R E S E A R C H
Open Access
Fixed points of Suzuki type generalized
multivalued mappings in fuzzy metric spaces
with applications
Naeem Saleem
1*, Basit Ali
2, Mujahid Abbas
3and Zahid Raza
1*Correspondence:
naeem.saleem2@gmail.com
1Department of Mathematics,
National University of Computer and Emerging Sciences, Lahore, Pakistan
Full list of author information is available at the end of the article
Abstract
The aim of this paper is to introduce a class of multivalued mappings satisfying a Suzuki type generalized contractive condition in the framework of fuzzy metric spaces and to present fixed point results for such mappings. Some examples are presented to support the results proved herein. As an application, a common fixed point result for a hybrid pair of single and multivalued mappings is obtained. We show the existence and uniqueness of a common bounded solution of functional equations arising in dynamic programming. Our results generalize and extend various results in the existing literature.
MSC: 47H10; 47H04; 47H07
Keywords: Suzuki type mapping; fuzzy metric space; fixed point; multivalued
mapping;t-norm
1 Introduction and preliminaries
One of the basic fixed point theorems with a wide range of applications in all types of anal-yses is the Banach (or Banach-Cassioppoli) contraction principle due to Banach []. The Banach contraction principle [] is a simple and powerful result including iterative meth-ods for solving linear, nonlinear, differential, integral, and difference equations. Due to its applications in mathematics and other related disciplines, this principle has been gen-eralized in many directions. Banach contractions are continuous mappings. Kannan [] proved fixed point theorems for mappings which are not necessarily continuous. Kikkawa and Suzuki [] further generalized Banach contractions and Kannan mappings. Suzuki [] obtained a variant of Banach contraction principle that characterizes metric completeness by using a different type of contractive condition. There are multivalued generalizations of Banach contraction principle (see [–]). Ðorić and Lazović [] extended a Suzuki type fixed point theorem and obtained fixed points of a multivalued mapping. They also gave applications of their results in dynamic programming.
On the other hand, the evolution of fuzzy mathematics commenced with an introduc-tion of the nointroduc-tion of fuzzy sets by Zadeh [] in as a new way to represent vagueness in everyday life. There are different notions of fuzzy metric spaces. Kramosil and Michalek [] introduced the notion of fuzzy metric space by using continuoust-norms, which gen-eralizes the concept of probabilistic metric space to fuzzy situation. Moreover, George and Veeramani [, ] modified the concept of fuzzy metric spaces and obtained a Hausdorff
topology for this kind of fuzzy metric spaces. Recently, Gregoriet al.[] gave applica-tions of fuzzy metrics to color image process and used the concept of fuzzy metric to filter noisy images and in other engineering problems of special interests. Fixed point theory in fuzzy metric spaces has been studied by a number of authors (compare for details [– ]). Rodríguez-López and Romaguera [] introduced Hausdorff fuzzy metric on a set of nonempty compact subsets of a fuzzy metric space.
In this paper, we obtain Suzuki type fixed point theorems for multivalued mappings in fuzzy metric spaces. Our results extend the comparable results given in [, , ] to fuzzy metric spaces.
The following definitions and well-known results will be needed in the sequel.
Definition .([]) A binary operation∗: [, ]→[, ] is called a continuoust-norm
if
() ∗is associative and commutative;
() ∗: [, ]→[, ]is continuous (note that at-norm is continuous if it is continuous
as a mapping under usual topology on[, ]);
() a∗ =afor alla∈[, ];
() a∗b≤c∗dwhenevera≤candb≤d.
Some basic examples of continuoust-norms are∧,·and∗L, where, for alla,b∈[, ],
a∧b=min{a,b},a·b=ab, and∗Lis the Lukasiewiczt-norm defined bya∗Lb=max{a+
b– , }.
It is easy to check that∗L≤ · ≤ ∧. In fact∗ ≤ ∧for all continuoust-norm∗.
Definition .([, ]) LetXbe a nonempty set and∗be a continuoust-norm. A fuzzy setMonX×X×(, +∞) is said to be a fuzzy metric onXif for anyx,y,z∈Xands,t> , the following conditions hold:
(i) M(x,y,t) > ;
(ii) x=yif and only ifM(x,y,t) = for allt> ; (iii) M(x,y,t) =M(y,x,t);
(iv) M(x,z,t+s)≥M(x,y,t)∗M(y,z,s)for allt,s> ; (v) M(x,y,·) : (, +∞)→(, ]is continuous.
The triplet (X,M,∗) is called a fuzzy metric space. Each fuzzy metricMonXgenerates Hausdorff topologyτM onXwhose base is the family of openM-balls{BM(x,ε,t) :x∈
X,ε∈(, ),t> }, where
BM(x,ε,t) =
y∈X:M(x,y,t) > –ε.
Note that a sequence {xn} converges to x ∈X (with respect to τM) if and only if
limn→∞M(xn,x,t) = for allt> .
It is well known that for eachx,y∈X,M(x,y,·) is a nondecreasing function on (, +∞) (see []). Moreover, every fuzzy metric spaceX(in the sense of George and Veeramani []) is metrizable, that is, there exists a metricdonXwhich induces a topology that agrees withτM(see []). Conversely if (X,d) is a metric space andMd:X×X×(, +∞)→(, ]
is defined as
Md(x,y,t) =
for allt> , then (X,Md,∧) is a fuzzy metric space called the standard fuzzy metric space
induced by the metricd(see []). The topologies induced by the standard fuzzy metric and the corresponding metric are the same [].
A sequence{xn}in a fuzzy metric spaceXis said to be a Cauchy sequence if for each ε∈(, ), there existsn∈Nsuch thatM(xn,xm,t) > –εfor alln,m≥n. A fuzzy metric
space Xis complete [] if every Cauchy sequence converges inX. A subsetAof Xis closed if for each convergent sequence{xn}inAwithxn→x, we havex∈A. A subsetA
ofXis compact if each sequence inAhas a convergent subsequence. We denote the set of all nonempty compact subsets ofXbyK(X). LetT:X→K(X) be a multivalued mapping andgbe a self-mapping onX. Further a pointx∈Xis called: () a fixed point ofTif and only ifx∈Tx, () a coincidence point ofTandgif and only ifgx∈Tx, () a common fixed point ofTandgif and only ifx=gx∈Tx.
We denote the set of all fixed points ofT byF(T) and the set of all coincidence points (common fixed points) of mappingsT andgbyC(g,T) (F(g,T)). A mappingg is called T-weakly commuting at some pointx∈X ifgx∈Tgx. The hybrid pair (T,g) is called w-compatible ifgTx⊆Tgxwheneverx∈C(g,T) (see [] and the references therein).
Definition .([]) A fuzzy metricMis said to be continuous onX×(,∞) if
lim
n→∞M(xn,yn,tn) =M(x,y,t)
whenever{(xn,yn,tn)}is a sequence inX×(,∞) which converges to a point (x,y,t)∈
X×(,∞).
Proposition .([]) Let(X,M,∗)be a fuzzy metric space.Then M is a continuous func-tion on X×X×(,∞).
Consistent with [], we recall the notion of Hausdorff fuzzy metric induced by a fuzzy metricMas follows: Forx∈X,A∈K(X) andA,B∈K(X) define
HM(A,B,t) =min
inf
a∈AM(a,B,t),binf∈BM(A,b,t)
for allt> , whereM(x,A,t) =supa∈AM(x,a,t). ThenHM is called the Hausdorff fuzzy
metric induced by the fuzzy metricM. The triplet (K(X),HM,∗) is called Hausdorff fuzzy
metric space.
Lemma .([]) Let X be a fuzzy metric space.Then,for each a∈X,B∈K(X)and t> , there is b∈B such that M(a,B,t) =M(a,b,t).
Lemma .([]) Let g be a self-mapping on a nonempty set X.Then there exists a subset E⊆X such that g(E) =g(X)and g:E→X is one-to-one.
Let={η: [, ]→[, ],ηis continuous, nondecreasing andη(t) >tfort∈(, )}. Note thatη() = andη() = , thenη(t)≥tfor allt∈[, ].
A sequence{tn}of positive real numbers is said to bes-increasing (for further details,
see []) if there existsn∈Nsuch that
for allm≥n. In a fuzzy metric space (X,M,∧), an infinite product (compare []) is
denoted by
M(x,y,t)∧M(x,y,t)∧ · · · ∧M(x,y,tn)∧ · · ·=
∞
i=
M(x,y,ti)
for allx,y∈X.
2 Fixed points of multivalued mappings in fuzzy metric space Suppose that
ϕ(r) =
, ≤r<, –r, ≤r< .
In this section, we obtain several fixed point theorems for multivalued mappings satisfying a Suzuki type contractive condition in the setup of fuzzy metric spaces. We start with the following result.
Theorem . Let (X,M,∧)be a complete fuzzy metric space and T :X→K(X)be a multivalued mapping.Suppose that there exists r in[, )such that for each x,y∈X and
η∈,
M(x,Tx,t)≥Mx,y,ϕ(r)t
implies
HM(Tx,Ty,rt)≥NTM(x,y),
where
NTM(x,y) =ηminM(x,y,t),M(x,Tx,t),M(y,Ty,t),M(x,Ty, t),M(y,Tx,t) .
Then F(T)=φprovided that the following conditions hold:
(c-) For eachε> and ans-increasing sequence{tn},there existsninNsuch that
∞
n≥nM(x,y,tn)≥ –εfor alln≥n.
(c-) If a sequence{un}converges tozin(X,M,∧),then there existsninNsuch that
M(un,x,t)≤M(z,x,t)for allz= x,for alln≥nand for allt> .
Proof Letube a given point inX. SinceTu∈K(X), we can chooseu∈Tusuch that
M(u,Tu,rt)≥HM(Tu,Tu,rt).
Asϕ(r) < , so we have
M(u,Tu,t)≥M
u,Tu,ϕ(r)t ≥M
Then, by a given assumption, it follows that
M(u,Tu,rt)
≥HM(Tu,Tu,rt)
≥ηminM(u,u,t),M(u,Tu,t),M(u,Tu,t),M(u,Tu, t),M(u,Tu,t)
≥ηminM(u,u,t),M(u,u,t),M(u,Tu,t),M(u,Tu, t),M(u,u,t)
≥ηminM(u,u,t),M(u,u,t),M(u,Tu,t),M(u,Tu, t) .
SinceTu is compact, so by Lemma . there existsu∈Tu such that M(u,Tu,t) =
M(u,u,t). Note that
M(u,Tu, t) =M(u,u, t)≥M(u,u,t)∧M(u,u,t).
Hence we obtain that
M(u,u,rt)≥η
minM(u,u,t),M(u,u,t) .
Ifmin{M(u,u,t),M(u,u,t)}=M(u,u,t), then
M(u,u,rt)≥η
M(u,u,t)
gives a contradiction. Thus we have
M(u,u,t)≥η
M
u,u,
t r
.
Continuing this way, we can obtain a sequence{un}inXsuch thatun+∈Tun, which
sat-isfies
M(un,un+,t)≥M
un–,un,
t r
≥M
un–,un–,
t r
≥ · · · ≥M
u,u,
t rn
.
Lett> ,ε> ,m,n∈Nsuch thatm>nandhi=i(i+)fori∈ {n,n+ , . . .m– }. Ashn+
hn++· · ·+hm–< , so we have
M(un,um,t)
≥Mun,um, (hn+hn++· · ·+hm–)t
≥M(un,un+,hnt)∧M(un+,un+,hn+t)∧ · · · ∧M(um–,um,hm–t)
≥M
u,u,
hn
rnt
∧M
u,u,
hn+
rn+t
∧ · · · ∧M
u,u,
hm–
rm–t
=M
u,u,
n(n+ )rnt
∧M
u,u,
(n+ )(n+ )rn+t
∧M
u,u,
m(m– )rm–t
≥
∞
n=
M
u,u,
t n(n+ )rn
=
∞
n=
M(u,u,tn),
wheretn=n(n+)t rn. Sincelimn→∞(tn+–tn) =∞, therefore{tn}is ans-increasing sequence.
Consequently, there existsn∈Nsuch that for eachε> , we have∞n=M(u,u,tn)≥
–εfor alln≥nand hence we obtainM(un,um,t)≥ –εfor alln,m≥n. Thus{un}
is a Cauchy sequence inX. AsX is a complete, so there exists a pointzinXsuch that
limn→∞M(un,z,t) = . Now we show that
M(z,Tx,rt)≥ηminM(z,x,t),M(x,Tx,t)
for allxinX\ {z}. Sincelimn→∞un=z, so there exists a positive integern∈Nsuch that
for alln≥n, we have
M
z,un,
t
≥M(z,x,t)
and by (c-) there existsninNsuch thatM(un,x,t)≤M(z,x,t) for allz=x,n≥nand
for allt> . Now if we taken=max{n,n}, then we obtain that
M(un,Tun,t)≥M
un,z,
t
∧M
z,Tun,
t
≥M
un,z,
t
∧M
z,un+,
t
≥M(z,x,t)∧M(z,x,t) =M(z,x,t)≥M(un,x,t)
≥Mun,x,ϕ(r)t
for alln≥n. Thus, we have
M(un+,Tx,rt)
≥HM(Tun,Tx,rt)
≥ηminM(un,x,t),M(un,Tun,t),M(x,Tx,t),M(un,Tx, t),M(x,Tun,t)
≥ηminM(un,x,t),M(un,un+,t),M(x,Tx,t),M(un,Tx, t),M(x,un+,t) ,
which on taking limit asn→ ∞gives
M(z,Tx,rt)≥ηminM(z,x,t),M(x,Tx,t),M(z,Tx,t),M(x,z,t)
≥ηminM(z,x,t),M(x,Tx,t),M(z,Tx, t)
≥ηminM(z,x,t),M(x,Tx,t),M(z,x,t)∧M(x,Tx,t) ≥ηminM(z,x,t),M(x,Tx,t) .
Hence
M(z,Tx,rt)≥ηminM(z,x,t),M(x,Tx,t) . ()
(a) Take ≤r<. Assume on the contrary thatz∈/ Tz. Leta∈Tz. Thena=zand it follows that
M(a,z,t)≥Ma,Tz, ( – r)t ∧M(z,Tz, rt)
≥Ma,a, ( – r)t ∧M(z,Tz, rt)
=M(z,Tz, rt).
Using () we have
M(z,Ta,rt)≥ηminM(z,a,t),M(a,Ta,t) . ()
Also,
M(z,Tz,t)≥Mz,Tz,ϕ(r)t ≥Mz,a,ϕ(r)t .
Thus, we have
HM(Tz,Ta,rt)≥η
minM(z,a,t),M(z,Tz,t),M(a,Ta,t),M(z,Ta, t),M(a,Tz,t)
≥ηminM(z,a,t),M(a,Ta,t),M(z,Ta, t)
≥ηminM(z,a,t),M(a,Ta,t),M(z,a,t)∧M(a,Ta,t) ≥ηminM(z,a,t),M(a,Ta,t) .
NowM(a,Ta,t)≥HM(Tz,Ta,t) implies that
HM(Tz,Ta,rt)≥η
M(z,a,t) >M(z,a,t).
Hence
M(z,Ta,rt) >M(z,a,t).
Sinceη(t) >tfort∈(, ), so fora=zwe have
M(z,Tz, rt)≥M(z,a, rt)≥M(z,Ta,rt)∧M(Ta,a,rt)
≥M(z,Ta,rt)∧HM(Ta,Tz,rt)
>M(z,a,t)∧ηminM(z,a,t),M(a,Ta,t)
≥M(z,a,t)∧ηM(z,a,t)
>M(z,a,t)∧M(z,a,t)
=M(z,a,t)≥M(z,Tz, rt).
That is,M(z,Tz, rt) >M(z,Tz, rt), a contradiction. Consequentlyz∈Tz. (b) Take ≤r< . We claim that
HM(Tx,Tz,rt)
for allx. Ifx=z, then the above inequality is immediate. Ifx=z, then we have
M(x,Tx,t)≥Mx,z, ( –r)t ∧M(z,Tx,rt)
≥Mx,z,ϕ(r)t ∧M(z,Tx,rt).
Using () we have
M(x,Tx,t)≥Mx,z,ϕ(r)t ∧ηminM(z,x,t),M(x,Tx,t)
≥Mx,z,ϕ(r)t ∧ηminM(z,x,t),M(x,Tx,t) .
Ifmin{M(z,x,t),M(x,Tx,t)}=M(z,x,t), then
M(x,Tx,t)≥Mx,z,ϕ(r)t ∧ηminM(z,x,t),M(x,Tx,t)
≥Mx,z,ϕ(r)t ∧ηM(z,x,t)
≥Mx,z,ϕ(r)t ∧M(z,x,t) =Mx,z,ϕ(r)t .
Ifmin{M(z,x,t),M(x,Tx,t)}=M(x,Tx,t), then
M(x,Tx,t)≥Mx,z,ϕ(r)t ∧ηminM(z,x,t),M(x,Tx,t)
≥Mx,z,ϕ(r)t ∧ηM(x,Tx,t).
IfM(x,z,ϕ(r)t)∧η(M(x,Tx,t)) =M(x,z,ϕ(r)t), then we obtain
M(x,Tx,t)≥Mx,z,ϕ(r)t .
If not, thenM(x,z,ϕ(r)t)∧η(M(x,Tx,t)) =η(M(x,Tx,t)) gives
M(x,Tx,t)≥ηM(x,Tx,t) >M(x,Tx,t),
a contradiction. Hence
M(x,Tx,t)≥Mx,z,ϕ(r)t .
Thus the claim follows. Usingx=unin () we obtain that
M(z,Tz,t)
= lim
n→∞M(un+,Tz,t)≥nlim→∞HM(Tun,Tz,t)
≥ lim
n→∞η
minM(un,z,t),M(un,Tun,t),M(z,Tz,t),M(un,Tz, t),M(z,Tun,t)
≥ lim
n→∞η
minM(un,z,t),M(un,un+,t),M(z,Tz,t),M(un,Tz, t),M(z,Tun,t)
≥ηM(z,Tz,t) .
Sinceη(t)≥t. Henceη(M(z,Tz,t)) =M(z,Tz,t) shows thatM(z,Tz,t) = . SinceTzis
Problem . Does Theorem . hold for an arbitrary continuoust-norm?
Now we present an example of fuzzy metric space that satisfies all the conditions of Theorem ..
Example . LetX={, , }andd:X×X→Rbe a metric defined by
d(, ) =d(, ) = , d(, ) =d(, ) = .,
d(, ) =d(, ) = , d(, ) =d(, ) =d(, ) = .
The fuzzy metricMonXis given by
M(x,y,t) = t
t+d(x,y).
DefineT:X×X→CB(X) as
T(x) =
{} ifx= , {, } ifx= .
Letr=
andη=t
∈. Note that for allx,y∈X, except atx= =y,
M(x,Tx,t)≥Mx,y,ϕ(r)t ()
holds true. It is sufficient to check the validity of
HM(Tx,Ty,rt)≥NTM(x,y) ()
for allx,yfor which () holds true. For this, we consider the following cases: (i) Ifx=y= , then we have
HM
T,T,
t
= .
So () holds in this case.
(ii) Ifx= ,y= orx= ,y= , then
NTM(x,y) =ηminM(, ,t),M(,T,t),M(,T,t),M(,T, t),M(,T,t)
=ηminM(, ,t),M,{, },t ,M,{},t ,M,{}, t ,M,{, },t
=η
min
t
t+d(, ),
t
t+d(, ),
t
t+d(, ),
t
t+d(, ),
t
t+d(, )
=η
min
t t+ , ,
t t+ , ,
t t+ .
=η
t
t+
HM
T,T, t
=min
inf
x∈TM
x,T,
t
, inf
y∈TM
T,T,
t =min inf
x∈TM
x,{},
t ,M {, }, , t =min inf M , , t ,M , , t ,M , , t =min inf , t t+
, = t t+
.
Note that
t t+
≥
t
t+
> t
t+ .
(iii) Ifx= ,y= orx= ,y= , thenHM(T,T,
t) = andN
M
T(x,y) =η( t
t+).
(iv) Ifx= ,y= orx= ,y= , then
NTM(x,y) =ηminM(, ,t),M(,T,t),M(,T,t),M(,T, t),M(,T,t)
=ηminM(, ,t),M,{},t ,M,{, },t ,M,{, }, t ,M,{},t
=η
min
t
t+d(, ),
t
t+d(, ),
t
t+d(, ),
t
t+d(, ),
t
t+d(, )
=η min t
t+ .,
t
t+ , ,
t
t+ .,
t
t+
=η
t
t+
,
HM
T,T,
t =min inf
x∈TM
x,T,
t
, inf
y∈TM
T,y,
t =min inf
x∈TM
x,{, },
t
, inf
y∈TM
{},y,
t =min M ,{, }, t , inf
y∈T
M , , t ,M , , t =min ,inf , t t+
=
t t+
. Note that t t+
≥
t t+
> t
Hence, for allx,y∈X,
HM
Tx,Ty,
t
≥ηNTM(x,y) >NTM(x,y).
Moreover, for anys-increasing sequence{tn},
∞
n=M(x,y,tn) is convergent (see [],
Ex-amples . and .), that is, (c-) is satisfied. Since only convergent sequences inXare constant sequences{un}, whereun=candc∈X, therefore (c-) holds true. Thus all the
conditions of Theorem . are satisfied.
Remark . If we consider the standard fuzzy metricMd(x,y,t) =t+dt(x,y) in the example
above, then all the conditions in Theorem . are satisfied except (c-). Indeed, for any s-increasing sequence{tn}withtn=n,
∞
n=Md(x,y,tn) converges to (see [],
Exam-ple .).
Remark . Let (X,M,∧) be a complete fuzzy metric space andT:X→K(X) be a mul-tivalued mapping. We modify condition (c-) in Theorem . as follows:
(c-) For eachε> and ans-increasing sequence{tn}, there existn∈N,u∈Xand
u∈Tusuch that
∞
n=M(u,u,tn)≥ –ε.
If we replace condition (c-) with (c-) in Theorem . and start the proof of the theo-rem with initial pointsu∈Xandu∈Tufor which (c-) holds, then the conclusion of
Theorem . remains the same.
Example . LetX=NandP:X×X×(,∞)→(, ] be a fuzzy metric defined by
P(x,y,t) =
ifx=y,
ifx=y.
Note that (X,P,∧) is a complete fuzzy metric space which induces the discrete topology onXand there does not exist any metricdonXsatisfying
P(x,y,t) = t t+d(x,y).
Moreover, for an s-increasing sequence{tn=n},∞n=P(x,y,tn) converge to (see [],
Examples . and .). HencePdoes not satisfy condition (c-) of Theorem .. Define T:X→CB(X) as
T(x) =
{} ifx= , {, } ifx= .
Ifu= andu= ∈Tu, then for anys-increasing sequence{tn},
∞
n=M(u,u,tn)
con-verges to . That is,Psatisfies condition (c-) in Remark .. Note that for anyr∈[, ) andx=y,
P(x,Tx,t)≥ =P
x,y,ϕ(r)t
is satisfied, whereas
HP(Tx,Ty,rt)≥
=N
holds true for allx,yinX. Moreover, conditions (c-) and (c-) are satisfied. Hence all the conditions of the variant of Theorem . are satisfied in the setup of (X,P,∧).
Corollary . Let(X,M,∧)be a complete fuzzy metric space and T:X→K(X)be a mul-tivalued mapping.Suppose that there exists r in[, )such that for each x,y∈X andη∈,
M(x,Tx,t)≥Mx,y,ϕ(r)t
implies
HM(Tx,Ty,rt)≥η
M(x,y,t) . ()
Then there exists a point z∈X such that z∈Tz provided that conditions(c-)and(c-)in Theorem.hold.
Corollary . Let(X,M,∧)be a complete fuzzy metric space and T:X→K(X)be a mul-tivalued mapping.Suppose that there exists r in[, )such that for each x,y∈X andη∈,
M(x,Tx,t)≥Mx,y,ϕ(r)t
implies
HM(Tx,Ty,rt)≥η
minM(x,y,t),M(x,Tx,t),M(y,Ty,t) . ()
Then there exists a point z∈X such that z∈Tz provided that conditions(c-)and(c-)in Theorem.hold.
Corollary . Let (X,M,∧)be a complete fuzzy metric space and T :X→K(X)be a multivalued mapping.Suppose that there exists r in[, )such that for each x,y∈X and
η∈,
M(x,Tx,t)≥Mx,y,ϕ(r)t
implies
HM(Tx,Ty,rt)≥η
minM(x,y,t),M(x,Tx,t),M(y,Tx,t) .
Then there exists a point z∈X such that z∈Tz provided that conditions(c-)and(c-)in Theorem.hold.
Corollary . Let(X,M,∧)be a complete fuzzy metric space and T:X→K(X)be a mul-tivalued mapping.Suppose that there exists r in[, )such that for each x,y∈X andη∈,
M(x,Tx,t)≥Mx,y,ϕ(r)t
implies
HM(Tx,Ty,rt)≥η
wherei=αi= andη∈.Then there exists a point z∈X such that z∈Tz provided that
conditions(c-)and(c-)in Theorem.hold.
Now, in the next theorem, we prove the existence of coincidence and common fixed points of a hybrid pair of single and multivalued mappings.
Theorem . Let(X,M,∧)be a complete fuzzy metric space,T :X→K(X)and g be a self-mapping on X such that T(X)⊆g(X).Suppose that there exists r in[, )such that for each x,y∈X,t> andη∈,
M(gx,Tx,t)≥Mgx,gy,ϕ(r)t
implies
HM(Tx,Ty,rt)≥η
minM(gx,gy,t),M(gx,Tx,t),M(gy,Tx,t) .
Then C(g,T)=φprovided that the following conditions hold:
(c-) For eachε> and ans-increasing sequence{tn},there existsninNsuch that
∞
n≥nM(gx,gy,tn)≥ –ε.
(c-) If a sequence{gun}converges togzinX,thenM(gun,gx,t)≤M(gz,gx,t)for all
z=x.
Further, if one of the following conditions holds:
(a) T andgarew-compatible,limn→∞gnx=yfor somex∈C(g,T),y∈Xandgis
continuous aty.
(b) gisT-weakly commuting for somex∈C(g,T)andgxis a fixed point ofg, that is,
gx=gx.
(c) gis continuous atxfor somex∈C(g,T)and for somey∈X,limn→∞gny=x.
ThenF(g,T)=φ.
Proof By Lemma ., there existsE⊆Xsuch thatg:X→Xis one-to-one andg(E) =g(X). We defineA:g(E)→K(X) by
A(gx) =T(x) forgx∈g(E).
Asgis one-to-one onE, soAis well defined. Further
Mgx,A(gx),t =M(gx,Tx,t)≥Mgx,gy,ϕ(r)t
implies
HM
A(gx),A(gy),rt =HM(Tx,Ty,rt)
≥ηminM(gx,gy,t),Mgx,A(gx),t ,Mgy,A(gx),t .
pointuing(E) andT(X)⊆g(X), so there existsv∈Xsuch thatgv=u. Hence
u=gv∈A(u) =A(gv) =T(v).
That is,v∈Xis a coincidence point ofT andg. HenceC(g,T) is nonempty. If (a) holds, then, for somex∈C(g,T),limn→∞gnx=y, wherey∈X. Sincegis continuous aty, soyis
a fixed point ofg. AsT andgarew-compatible, sognx∈C(g,T) for alln≥. That is, for
alln≥,gnx∈T(gn–x). Now
M(gy,Ty,t)
≥M
gy,gnx,t
∧M
gnx,Ty, t
≥M
gy,gnx,t
∧HM
Tgn–x ,Ty,t
≥M
gy,gnx,t
∧HM
Tgn–x ,Ty,rt
≥M
gy,gnx,t ∧η M
gnx,gy,t
,M
gnx,Tgn–x ,t
,M
gy,Tgn–x ,t
≥M
gy,gnx,t ∧η M
gnx,gy,t
,M
gnx,gnx,t
,M
gy,gnx,t
≥M
gy,gnx,t ∧η M
gnx,gy,t
,M
gy,gnx, t
≥M
gy,gnx,t ∧η M
gnx,gy,t
.
On taking limit asn→ ∞, we get that
M(gy,Ty,t)≥M
y,y,t ∧η M y,y,t
≥∧η
M
y,Ty, t
≥M
y,y,t
= .
Hencegy∈T(y). Consequently,y=gy∈Tyandyis a common fixed point ofTandg. If (b) holds, that is, for somex∈C(g,T),gisT-commuting andgx=gx, thengx=gx∈T(gx).
Hence,gxis a common fixed point ofTandg. If (c) holds, then, by the continuity ofgat x, we obtain thatx=gx∈T(x). Hence,xis a common fixed point ofTandg.
Corollary . Let(X,M,∧)be a complete fuzzy metric space,T:X→K(X)and g be a self-mapping on X such that T(X)⊆g(X).Suppose that there exists r in[, )such that for each x,y∈X,t> andη∈,
implies
HM(Tx,Ty,rt)≥η
M(gx,gy,t).
Then T and g have a coincidence point provided that conditions(c-)and(c-)in Theo-rem.are satisfied.Moreover,F and g have a common fixed point if conditions(a)-(c)in Theorem.hold.
Now we prove the following corollary for a pair of single-valued self-mappings.
Corollary . Let(X,M,∧)be a complete fuzzy metric space, T,g:X→X such that T(X)⊆g(X).Suppose that there exists r in [, )such that for each x,y∈X,t> and
η∈,
M(gx,Tx,t)≥Mgx,gy,ϕ(r)t
implies
M(Tx,Ty,rt)≥ηM(gx,gy,t) .
Then T and g have a coincidence point provided that conditions(c-)and(c-)in Theo-rem.hold.Further,if T and g commute at x for x∈C(g,T),then F and g have a unique common fixed point.
Proof By Corollary .,C(g,T) is nonempty. Letx∈C(g,T). Now, we claim thatgx=
Tgx. To prove the result, we need to show thatgx=gx. Ifgx=gx, thenM(gx,Tx,t) = ≥
M(gx,gy,ϕ(r)t). Note that
Mgx,gx,t >Mgx,gx,rt =M(Tx,Tgx,rt)
≥ηMgx,gx,t >Mgx,gx,t ,
a contradiction. Hencegx=gx=Tgximplies thatgx∈F(g,T). To prove the uniqueness,
assume on the contrary that there existuandwsuch thatu=gu=Tu,w=gw=Twand u=w. SinceM(gu,Tu,t) = ≥M(gu,gw,ϕ(r)t), by a given assumption we have
M(gu,gw,t) =M(Tu,Tw,t)≥M(Tu,Tw,rt)
≥ηM(gu,gw,t) >M(gu,gw,t),
a contradiction. HenceF(g,T) is singleton.
3 Application in dynamic programming
Now we present an application of our result in solving functional equations arising in dynamic programming.
is desirable to find an optimal decision in the given state space using dynamic program-ming. Dynamic programming provides useful tools for mathematical optimization and computer programming as well. In particular, the problem of dynamic programming re-lated to multistage process reduces to the problem of solving the functional equation
⎧ ⎪ ⎨ ⎪ ⎩
p(x) =supy∈D{g(x,y) + (x,y,p(τ(x,y)))} forx∈W, q(x) =supy∈D{h(x,y) +(x,y,q(τ(x,y)))} forx∈W,
whereτ:W×D→W,g,h:W×D→Rand ,:W×D×R→R.
()
However, for detailed background of the problem, we refer to [–].
Now, we study the existence and uniqueness of a bounded solution of the functional equation. For this we proceed as follows.
LetB(W) be the set of all bounded real-valued functions onW. Forh,k∈B(W), define
M(h,k,t) =e–d(ht,k),
where
d(h,k) =sup
x∈W
h(x) –k(x)=h–k.
Then (B(W),M,·) is a complete fuzzy metric space. For every (x,y)∈W×D,h,k∈B(W) andx∈W, define
Th(x) =sup
y∈D
g(x,y) + x,y,hτ(x,y) ,
Sh(x) =sup
y∈D
h(x,y) +x,y,hτ(x,y) .
Suppose that the following conditions hold:
(D-) ,,gandhare bounded.
(D-) There existr∈[, )andη∈such that for every(x,y)∈W×D,h,k∈B(W)
andx∈W, we have
e–(d( (x,y,h(τ(x,y))),rt (x,y,k(τ(x,y)))))≥ηe–(
d(Sh(x),Sk(x))
t ) .
(D-) For anyh∈B(W)andx∈W, there existsk∈B(W)such thatTh(x) =Sk(x). (D-) There existsh∈B(W)such thatTh(x) =Sh(x)implies thatSTh(x) =TSh(x).
Theorem . If conditions(D-)-(D-)are satisfied,then the system of functional equa-tions()has a unique bounded and common solution in B(W).
Proof Note thatTis a self-map onB(W) and (B(W),M,·) is a complete fuzzy metric space. Ifh,k∈B(W), then for every real numberαandx∈W,there existy,y∈Dsuch that
Th(x) <g(x,y) +
x,y,h(τ) +α and ()
Tk(x) <g(x,y) +
x,y,k(τ) +α, ()
Then we have
Th(x) ≥g(x,y) +
x,y,h(τ) , ()
Tk(x) ≥g(x,y) +
x,y,k(τ) . ()
Thus, from () and (), we obtain
Th(x) –Tk(x) < x,y,h(τ) –
x,y,k(τ) +α
≤ x,y,h(τ) –
x,y,k(τ) +α.
Also, from () and (), we have
Tk(x) –Th(x) < x,y,k(τ) –
x,y,h(τ) +α
≤ x,y,k(τ) –
x,y,h(τ) +α.
Sinceα> was taken as an arbitrary number, we obtain
dTh(x),Tk(x) =Th(x) –Tk(x)
≤ x,y,k(τ) –
x,y,h(τ) .
Thus
e–d(T(h(x)),rtT(k(x))) ≥e–
| (x,y,k(τ))– (x,y,h(τ))|
rt
≥e–(d( (x,y,k(τ)),rt (x,y,h(τ))))
≥ηe–d(Sh(x),tSk(x)) .
Then it follows that
MTh(x) ,Tk(x) ,rt ≥ηMSh(x),Sk(x),t .
Therefore all the conditions of Corollary . are satisfied for mappingsT andS. Hence the system of functional equations () has a unique bounded and common solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, National University of Computer and Emerging Sciences, Lahore, Pakistan.2Department of
Mathematics, School of Science and Technology, University of Management and Technology, C-II, Johar town, Lahore, Pakistan. 3Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria, 0002,
South Africa.
Acknowledgements
We appreciate the reviewer’s careful reading and remarks which helped us to improve the paper.
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