R E S E A R C H
Open Access
Suzuki-type fixed point theorem for fuzzy
mappings in ordered metric spaces
Basit Ali
1*and M Abbas
1,2*Correspondence:
[email protected] 1Department of Mathematics,
School of Science and Engineering, Lahore University of Management Sciences, Lahore, 54792, Pakistan Full list of author information is available at the end of the article
Abstract
In this paper, a Suzuki-type fixed fuzzy point result for fuzzy mappings in complete ordered metric spaces is obtained. As an application, we establish the existence of coincidence fuzzy points and common fixed fuzzy points for a hybrid pair of a single-valued self-mapping and a fuzzy mapping. An example is also provided to support the main result presented herein.
MSC: 47H10; 47H04; 47H07
Keywords: fixed fuzzy point; fuzzy mapping; fuzzy set; approximate quantity
1 Introduction and preliminaries
LetXbe a space of points with generic elements ofXdenoted byxandI= [, ]. A fuzzy subset ofX is characterized by a membership function such that each element inXis associated with a real number in the intervalI. Let (X,d) be a metric space and a fuzzy setAinXis characterized by a membership functionA. Thenα-level set ofA, denoted byAα, is defined as
Aα=x:A(x)≥α
forα∈(, ] and forα= , we have
A=
x:A(x) > ,
whereBdenotes the closure of the non-fuzzy setB. A fuzzy setAinXis said to be an approximate quantity if and only if forα∈[, ],Aαis a compact, convex subset ofXand
sup x∈X
A(x) = .
LetW(X) be a family of all approximate quantities inX. A fuzzy setAis said to be more accurate than a fuzzy setBdenoted byA⊂B(that is,BincludesA) if and only ifA(x)≤ B(x) for eachxinX, whereA(x) andB(x) denote the membership function ofAandB, respectively. It is easy to see that if <α≤β≤, thenAα⊆Aβ.
Corresponding to eachα ∈[, ] andx∈X, the fuzzy point xα of Xis the fuzzy set xα:X→[, ] given by
xα(y) =
⎧ ⎨ ⎩
α ifx=y,
otherwise.
Forα= , we have
x(y) =
⎧ ⎨ ⎩
ifx=y,
otherwise={x}.
LetIXbe a collection of all fuzzy subsets ofXandW(X) be a subcollection of all approx-imate quantities. ForA,B∈W(X) andα∈[, ], define
pα(A,B) =infd(x,y),x∈Aα,y∈Bα,
Dα(A,B) =max
sup x∈Aα
d(x,Bα),sup y∈Bα
d(y,Aα)
and
D(A,B) =sup α
Dα(A,B).
Note thatpαis a nondecreasing function ofαandDis a metric onW(X). Letα∈[, ]. De-fineWα(X) ={A∈IX:Aαis nonempty, convex and compact}. Let (X,d) be a metric space andY be an arbitrary set. A mappingF:Y→Wα(X) is called a fuzzy mapping, that is, Fy∈Wα(X) for eachyinY. Thus, if we characterize a fuzzy setFyin a metric spaceXby a membership functionFy, thenFy(x) is the grade of membership ofxinFy. Therefore, a fuzzy mappingFis a fuzzy subset ofY×Xwith a membership functionFy(x).
In a more general sense than that given in [], a mappingF:X→IXis a fuzzy mapping overX[] and (F(x)x) is the fixed degree ofxinF(x).
Definition ([]) A fuzzy pointxαinXis called a fixed fuzzy point of the fuzzy mapping
Fifxα⊂Fx, that is, (Fx)x≥αorx∈(Fx)α. That is, the fixed degree ofxinFxis at leastα. If{x} ⊂Fx, thenxis a fixed point of a fuzzy mappingF.
LetF:X→Wα(X) andg:X→X.
A fuzzy point xα in Xis called a coincidence fuzzy point of the hybrid pair {F,g} if (gx)α ⊂Fx, that is, (Fx)gx≥α orgx∈(Fx)α. That is, the fixed degree ofgxin Fxis at leastα. A fuzzy pointxαinXis called a common fixed fuzzy point of the hybrid pair{F,g} ifxα= (gx)α⊂Fx, that is,x=gx∈(Fx)α(the fixed degree ofxandgxinFxis the same and is at leastα).
We denote byCα(F,g) andFα(F,g) the set of all coincidence fuzzy points and the set of all common fixed fuzzy points of the hybrid pair{F,g}, respectively.
Lemma ([]) Let X be a nonempty set and g:X→X.Then there exists a subset E⊆X such that g(E) =g(X)and g:E→X is one-to-one.
Definition LetXbe a nonempty set. Then (X,d,) is called an ordered metric space if (X,d) is a metric space and (X,) is partially ordered.
Let (X,) be a partially ordered set. Thenx,y∈Xare said to be comparable ifxyor yxholds.
Define
∇=(x,y)∈X×X:xyoryx.
An ordered metric space is said to satisfy the order sequential limit property if (un,z)∈ ∇ for alln, whenever a sequenceun→zand (un,un+)∈ ∇for alln.
A mappingF:X→Wα(X) is said to be an ordered fuzzy mapping if the following con-ditions are satisfied:
(a) y∈F(x)αimplies that(y,x)∈ ∇.
(b) (x,y)∈ ∇implies that(u,v)∈ ∇wheneveru∈(Fx)αandv∈(Fy)α.
The following lemmas are needed in the sequel.
Lemma (Heilpern []) Let(X,d)be a metric space,x,y∈X and A,B∈W(X):
. ifpα(x,A) = ,thenxα⊂A; . pα(x,A)≤d(x,y) +pα(y,A); . ifxα⊂A,thenpα(x,B)≤Dα(A,B).
Lemma (Lee and Cho []) Let(X,d)be a complete metric space and F be a fuzzy mapping from X into W(X)and x∈X.Then there exists an x∈X such that{x} ⊂Fx.
Zadeh [] introduced the concept of a fuzzy set. Heilpern [] introduced the concept of fuzzy mappings in a metric space and proved a fixed point theorem for fuzzy contraction mappings as a generalization of the fixed point theorem for multivalued mappings given by Nadler []. Estruch and Vidal [] proved a fixed point theorem for fuzzy contraction mappings in complete metric spaces which in turn generalizes the Heilpern fixed point theorem. Further generalizations of the result given in [] were proved in [, ]. Recently, Suzuki [] generalized the Banach contraction principle and characterized the metric completeness property of an underlying space. Among many generalizations (see [–]) of the results given in [], Dorić and Lazović [] obtained Suzuki-type fixed point results for a generalized multivalued contraction in complete metric spaces.
On the other hand, the existence of fixed points in ordered metric spaces has been in-troduced and applied by Ran and Reurings []. Fixed point theorems in partially ordered metric spaces are hybrid of two fundamental principles: Banach contraction theorem with a contractive condition for comparable elements and a selection of an initial point to gen-erate a monotone sequence. For results concerning fixed points and common fixed points in partially ordered metrics spaces, we refer to [–].
common fixed fuzzy point of the hybrid pair of a single-valued self-mapping and a fuzzy mapping are obtained. We provide an example to support the result.
Throughout this paper, letσ: [, )→(, ] be the nonincreasing function defined by
σ(r) =
⎧ ⎨ ⎩
if ≤r<, –r if
≤r< .
()
2 Main results
The following theorem is the main result of the paper and is a generalization of [, The-orem .] for fuzzy mappings in ordered metric spaces.
Theorem Let(X,d,)be a complete ordered metric space.If an ordered fuzzy mapping F:X→Wα(X)satisfies
σ(r)pα(x,Fx)≤d(x,y) implies Dα(Fx,Fy)≤rMα(F) ()
for all(x,y)∈ ∇,where
Mα(F) =max d(x,y),pα(x,Fx),pα(y,Fy),pα(x,Fy) +pα(y,Fx)
.
Then there exists a point x∈X such that xα⊂Fx provided that X satisfies the order se-quential limit property.
Proof Letrbe a real number such that ≤r<r< andu∈X. Since (Fu)αis nonempty
and compact, there existsu∈(Fu)αsuch that
d(u,u) =pα(u,Fu).
By the given assumption, we have (u,u)∈ ∇. Since (Fu)α is nonempty and compact,
there existsu∈(Fu)αsuch that
d(u,u) =pα(u,Fu)≤Dα(Fu,Fu).
Also, (u,u)∈ ∇. Sinceσ(r) < , we obtain
σ(r)ρα(u,Fu)≤pα(u,Fu) =d(u,u).
That is,
σ(r)pα(u,Fu)≤d(u,u).
So, we have
d(u,u)≤Dα(Fu,Fu)
≤rmax d(u,u),pα(u,Fu),pα(u,Fu),
pα(u,Fu) +pα(u,Fu)
≤rmax d(u,u),d(u,u),d(u,u),
d(u,u) +pα(u,Fu)
≤rmax d(u,u),d(u,u),
d(u,u) +d(u,u)
.
Note thatd(u,u)≤d(u,u). If not, then the above inequality gives
d(u,u)≤rmax d(u,u),d(u,u),
d(u,u) +d(u,u)
=rd(u,u) <d(u,u) asr< ,
a contradiction. Hence,d(u,u)≤rd(u,u). Continuing this process, we construct a
sequence{un}inXsuch thatun+∈(Fun)αandun+∈(Fun+)αwith
d(un+,un+) =pα(un+,Fun+)≤Dα(Fun,Fun+).
By the given assumption, we have (un,un+)∈ ∇and (un+,un+)∈ ∇. Asσ(r) < , so
σ(r)pα(un,Fun)≤pα(un,Fun) =d(un,un+).
Therefore,
d(un+,un+)≤Dα(Fun,Fun+)
≤rmax d(un,un+),pα(un,Fun),pα(un+,Fun+),
pα(un,Fun+) +pα(un+,Fun)
≤rmax d(un,un+),d(un,un+),d(un+,un+),
d(un,un+) +pα(un+,Fun+)
≤rmax d(un,un+),d(un+,un+),
d(un,un+) +d(un+,un+)
.
We claim thatd(un+,un+)≤d(un,un+). If not, then by the above inequality, we obtain
d(un+,un+)≤rmax d(un+,un+),d(un+,un+),
d(un+,un+) +d(un+,un+)
≤rd(un+,un+) <d(un+,un+),
a contradiction asr< . So, we have
d(un+,un+)≤rd(un,un+)≤ · · · ≤(r)nd(u,u)
and
∞
n=
d(un+,un+)≤
∞
n=
Hence,{un}is a Cauchy sequence inX. SinceXis complete, there is some pointz∈Xsuch thatlimn→∞un=z. As (un,un+)∈ ∇for alln, then by the assumption, (un,z)∈ ∇. Now,
we show that for every pair (x,z)∈ ∇withx=z, the following inequality holds:
pα(z,Tx)≤rmaxd(z,x),pα(x,Fx).
Aslimn→∞un=z, there exists a positive integern∈Nsuch that for alln≥n, we have
d(z,un)≤
d(z,x). ()
Now, for alln≥n,
σ(r)pα(un,Fun)≤pα(un,Fun)
≤d(un,z) +pα(z,Fun)≤d(un,z) +d(z,un+)
≤
d(z,x) +
d(z,x)≤ d(z,x)
=d(z,x) – d(z,x)
≤d(z,x) –d(un,z)≤d(un,x)
implies that
pα(un+,Fx)≤Dα(Fun,Fx)
≤rmax d(un,x),pα(un,Fun),pα(x,Fx),pα(un,Fx) +pα(x,Fun)
≤rmax d(un,x),d(un,un+),pα(x,Fx),
pα(un,Fx) +d(x,un+)
,
which on taking limit asn→ ∞gives
pα(z,Fx)≤rmax d(z,x),pα(x,Fx),pα(z,Fx) +d(x,z)
.
If
max d(z,x),pα(x,Fx),pα(z,Fx) +d(x,z)
=pα(z,Fx) +d(x,z)
,
then
pα(z,Fx)≤rpα(z,Fx) +d(x,z)
≤
r
pα(z,Fx) + r d(x,z),
pα(z,Fx)≤ r
–rd(x,z)≤ r
–rd(x,z)≤rd(x,z).
Hence,
Now, we show thatzα⊂Fzfor eachα∈[, ]. First, consider the case ≤r< /. Assume on the contrary thatzαFz, that is,z∈/ (Fz)α. Leta∈(Fz)α, as (Fz)α is nonempty and compact, so for eachα∈[, ], we have
rd(a,z) <pα(z,Fz). ()
Now,a∈(Fz)αimplies (a,z)∈ ∇anda=z. From () we have
pα(z,Fa)≤rmaxd(z,a),pα(a,Fa). ()
Now,
σ(r)pα(z,Fz)≤pα(z,Fz) =d(z,a)
implies that
Dα(Fz,Fa)≤rmax d(z,a),pα(z,Fz),pα(a,Fa),pα(z,Fa) +pα(a,Fz)
≤rmax d(z,a),pα(z,Fz),pα(a,Fa),pα(z,Fa)
≤rmax d(z,a),pα(z,Fz),pα(a,Fa),d(z,a) +pα(a,Fa)
≤rmaxd(z,a),pα(z,Fz),pα(a,Fa) ≤rmaxd(z,a),pα(a,Fa).
Hence,
Dα(Fz,Fa)≤rmaxd(z,a),pα(a,Fa),
which further implies that
pα(a,Fa)≤rmaxd(z,a),pα(a,Fa) aspα(a,Fa)≤Dα(Fz,Fa).
We claim thatpα(a,Fa)≤d(z,a). If not, then the above inequality becomes
pα(a,Fa)≤rpα(a,Fa) <pα(a,Fa) asr< ,
a contradiction, so we deduce thatpα(a,Fa)≤rd(z,a). From inequality (), we have
pα(z,Fa)≤rd(z,a).
Therefore,
pα(z,Fz)≤pα(z,Fa) +Dα(Fz,Fa)
≤rd(z,a) +rmaxd(z,a),pα(a,Fa) ≤rd(z,a) +rd(z,a)≤rd(z,a) <pα(z,Fz),
Now, when /≤r< , we first prove that
Dα(Fx,Fz)≤rmax d(x,z),pα(x,Fx),pα(z,Fz),pα(x,Fz) +pα(z,Fx)
()
for all (x,z)∈ ∇. Ifx=z, then () holds trivially. So, assume thatx=z. For everyn∈N, one may find a sequenceyn∈(Fx)αsuch that
d(z,yn)≤pα(z,Fx) + nd(x,z).
Asyn∈(Fx)α, this implies (yn,x)∈ ∇. Using () we have
pα(x,Fx)≤d(x,yn)
≤d(x,z) +d(z,yn)
≤d(x,z) +pα(z,Fx) + nd(x,z)
≤d(x,z) +rmaxd(z,x),pα(x,Fx)+ nd(x,z)
for alln∈N. Ifd(x,z)≥pα(x,Fx), then
pα(x,Fx)≤d(x,z) +rd(z,x) + nd(x,z)
≤d(x,z) +rd(z,x) +
nd(x,z)≤
+r+ n
d(x,z).
This implies that
( +r)pα(x,Fx)≤
+ ( +r)n
d(x,z).
Hence, for≤r< , we obtain
σ(r)pα(x,Fx) = ( –r)pα(x,Fx)
≤
( +r)pα(x,Fx)≤
+ ( +r)n
d(x,z).
On taking the limit asn→ ∞, we have
σ(r)pα(x,Fx)≤d(x,z).
Ifd(x,z)≤pα(x,Fx), then
pα(x,Fx)≤d(x,z) +rpα(x,Fx) + nd(x,z),
pα(x,Fx) –rpα(x,Fx)≤d(x,z) + nd(x,z),
On taking the limit asn→ ∞, we have
σ(r)pα(x,Fx)≤d(x,z).
By the given assumption, we have
Dα(Fx,Fz)≤rmax d(x,z),pα(x,Fx),pα(z,Fz),pα(x,Fz) +pα(z,Fx)
.
Thus, for anyx=z, () holds true. Putx=unin the above inequality to obtain
pα(z,Fz)≤ lim
n→∞pα(un+,Fz)≤nlim→∞Dα(Fun,Fz)
≤ lim
n→∞rmax d(un,z),d(un,un+),pα(z,Fz),
pα(un,Fz) +pα(z,Fun)
=rpα(z,Fz)
asr< , we getpα(z,Fz) = . Hence by Lemma ,zα⊂Fz.
Corollary Let(X,d,)be a complete ordered metric space.If an ordered fuzzy mapping F:X→Wα(X)satisfies
σ(r)pα(x,Fx)≤d(x,y) implies Dα(Fx,Fy)≤rMα(F)
for all(x,y)∈ ∇,where
Mα(F) =maxd(x,y),pα(x,Fx),pα(y,Fy).
Then there exists a point x∈X such that xα⊂Fx provided that X satisfies the order se-quential limit property.
Corollary Let(X,d,)be a complete ordered metric space.If an ordered fuzzy mapping F:X→Wα(X)satisfies
σ(r)pα(x,Fx)≤d(x,y) implies Dα(Fx,Fy)≤λMα(F)
for all(x,y)∈ ∇,where
Mα(F) =d(x,y) +pα(x,Fx) +pα(y,Fy)}
andλ∈[,),r= λ.Then there exists a point x∈X such that xα⊂Fx provided that X satisfies the order sequential limit property.
3 An application
LetF:X→Wα(X) andg:X→X. A pair{F,g}is said to be an ordered fuzzy hybrid pair if the following conditions are satisfied:
(c) gy∈F(x)αimplies that(y,x)∈ ∇.
Theorem Let(X,d,)be a complete ordered metric space.If an ordered fuzzy hybrid pair{F,g}satisfies
σ(r)pα(gx,Fx)≤d(gx,gy) implies Dα(Fx,Fy)≤rMα(F,g) ()
for all(x,y)∈ ∇,where
Mα(F,g) =max d(gx,gy),pα(gx,Fx),pα(gy,Fy),pα(gx,Fy) +pα(gy,Fx)
.
Then Cα(F,g)=φ provided that X satisfies the order sequential limit property and (F(X))α⊆g(X)for eachα. Moreover, F and g have a common fixed fuzzy point if any of the following conditions holds:
(f ) Fandgarew-fuzzy compatible,limn→∞gnx=uandlimn→∞gny=vfor some
x∈Cα(F,g),u∈Xandgis continuous atu.
(g) gisF-fuzzy weakly commutingfor somex∈Cα(g,F)and is a fixed point ofg,that is,
gx=gx.
(h) gis continuous atxfor somex∈Cα(g,F)and for someu∈Xsuch that limn→∞gnu=x.
Proof By Lemma , there existsE⊆Xsuch thatg:E→Xis one-to-one andg(E) =g(X). Define a mappingA:g(E)→Wα(X) by
Agx=Fx for allgx∈g(E). ()
Asgis one-to-one onE,Ais well defined. Also,
σ(r)pα(gx,Fx)≤d(gx,gy) implies Dα(Fx,Fy)≤rMα(F,g) ()
for all (x,y)∈ ∇. Therefore,
σ(r)pα(gx,Agx)≤d(gx,gy) implies Dα(Agx,Agy)≤rMα(F,g)
for all (gx,gy)∈ ∇. Hence,Asatisfies () and all the conditions of Theorem . Using Theo-rem with a mappingA, it follows thatAhas a fixed fuzzy pointu∈g(E). Now, it is left to prove thatFandghave a coincidence fuzzy point. SinceAhas a fixed fuzzy pointuα⊂Au, we getu∈(Au)α. As (F(X))α⊆g(X), so there existsu∈Xsuch thatgu=u, thus it
fol-lows thatgu∈(Agu)α= (Fu)α. This implies thatu∈Xis a coincidence fuzzy point of
Fandg. Hence,Cα(F,g)=φ. Suppose now that (f ) holds. Then for somexα∈Cα(F,g), we havelimn→∞gnx=u, whereu∈X. Thus (gn–x,u)∈ ∇. Sincegis continuous atu, we have
thatuis a fixed point ofg. AsFandgarew-fuzzy compatible, and (gnx)α∈Cα(F,g) for all n≥. That is,gnx∈F(gn–x)αfor alln≥. Now,
implies that
pα(gu,Fu)≤pαgu,gnx+pαgnx,Fu ≤pαgu,gnx+DαFgn–x,Fu
≤pαgu,gnx+rmax dggn–x,gu,pαgnx,Fgn–x,pα(gu,Fu),
pα(gu,Fgn–x) +pα(gnx,Fu)
.
On taking limit asn→ ∞, we getpα(gu,Fu)≤rpα(gu,Fu) and thereforepα(gu,Fu) = . By Lemma we obtaingu∈(Fu)α. Consequently,u=gu∈(Fu)α. Hence,uαis a common fixed fuzzy point ofFandg. Suppose now that (g) holds. If for somexα∈Cα(F,g),gisF-fuzzy weakly commutingandgx=gx, thengx=gx∈(Fgx)α. Hence, (gx)αis a common fixed
fuzzy point ofFandg. Suppose now that (h) holds and assume that for somexα∈Cα(F,g) and for someu∈X,limn→∞gnu=xandlimn→∞gnv=y. By the continuity ofgatxandy,
we getx=gx∈(Fx)α. The result follows.
Example LetX= [, ] be endowed with the usual metric. Letα∈(,) andr=, then
σ(r) =. Define a fuzzy mappingFfromXintoWα(X) as
(F)(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifx= ,
α ifx∈(,], α
ifx∈( , ]
and (F)(x) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifx= ,
α ifx∈(,], α
ifx∈( , ]
and forz∈(, ),
(Fz)(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifx= ,
α ifx∈(,], ifx∈(, ].
Define a self-mapg:X→Xbyg(x) =x. Then
(F)= (F)= (Fz)={}, (F)α= (F)α= (Fz)α=
, , (F)α
= (F)α = [, ] and (Fz)α =
,
.
Note that for allx,y∈X, we have
D(Fx,Fy) =H
(Fx), (Fy)
=Dα(Fx,Fy) =H(Fx)α, (Fy)α= .
Also, for allx,y∈ {, }, we have
Dα
(Fx,Fy) =H
(Fx)α
, (Fy)α
And
Dα
(Fx,Fy) =H
(Fx)α
, (Fy)α
= for allx,y∈(, ).
Ifx∈ {, }andy∈(, ), thenDα
(Fx,Fy) =H((Fx)α, (Fy)α) =
. So, for allx,y∈X, with
σ(r)pα(x,Fx)≤d(x,y), we haveDα(Fx,Fy) = . Hence, for allx,y∈X,
Dα(Fx,Fy)≤rMα(F) and Dα(Fx,Fy)≤rMα(F,g)
hold true, where
Mα(F) =max d(x,y),pα(x,Fx),pα(y,Fy),pα(x,Fy) +pα(y,Fx)
and
Mα(F,g) =max d(gx,gy),pα(gx,Fx),pα(gy,Fy),pα(gx,Fy) +pα(gy,Fx)
.
Hence, all the conditions of Theorem are satisfied. Moreover, for eachx∈[,], we have xα⊂F(x) and (gx)α⊂F(x). Forα= , we have{}={g} ⊂(F).
4 Conclusion
The Banach contraction principle has become a classical tool to show the existence of so-lutions of functional equations in nonlinear analysis (see for details [–]). Suzuki-type fixed point theorems [, ] are the generalizations of the Banach contraction principle that characterize metric completeness of underlying spaces. Fuzzy sets and mappings play important roles in the process of fuzzification of systems. Suzuki-type fixed point theo-rems for fuzzy mappings obtained in this article can further be used in the process of finding the solutions of functional equations involving fuzzy mappings in fuzzy systems. In the main result, we not only extended the mapping to a fuzzy mapping, but also the underlying metric space has been replaced with ordered metric spaces. In this article, we defined coincidence fuzzy points and common fixed fuzzy points of the hybrid pair of a single-valued self-mapping and a fuzzy mapping and applied our main result to obtain the existence of coincidence fuzzy points and common fixed fuzzy points of the hybrid pair.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics, School of Science and Engineering, Lahore University of Management Sciences, Lahore,
54792, Pakistan.2Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Pretoria, South Africa.
Acknowledgements
The authors are thankful to the referees for their critical remarks which helped to improve the presentation of this paper.
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doi:10.1186/1687-1812-2013-9
Cite this article as:Ali and Abbas:Suzuki-type fixed point theorem for fuzzy mappings in ordered metric spaces.
