x1X, there exists αL(x1)L{0L} such that [Tx1]αL(x1) is a nonempty closed bounded subset of X. Since a1+a3+a5>0, by Lemma 2, there exists x2[Tx1]αL(x1) such that (26)d(x1,x2)H([Sx0]αL(x0),[Tx1]αL(x1))+a1+a3+a5β(x0,x1)[H([Sx0]αL(x0),[Tx1]αL(x1))]+a1+a3+a5max{β(x0,x1),β(x1,x0)}×[H([Sx0]αL(x0),[Tx1]αL(x1))]+a1+a3+a5a1d(x0,[Sx0]αL(x0))+a2d(x1,[Tx1]αL(x1))+a3d(x0,[Tx1]αL(x1))+a4d(x1,[Sx0]αL(x0))+a5d(x0,x1)+a1+a3+a5(a1+a5)d(x0,x1)+a2d(x1,x2)+a3d(x0,x2)+a1+a3+a5(a1+a3+a5)d(x0,x1)+(a2+a3)d(x1,x2)+a1+a3+a5. This implies that (27)d(x1,x2)λd(x0,x1)+λ. By the same argument, for x2X, there exists αL(x2)L{0L} such that [Sx2]αL(x2) is a nonempty closed bounded subset of X. Since a2+a4+a5>0, by Lemma 2, there exists x3[Sx2]αL(x2) such that (28)d(x2,x3)H([Tx1]αL(x1),[Sx2]αL(x2))+λ(a2+a4+a5). By condition (c), for x0X and x1[Sx0]αL(x0) such that β(x0,x1)1, we have β(x1,x2)1 for x2[Tx1]αL(x1). So we have (29)d(x2,x3)H([Tx1]αL(x1),[Sx2]αL(x2))+λ(a2+a4+a5)=H([Sx2]αL(x2),[Tx1]αL(x1))+λ(a2+a4+a5)β(x1,x2)[H([Sx2]αL(x2),[Tx1]αL(x1))]+λ(a2+a4+a5)max{β(x1,x2),β(x2,x1)}×[H([Sx2]αL(x2),[Tx1]αL(x1))]+λ(a2+a4+a5)a1d(x2,[Sx2]αL(x2))+a2d(x1,[Tx1]αL(x1))+a3d(x2,[Tx1]αL(x1))+a4d(x1,[Sx2]αL(x2))+a5d(x2,x1)+λ(a2+a4+a5)a1d(x2,x3)+(a2+a5)d(x1,x2)+a4d(x1,x3)+λ(a2+a4+a5)(a2+a4+a5)d(x1,x2)+(a1+a4)d(x2,x3)+λ(a2+a4+a5). This implies that (30)d(x2,x3)μd(x1,x2)+λμ. By repeating the above process, for x3X, there exists αL(x3)L{0L} such that [Tx3]αL(x3) is a nonempty closed bounded subset of X. From Lemma 2, there exists x4[Tx3]αL(x3) such that (31)d(x3,x4)H([Sx2]αL(x2),[Tx3]αL(x3))+λμ(a1+a3+a5). By condition (c), for x1X and x2[Tx1]αL(x1) such that β(x1,x2)1, we have β(x2,x3)1 for x3[Sx2]αL(x2). So we have (32)d(x3,x4)H([Sx2]αL(x2),[Tx3]αL(x3))+λμ(a1+a3+a5)β(x2,x3)[H([Sx2]αL(x2),[Tx3]αL(x3))]+λμ(a1+a3+a5)max{β(x2,x3),β(x3,x2)}×[H([Sx2]αL(x2),[Tx3]αL(x3))]+λμ(a1+a3+a5)a1d(x2,[Sx2]αL(x2))+a2d(x3,[Tx3]αL(x3))+a3d(x2,[Tx3]αL(x3))+a4d(x3,[Sx2]αL(x2))+a5d(x2,x3)+λμ(a1+a3+a5)(a1+a5)d(x2,x3)+a2d(x3,x4)+a3d(x2,x4)+λμ(a1+a3+a5)(a1+a3+a5)d(x2,x3)+(a2+a3)d(x3,x4)+λμ(a1+a3+a5). This implies that (33)d(x3,x4)λd(x2,x3)+λ(λμ). By induction, we produce a sequence {xn} in X such that (34)x2k+1[Sx2k]αL(x2k),x2k+2[Tx2k+1]αL(x2k+1),k=0,1,2,,β(xn-1,xn)1,n. Now, we have (35)d(x2k+1,x2k+2)H([Sx2k]αL(x2k),[Tx2k+1]αL(x2k+1))+(λμ)k(a1+a3+a5)β(x2k,x2k+1)×[H([Sx2k]αL(x2k),[Tx2k+1]αL(x2k+1))]+(λμ)k(a1+a3+a5)max{β(x2k,x2k+1),β(x2k+1,x2k)}×[H([Sx2k]αL(x2k),[Tx2k+1]αL(x2k+1))]+(λμ)k(a1+a3+a5)a1d(x2k,[Sx2k]αL(x2k))+a2d(x2k+1,[Tx2k+1]αL(x2k+1))+a3d(x2k,[Tx2k+1]αL(x2k+1))+a4d(x2k+1,[Sx2k]αL(x2k))+a5d(x2k,x2k+1)+(λμ)k(a1+a3+a5)(a1+a5)d(x2k,x2k+1)+a2d(x2k+1,x2k+2)+a3d(x2k,x2k+2)+(λμ)k(a1+a3+a5)(a1+a3+a5)d(x2k,x2k+1)+(a2+a3)d(x2k+1,x2k+2)+(λμ)k(a1+a3+a5). This implies that (36)d(x2k+1,x2k+2)λd(x2k,x2k+1)+λ(λμ)k. Similarly, (37)d(x2k+2,x2k+3)H([Sx2k+2]αL(x2k+2),[Tx2k+1]αL(x2k+1))+(λμ)kλ(a2+a4+a5)β(x2k+1,x2k+2)×[H([Sx2k+2]αL(x2k+2),[Tx2k+1]αL(x2k+1))]+(λμ)kλ(a2+a4+a5)max{β(x2k+1,x2k+2),β(x2k+2,x2k+1)}×[H([Sx2k+2]αL(x2k+2),[Tx2k+1]αL(x2k+1))]+(λμ)kλ(a2+a4+a5)a1d(x2k+2,[Sx2k+2]αL(x2k+2))+a2d(x2k+1,[Tx2k+1]αL(x2k+1))+a3d(x2k+2,[Tx2k+1]αL(x2k+1))+a4d(x2k+1,[Sx2k+2]αL(x2k+2))+a5d(x2k+2,x2k+1)+(λμ)kλ(a2+a4+a5)a1d(x2k+2,x2k+3)+(a2+a5)d(x2k+1,x2k+2)+a4d(x2k+1,x2k+3)+(λμ)kλ(a2+a4+a5)(a2+a4+a5)d(x2k+1,x2k+2)+(a1+a4)d(x2k+2,x2k+3)+(λμ)kλ(a2+a4+a5). This implies that (38)d(x2k+2,x2k+3)μd(x2k+1,x2k+2)+(λμ)k+1. From (36) and (38), it follows that, for each k=0,1,2,, (39)d(x2k+1,x2k+2)λd(x2k,x2k+1)+λ(λμ)kλ[μd(x2k-1,x2k)+(λμ)k]+λ(λμ)k=(λμ)d(x2k-1,x2k)+2λ(λμ)k(λμ)[λd(x2k-2,x2k-1)+λ(λμ)k-1]+2λ(λμ)k=(λμ)λd(x2k-2,x2k-1)+3λ(λμ)kλ(λμ)kd(x0,x1)+(2k+1)λ(λμ)k,d(x2k+2,x2k+3)μd(x2k+1,x2k+2)+(λμ)k+1(λμ)k+1d(x0,x1)+(2k+2)(λμ)k+1. Then for m<n, we have (40)d(x2m+1,x2n+1)d(x2m+1,x2m+2)+d(x2m+2,x2m+3)+d(x2m+3,x2m+4)++d(x2n,x2n+1)[λi=mn-1(λμ)i+i=m+1n(λμ)i]d(x0,x1)+λi=mn-1(2i+1)(λμ)i+i=m+1n2i(λμ)i. Similarly, we obtain that (41)d(x2m,x2n+1)[i=mn(λμ)i+λi=mn-1(λμ)i]d(x0,x1)+i=mn2i(λμ)i+λi=mn-1(2i+1)(λμ)i,d(x2m,x2n)[i=mn-1(λμ)i+λi=mn-1(λμ)i]d(x0,x1)+i=mn-12i(λμ)i+λi=mn-1(2i+1)(λμ)i,d(x2m+1,x2n)[λi=mn-1(λμ)i+i=m+1n-1(λμ)i]d(x0,x1)+λi=mn-1(2i+1)(λμ)i+i=m+1n-12i(λμ)i. Since 0<λμ<1, so by Cauchy’s root test, we get (2i+1)(λμ)i and 2i(λμ)i are convergent series. Therefore, {xn} is a Cauchy sequence in X. Now, from the completeness of X, there exists zX such that xnz as n. By condition (d), we have β(xn-1,z)1 for all n. Now, we have (42)d(x2n,[Sz]α(z))H([Tx2n-1]αL(x2n-1),[Sz]α(z))=H([Sz]α(z),[Tx2n-1]αL(x2n-1))β(x2n-1,z)H([Sz]α(z),[Tx2n-1]αL(x2n-1))max{β(x2n-1,z),β(z,x2n-1)}×H([Sz]α(z),[Tx2n-1]αL(x2n-1))a1d(z,[Sz]αL(z))+a2d(x2n-1,[Tx2n-1]αL(x2n-1))+a3d(z,[Tx2n-1]αL(x2n-1))+a4d(x2n-1,[Sz]αL(z))+a5d(z,x2n-1)a1d(z,[Sz]αL(z))+a2d(x2n-1,x2n)+a3d(z,x2n)+a4d(x2n-1,[Sz]αL(z))+a5d(z,x2n-1). Since (43)d(z,[Sz]αL(z))d(z,x2n)+d(x2n,[Sz]αL(z)), so we get (44)d(z,[Sz]αL(z))d(z,x2n)+a1d(z,[Sz]αL(z))+a2d(x2n-1,x2n)+a3d(z,x2n)+a4d(x2n-1,[Sz]αL(z))+a5d(z,x2n-1)(1+a3)d(z,x2n)+(a4+a5)d(z,x2n-1)+a2d(x2n-1,x2n)+(a1+a4)d(z,[Sz]αL(z)). This implies that (45)d(z,[Sz]αL(z))(1+a31-a1-a4)d(z,x2n)+(a4+a51-a1-a4)d(z,x2n-1)+(a21-a1-a4)d(x2n-1,x2n). Letting n, we have d(z,[Sz]αL(z))=0. It implies that z[Sz]αL(z). Similarly, by using (46)d(z,[Tz]αL(z))d(z,x2n+1)+d(x2n+1,[Tz]αL(z)), we can show that z[Tz]αL(z). Therefore, z[Sz]αL(z)[Tz]αL(z). This completes the proof.

Next, we give an example to support the validity of our result.

Example 15.

Let X=[0,1], d(x,y)=|x-y|, whenever x,yX; then (X,d) is a complete metric space. Let L={δ,ω,τ,κ} with δLωLκ, δLτLκ, ω and τ are not comparable; then (L,L) is a complete distributive lattice. Define a pair of mappings S,T:XL(X)    as follows: (47)S(x)(t)={κ,if0tx6,ω,ifx6<tx3,τ,ifx3<tx2,δ,ifx2<t1,T(x)(t)={κ,if0tx12,δ,ifx12<tx8,ω,ifx8<tx4,τ,ifx4<t1. Define β:X×X[0,) as follows: (48)β(x,y)={1|x-y|,xy,1,x=y. For all xX, there exists αL(x)=κ, such that (49)[Sx]κ=[0,x6],[Tx]κ=[0,x12], and all conditions of the above theorem are satisfied. Hence, there exists 0X, such that 0[S0]αL(0)[T0]αL(0).

Corollary 16.

Let (X,d) be a complete metric space, β:X×X[0,), and S, T fuzzy mappings from X into (X) satisfying the following conditions.

For each xX, there exists α(x)(0,1] such that [Sx]α(x), [Tx]α(x) are nonempty closed bounded subsets of X and for x0X, there exists x1[Sx0]α(x0) with β(x0,x1)1.

For all x,yX, we have (50)max{β(x,y),β(y,x)}[H([Sx]α(x),[Ty]α(y))]a1(dx,[Sx]α(x))+a2(dy,[Ty]α(y))+a3(dx,[Ty]α(y))+a4(dy,[Sx]α(x))+a5d(x,y),

where a1, a2, a3, a4, and a5 are nonnegative real numbers and i=15ai<1 and either a1=a2 or a3=a4.

(S,T) is β-admissible pair.

If {xn}X, such that β(xn,xn+1)1 and xnx then β(xn,x)1.

Then there exists zX such that z[Sz]α(z)[Tz]α(z).

Proof.

Consider an L-fuzzy mapping A:XL(X) defined by (51)Ax=χLTx. Then for αLL{0L}, we have (52)[Ax]αL=Tx. Hence by Theorem 14, we follow the result.

If we set β(x,y)=1 for all x,yX in Corollary 16, we get the following result.

Corollary 17 (see [<xref ref-type="bibr" rid="B7">26</xref>]).

Let (X,d) be a complete metric space and S, T fuzzy mappings from X into (X) satisfying the following conditions:

for each xX, there exists α(x)(0,1] such that [Sx]α(x), [Tx]α(x) are nonempty closed bounded subsets of X;

for all x,yX, we have (53)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),

where a1, a2, a3, a4, and a5 are nonnegative real numbers and i=15ai<1 and either a1=a2 or a3=a4.

Then there exists zX such that z[Sz]α(z)[Tz]α(z).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Heilpern S. Fuzzy mappings and fixed point theorem Journal of Mathematical Analysis and Applications 1981 83 2 566 569 2-s2.0-0000388180 Nadler B. Multivalued contraction mappings Pacific Journal of Mathematics 1969 30 475 488 Azam A. Waseem M. Rashid M. Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces Fixed Point Theory and Applications 2013 2013, article 27 10.1186/1687-1812-2013-27 Azam A. Arshad M. Beg I. Fixed points of fuzzy contractive and fuzzy locally contractive maps Chaos, Solitons and Fractals 2009 42 5 2836 2841 2-s2.0-67651202358 10.1016/j.chaos.2009.04.026 Azam A. Arshad M. A note on “Fixed point theorems for fuzzy mappings” by P. Vijayaraju and M. Marudai Fuzzy Sets and Systems 2010 161 8 1145 1149 2-s2.0-76749135570 10.1016/j.fss.2009.10.016 Azam A. Arshad M. Vetro P. On a pair of fuzzy φ-contractive mappings Mathematical and Computer Modelling 2010 52 1-2 207 214 2-s2.0-77953130294 10.1016/j.mcm.2010.02.010 Azam A. Beg I. Common fixed points of fuzzy maps Mathematical and Computer Modelling 2009 49 7-8 1331 1336 2-s2.0-60949086291 10.1016/j.mcm.2008.11.011 Bose R. K. Sahani D. Fuzzy mappings and fixed point theorems Fuzzy Sets and Systems 1987 21 1 53 58 2-s2.0-38249038185 Román-Flores H. Flores-Franulic A. Rojas-Medar M. Bassanezi R. C. Stability of fixed points set of fuzzy contractions Applied Mathematics Letters 1998 11 4 33 37 2-s2.0-0039049319 Soo Lee B. Jin Cho S. A fixed point theorem for contractive-type fuzzy mappings Fuzzy Sets and Systems 1994 61 3 309 312 2-s2.0-0039403857 Lee B. S. Lee G. M. Cho S. J. Kim D. S. Generalized common fixed point theorems for a sequence of fuzzy mappings International Journal of Mathematics and Mathematical Sciences 1994 17 3 437 440 Park J. Y. Jeong J. U. Fixed point theorems for fuzzy mappings Fuzzy Sets and Systems 1997 87 1 111 116 2-s2.0-0031124416 Rashwan R. A. Ahmad M. A. Common fixed point theorems for fuzzy mappings Archivum Mathematicum 2002 38 219 226 Rhoades B. E. A common fixed point theorem for sequence of fuzzy mappings International Journal of Mathematics and Mathematical Sciences 1995 8 447 450 Som T. Mukherjee R. N. Some fixed point theorems for fuzzy mappings Fuzzy Sets and Systems 1989 33 2 213 219 2-s2.0-38249004629 Vijayaraju P. Marudai M. Fixed point theorems for fuzzy mappings Fuzzy Sets and Systems 2003 135 3 401 408 2-s2.0-0037401499 10.1016/S0165-0114(02)00367-6 Vijayaraju P. Mohanraj R. Fixed point theorems for sequence of fuzzy mappings Southeast Asian Bulletin of Mathematics 2004 28 735 740 Zadeh L. A. Fuzzy sets Information and Control 1965 8 3 338 353 2-s2.0-34248666540 Goguen J. A. L-fuzzy sets Journal of Mathematical Analysis and Applications 1967 18 145 174 Wang G. J. Theory of L-Fuzzy Topological Spaces 1988 Xi'an, China Shaanxi Normal University Press (Chinese) Wang G.-J. He Y.-Y. Intuitionistic Fuzzy Sets and Systems 2000 110 2 271 274 2-s2.0-0242521093 Zhao D. S. The N-compactness in L-fuzzy topological spaces Journal of Mathematical Analysis and Applications 1987 128 1 64 79 2-s2.0-45949113218 Samet B. Vetro C. Vetro P. Fixed point theorems for α-ψ-contractive type mappings Nonlinear Analysis, Theory, Methods and Applications 2012 75 4 2154 2165 2-s2.0-84655168090 10.1016/j.na.2011.10.014 Asl J. H. Rezapour S. Shahzad N. On fixed points of α-ψ-contractive multifunctions Fixed Point Theory and Applications 2012 2012, article 212 Mohammadi B. Rezapour S. Shahzad N. Some results on fixed points of α-ψ-Ciric generalized multifunctions Fixed Point Theory and Applications 2013 2013, article 24 Azam A. Beg I. Common fuzzy fixed points for fuzzy mappings Fixed Point Theory and Applications 2013 2013, article 14