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On the α−ψ− Contractive Mappings in C∗-Algebra Valued b-Rectangular Metric Spaces and

Fixed Point Theorems

Mohamed Rossafi1,∗, Abdelkarim Kari2, Hafida Massit3
1LaSMA Laboratory Department of Mathematics, Faculty of Sciences Dhar El Mahraz,

University Sidi Mohamed Ben Abdellah, B. P. 1796 Fes Atlas, Morocco
mohamed.rossafi@usmba.ac.ma

2AMS Laboratory Faculty of Sciences Ben M’Sik, Hassan II University, Casablanca, Morocco
abdkrimkariprofes@gmail.com

3Laboratory of Partial Differential Equations, Spectral Algebra and Geometry Department of Mathematics,
Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco

massithafida@yahoo.fr
∗Correspondence: mohamed.rossafi@usmba.ac.ma

Abstract. This present paper extends a version of α−ψ−contraction in C∗-algebra valued rectangu-lar b-metric spaces and establishing the existence and uniqueness of fixed point for them. Non-trivialexamples are further provided to support the hypotheses of our results.

1. Introduction
A C∗-algebra valued metric spaces were introduced by Ma et al. [6] as a generalization of metricspaces they proved certain fixed point theorems, by giving the definition of C∗-algebra valuedcontractive mapping analogous to Banach contraction principle. Many mathematicians worked onthis interesting space.Various fixed point results were established on such spaces, see [1–3] and references therein.Combining conditions used for definitions of C∗-algebra valued metric and generalized metricspaces, G Kalapana and Tasneem [4] announced the notions of C∗-algebra valued metric space andestablish nice results of fixed point on such space.In this paper, inspired by the work done in [9], we introduce the notion of α − ψ−contractionand establish some new fixed point theorems for mappings in the setting of complete C∗-algebravalued rectangular b- metric spaces.Moreover, an illustrative examples is presented to support the obtained results.
Received: 18 Dec 2021.
Key words and phrases. fixed point; C∗-algebra valued metric spaces; α − ψ− contraction; α − ψ − C∗ valuedcontraction. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.2.11


Eur. J. Math. Anal. 10.28924/ada/ma.2.11 22. preliminaries
Throughout this paper, we denote A by an unital (i.e ,unity element I) C∗-algebra with linearinvolution ∗, such that for all x, y ∈ A,

(xy)∗ = y∗x∗,and x∗∗ = x .We call an element x ∈ A a positive element, denote it by x � θif x ∈ Ah = {x ∈ A : x = x∗} and σ(x) ⊂ R+,where σ(x) is the spectrum of x .Using positiveelement ,we can define a partial ordering � on Ah as follows :
x � y if and only if y − x � θwhere θ means the zero element in A.

we denote the set x ∈ A : x � θ by A+ and |x | = (x∗x)

1

2 .and A′ will denote the set {a ∈ A+; ab = ba, ∀b ∈ A}

Lemma 2.1. [8] Suppose that A is a unital C∗-algebra with a unit I,(1) for any x ∈ A+ we have x � I ⇐⇒ ‖x‖ ≤ 1(2) If a ∈ A+ with ‖a‖ < 1

2
then I − a is unvertible and ‖a(1− a)−1‖ < 1(3) Suppose that a, b ∈ A+ and ab = ba, then ab � θ(4) Let a ∈ A′, if b, c ∈ A, with b � c � θ, and I − a ∈ A′+ is invertible operator, then

(I − a)−1b � (I − a)−1c

Definition 2.2. [4] Let X be a non-empty set and b ∈ A such that b � I . supposa the mapping
d : X ×X → A+ satisfies:(i) d(x, y) = θ if and only if x = y ;(ii) d(x, y) = d(y , x) for all distinct points x, y ∈ X;(iii) d(x, y) � b[d(x, u) + d(u, v) + d(v , y)] for all x, y ∈ X and for all distinct points u, v ∈

X − {x, y}.Then (X,A+, d) is called a C∗-algebra valued rectangular b−metric space.
Example 2.3. Let X = R and A = M2(R). Define d(x, y) = diag(|x − y |, 2|x − y |) where
x, y ∈ R. It is easy to verify d is a C∗− algebra-valued rectangular b− metric and (X,M2(R), d)is a copmlete C∗-algebra valued rectangular b−metric space.
Definition 2.4. [9] If ψ : A → B is a linear mapping in C∗-algebra, it is said to be positive if
ψ(A+) ⊆ B+. In this case ψ(Ah) ⊆ Bh, and the restriction map ψ : Ah → Bh is increasing.
Definition 2.5. [9] Suppose that A and B are C∗-algebra .A mapping ψ : A→ B is said to be C∗- homomorphism if :(i) ψ(ax + by) = aψ(x) + bψ(y) for all a, b ∈ C and x, y ∈ A(ii) ψ(xy) = ψ(x)ψ(y) for all x, y ∈ A

https://doi.org/10.28924/ada/ma.2.11


Eur. J. Math. Anal. 10.28924/ada/ma.2.11 3(iii) ψ(x∗) = ψ(x)∗ for all x ∈ A(iv) ψ maps the unit in A to the unit in B.
Definition 2.6. [9] Let A and B be C∗-algebra spaces and let ψ : A→ B be a homomorphismthen ψ is called an ∗− homomorphism if it is one to one ∗− homomorphism.A C∗-algebra A is ∗−isomorphic to a C∗-algebra B if there exists ∗− isomorphism of A onto B.
Definition 2.7. [9] Let Ψ be the set of positive functions ψ : A+ → A+ satisfying the followingconditions :(i) ψ is continous and nondecrasing(ii) ψ(a) = θ if and only if a = θ(iii) l imn−→∞ψn(a) = θ, (a � θ) ,∑∞n=1 ψn(a) <∞(iv) The series ∑∞k=1 bkψk(a) <∞ for a � θ is increasing and continuous at θ.
Corollary 2.8. [9] Every C∗− homomorphism is contractive and hence bounded.

Lemma 2.9. Every ∗− homomorphism is positive.

Definition 2.10. [9] Let X be a nonempty set and α : X ×X → A′+ be a function, wesay that theself map T is α− admissible if
(x, y) ∈ X ×X,α(x, y) � I ⇒ α(Tx, T y) � I ,where I the unit of A.

Definition 2.11. [9] Let (X,A, d) be a C∗-algebra valued b− metric space and T : X → X ismapping, we say that T is an α−ψ− contractive mapping if there exist two functions α : X×X →
A+ and ψ ∈ Ψ such that

α(x, y)d(Tx, T y) � ψ(d(x, y)), for all x, y ∈ X
3. Main result

In [9] introduced the concept of α−ψ− contractive mappings in a unital C∗-algebra valued b−metric space. In this paper we will develop the definitions in case of unital C∗-algebra valuedrectangular b− metric space and give some Banach fixed point theorems.
Definition 3.1. Let (X,A, d) be a C∗-algebra valued b− rectangular metric space and T : X → Xis mapping, we say that T is an α − ψ− contractive mapping if there exist two functions α :

X ×X → A+ and ψ ∈ Ψ such that
α(x, y)d(Tx, T y) � ψ(d(x, y)), f oral lx, y ∈ X (3.1)

Theorem 3.2. Let (X,A, d) be a complete C∗-algebra valued rectangular b− metric space and let
T : X → X be a α,ψ− contractive mapping satisfying the following conditions:(i) T is α− admissible

https://doi.org/10.28924/ada/ma.2.11


Eur. J. Math. Anal. 10.28924/ada/ma.2.11 4(ii) There exists x0 ∈ X such that α(x0, T x0) � I(iii) for all x, y ∈ X ,there exists z ∈ X such that α(x, z) � I and α(y , z) � I(iv) T is continuous

Then, T has a unique fixed point in X .

Proof. Let x0 ∈ X such that α(x0, T x0) � I and define a sequence {xn} ∈ X such that xn+1 = Txn,
∀n ∈ N. Suppose that there exists n ∈ N such that xn = Txn. Then xn is a fixed point of T andthe proof is finished.Hence, we assume that xn 6= Txn+1, ∀n ∈ N, since T is α−admissible, we get

α(x0, x1) = α(x0, T x0) � I ⇒ α(Tx0, T
2x0) = α(x1, x2) � I .

Continuing this process, we have
α(xn, xn+1) � I ∀n ∈ N. (3.2)

By 3.1 and 3.2, we get
d(xn, xn+1) = d(Txn−1, T xn) � α(xn−1, xn)d(Txn−1, T xn)

� ψ(d(xn−1, xn))

�

� ψn(d(x0, x1)).

For m ≥ 1 and p ≥ 1, it follows that
d(xm+p, xm) � b[d(xm+p, xm+p−1) + d(xm+p−1, xm+p−2) + d(xm+p−2, xm)]

� bd(xm+p, xm+p−1) + bd(xm+p−1, xm+p−2) + b[b[d(xm+p−2, xm+p−3) + d(xm+p−3, xm+p−4) +

d(xm+p−4, xm)]]

= bd(xm+p, xm+p−1) + bd(xm+p−1, xm+p−2) + b2d(xm+p−2, xm+p−3) + b2d(xm+p−3, xm+p−4) +

b2d(xm+p−4, xm)

� bd(xm+p, xm+p−1) + bd(xm+p−1, xm+p−2) + b2d(xm+p−2, xm+p−3) + b2d(xm+p−3, xm+p−4) +

....+ b
p−1
2 d(xm+3, xm+2) + b

p−1
2 d(xm+2, xm+1) + b

p−1
2 d(xm+1, xm)

� bψm+p−1(d(x0, x1)) + bψm+p−2(d(x0, x1)) + ...+ b
p−1
2 d(x0, x1)Since b � I , using definition 2.6 we have

d(xm, xm+p) � bψm+p−1(d(x0, x1))+bψm+p−2(d(x0, x1))+ ...+b
p−1
2 d(x0, x1)→ θ as n → +∞Therefore {xn} is a Cauchy sequence in X . By the completeness of (X,A, d) there exists an

x ∈ X such that
l imn→∞xn = l imn→∞Txn−1 = x .

From continuity of T and by uniqueness of the limit, we get Tx = x , ie. x is a fixed point of T .Now suppose that y 6= x is another fixed point of T .

https://doi.org/10.28924/ada/ma.2.11


Eur. J. Math. Anal. 10.28924/ada/ma.2.11 5From (i i i), there exists z ∈ X such that α(x, z) � I and α(y , z) � I .Since T is α− admissible, we have
α(x, T nz) � I and α(y , T nz) � I for all n ∈ N

Using (1), we obtain
d(x, T nz) = d(Tx, T (T n−1z))

� α(x, T n−1z)d(Tx, T (T n−1z))

� ψn(d(x, z))→ θ as n →∞.
Thus, T nz = x . Similary T nz = y as n →∞ So, the uniqueness of the limit we obtain x = y . �
Example 3.3. Let X = R and A = M2(R) as given in Example 2.3, define T : X → X , by Tx =

x

3and α : X ×X → M2(R) such that
α(x, y) =

(
|x − y | 0

0 0

) thus, T is α− admissible,
and ψ : M2(R)+ → M2(R)+ , ψ(a) =

(
a2 0

0 a2

)
∀a ∈ (R)+.

This is clear that T is α− ψ− contractive mapping and satisfies
α(x, y)d(Tx, T y) � ψ(d(x, y)), for all x, y ∈ X

Theorem 3.4. Let (X,A, d) be a complete C∗-algebra valued rectangular b− metric space and let
T : X → X be a α,ψ− contractive mapping of Kannan type ie,

α(x, y)d(Tx, T y) � ψ(d(Tx, x) + d(Ty, y)) (3.3)
for all x, y ∈ X Where ψ ∈ Ψ and α : X ×X → A+

and the following conditions holds:(i) T is α− admissible(ii) There exists x0 ∈ X such that α(x0, T x0) � I(iii) T is continuous

Then, T has a fixed point in X .

Proof. By (3.3), we obtain
d(xn, xn+1) = d(Txn−1, T xn) � α(xn−1, xn)d(Txn−1, T xn)

� ψ(d(Txn−1, xn−1) + d(Txn, xn)) = ψ(d(xn, xn−1) + d(xn+1, xn))

= ψ(d(xn, xn−1)) + ψ(d(xn+1, xn))

(I − ψ)(d(xn, xn−1)) � ψ(d(xn, xn−1))

from Lemma 2.1 and Definition 2.6, we obtain

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Eur. J. Math. Anal. 10.28924/ada/ma.2.11 6

d(xn, xn+1) � (I − ψ)−1ψ(d(xn, xn−1)) = Φ(d(xn, xn−1)) where
Φ = (I − ψ)−1ψ

Therefore
d(xn, xn+1) � Φn(d(x0, x1))∀n ∈ N

For any m ≥ 1 and p ≥ 1 similary in Theorem 3.1 we have
d(xm, xm+p) � bψm+p−1(d(x0, x1))+bψm+p−2(d(x0, x1))+...+b

p−1
2 d(x0, x1)→ θ as n → +∞.Thus {xn} is a Cauchy sequence in X . By the completeness of (X,A, d), there exists x ∈ Xsuch that

l imn→∞xn = l imn→∞Txn−1 = x .
the continuity of T gives that x is a fixed point of T .To prove that x is the unique fixed point, we suppose that y ∈ X is another fixed point of T .Then

θ � d(x, y) = d(Tx, T y)

� α(x, y)d(Tx, T y)

� ψ(d(Tx, x) + d(Ty, y))

= ψ(d(x, x) + d(y , y)) = θ

Hence x = y .Therefore the fixed point is unique. �

Theorem 3.5. Let (X,A, d) be a complete C∗-algebra valued rectangular b− metric space and let
T : X → X be a α,ψ− contractive mapping of Banach-Kannan type ie,

α(x, y)d(Tx, T y) � ψ(d(x, y) + d(Tx, x) + d(Ty, y)) (3.4)
for all x, y ∈ X Where ψ ∈ Ψ and α : X ×X → A+ such that ψ(1− ψ)−1 �

1

2I
,

and the following conditions holds:(i) T is α− admissible(ii) There exists x0 ∈ X such that α(x0, T x0) � I(iii) T is continuous

Then, T has a fixed point in X

Proof. Using (3.4), we get
d(xn, xn+1) = d(Txn−1, T xn) � α(xn−1, xn)d(Txn−1, T xn)

� ψ(d(xn−1, xn) + d(Txn−1, xn−1) + d(Txn, xn))

= ψ(d(xn−1, xn)2I + d(xn, xn+1)) ⇒ (I − ψ)(d(xn, xn+1)) � 2Iψ(d(xn, xn−1))

⇒ d(xn, xn+1) � 2I(I − ψ)−1ψ(d(xn, xn−1)) � Φ(d(xn, xn−1)).Where

https://doi.org/10.28924/ada/ma.2.11


Eur. J. Math. Anal. 10.28924/ada/ma.2.11 7

ϕ = 2I(I − ψ)−1ψ.
Then

d(xn, xn+1) � Φn(d(x0, x1).
We refer to the proof of the Theorem 3.1 we get that x is a fixed point of T . Now, if y 6= x isanother fixed point of T , we have

θ � d(x, y) = d(Tx, T y)

� α(x, y)d(Tx, T y)

� ψ(d(x, y) + d(Tx, x) + d(Ty, y))

= ψ(d(x, y) + d(x, x) + d(y , y)) = ψ(d(x, y).
So d(x, y) = θ ; ie x = y . �

4. Applications
As application of α − ψ contractive in unital C∗-algebra valued rectangular b− metric spaces,existence and uniqueness results for a type of operator equation is given.

Example 4.1. Suppose that H is a Hilbert space, B(H) is the set of linear bounded operators on
H. Let A1, A2, ..., An, ... ∈ B(H)which satisfy ∑∞n=1 ‖An‖ < 1 and Q ∈ B(H)+.Then the operator equation X −∑∞n=1 A∗nXAn = Q has a unique solution in B(H).
Proof. Set a = (

∑∞
n=1 ‖An‖)p with p ≥ 1, then ‖a‖ < 1.Without loss of generality, one can supposethat a > 0.Choose a positive operator M ∈ B(H).For X, Y ∈ B(H) and p ≥ 1, set

d(X, Y ) = ‖X − Y ‖pM .Then d(X, Y ) is a C∗-algebra valued rectangular b− metric.Suppose that X, Y, Z,W ∈ B(H) we have
‖X − Y ‖p � 2P (‖X − Z‖p + ‖Z −W‖p + ‖W − Y ‖p).Which implies that d(X, Y ) � A[d(X,Z) + d(Z,W ) + d(W, Y )]Where A = 2pI . Consider the map
T : B(H)→ B(H) such that T (X) =

∑∞
n=1 A

∗
nXAn +Q.Then

d(T (X), T (Y )) = ‖T (X), T (Y )‖pM
= ‖

∑∞
n=1 A

∗
n(X − Y )An‖pM

�
∑∞
n=1 ‖An‖2p‖X − Y ‖pM
� a2d(X, Y )

https://doi.org/10.28924/ada/ma.2.11


Eur. J. Math. Anal. 10.28924/ada/ma.2.11 8Let α : B(H)× B(H)→ B(H)+ defined by α(X, Y ) = (X, Y )Iand ψ : B(H)+ → B(H)+ defined by ψ(X) = X .We get α(X, Y )d(TX, TY ) � ψ(d(X, Y )).Using Theorem 3.1, there exists a unique fixed point X in B(H). �

5. Acknowledgments
It is our great pleasure to thank the referee for his careful reading of the paper and for severalhelpful suggestions.

References
[1] H.H. Alsulami, R.P. Agarwal, E. Karapınar, F. Khojasteh, A short note on C∗-valued contraction mappings, J. Inequal.Appl. 2016 (2016) 50. https://doi.org/10.1186/s13660-016-0992-5.[2] S. Chandok, D. Kumar, C. Park, C∗−algebra-valued partial metric space and fixed point theorems. Proc. Math. Sci.129 (2019) 37. https://doi.org/10.1007/s12044-019-0481-0.[3] M. Jleli, B. Samet, A new generalization of the Banach contraction principle. J. Inequal. Appl. 2014 (2014), 38.

https://doi.org/10.1186/1029-242X-2014-38.[4] G. Kalapana, Z.S. Tasneem C∗−algebra-valued rectangular b-metric spaces and some fixed point theorems, Commun.Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (2019) 2198-2208. https://doi.org/10.31801/cfsuasmas.598146.[5] W. A. Kirk, N. Shahzad, Generalized metrics and Caristi’s theorem, Fixed Point Theory Appl. 2013 (2013) 129.
https://doi.org/10.1186/1687-1812-2013-129.[6] Z. Ma, L. Jiang, H. Sun, C∗-algebra-valued metric spaces and related fixed point theorems, Fixed Point TheoryAppl. (2014) 2014, 206. https://doi.org/10.1186/1687-1812-2014-206.[7] H. Massit, M. Rossafi, Fixed point for ψ− contractive mapping in C∗− algebra valued rectangular b-metric, J. Math.Comput. Sci. 11(2021) 6507-6521. https://doi.org/10.28919/jmcs/6363.[8] G.J. Murphy, C∗-Algebras and operator theory, Academic Press, London, UK, 1990.[9] S. Omran, I. Masmali, On the (α− ψ)-contractive mappings in C∗-algebra valued b-metric spaces and fixed pointtheorems, J. Math. 2021 (2021) 7865976. https://doi.org/10.1155/2021/7865976.[10] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α− ψ−contractive type mappings, Nonlinear Anal.: TheoryMethods Appl. 75 (2012) 2154–2165. https://doi.org/10.1016/j.na.2011.10.014.

https://doi.org/10.28924/ada/ma.2.11
https://doi.org/10.1186/s13660-016-0992-5
https://doi.org/10.1007/s12044-019-0481-0
https://doi.org/10.1186/1029-242X-2014-38
https://doi.org/10.31801/cfsuasmas.598146
https://doi.org/10.1186/1687-1812-2013-129
https://doi.org/10.1186/1687-1812-2014-206
https://doi.org/10.28919/jmcs/6363
https://doi.org/10.1155/2021/7865976
https://doi.org/10.1016/j.na.2011.10.014

	1. Introduction 
	2. preliminaries
	3. Main result
	4. Applications
	5. Acknowledgments
	References

