# On theα−ψ-Contractive Mappings in C∗-Algebra Valued b-Metric Spaces and Fixed Point Theorems

Authors: Saleh Omran; Ibtisam Masmali
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, for a unitalC∗-algebra A, we introduce a version of α-ψ-contractive mappings in C∗-algebra valued b-metric spaces, and we prove some Banach fixed point theorems and give some examples to illustrate our results.


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## Body

## 1. Introduction

Ma et al. [1] introduced the notion of C∗-algebra valued metric spaces, where the set of real number was replaced by the positive cone of a unital C∗-algebra. Later in [2], the class of C∗-algebra valued b-metric spaces is considered. Many results are introduced in this direction (see [3–10]). The notion of α-ψ-contractive mappings in metric spaces was introduced by Samet et al. [11]. Later in [12], Samet developed the notion of α-ψ-contractive mappings in b-metric spaces. Several results have been introduced in some related studies of α-admissible and α−ψ-contractive mappings and related fixed point theorems [13–22]. In this present work, we introduced a version of α−ψ-contractive mapping in a unital C∗-algebra valued b-metric spaces and proved some basic Banach fixed point theorems.Some nontrivial examples are given to support our results. Suppose thatA is a unital C∗-algebra with a unit IA. Set Ah=x∈A:x=x∗. An element x∈A is a positive element, if x=x∗ and σx⊂ℝ+ is the spectrum of x. We define a partial ordering ⪯¯ on A as x⪯¯y if 0A⪯¯y−x, where 0A means the zero element in A, and we let A+ denote the x∈A:x⪰0A and x=x∗x1/2.Lemma 1.
Suppose thatA is a unital C∗-algebra with unit IA. The following holds:(1)
Ifa∈A, with a<1/2, then 1−a is invertible and a1−a−1<1(2)
For anyx∈A and a,b∈A+, such that a⪯¯b, we have x∗ax and x∗bx which are positive element and x∗ax⪯¯x∗bx(3)
If0A⪯¯a⪯¯b, then a≤b(4)
Ifa,b∈A+ and ab=ba, then a.b⪰0A(5)
LetA′ denote the set a∈A:ab=ba∀b∈A and let a∈A′, if b,c∈A with b⪰c⪰0A and 1−a∈A′+ is an invertible element, then IA−a−1b⪯IA−a−1c
We refer [23] for more C∗ algebra details.Definition 1.
LetX be a nonempty set and b⪰IA, b∈A′, suppose the mapping dA:X×X⟶A satisfies the following:(1)
dAx,y⪰0A for all x,y∈X and dAx,y=0A⇔x=y.(2)
dAx,y=dAy,x for all x,y∈X.(3)
dAx,z⪯¯bdAx,y+dAy,z for all x,y,z∈X, where 0A is zero element in A and IA is the unit element in A. Then, dA is called a C∗-algebra valued b-metric on X and X,A,dA is called C∗-algebra valued b-metric space.Definition 2.
LetX,A,dA be a C∗-algebra valued b-metric space, x∈X, and xnn=1∞ be a sequence in X, then(i)
xnn=1∞ is convergent to x whenever, for every c∈A with c≻0A, there is a natural number N∈ℕ such that(1)dAxn,x⪯c,for all n>N. We denote this by limn⟶∞xn=x or xn⟶x as n⟶+∞.(ii)
xnn=1∞ is said to be a Cauchy sequence whenever, for every c∈A with c≻0A, there is a natural number N∈ℕ such that(2)dAxn,xm⪯c,for all n,m>N.Lemma 2.
(i)
xnn=1∞ is a convergence sequence in X if for any element ε>0 there is N∈ℕ such that for all n>N, dxn,x≤ε.(ii)
xnn=1∞ is a Cauchy sequence in X, if for any ϵ>0 there is N∈ℕ such that dAxn,xm≤ε, for all n,m>N. We say that X,A,dA is a complete C∗-algebra valued b-metric space if every Cauchy sequence is convergent with respect to A.Example 1.
LetX=ℝ and A=M2ℂ be the set of all 2×2 matrices with entries in ℂ, and M2ℂ is a C∗-algebra with the matrix norm. Define(3)dAa,b=λ1a11−b11p00λ1a11−b11p,where a=aiji,j=12 and b=biji,j=12 are two 2×2-matrices, aij,bij∈ℂ, for all i,j=1,2, λ1,λ2>0.
One can define a partial ordering⪯¯ on M2ℂ as following a⪯¯b if and only if aij≤bij∀i,j=1,2.
And an elementa⪰0 is positive in M2ℂ if and only if aij≥0 for all i,j=1,2, we denote M2ℂ+ the set of all positive element in M2ℂ. Then, X,M2ℂ,dM2ℂ is C∗-algebra valued b-metric space.Definition 4.
Ifψ:A⟶B is a linear mapping in C∗-algebra, it is said to be positive if ψA+⊆B+. In this case, ψAh⊆Bh, and the restriction map: ψ:Ah⟶Bh is increasing.Definition 5.
Suppose thatA and B are C∗-algebras. A mapping ψ:A⟶B is said to be C∗-homomorphism if(a)
ψax+by=aψx+bψy for all a,b∈ℂ and x,y∈A(b)
ψxy=ψxψy∀x,y∈A(c)
ψx∗=ψx∗∀x∈A(d)
ψ maps the unit in A to the unit in BDefinition 6.
LetΨA be the set of positive functions ψA:A+⟶A+ satisfying the following conditions:(a)
ψAa is continuous and nondecreasing(b)
ψAa=0 iff a=0(c)
∑n=1∞ψAna<∞, limn⟶∞ψAna=0 for each a≻0, where ψAn is nth iterate of ψA(d)
The series∑k=0∞bkψAka<∞ for a≻0 is increasing and continuous at 0Corollary 1.
EveryC∗-homomorphism is contractive and hence bounded.Lemma 3.
Every∗-homomorphism is positive.


## 2. Main Results

In [11] Samet et al. and in [12] Samet introduced the concept of α-ψ-contractive mappings in metric space and α-ψ-contractive mappings in b-metric space, respectively. Here, we will develop the definitions in case of unital C∗-algebra and study some Banach fixed point theorems.Definition 7.
(see [11]). Let T:X⟶X be self map and α:X×X⟶0,+∞. Then, T is called α-admissible if for all x,y∈X with αx,y≥1 implies αTx,Ty≥1.Definition 8.
LetX be a nonempty set and αA:X×X⟶A′+ be a function, we say that the self map T is αA-admissible if x,y∈X×X, αAx,y⪰IA⇒αATx,Ty⪰IA, where IA the unit of A.Definition 9.
LetX,A,dA be a C∗-algebra valued b-metric space and T:X⟶X is mapping, we say that T is an αA-ψA-contractive mapping if there exist two functions αA:X×X⟶A+ and ψA∈ΨA such that(4)αAx,ydATx,Ty⪯¯ψAdAx,y,for all x,y∈X.Theorem 1 (Banach version fixed point).
LetX,A,dA be a complete C∗-algebra valued b-metric space and T:X⟶X be an αA-ψA-contractive mapping satisfying the following conditions:(i)
T is αA-admissible(ii)
There existsx0∈X such that αAx0,Tx0⪰IA(iii)
T is continuous
Then,T has a fixed point in X.Proof.
Letx0∈X such that αAx0,Tx0⪰IA and define a sequence xnn=0∞ in X such that xn+1=Txn for all n∈ℕ. If xn=xn+1 for some n∈ℕ, then xn is a fixed point for T.
Suppose thatxn≠xn+1 for all n∈ℕ, since T is αA-admissible, we get(5)αAx0,x1=αAx0,Tx0⪰IA⇒αATx0,T2x0=αAx1,x2⪰IA.
By induction, we have(6)αAxn,xn+1⪰IA,for alln∈ℕ.
By inequalities (4) and (6), we get(7)dAxn,xn+1=dATxn−1,Txn,⪯¯αAxn−1,xndATxn−1,Txn⪯¯ψAdAxn−1,xn.
By induction, we obtain(8)dAxn,xn+1⪯¯ψAndAx0,x1,for alln∈ℕ.
Form≥1 and p≥1, it follows that(9)dAxm,xm+p⪯¯bdAxm,xm+1+dAxm+1,xm+p⪯¯bdAxm,xm+1+b2dAxm+1,xm+2+⋯+bp−1dAxm+p−2,xm+p−1+bpdAxm+p−1,xm+p⪯¯bψAmdAx0,x1+b2ψAm+1dAx0,x1+bp−1ψAm+p−2dAx0,x1+bpψAm+p−1dAx0,x1=∑k=1p−1bkψAm+k−1dAx0,x1+bpψAm+p−1dAx0,x1.
Sinceb⪰IA, using Definition 6, we obtain(10)dAxm,xm+p⪯¯∑k=1p−1bkψAm+k−1dAx0,x1+bpψAm+p−1dAx0,x1⟶0A,asn⟶+∞.
Thus,xnn=0∞ is a Cauchy sequence in X. Since X,A,dA is complete, there exists x∈X such that xn⟶x as n⟶+∞, from continuity of T if it follows that xn+1=Txn⟶Tx as n⟶+∞. And by uniqueness of the limit, we get Tx=x, that is, x is a fixed point of T. To prove the uniqueness of the fixed point, we will consider the following condition. HA: for all x,y∈X, there exists z∈X such that αAx,z⪰IA and αAy,z⪰IA.Theorem 2.
Adding conditionHA to the hypothesis of Theorem 1, we obtain the uniqueness of the fixed point of T.Proof.
Suppose thatx and y are two fixed points of T. From HA, there exists z∈X such that(11)αAx,z⪰IA,αAy,z⪰IA.
SinceT is αA-admissible, we get (12)αAx,Tnz⪰IA,αAy,Tnz⪰IA,for alln∈ℕ.
Using (4) and (12), we obtain(13)dAx,Tnz=dATx,TTn−1z,⪯¯αAx,Tn−1zdATx,TTn−1z⪯¯ψAndAx,z,for alln∈ℕ⟶0Aasn⟶+∞.
Thus,Tnz=x. Similarly Tnz=y as n⟶+∞. So, the uniqueness of the limit gives x=y. This completes the proof.Theorem 3 (Kannan version fixed point).
.LetX,A,dA be a complete C∗-algebra valued b-metric space and T:X⟶X be a mapping satisfying(14)αAx,ydATx,Ty⪯¯ψAdATx,x+dATy,y,forx,y∈X, where(15)αA:X×X⟶A+,andψA∈ΨA,and the following conditions holds:(i)
T is αA-admissible(ii)
There existsx0∈X such that αAx0,Tx0⪰IA(iii)
T is continuous
Then,T has a fixed point in X.Proof.
Following the proof of Theorem1, we get (16)αAxn,xn+1⪰IA,for all n∈ℕ.
By inequalities (14) and (16), we obtain(17)dAxn,xn+1=dATxn−1,Txn,⪯¯αAxn−1,xndATxn−1,Txn⪯¯ψAdATxn−1,xn−1+dATxn,xn=ψAdAxn,xn−1+dAxn+1,xn=ψAdAxn,xn−1+ψAdAxn,xn+11−ψAdAxn,xn+1⪯¯ψAdAxn,xn−1dAxn,xn+1⪯¯ψA1−ψA−1dAxn,xn−1,from Lemma 1 and Definition 6, and let φ=ψA1−ψA−1=φA∑n=0∞ψAn=∑n=0∞ψAn<∞.
So, we getdAxn,xn+1⪯¯φAdAxn,xn+1.
By induction, we obtain(18)dAxn,xn+1⪯¯φAndAx0,x1,for alln∈ℕ.
Form≥1 and p≥1, it follows by similar calculation in Theorem 1 that(19)dAxm,xm+p=∑k=1p−1bkφAm+k−1dAx0,x1+bpφAm+p−1dAx0,x1.
Sinceb⪰IA, using Definition 6, we obtain(20)dAxm,xm+p⪯¯∑k=1p−1bkφAm+k−1dAx0,x1+bpφAm+p−1dAx0,x1⟶0A,asn⟶+∞.
Thus,xnn=0∞ is a Cauchy sequence in X. Since X,A,dA is complete, there exists x∈X such that xn⟶x as n⟶+∞, and from continuity of T, it follows that xn+1=Txn⟶Tx as n⟶+∞.
And by uniqueness of the limit, we getTx=x; that is, x is a fixed point of T.
Now, ify≠x is another fixed point of T, then(21)0A⪯¯dAx,y=dATx,Ty,⪯¯αAx,ydATx,Ty⪯¯psiAdATx,x+dATy,y=ψAdAx,x+dAy,y=psiA0=0A.
This implies thatdAx,y=0A. That is, x=y, and this complete the proof.Theorem 4 (Banach–Kannan version fixed point).
LetX,A,dA be a complete C∗-algebra valued b-metric space and T:X⟶X be a mapping satisfying(22)αAx,ydATx,Ty⪯¯ψAdAx,y+dATx,x+dATy,y,forx,y∈X, where(23)αA:X×X⟶A+,andψA∈ΨA,such that ψA1−ψA−1⪯¯1/2IA, and the following conditions hold:(i)
T is αA-admissible(ii)
There existsx0∈X such that αAx0,Tx0⪰IA(iii)
T is continuous
Then,T has a fixed point in X.Proof.
Following the proof of Theorem1, we get(24)αAxn,xn+1⪰IA,for alln∈ℕ.
By using inequalities (22) and (24), we have(25)dAxn,xn+1=dATxn−1,Txn,⪯¯αAxn−1,xndATxn−1,Txn⪯¯ψAdAxn−1,xn+dATxn−1,xn−1+dATxn,xn=ψAdAxn,xn−12IA+dAxn,xn+1,
SinceψA is additive, we get(26)1−ψAdAxn,xn+1⪯¯2IAψAdAxn,xn−1dAxn,xn+1⪯¯2IA1−ψA−1ψAdAxn,xn−1,and putting 1−ψA−1ψA=φAIA/2, we get(27)dAxn,xn+1⪯¯φAndAx0,x1,for all n∈ℕ; for m≥1 and p≥1 and by similar calculation as in the proof of Theorem 1, we get(28)dAxm,xm+p⪯¯∑k=1p−1bnφAm+k+1dAx0,x1+bpφAm+p−1dAx0,x1⟶0A,as n⟶+∞. This x is a fixed point of T.
To prove the uniqueness part of the fixed point ofT, if y≠x is another fixed point of T, we have(29)0A⪯¯dAx,y=dATx,Ty,⪯¯αAx,ydATx,Ty⪯¯psiAdAx,y+dATx,x+dATy,y=ψAdAx,y+dAx,x+dAy,y⪯¯ψAdAx,y⪯¯dAx,y.
This is a contraction, sodAx,y=0A, and this gives x=y. This completes the proof.Example 2.
LetX=ℝ and A=M2ℂ as given in Example 1, define T:X⟶X, by Tx=x/2, and αM2ℂ:X×X⟶M2ℂ+ and αM2ℂx,y=IM2ℂ, so αM2ℂTx,Ty=αM2ℂx/2,y/2=IM2ℂ; thus, T is αM2ℂ−admissible, where M2ℂ+ is the set of all positive elements in M2ℂ. Define ψM2ℂ:M2ℂ+⟶M2ℂ+, ψM2ℂa=a/2. This is clear that αM2ℂ−ψM2ℂ-contractive mapping and satisfies αM2ℂx,y.dM2ℂTx,Ty⪯¯ψM2ℂdM2ℂx,y for all x,y∈X.


## 3. Applications

In this section, we shall apply Theorem1 to prove the existence and uniqueness of solution an integral equation in C∗-algebra.Example 3.
LetE be a compact Hausdorff space, we denote by CE the algebra of all complex-valued continuous functions on E with pointwise addition and multiplication. The algebra CE with the involution defined by f∗t=ft¯ for each f∈CE, t∈E and with the norm f∞=supft,t∈E is a commutative C∗-algebra, with unit ICE is the constant function. Let C+E=f∈CE:ft¯=ft,ft≥0 denote the positive Cone of CE, with partial order relation f≤g if and only if ft≤gt. Put dCE:CE×CE⟶CE as dCEf,g=supt∈Eft−gtp.ICE. It is clear that CE,CE,dCE is a complete C∗-algebra valued b-metric space.Theorem 5 (Application).
Consider the integral equation(30)xt=∫EFt,xsds+ht,where E is the compact topological Hausdorff space. Suppose(1)
F:E×ℝ⟶ℝ.(2)
There exists a continuous functionϕ:E×E⟶ℝ and k∈0,1 such that(31)Ft,fs−Ft,gs≤kϕt,sfs−gs,for allf,g∈CE, t,s∈E.(3)
supt∈E∫Eϕt,sds≤1, then the integral equation (30) has a unique solution x∗=x¯∈CE.Proof.
LetX=CE and A=CE,dCE as in the Example3, CE,CE,dCE is a complete C∗-algebra valued b-metric space, and let T:CEE⟶CE given by Txt=∫EFt,xsds+ht, x,h∈CE,t,s∈E. αCE:CE×CE⟶C+E defined by αCEf,g=f,g.IE. And ψCE:C+E⟶C+E defined by ψCEf=f.
Now,(32)dCETf,Tg=supt∈ETft−Tgtp.ICE,=supt∈E∫EFt,fsds−Ft,gsdsp.ICEdt≤supt∈E∫Eϕt,sdspfs−gsp.ICEdt≤kpdCEf,g.
PutA=kp since k∈0,1, this gives A≤1, and we get(33)αCEdCETf,Tg≤AψCEdCEf,g≤ψCEdCEf,g,for all f,g∈CE.
Thus,T is an αCE−ψCE-contractive mapping and satisfies Theorem 1. So, T has a unique fixed point, and the integral equation (30) has a unique solution x∗=x¯∈CE.


## 4. Conclusions

In this paper, we define a new version ofαA-ψA-admissible in the case of self mappings T:A⟶A. We prove the principal Banach fixed point theorem, Kannan fixed point theorem, and Banach–Kannan fixed point theorem in the C∗-algebra valued b-metric space, which generalized the given results in [1, 2, 11, 12, 24].



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