International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 1 (2016), 58-63
http://www.etamaths.com

FIXED POINT RESULTS OF ALTMAN INTEGRAL TYPE MAPPINGS IN

S-METRIC SPACES

MUJEEB UR RAHMAN1,∗, MUHAMMAD SARWAR1 AND MUHIB UR RAHMAN2

Abstract. In this article, we introduce the concept of ϕ-weakly commuting self-mappings pairs in

S-metric space. Using this idea we establish a common fixed point theorem of Altman integral type

for four self-mappings in the context of S-metric space. Example is constructed to support our result.

1. Introduction and preliminaries

Fixed point theory is one of the most dynamic research subject in nonlinear analysis. In the field of
metric fixed point theory the first important and significant result was proved by Banach in 1922 for
contraction mapping in complete metric space. The well known Banach contraction theorem may be
stated as follows: ”Every contraction mapping of a complete metric space X into itself has a unique
fixed point”(Bonsall 1962).

In [1] Altman proved a fixed point theorem for a single self-mapping in compact metric space
satisfying the following contractive condition:

d(Tx,Ty) ≤ Q(d(x,y)) ∀ x,y ∈ X
where Q : [0,∞) → [0,∞) is an increasing function satisfies the following conditions:

(1) 0 < Q(t) < t, t ∈ (0,∞);
(2) ρ(t) = t

t−Q(t) is a decreasing function;

(3)
t1∫
0

ρ(t)dt < +∞ for some positive number t1.

Remark 1.1. By (1) and that Q is increasing we have Q(0) = 0 also Q(t) = t ⇐⇒ t = 0.

Common fixed point for Altman type mapping has been discussed by Garbone and Singh [2] and Li
and Gu [3] in metric spaces. In 2006, Mustafa and Sims [4] introduced a new structure of generalized
metric space called G-metric space. Gu and Ye [5] obtained a common fixed point theorem for Altman
integral type mapping in complete G-metric space. Recently, Sedghi et al. [6] initiated the idea of
S-metric space as a generalization of G-metric space. While in [7] Sedghi proved fixed point theorems
for implicit relation in complete S-metric space. In this paper, we derive a common fixed point Altman
integral type mapping for two pairs of ϕ-weakly commuting self-mappings in complete S-metric space.

We begin with the following definitions and results in the framework of S-metric space which can
be found in [6, 7].

Definition 1.2. Let X be a non-empty set. An S-metric is a function S : X × X × X → [0,∞)
satisfying the following conditions for all x,y,z,a ∈ X
S1) S(x,y,z) = 0 if and only if x = y = z;
S2) S(x,y,z) ≤ S(x,a,a) + S(y,a,a) + S(z,a,a).

The pair (X,S) is called S-metric space.

Example 1.3. Let X=(−∞, +∞) the distance function S : X ×X ×X → [0,∞) is defined by
S(x,y,z) = |x−z| + |y −z| for all x,y,z ∈ X.

2010 Mathematics Subject Classification. Primary 39B82; Secondary 44B20, 46C05.
Key words and phrases. Altman type mapping; common fixed point; self-mapping; ϕ-weakly commuting self-

mappings.

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.

58



ALTMAN INTEGRAL TYPE MAPPINGS 59

Then (X,S) is a complete S-metric space.

Definition 1.4. Let (X,S) be an S-metric space. A sequence {xn} in X converges to x ∈ X if
S(xn,xn,x) → 0 as n →∞.

We write xn → x for brevity.

Definition 1.5. Let (X,S) be an S-metric space. A sequence {xn} in X is called Cauchy sequence if
for � > 0, there exists n0 ∈ N such that for all n,m ≥ n0 we have

S(xn,xn,xm) < �.

Definition 1.6. An S-metric space (X,S) is said to be complete if every Cauchy sequence in X
converges in X.

Lemma 1.7. Limit of the convergent sequence in S-metric space is unique.

Lemma 1.8. S-metric is jointly continuous on all three variables.

Lemma 1.9. In an S-metric space, we have

S(x,x,y) = S(y,y,x) for all x,y ∈ X.

Now we introduce the concept of ϕ-weakly commuting pairs of self-mappings in S-metric space as
follows:

Definition 1.10. A pair of self-mappings (S,T) on S-metric space is called ϕ-weakly commuting. If
there exist a continuous function ϕ : [0,∞) → [0,∞), ϕ(0) = 0 such that

S(STx,STx,TSx) ≤ ϕ(S(Sx,Sx,Tx)) ∀ x ∈ X.

Example 1.11. Let X = [0,∞), S(x,y,z) = |x− z| + |y − z| for all x,y,z ∈ X. Let S,T : X → X
are defined by Sx = x

8
and Tx = x

2
then

S(STx,STx,TSx) = S(
x

16
,
x

16
,
x

16
) ≤

1

2

3

4
x =

1

2
S(Sx,Sx,Tx)

S(STx,STx,TSx) ≤ ϕ(S(Sx,Sx,Tx)).

Lemma 1.12. [5]. Let ρt ba a Lebesgue integrable function and ρ(t) > 0 for all t > 0. Let F(x) =
x∫
0

ρ(t)dt, then F(x) is an increasing function in [0, +∞).

Definition 1.13. [8]. Let S and T be two self-mappings on a set X. Any point x ∈ X is called
coincidence point of S and T if Sx = Tx for some x ∈ X and we called u = Sx = Tx is a point of
coincidence of S and T .

Definition 1.14. A function φ : [0,∞) → [0,∞) is called contractive modulus if it satisfy φ(t) ≤ t for
all t ≥ 0.

2. Main results

Theorem 2.1. Let (X,S) be a complete S-metric space and P,T,f,g : X → X be self-mappings.
If there exists an increasing function Q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) also the
following conditions holds:

(4) P(X) ⊆ g(X) and T(X) ⊆ f(X);

(5)
S(Px,Px,Ty)∫

0

ρ(t)dt ≤ φ(
Q(S(fx,fx,gy))∫

0

ρ(t)dt) for all x,y ∈ X and φ is contractive modulus where

ρ(t) is a Lebesgue integrable function which is summable nonnegative and such that

δ∫
0

ρ(t)dt > 0 ∀ δ > 0.

(6) If (P,f) and (T,g) are two pairs of continuous ϕ-weakly commuting mappings.
Then P,T,f and g have a unique common fixed point in X.



60 RAHMAN, SARWAR AND RAHMAN

Proof. Since P(X) ⊆ g(X) and T(X) ⊆ f(X) so we define two sequences {xn} and {yn} in X by the
rule

y2n+1 = Px2n = gx2n+1 and y2n+2 = Tx2n+1 = fx2n+2 n = 0, 1, 2, .....

Now consider
S(y2n+1,y2n+1,y2n+2)∫

0

ρ(t)dt =

S(Px2n,Px2n,Tx2n+1)∫
0

ρ(t)dt.

Using (5) we have

≤ φ
( Q(S(fx2n,fx2n,gx2n+1))∫

0

ρ(t)dt

)
= φ

( Q(S(y2n,y2n,y2n+1))∫
0

ρ(t)dt

)
.

Using the property of φ we have

≤
Q(S(y2n,y2n,y2n+1))∫

0

ρ(t)dt.

Let tn = S(yn,yn+1) then the above inequality take the form

t2n+1∫
0

ρ(t)dt ≤
Q(t2n)∫
0

ρ(t)dt.

Now by the property of Q and Lemma 1.12 we have

t2n+1 ≤ Q(t2n) < t2n.
Similarly we can show that

t2n ≤ Q(t2n−1) < t2n−1.
Hence {tn} is a nonnegative strictly decreasing sequence and hence convergent. Thus tn+1 ≤ Q(tn) <
tn for all n = 0, 1, 2, 3, .....

Now to prove that {yn} is a Cauchy sequence consider for m ≥ n and by triangle inequality we have

S(yn,yn,ym) ≤ 2
m−1∑
i=n

S(yi,yi,yi+1) = 2

m−1∑
i=n

ti

= 2

m−1∑
i=n

ti(ti − ti+1)
(ti − ti+1)

≤ 2
m−1∑
i=n

ti(ti − ti+1)
(ti −Q(ti))

≤ 2
m−1∑
i=n

ti∫
ti+1

t

(t−Q(t))
dt = 2

tn∫
tm

t

(t−Q(t))
dt = 2

tn∫
tm

P(t)dt.

It follows from the convergence of the sequence {tn} and by condition (3) we have

lim
n→∞

tn∫
tm

P(t)dt = 0.

Thus {yn} is a Cauchy sequence in X. Since X is complete so there must exists u ∈ X such that
lim
n→∞

yn = u.

Also the subsequences {y2n+1} and {y2n+2} converges to u. Therefore
lim
n→∞

y2n+1 = lim
n→∞

Px2n = lim
n→∞

gx2n+1 = u

lim
n→∞

y2n+2 = lim
n→∞

Tx2n+1 = lim
n→∞

fx2n+2 = u.

Since (P,f) are continuous ϕ-weakly commuting pair so

S(Pfx2n,Pfx2n,fPx2n) ≤ ϕ(S(Px2n,Px2n,fx2n)).



ALTMAN INTEGRAL TYPE MAPPINGS 61

Taking limit n →∞ and since (P,f) is continuous pair of mappings thus

S(Pu,Pu,fu) ≤ ϕ(S(u,u,u)) = ϕ(0) = 0.

Which implies that Pu = fu. Similarly from continuous ϕ-weakly commuting pair (T,g) we can show
that Tu = gu.

Now by condition (5) and using other given information we have

S(Pu,Pu,Tu)∫
0

ρ(t)dt ≤ φ
( Q(S(fu,fu,gu))∫

0

ρ(t)dt

)
≤

Q(S(fu,fu,gu))∫
0

ρ(t)dt

S(Pu,Pu,Tu) ≤ Q(S(fu,fu,gu)) ≤ S(fu,fu,gu)
≤ S(fu,fu,Pu) + S(fu,fu,Pu) + S(gu,gu,Pu)
≤ S(gu,gu,Tu) + S(gu,gu,Tu) + S(Pu,Pu,Tu)
= S(Pu,Pu,Tu).

Which implies that fu = gu. Thus fu = gu = Pu = Tu and let z = fu = gu = Pu = Tu. Therefore
u is the common coincidence point of mappings P,T,f and g.

Again since (P,f) are ϕ-weakly commuting pair so

S(Pz,Pz,fz) = S(Pfu,Pfu,fPu) ≤ ϕ(S(Pu,Pu,fu)) = ϕ(0) = 0.

Implies that Pz = fz. Similarly we can show that Tz = gz. Thus

Pfu = fPu and Tgu = gTu.

Again by condition (5) we have

S(Pz,Pz,z)∫
0

ρ(t)dt =

S(PPu,PPu,Tu)∫
0

ρ(t)dt

≤ φ
( Q(S(fPu,fPu,gu))∫

0

ρ(t)dt

)
≤

Q(S(fPu,fPu,gu))∫
0

ρ(t)dt.

By Lemma 1.12 and using the property of Q we have

S(Pz,Pz,z) ≤ Q(S(fPu,fPu,gu)) ≤ S(fPu,fPu,gu)

= S(Pfu,Pfu,gu) = S(Pz,Pz,z).

Which implies Pz = z but Pz = fz therefore Pz = fz = z. Similarly we can prove that Tz = gz = z.
Hence Pz = fz = gz = Tz = z. Thus z is a common fixed point of mappings P,T,f and g.

Uniqueness. Assume that common fixed point of P,T,f and g is not unique i.e z 6= w be two
distinct fixed points of P,T,f and g. Then using condition (5) we have

S(z,z,w)∫
0

ρ(t)dt =

S(Pz,Pz,Tw)∫
0

ρ(t)dt

≤ φ
( Q(S(fz,fz,gw))∫

0

ρ(t)dt

)
≤

Q(S(fz,fz,gw))∫
0

ρ(t)dt.

By Lemma 1.12 and using the property of Q we have

S(z,z,w) ≤ Q(S(fz,fz,gw)) ≤ S(z,z,w).

Which is a contradiction hence z = w. Therefore, fixed point of P,T,f and g is unique. �

Remark 2.2. If we take (1) P = T (2) f = g (3) P = T and f = g = I (4) φ = I in Theorem 2.1.
Then we obtain several new results in the setting of S-metric space.



62 RAHMAN, SARWAR AND RAHMAN

Corollary 2.3. Let (X,S) be a complete S-metric space and P,T,f,g : X → X be self-mappings.
If there exists an increasing function Q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) also the
following conditions holds:

(4) P(X) ⊆ g(X) and T(X) ⊆ f(X);
(5) S(Px,Px,Ty) ≤ φ(Q(S(fx,fx,gy))) for all x,y ∈ X where φ is contractive modulus;
(6) If (P,f) and (T,g) are two pairs of continuous ϕ-weakly commuting mappings.

Then P,T,f and g have a unique common fixed point in X.

Proof. Putting ρ(t) = I in Theorem 2.1, one can easily obtain the proof of Corollary 2.3 from Theorem
2.1. �

Now we construct an example to support Corollary 2.3.

Example 2.4. Let X = [0,∞)) and S(x,y,z) = |x − y| + |y − z| for all x,y,z ∈ X with self-
mappings defined on X is given by Px = x

8
,fx = x,Tx = x

16
and gx = x

2
. Clearly P(X) ⊆ g(X) and

T(X) ⊆ f(X). Also we have

S(Px,Px,Ty) = S

(
x

8
,
x

8
,
y

16

)
= |

x

8
−

y

16
| + |

x

8
−

y

16
|

=
1

8

(
|x−

y

2
| + |x−

y

2
|
)

=
1

8
S(fx,fx,gy)

S(Px,Px,Ty) =
1

8
S(fx,fx,gy).

Let φ(t) = 3
4
t and Q(t) = 1

2
t. Then φ(t) ≤ t and Q(t) satisfies conditions (1)-(3). Then we have

S(Px,Px,Ty) =
1

8
S(fx,fx,gy) ≤

3

4
·

1

2
S(fx,fx,gy)

=
3

4
Q(S(fx,fx,gy)).

On the other side if ϕ(t) = 1
2
t for all t ∈ [0,∞). Then one can easily show that (P,f) and (T,g) are

two pairs of continuous ϕ-weakly commuting mappings in X. So that all the conditions of Corollary
2.3 are satisfied. Therefore, 0 is the unique common fixed point of P,T,f and g.

Corollary 2.5. Let (X,S) be a complete S-metric space and P,T : X → X be self-mappings. If
there exists an increasing function Q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) and φ is a
contractive modulus also the following condition holds:

S(Px,Px,Ty) ≤ φ(Q(S(x,x,y))) for all x,y ∈ X.
Then P and T have a unique common fixed point in X.

Corollary 2.6. Let (X,S) be a complete S-metric space and P : X → X be self-mappings. If
there exists an increasing function Q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) and φ is a
contractive modulus also the following condition holds:

S(Px,Px,Py) ≤ φ(Q(S(x,x,y))) for all x,y ∈ X.
Then P has a unique fixed point in X.

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ALTMAN INTEGRAL TYPE MAPPINGS 63

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1Department of Mathematics, University of Malakand,Chakdara Dir(L), Khyber Pukhtunkhwa, Pakistan

2Department of Electrical (Telecom), MCS, Nust, Rawalpindi, Pakistan

∗Corresponding author: mujeeb846@yahoo.com