\(C^{*}\)-Valued contractive type mappings

R E S E A R C H

Open Access

C

∗

-Valued contractive type mappings

Samina Batul

_{and Tayyab Kamran}

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Quaid-i-Azam University, Islamabad, Pakistan

Full list of author information is available at the end of the article

Abstract

In this paper we generalize the notion ofC∗-valued contraction mappings, recently introduced by Maet al., by weakening the contractive condition introduced by them. Using the new notion ofC∗-valued contractive type mappings, we establish a fixed point theorem for such mappings. Our result generalizes the result by Maet al.and those contained therein except for the uniqueness.

Keywords: C∗-algebra; contractions; fixed point theorems; orbits

1 Introduction and preliminaries

Let (X,d) be a metric space. A mappingT:X→Xis said to be a contraction if∃α∈(, ) such that

d(Tx,Ty)≤αd(x,y). ()

One of the most important tools used for the existence of solutions of many nonlinear problems arising in physics and engineering sciences is the Banach fixed point theorem which asserts that every contraction on a complete metric space has a unique fixed point. This theorem is also known as the Banach contraction principle (BCP), and it first ap-peared in its explicit form in Banach’s PhD Thesis []. The strength of BCP lies in the fact that the underlying space is a quite general, complete metric space, while the conclusion is very strong, including even error estimates. Note that a mappingT :X→X satisfy-ing () is uniformly continuous onX. Therefore, the Banach contraction condition forces the mappingTto be continuous. Given a mappingT:X→Xandx∈X, the sequence of pointsOT(x) ={x,Tx,Tx, . . .}is called the orbit ofxwith respect toT. Hicks and Rhoades

[] showed that if a mappingT:X→Xsatisfies the following contractive condition

dTy,Ty≤hd(y,Ty) ()

for someh∈(, ) and everyy∈OT(x), thenThas a fixed point. Note that the

contrac-tive condition () is weaker than condition (). Moreover, condition () does not force the mappingTto be continuous []. In contrast to the Banach contraction principle, the Hicks and Rhodes theorem [] does not guarantee the uniqueness of the fixed point ofT.

Recently, Maet al.[] introduced the notion ofC∗-valued metric spaces and, analogous to the Banach contraction principle, established a fixed point theorem forC∗-valued con-traction mappings. In this paper, we first introduce the notion of continuity in the context

ofC∗-valued metric spaces and show that aC∗-valued contraction map is continuous with respect to our notion of continuity. Then we introduce aC∗-valued contractive type con-dition and establish a fixed point theorem analogous to the results presented in []. We also show that aC∗-valued contractive type map need not be continuous in the context of

C∗-valued metric.

We now recollect some basic definitions, notations, and results that will be used sub-sequently. For details, we refer to [, ]. An algebra Atogether with a conjugate lin-ear involution map ∗:A→A, defined bya→a∗ such that for all a,b∈Awe have (ab)∗=b∗a∗ and (a∗)∗=a, is called a∗-algebra. Moreover, ifAcontains an identity ele-ment A, then the pair (A,∗) is called a unital∗-algebra. A unital∗-algebra (A,∗) together

with a complete sub multiplicative norm satisfyinga∗=afor alla∈Ais called a Banach∗-algebra. AC∗-algebra is a Banach∗-algebra (A,∗) such thata∗a=a_for

alla∈A. An elementa∈Ais called a positive element ifa=a∗ andσ(a)⊂R+, where

σ(a) ={λ∈R:λI–ais non-invertible}. Ifa∈Ais positive, we write it asaA. Using

positive elements, one can define a partial ordering onAas follows:baif and only if

b–aA. Each positive elementaof aC∗-algebraAhas a unique positive square root.

Subsequently,Awill denote a unitalC∗-algebra with the identity element A. Further,A+

is the set{a∈A:aA}of positive elements ofAand (a∗a)/_{=|a|. Using the concept of}

positive elements inA, aC∗-algebra-valued metric space is defined in the following way.

Definition .[] LetXbe a nonempty set. AC∗-algebra-valued metric onXis a mapping

d:X×X→Asatisfying the following conditions:

(i) Ad(x,y)for allx,y∈Xandd(x,y) = A⇔x=y, (ii) d(x,y) =d(y,x)∀x,y∈X,

(iii) d(x,y)d(x,z) +d(z,y)∀x,y,z∈X.

The triplet (X,A,d) is called aC∗-algebra-valued metric space.

A sequence{xn}in (X,A,d) is said to converge tox∈Xwith respect toAif for any>  there existsN∈Nsuch thatd(xn,x)< for alln>N. We write it aslimn→∞xn=x.

A sequence{xn}is called a Cauchy sequence with respect toAif for any>  there exists

N∈Nsuch thatd(xn,xm)<for alln,m>N. The triplet (X,A,d) is said to be a complete C∗-valued metric space if every Cauchy sequence with respect toAis convergent. Now we state the definition and result from [], for convenience.

Definition .[] Let (X,A,d) be aC∗-algebra-valued metric space. A mappingT:X→

Xis said to be aC∗-algebra-valued contraction mapping onXif there existsa∈Awith

a<  such that

d(Tx,Ty)a∗d(x,y)a for allx,y∈X. ()

Theorem .[] Let(X,A,d)be a C∗-algebra-valued complete metric space and T:X→

X satisfy(),then T has a unique fixed point in X.

2 Main results

We begin this section by introducing the notion of continuity in the context ofC∗-valued metric spaces.

Definition . Let (X,A,d) be aC∗-valued metric space. A mappingT :X→Xis said to be continuous atx with respect toAif given any >  there existsδ>  such that

d(Tx,Tx)<wheneverd(x,x)<δ.Tis said to be continuous onXwith respect to

Aif it is continuous for everyx∈X.

Example . LetA=R_{, thenAis aC}∗_{-algebra with pointwise operations of addition,}

multiplication, and scaler multiplication. The norm onAis defined by

(x,y)=max|x|,|y|, ()

where ordering onAis given by

(a,b)(c,d) ⇔ a≤candb≤d. ()

LetX= [, ], define aC∗-valued metricd:X×X→AonXby

d(x,y) =|x–y|, . ()

ThenT:X→X, given byT(x) = x_, is continuous with respect toAsince

d(Tx,Ty)=d

x

,

y



=x –

y



< wheneverx–y<δ= .

Remark . Note that every continuous self-map is continuous with respectA=Rand a

C∗-valued contraction map is continuous with respect to theC∗-algebraA.

Definition . A functionf :X→Ais said to beT-orbitally lower semicontinuous atξ

with respect toAif there exist a mappingT:X→Xand a sequence{xn}inOT(x), for

somex∈X, such thatlimn→∞xn=ξwith respect toAimplies

f(ξ)≤lim inff(xn). ()

Remark . IfA=R, then our definition coincides with the usual definition ofT-orbitally lower semicontinuous as defined by [].

Example . Consider theC∗-algebraA=R_{as defined in Example .. LetX= [–, ]}

and definef:X→Aby

f(x) = ⎧ ⎨ ⎩

(_x, ) ifx≥,

(|x– |, ) ifx< .

By takingT:X→X,Tx=x

, we see that, for 

∈[–, ], we have

OT

 

=

 ,

 ,

 ,

 , . . .

and any sequence{xn}inXconverges to . Further,

f()=(, )=lim inff(xn).

Thusf isT-orbitally lower semicontinuous atx= .

Definition . Let (X,A,d) be aC∗-valued metric space. A mappingT:X→Xis said to be aC∗-valued contractive type mapping if∃x∈Xanda∈Asuch that

dTy,Tya∗d(y,Ty)a witha<  for everyy∈OT(x). ()

Remark . AC∗-valued contraction mapping is aC∗-valued contractive type mapping, but the converse is not true as shown in the following example.

Example . LetX= [–, ] andA=M×(R) withA=max{|a|,|a|,|a|,|a|}, where

ai’s are the entries of the matrixA∈M×(R). Then (X,A,d) is aC∗-algebra-valued metric

space, where

d(x,y) =

|x–y|   |x–y|

,

and partial ordering onAis given as

a a

a a

b b

b b

⇔ ai≥bi fori= , , , .

Define a mappingT:X→Xby

T(x) = ⎧ ⎨ ⎩

x

 ifx≥,

 ifx< .

Then, fory∈OT(x),x≥,

dTy,Ty=

|y

–

y

| 

 |y_–_y|

= _

√

 

 √



|y

| 

 |y_|



√

 

 √



=a∗d(y,Ty)a,

wherea= _

√

 

 √



anda=√ .

ThusTis aC∗-valued contractive type mapping. Note thatT is not continuous with re-spect to theC∗-algebraAand hence not aC∗-valued contraction mapping.

Lemma . LetAbe a C∗-algebra with the identity elementAand x be a positive element

ofA.If a∈Ais such thata< ,then for m<n we have

lim

n→∞

n

k=m

a∗kxak_{= }

A(x)/

am

 –a

()

and

n

k=m

a∗kxak_−→

A as m−→ ∞. ()

Proof Sincexis a positive element ofA, we have

n

k=m

a∗kxak =

n

k=m

a∗k(x)/(x)/ak

=

n

k=m

(x)/ak∗(x)/ak

=

n

k=m

(x)/ak

A

n

k=m

(x)/ak

A n

k=m

(x)/ak

= A(x)/ n

k=m

ak.

Sincea<  andm<n, thereforem−→ ∞implies thatn−→ ∞. The proof of () follows from the fact thatn_k=mak is a geometric series. Moreover,m−→ ∞ ⇒ am−→

and hence () follows from ().

We are now ready to state and prove our main result.

Theorem . Let(X,A,d)be a complete C∗-valued metric space and T:X→X be a C∗-valued contractive type mapping.Then

(A) ∃x∈Xsuch that the sequenceTnxconverges tox,

(A) d(Tn_x,x

)a

n

–ad(x,Tx)  _A,

(A) xis a fixed point ofT if and only ifG(x) =d(x,Tx)isT-orbitally lower

semicontinuous atxwith respect toA.

Proof IfA={A}, then there is nothing to prove. Assume thatA={A}.

(A): Letx∈Xand consider the orbitOT(x). Since condition () holds for each element

dTx,Tx=dT(Tx),T(Tx)

a∗dTx,T(Tx)a

=a∗dTx,Txa

a∗a∗d(x,Tx)aa

=a∗d(x,Tx)a.

Continuing in this way, one can show that

dTnx,Tn+xa∗nd(x,Tx)an. ()

Let{Tn_{x}be a sequence inO}

T(x). Then, form<n, from the triangle inequality and ()

we have

dTn+x,TmxdTmx,Tm+x+dTm+x,Tm+x+· · ·+dTnx,Tn+x

a∗md(x,Tx)am+a∗m+d(x,Tx)am++· · ·+a∗nd(x,Tx)an

=

n

k=m

a∗kd(x,Tx)ak−→A asm−→ ∞

using () of Lemma .. This shows that{Tn_{x}is a Cauchy sequence inO}

T(x)⊂Xwith

respect toA. Since (X,A,d) is a completeC∗-valued metric space, there existsx∈Xsuch

thatTn_x−→x

. This completes the proof of (A).

(A): It follows again from the triangle inequality and () that

dTnx,Tn+mxdTnx,Tn+x+dTn+x,Tn+x+· · ·+dTn+m–x,Tn+mx

a∗nd(x,Tx)an+a∗n+d(x,Tx)an++· · ·

+a∗n+m–d(x,Tx)an+m–

=

n+m–

k=n

a∗kd(x,Tx)ak.

Sinced(x,Tx) is a positive element ofA, using () of Lemma . and lettingm−→ ∞, we conclude (A).

(A): To prove (A), ifTx=xand{xn}is a sequence in OT(x) withxn−→x with

respect to A, then G(x)=d(Tx,x)= ≤lim infG(xn). Conversely, ifGis T

-orbitally lower semicontinuous atx, then

G(x)=d(x,Tx)≤lim infG

Tnx

=lim infdTnx,Tn+x

≤lim inf a

n

 –ad(x,Tx)

 _{= .}

Remark . Note that:

() By takingA=R, we see that the main result of [] follows immediately from

Theorem ..

() Theorem . is a special case of Theorem . except for the uniqueness of a fixed point of the mapping involved.

The following example shows that our result properly generalizes Theorem ..

Example . Consider theC∗-algebraA=R_{with component-wise operations where}

norm and ordering are given by () and (), respectively. LetX= [–, ]×[–, ] and define theC∗-valued metricd:X×X→Rbyd(x,y) = (|x–y|,|x–y|) for allx= (x,x),y=

(y,y)∈X. DefineT:X→Xby

T(x,x) =

⎧ ⎨ ⎩

(x

,

x

) ifx,x≥,

(, ) otherwise.

Taking (u,u)∈Xsuch that  <u,u< , we have

OT

(u,u)

=

(u,u),

u

,

u



,

u

,

u



, . . .

.

For anyun= (_un– ,

u

n–)∈OT((u,u)), we have

dTun,Tun

=a∗d(un,Tun)a,

wherea= (_√ ,



√

). Note thatun→(, ). Further,G:X→Adefined byG(x) =d(x,Tx) is

T-orbitally lower semicontinuous at (, ). Therefore, all conditions of Theorem . are satisfied and (, ) is the fixed point ofT. Note that Theorem . is not applicable here sinceTis not continuous at (, ) with respect toA.

3 Application

In this section we provide the existence result for an integral equation as an application ofC∗-valued contractive type mappings on completeC∗-valued metric spaces. LetEbe a Lebesgue measurable set,X=L∞(E), andH=L_{(E). We denote the set of all bounded}

linear operators on a Hilbert spaceHbyL(H). With the usual operator norm,L(H) is aC∗ -algebra. ForS,T∈X, defined:X×X→L(H) byd(T,S) =π|T–S|, whereπh:H→His the

multiplication operator given byπh(φ) =h·φforφ∈H. Then (X,L(H),d) is a complete C∗-valued metric space [].

Example . LetE,X,H, and the metricdbe as above. Suppose that

() K:E×E×R→R, and letTbe a self-mapping onX,

() there exists a continuous functionφ:E×E→Randα∈(, )such that for every x∈X,y∈OT(x), andt,s∈E, we have

Kt,s,x(s)–Kt,s,y(s)≤αφ(t,s)x(s) –y(s). ()

Then the integral equation

x(t) =

E

Kt,s,x(s)ds, t∈E

has a solution.

Proof Here (X,L(H),d) is a completeC∗-valued metric space with respect toL(H). LetT:X→Xbe

Tx(t) =

E

Kt,s,x(s)ds, t∈E.

LetTx=y, then

dTx,Tx=d(Tx,Ty)

=π|Tx–Ty|

= sup

h=

π|Tx–Ty|h,h for anyh∈H

= sup

h=

E

_EKt,s,x(s)–Kt,s,y(s)ds

h(t)h(t)dt

≤ sup

h=

E

_EKt,s,x(s)–Kt,s,y(s)dsh(t)dt

≤ sup

h=

E

E|

kφ(t,s)x(s) –y(s)|dsh(t)dt

≤ksup

h=

E

E

φ(t,s)dsh(t)dt· x–y∞

≤ksup

t∈E

E

φ(t,s)ds· sup

h=

E

h(t)dt· x–y∞

≤kx–y∞

=ad(x,y)=ad(x,Tx).

Settinga=kI, we havea∈L(H)+anda=k. Thus all the conditions of Theorem .

hold and hence the conclusion.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Department of Mathematics, Mohammad Ali Jinnah University, Islamabad, Pakistan.}2_{Department of Mathematics,}

Quaid-i-Azam University, Islamabad, Pakistan.

Acknowledgements

The authors are thankful to the reviewers for their useful comments and suggestions.

References

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3, 133-181 (1922)

2. Hicks, TL, Rhoades, BE: A Banach type fixed point theorem. Math. Jpn.24, 327-330 (1979)

3. Ma, Z, Jiang, L, Sun, H:C∗-Algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory Appl.

2014, 206 (2014). doi:10.1186/1687-1812-2014-206