International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 1 (2016), 23-27
http://www.etamaths.com

COMMON FIXED POINT THEOREMS FOR G–CONTRACTION IN

C∗–ALGEBRA–VALUED METRIC SPACES

AKBAR ZADA1,∗, SHAHID SAIFULLAH1 AND ZHENHUA MA2,3

Abstract. In this paper we prove the common fixed point theorems for two mappings in complete

C∗–valued metric space endowed with the graph G = (V,E), which satisfies G-contractive condition.
Also, we provide an example in support of our main result.

1. Introduction and Preliminaries

The Banach contraction principle [5] plays an important role in solving non linear problems. The
Banach contraction principle says that: if (X,d) be a complete metric space and f is a self mapping
on X with the condition that there exists λ ∈ (0, 1) such that

d(fx,fy) ≤ λd(x,y) for all x,y ∈ X,

then f has a unique fixed point in X. Since then a lot of publications are devoted to the study
and solutions of many practical and theoretical problems by using this condition. Due to a numerous
applications of the fixed point theory, from the last few decades this theory is a central topic of research.
In this theory one of the approach is the common fixed point theorems. The concept of the common
fixed point theorems was investigated by Jungck [1]. Many authors studied the fixed and common fixed
point theorems for different spaces, like in cone metric spaces [8], non-commutative Banach spaces [22],
fuzzy metric spaces [14] and uniform metric spaces [21]. For more information about this topic see
([1, 6, 7, 9, 17, 18, 23]).

On the other hand the concept of C∗–algebra is well developed. Here we recall some basic definitions,
notations and results of C∗–algebra that may be found in [13]. A ∗-algebra A is a complex algebra
with linear involution ∗ such that x∗∗ = x and (xy)∗ = y∗x∗, for any x,y ∈ A. If ∗-algebra together
with complete sub multiplicative norm satisfying ||x∗|| = ||x|| for all x ∈ A, then ∗-algebra is said to
be a Banach ∗-algebra. A C∗–algebra is a Banach ∗-algebra such that ||x∗x|| = ||x||2 for all x ∈ A.
An element of A is called positive element, if A+ = {x∗ = x|x ∈ A} and σ(x) ⊂ R+, where σ(x) is
the spectrum of an element x ∈ A, i.e., σ(x) = {λ ∈ C : λI −x is not invertible}. There is a natural
partial ordering on A+ given by x � y if and only if x − y ∈ A+. In [12] Z. Ma et al., introduced
the notion of C∗-algebra valued metric space and proved fixed point theorems for C∗-algebra valued
contractive mapping.

Many researchers tried to obtain some fixed point theorems of Banach type contraction endowed
with the graph G, we recommend [2, 3, 4, 15, 16, 20]. Recently, T. Kamran et al., in [19] extended
the results of Ma et al., which was given in[12], by using C∗–valued metric spaces and G-contraction
principles.

Now we give some definitions of graph theory which is found in any text on graph theory, for example
[11]. Following Jachymski [10], let ∆ denote the diagonal of the X ×X in a metric space (X,d), and
consider a directed graph G = (V (G),E(G)) = (V,E) the set in which V of its vertices and E of its
edges, and ∆ ⊆ E. Assume that G has no parallel edges. We may treat G as a weighted graph by
assigning to each edge the distance between its vertices.

In this paper we will continue to study common fixed points in the C∗–valued metric space endowed
with the graph G under G–contractive condition.

2010 Mathematics Subject Classification. 47H10, 47A56.
Key words and phrases. metric space; C∗–algebra valued metric spaces; G-contraction; common fixed point.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

23



24 ZADA, SAIFULLAH AND MA

Definition 1.1. Let X be a nonempty set, and the mapping d : X ×X →A endowed with the graph
G = (V,E), if it satisfies the following conditions:
(1) d(x,y) ≥ 0 for all x,y ∈ X and d(x,y) = 0 ⇔ x = y;
(2) d(x,y) = d(y,x) for all x,y ∈ X;
(3) d(x,y) ≤ d(x,z) + d(z,y) for all x,y,z ∈ X.
Then d is called a C∗–valued metric on X, and (X,d,A) is called C∗–valued metric space.

Definition 1.2. Suppose that (X,d,A) is a C∗–valued metric space. Let x ∈ (X,d,A) and {xn} be
a sequence in X. The sequence {xn} is said to be convergent, if for any � > 0 there exists a positive
integer N such that

||d(xn,x)|| ≤ � for all n ≥ N.

The sequence {xn} is said to be Cauchy, if for any � > 0 there exists a positive integer N such that

||d(xn,xm)|| ≤ � for all n,m ≥ N.

If every Cauchy sequence is convergent in (X,d,A), then (X,d,A) is said to be complete C∗–valued
metric space.

Example 1.3. Let X = R and A = M2(R). Define d : X ×X →A such that

d(x,y) =

(
|x−y| 0

0 α|x−y|

)
for all x,y ∈ R and α ≥ 0.

It is essay to verify that d is a C∗–algebra valued metric space and (X,d,M2(R)) is a complete C∗–
algebra valued metric space.

Definition 1.4. Let (X,d,A) be a C∗–valued metric space. A mapping f : X → X is said to be a
C∗–algebra–valued contraction mapping on X if there exists an a ∈A with ||a|| < 1 such that

(1.1) d(fx,fy) ≤ a∗d(x,y)a, for all x, y ∈ X.

Theorem 1.5. [12] Let (X,d,A) be a complete C∗–algebra-valued metric space and f satisfies (1.1),
then f has a unique fixed point in X.

Property 1.6. [12]

(1) For any {xn} ∈ X such that xn converges to x with (xn+1,xn) ∈ E for all n ≥ 1 there exists
a subsequence {xnk} of {xn} such that (x,xnk ) ∈ E.

(2) For any {fnx} ∈ X such that fnx converges to x ∈ X with (fn+1x,fnx) ∈ E there exists a
subsequence {fnkx} and n0 ∈ N such that (x,fnkx) ∈ E for all k ≥ n0.

2. Main Result

In this section, we prove common fixed point theorems for two mappings satisfying G–contractive
condition in a complete C∗–valued metric space endowed with the graph G = (V,E).

Definition 2.1. Let (X,d,A) be a C∗–valued metric space endowed with the graph G = (V,E). The
mappings f,g : X → X are said to be C∗–valued G–contractive on X, if there exists an a ∈ A with
||a|| < 1 such that

(2.1) d(fx,gy) ≤ a∗d(x,y)a, for all (x,y) ∈ E.

Theorem 2.2. Let (X,d,A) is a complete C∗–valued metric space endowed with the graph G = (V,E).
Suppose that the mappings f,g : X → X are C∗–valued G–contractive mappings on X satisfying the
Property 1.6 (2) and the following conditions
(1) if (x,y) ∈ E then (fx,gy) ∈ E,
(2) there exists z0 ∈ X such that (z0,fz0), (z0,gz0) ∈ E.
Then f and g has a unique common fixed point in X.



G–CONTRACTION IN C∗–ALGEBRA–VALUED METRIC SPACES 25

Proof. Let z1 ∈ X, and construct sequence {zn}∈ X, such that z2n+1 = fz2n, z2n+2 = gz2n+1, and
(z2n−1,z2n) ∈ E for all n ∈ N. We have

d(z2n+1,z2n+2) = d(gz2n+1,fz2n)

≤ a∗d(z2n+1,z2n)a
≤ (a∗)2d(z2n,z2n−1)(a)2

.

.

.

≤ (a∗)2n+1d(z1,z0)(a)2n+1.

Similarly,

d(z2n+1,z2n) = d(fz2n,gz2n−1)

≤ a∗d(z2n,z2n−1)a
.

.

.

≤ (a∗)2nd(z1,z0)(a)2n

= (a∗)2nQ(a)2n.

Let us denote d(z1,z0) by Q ∈A. Then for any n ∈ N

d(zn+1,zn) = (a
∗)nd(z1,z0)(a)

n

= (a∗)nQ(a)n,

then for any q ∈ N and applying the triangular inequality (3) for the C∗–valued metric spaces,

d(zn+q,zn) = d(zn+q,zn+q−1) + d(zn+q−1,zn+q−2) + · · · + d(zn+1,zn)

≤
n+q−1∑
j=n

(a∗)jd(z1,z0)(a)
j

=

n+q−1∑
j=n

(a∗)jQ(a)j

=

n+q−1∑
j=n

(a∗)jQ
1
2 Q

1
2 (a)j

=

n+q−1∑
j=n

(Q
1
2 aj)∗(Q

1
2 aj)

=

n+q−1∑
j=n

|Q
1
2 aj|2

≤
n+q−1∑
j=n

|| |Q
1
2 aj|2||.I

= ||Q
1
2 ||2

n+q−1∑
j=n

||a2j||.

Since ||a|| < 1, thus d(zn+q,zn) → 0 as n →∞. Thus we conclude that the sequence {zn} is a Cauchy
sequence, with respect to A. Using the completeness of X, there exists an element z0 ∈ X = V, such
that zn → z0 as n →∞.



26 ZADA, SAIFULLAH AND MA

On the other hand, using the triangular inequality, we get

d(z0,fz0) = d(z0,z2n+1) + d(z2n+1,fz0)

= d(z0,z2n+1) + d(gz2n,fz0)

≤ d(z0,z2n+1) + a∗d(z2n,z0)a.

Thus if n → ∞, then d(z0,fz0) → 0 i.e. fz0 = z0. Similarly we can prove that gz0 = z0. Now we
will show the uniqueness of common fixed points in X. For this we assume that there is another point
z∗ ∈ X = V, such that(z0,z∗) ∈ E. Consider

d(z0,z
∗) = d(fz0,gz0) ≤ a∗d(z0,z∗)a.

Since ||a|| < 1, then the above inequality yields that

0 ≤ ||d(z0,z∗)|| ≤ ||a||2||d(z0,z∗)|| < ||d(z0,z∗)||.

Which is a contradiction. Thus, ||d(z0,z∗)|| = 0 which implies that d(z0,z∗) = 0 i.e. z0 = z∗. Thus
the proof is complete.

Corollary 2.3. Suppose that (X,d,A) is a C∗–valued metric space endowed with the graph G, and
suppose that the mappings f, g : X → X are G–contractive, satisfying

||d(fx,gy)|| ≤ ||a||||d(x,y)||, for all (x,y) ∈ E,

where a ∈A with ||a|| < 1. Then f and g have a unique common fixed point in X.

Corollary 2.4. Let (X,d,A) is a C∗–valued metric space endowed with the graph G, and suppose that
the mapping f : X → X is G–contractive, satisfying

||d(fmx,fny)|| ≤ a∗d(x,y)a, for all (x,y) ∈ E,

where a ∈A with ||a|| < 1 and m, n are positive integers. Then f has a unique fixed point in X.

Remark 2.5. In Theorem 2.2, if g = f, then we have

(2.2) d(fx,fy) ≤ a∗d(x,y)a, for all (x,y) ∈ E.

In this case we have the following corollary, which can also be found in [12].

Corollary 2.6. Let (X,d,A) be a complete C∗–valued metric space, and consider the mapping f :
X → X such that it satisfies (2.2), then f has a unique fixed point in X.

Example 2.7. Consider, A = M2×2(R), of all 2 × 2 matrices with the usual operation of addition,
scalar multiplication, and matrix multiplication. Thus A becomes C∗–algebra. Let us define d : R×R →
A by

d(x,y) =

(
|x−y| 0

0 |x−y|

)
.

It is essay to check that d satisfies all the conditions of Definition 1.1. Therefore (R,A,d) is C∗–valued
metric space. Define f,g : R → R by

f(x) =
x2

4
and g(x) =

x2

3
,

and consider the graph G = (V,E), where V = R and

E =
{( 1

4m
,

1

32m+1

)
; m = 1, 2, . . .

}
∪
{( 1

4m
, 0
)

; m = 1, 2, . . .
}
∪{(x,x); x ∈ R}.

Note that, for each m ∈ N, (
f(

1

4m
),g(

1

32m+1
)
)

=
( 1

42n+1
,

1

34n+3

)
∈ E,

and (
f(

1

4m
),g(0)

)
=
( 1

42m+1
, 0
)
∈ E.



G–CONTRACTION IN C∗–ALGEBRA–VALUED METRIC SPACES 27

Also, (fx,gx) = ( x
2

4
, x

2

3
), for each x ∈ R, which is again in E. Moreover, by taking A =

(
1√
2

0

0 1√
2

)
,

we have ||A|| < 1, so all the conditions of Theorem 2.2 are satisfied and thus the common fixed point
of f and g is 0.

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1Department of Mathematics, University of Peshawar, Peshawar, Pakistan

2School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

3Department of Mathematics and Physics, Hebei Institute of Architecture and Civil Engineering, Zhangji-

akou, 075024, China

∗Corresponding author: zadababo@yahoo.com