/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 11, No. 1, 2018, 331-351
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

n-tupled fixed point results with rational type

contraction in b-metric spaces

Saddam Hussain1, Muhammad Sarwar1,∗, Yongjin Li2

1 Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan
2 Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China.

Abstract. In this manuscript, using rational type contractive conditions the existence and unique-
ness of common n-tupled fixed point for a pair of mappings in complete b-metric spaces are studied.
Using the derived results some fixed theorems can be deduced in b−metric spaces.

2010 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Complete b-metric space, n-tupled fixed point, n-tupled coincidence
point, rational type contractive mappings

1. Introduction and preliminaries

The Bananch contraction theorem is the most important technique for solving non-
linear integral equations, differential equations and functional equations etc. It has fruitful
applications within as well as outside mathematics. Many authors have extended this
theorem employing relatively more general contractive conditions ensuring the existence
and uniqueness of a fixed point.

To solve the problem of the convergence of measurable functions with respect to a
measure, Bakhtin [2] and Czerwik [7] introduced the concept of b-metric spaces also called
metric type space [18]. Using this concept Czerwik, generalized the Banach contraction
principle in b-metric spaces (see [7, 8, 20, 21]). Yamaod and Sintunavarat [27] introduced
the concept of (α, β)-(ψ, φ)-contractive mapping in b-metric spaces, and established some
fixed point results for such mappings in b-metric spaces. Yamaod et al. [19] studied the
existence of a common solution for a system of nonlinear integral equations via fixed point
methods in b-metric space.

The concept of coupled fixed point was introduced by Gou and Lakshmikantham [9] for
partially ordered set. Bhaskar and Lakshmikantham [5] studied the existence and unique-
ness of a coupled fixed point results in partially ordered metric space. Lakshmikantham

∗Corresponding author.

Email addresses: saddamuom008@gmail.com (S. Hussain), sarwarswati@gmail.com (M. Sarwar),
stslyj@mail.sysu.edu.cn (Y. Li)

http://www.ejpam.com 331 c© 2018 EJPAM All rights reserved.



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 332

and Ciric in [12] defined mixed g-monotone property and studied coupled coincidence point
in partially ordered metric spaces. Samet [24] investigated coupled fixed point results in
the setting of partial ordered metric spaces for a generalized Meir- Keeler type contractive
condition. Very recently, Sarwar et al. [15] studied the common coupled fixed point re-
sults satisfying rational type contractive conditions in b-metric spaces. Many researchers
studied the coupled fixed point and discussed it’s application. (see [3, 9, 25, 26]).

Berinde and Borcut [4] introduced the notion of tripled fixed point and established
some results in the setting of partially ordered metric spaces. Karapinar [11] studied some
quadruple fixed point results for non-linear contraction partially ordered metric spaces.

Imdadet al.[13] introduced the concept of n-tupled coincidence as well as n-tupled fixed
point (for even n) and obtained n-tupled coincidence as well as n-tupled common fixed
point theorems for nonlinear φ-contraction. Paknazar et al. [14] introduced the concept
of a new g-monotone mapping and defined the notions of n-fixed point and n-coincidence
point and proved some related theorems for nonlinear contractive mappings in partially
ordered complete metric spaces. Soliman et al. [1] proved some n-tupled coincidence point
theorems for nonlinear φ-contraction mappings in partially ordered complete asymptoti-
cally regular metric spaces. In [22] the authors introduced the notion of compatibility for
n-tupled coincidence points and proved n-tupled fixed point for compatible mappings sat-
isfying contractive type conditions in partially ordered metric spaces. Murthy et al. [17]
introduced n-tupled fixed points (for all positive integers) and proved n-tupled fixed points
theorems for contractive type mappings in fuzzy metric spaces. Husain et al. [23] present
some n-tupled coincidence point results for a pair of mappings without mixed monotone
property satisfying a rational type contractive condition in metric spaces equipped with
a partial ordering as well as present results on the existence and uniqueness of n-tupled
common fixed points.

The aim of this manuscript is to study n-tupled fixed point results via rational type
contraction in complete b-metric spaces. The established result generalizes some recent
results ( particularly the result of Sarwar et al. [15] and Malhotra and Bansal [16] ) from
the existing literature in b-metric spaces.

Throughout this paper R is the set of real and R
+ is a set of non-negative real numbers.

Definition 1. [10] Let X be a non empty set and s ≥ 1, s ∈ R. A function d : X×X → R
+

is called b-metric if for each x, y, z ∈ X, the following conditions are satisfied:

(i) d(x, y) = 0 ⇔ x = y;

(ii) d(x, y) = d(y, x);

(iii) d(x, z) ≤ s[d(x, y) + d(y, z)].

Then the pair (X, d) with parameter s is called b-metric space.

Example 1. [6] The lp space, 0 < p < 1, lp = {(xn) ∈ R :
∑

|xn|
p <∞}, and function is

defined as d : lp× lp → R by d(x, y) = (
∑

|xn−yn|
p)

1

p , x = (xn), y = (yn) ∈ lp, then (X, d)

is called b-metric space with parameter s = 2
1

2 provided that d(x, z) ≤ 2
1

2 [d(x, y)+d(y, z)].



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 333

Definition 2. [6] Let (X, d) be a b-metric space. Then a sequence {xn} is said be converge
to x ∈ X if for each ǫ > 0 there exists j(ǫ) ∈ N, such that d(xn, x) < ǫ ∀ n ≥ j(ǫ).

Definition 3. [6] Let (X, d) be a b-metric space. Then a sequence {xn} is said be a
Cauchy sequence if for each ǫ > 0 there exists j(ǫ) ∈ N , such that d(xn, xm) < ǫ ∀ n,
m ≥ j(ǫ).

Definition 4. [13] Let X be a non empty set. An element (x1, x2, · · · , xn) ∈ Xn is called
an n-tupled fixed point of a given mapping T : Xn → X if

x1 = T (x1, x2, · · · , xn),
x2 = T (x2, x3, · · · , xn, x1),
x3 = T (x3, · · · , xn, x1, x2),
...
xn = T (xn, x1, x2 · · · , xn−1).

Definition 5. [13] Let X be a non empty set. An element (x1, x2, · · · , xn) ∈ Xn is called
an n-tupled coincidence point of the given mappings S, T : Xn → X if S(x1, x2, · · · , xn) =
T (x1, x2, · · · , xn),

S(x2, x3, · · · , xn, x1) = T (x2, x3, · · · , xn, x1),
S(x3, · · · , xn, x1, x2) = T (x3, · · · , xn, x1, x2),
...
S(xn, x1, x2, · · · , xn−1) = T (xn, x1, x2, · · · , xn−1).

Example 2. Suppose X = R and S, T : Xn → X be defined by
S(x1, x2, x3, · · · , xn) = x1+x2+x3+···+xn

n
and T (x1, x2, x3, · · · , xn) = x1x2x3 · · ·xn for

each x1, x2, x3,· · · , xn ∈ Xn. Then clearly (0, 0, 0, · · · , 0) and (1, 1, 1, · · · , 1) are n-tupled
coincidence points of S and T .

Definition 6. [1] Let X be a non empty set. An element (x1, x2, · · · , xn) ∈ Xn is called
an n-tupled common fixed point of the mappings S, T : Xn → X if

x1 = S(x1, x2, · · · , xn) = T (x1, x2, · · · , xn),
x2 = S(x2, x3, · · · , xn, x1) = T (x2, x3, · · · , xn, x1),
x3 = S(x3, · · · , xn, x1, x2) = T (x3, · · · , xn, x1, x2),
...
xn = S(xn, x1, x2, · · · , xn−1) = T (xn, x1, x2, · · · , xn−1).

2. Main results

We begin with the following theorem.

Theorem 1. Let (X, d) be a complete b-metric space with parameter s ≥ 1 and let the
mapping S, T : Xn → X satisfy:

d(S(x1, x2, · · · , xn), T (y1, y2, · · · , yn)) ≤ α1

d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

n



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 334

+ α2

d(x1, S(x1, x2, · · · , xn))d(y1, T (y1, y2, · · · , yn))

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α3

d(y1, S(x1, x2, · · · , xn))d(x1, T (y1, y2, · · · , yn))

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α4

d(S(x1, x2, · · · , xn), T (y1, y2, · · · , yn))d(x1, y1)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α5

d(S(x1, x2, · · · , xn), T (y1, y2, · · · , yn))d(x2, y2)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α6

d(y1, T (y1, y2, · · · , yn))d(x2, y2)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α7

d(y1, S(x1, x2, · · · , xn))d(x1, y1)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α8

d(y1, S(x1, x2, · · · , xn))d(x2, y2)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α9

d(y1, T (y1, y2, · · · , yn))d(xn, yn)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+ α10

d(y1, S(x1, x2, · · · , xn))d(xn, yn)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)
.

(1)

For all x1, x2, x3,· · · , xn and y1, y2,y3,· · · , yn ∈ X and αi ≥ 0, i = 1, 2, · · · , 10 with the
conditions sα1 + α2 + α4 + α5 + α6 + α9 < 1 and α1 + α3 + α4 + α5 + α7 + α8 + α10 < 1.
Then S and T have unique common n-fixed point in X.

Proof. Taking “n” arbitrary points x10, x
2
0, x

3
0,· · · , x

n
0 , in X, define the sequence by the

following rules
x12k+1 = S(x12k, x

2
2k, x

3
2k, · · · , x

n
2k),

x22k+1 = S(x22k, x
1
2k, x

3
2k, · · · , x

n
2k),

x32k+1 = S(x32k, x
2
2k, x

1
2k, · · · , x

n
2k)

...
xn2k+1 = S(xn2k, x

n−1
2k , xn−2

2k , · · · , x22k, x
1
2k),

and
x12k+2 = T (x12k+1, x

2
2k+1, x

3
2k+1, · · · , x

n
2k+1),

x22k+2 = T (x22k+1, x
1
2k+1, x

3
2k+1, · · · , x

n
2k+1),

x32k+2 = T (x32k+1, x
2
2k+1, x

1
2k+1, · · · , x

n
2k+1),

...
xn2k+2 = T (xn2k+1, x

n−1
2k+1

, xn−2
2k+1

, · · · , x22k+1, x
1
2k+1) for k=0,1,2,· · · .

Consider

d(x12k+1, x
1
2k+2) = d(S(x12k, x

2
2k, x

3
2k, · · · , x

n
2k), T (x

1
2k+1, x

2
2k+1, x

3
2k+1, · · · , x

n
2k+1)).



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 335

Then by using contractive condition (1) of Theorem 1, we have

d(x12k+1, x
1
2k+2) ≤ α1

d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)

n

+α2

d(x12k, S(x
1
2k, x

2
2k, · · · , x

n
2k))d(x

1
2k+1, T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α3

d(x12k+1, S(x
1
2k, x

2
2k, · · · , x

n
2k))d(x

1
2k, T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α4

d(S(x12k, x
2
2k, · · · , x

n
2k), T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1))d(x

1
2k, x

1
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α5

d(S(x12k, x
2
2k, · · · , x

n
2k), T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1))d(x

2
2k, x

2
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

α6

d(x12k+1, T (x
1
2k+1, x

2
2k+1, · · · , x

n
2k+1))d(x

2
2k, x

2
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α7

d(x12k+1, S(x
1
2k, x

2
2k, · · · , x

n
2k))d(x

1
2k, x

1
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α8

d(x12k+1, S(x
1
2k, x

2
2k, · · · , x

n
2k))d(x

2
2k, x

2
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α9

d(x12k+1, T (x
1
2k+1, x

2
2k+1, · · · , x

n
2k+1))d(x

n
2k, x

n
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α10

d(x12k+1, S(x
1
2k, x

2
2k, · · · , x

n
2k))d(x

n
2k, x

n
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

= α1

d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)

n

+α2

d(x12k, x
1
2k+1)d(x

1
2k+1, x

1
2k+2)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α3

d(x12k+1, x
1
2k+1)d(x

1
2k, x

1
2k+2)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α4

d(x12k+1, x
1
2k+2)d(x

1
2k, x

1
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α5

d(x12k+1, x
1
2k+2)d(x

2
2k, x

2
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 336

+α6

d(x12k+1, x
1
2k+2)d(x

2
2k, x

2
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α7

d(x12k+1, x
1
2k+1)d(x

1
2k, x

1
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α8

d(x12k+1, x
1
2k+1)d(x

2
2k, x

2
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α9

d(x12k+1, x
1
2k+2)d(x

n
2k, x

n
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

+α10

d(x12k+1, x
1
2k+1)d(x

n
2k, x

n
2k+1)

1 + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)

≤
α1

n
[d(x12k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)] + α2d(x

1
2k+1, x

1
2k+2)

+α4d(x
1
2k+1, x

1
2k+2) + α5d(x

1
2k+1, x

1
2k+2) + α6d(x

1
2k+1, x

1
2k+2) + α9d(x

1
2k+1, x

1
2k+2).

Which implies that
(1− α2 − α4 − α5 − α6 − α9)d(x

1
2k+1, x

1
2k+2) ≤

α1

n
[d(x12k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)]

d(x12k+1, x
1
2k+2) ≤

α1[d(x
1
2k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)]

n(1− (α2 + α4 + α5 + α6 + α9))
. (A1)

Similarly

d(x22k+1, x
2
2k+2) ≤

α1[d(x
1
2k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)]

n(1− (α2 + α4 + α5 + α6 + α9))
. (A2)

Proceeding n-times, one can write

d(xn2k+1, x
n
2k+2) ≤

α1[d(x
1
2k, x

1
2k+1) + d(x12k, x

1
2k+1) + · · ·+ d(xn2k, x

n
2k+1)]

n(1− (α2 + α4 + α5 + α6 + α9))
. (An)

Adding (A1), (A2), · · · , and (An), we get

d(x12k+1, x
1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2) ≤

α1[d(x
1
2k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)]

1− (α2 + α4 + α5 + α6 + α9)

= h[d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)].

Where
h =

α1

1− (α2 + α4 + α5 + α6 + α9)
< 1.



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 337

Also,

d(x12k+2, x
1
2k+3) ≤

α1[d(x
1
2k+1, x

1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]

n(1− (α2 + α4 + α5 + α6 + α9))
. (B1)

d(x22k+2, x
2
2k+3) ≤

α1[d(x
1
2k+1, x

1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]

n(1− (α2 + α4 + α5 + α6 + α9))
. (B2)

...

d(xn2k+2, x
n
2k+3) ≤

α1[d(x
1
2k+1, x

1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]

n(1− (α2 + α4 + α5 + α6 + α9))
. (Bn)

Adding equations, (B1), (B2), · · · , and (Bn), we get

d(x12k+2, x
1
2k+3) + d(x22k+2, x

2
2k+3) + · · ·+ d(xn2k+2, x

n
2k+3) ≤

α1[d(x
1
2k+1, x

1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]

1− (α2 + α4 + α5 + α6 + α9)

= h[d(x12k+1, x
1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]

≤ h2[d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)].

Therefore one can write,

d(x1n, x
1
n+1) + d(x2n, x

2
n+1) + · · ·+ d(xnn, x

n
n+1)

≤ h[d(x1n−1, x
1
n) + d(x2n−1, x

2
n) + · · ·+ d(xnn−1, x

n
n)]

≤ h2[d(x1n−2, x
1
n−1) + d(x2n−2, x

2
n−1) + · · ·+ d(xnn−2, x

n
n−1)]

≤ · · · ≤ hn[d(x10, x
1
1) + d(x20, x

2
1) + · · ·+ d(xn0 , x

n
1 )].

If we set d(x1n, x
1
n+1) + d(x2n, x

2
n+1) + · · ·+ d(xnn, x

n
n+1) = ψn.

Then ψn ≤ hψn−1 ≤ h2ψn−2 ≤ · · · ≤ hnψ0.

For m > n,
[d(x1n, x

1
m) + d(x2n, x

2
m) + · · ·+ d(xnn, x

n
m)] ≤

s[d(x1n, x
1
n+1) + d(x2n, x

2
n+1) + · · ·+ d(xnn, x

n
n+1)]

+s2[d(x1n+1, x
1
n+2) + d(x2n+1, x

2
n+2) + · · ·+ d(xnn+1, x

n
n+2)]

+ · · ·+ sm−n[d(x1m−1, x
1
m) + d(x2m−1, x

2
m) + · · ·+ d(xnm−1, x

n
m)]

≤ shnψ0 + s2hn+1ψ0 + · · · sm−nhm−1ψ0

< shn[1 + sh+ (sh)2 + · · · ]ψ0

= shn

1−sh
−→ 0 as n −→ ∞.

This shows that {x1n}, {x
2
n}, · · · , {x

n
n} are Cauchy sequences in X. As X is complete



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 338

b-metric space, so there exists x1, x2, x3, · · · , xn ∈ X such that x1n−→ x1, x2n−→ x2 ,· · · ,
xnn−→ xn as n−→∞.

Now we will prove that x1 = S(x1, x2, · · · , xn), x2 = S(x2, x3, x4, · · · , xn, x1) ,· · · ,
xn = S(xn, x1, x2 · · · , xn−1).

Suppose on contrary that
x1 6= S(x1, x2, · · · , xn), x2 6= S(x2, x3, x4, · · · , xn, x1), · · · , xn 6= S(xn, x1, x2 · · · , xn−1).
Then
d(x1, S(x1, x2, · · · , xn)) = l1 > 0, d(x2, S(x2, x3, x4, · · · , xn, x1)) = l2 > 0,· · · , d(xn, S(xn, x1, x2 · · · , xn−1)) =

l3 > 0.
Consider the following and using condition (1) of Theorem 1, we get

l1 = d(x1, S(x1, x2, · · · , xn)) ≤ s[d(x1, x12k+2) + d(x12k+2, S(x
1, x2, · · · , xn))]

= sd(x1, x12k+2) + sd(S(x1, x2, · · · , xn), x12k+2)

= sd(x1, x12k+2) + sd(S(x1, x2, · · · , xn), T (x12k+1, x
2
2k+1, · · · , x

n
2k+1))

≤ sd(x1, x12k+2) + sα1

d(x1, x12k+1) + d(x2, x22k+1) + · · ·+ d(xn, xn2k+1)

n

+sα2

d(x1, S(x1, x2, · · · , xn))d(x12k+1, T (x
1
2k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα3

d(x12k+1, S(x
1, x2, · · · , xn))d(x1, T (x12k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα4

d(S(x1, x2, · · · , xn), T (x12k+1, x
2
2k+1, · · · , x

n
2k+1))d(x

1, x12k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα5

d(S(x1, x2, · · · , xn), T (x12k+1, x
2
2k+1, · · · , x

n
2k+1))d(x

2, x22k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα6

d(x12k+1, T (x
1
2k+1, x

2
2k+1, · · · , x

n
2k+1))d(x

2, x22k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα7

d(x12k+1, S(x
1, x2, · · · , xn))d(x1, x12k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα8

d(x12k+1, S(x
1, x2, · · · , xn))d(x2, x22k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα9

d(x12k+1, T (x
1
2k+1, x

2
2k+1, · · · , x

n
2k+1))d(x

n, xn2k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα10

d(x12k+1, S(x
1, x2, · · · , xn))d(xn, xn2k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 339

= sd(x1, x12k+2) + sα1

d(x1, x12k+1) + d(x2, x22k+1) + · · ·+ d(xn, xn2k+1)

n

+sα2

d(x1, S(x1, x2, · · · , xn))d(x12k+1, x
1
2k+2)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα3

d(x12k+1, S(x
1, x2, · · · , xn))d(x1, x12k+2)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα4

d(S(x1, x2, · · · , xn), x12k+2)d(x
1, x12k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα5

d(S(x1, x2, · · · , xn), x12k+2)d(x
2, x22k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα6

d(x12k+1, x
1
2k+2)d(x

2, x22k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα7

d(x12k+1, S(x
1, x2, · · · , xn))d(x1, x12k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα8

d(x12k+1, S(x
1, x2, · · · , xn))d(x2, x22k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα9

d(x12k+1, x
1
2k+2)d(x

n, xn2k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)

+sα10

d(x12k+1, S(x
1, x2, · · · , xn))d(xn, xn2k+1)

1 + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)
.

Using the concept that {x1n}, {x
2
n},· · · , {x

n
n} are convergent sequences, so it’s subsequences.

Taking limit as k → ∞ we get l1 ≤ 0.
Which implies that
d(x1, S(x1, x2, · · · , xn)) = 0, so x1 = S(x1, x2, · · · , xn).
Similarly we can prove that
x2 = S(x2, x3, x4, · · · , xn, x1) ,· · · , xn = S(xn, x1, x2 · · · , xn−1).
Analogously we have
x1 = T (x1, x2, · · · , xn), x2 = T (x2, x3, · · · , xn, x1) ,· · · , xn = T (xn, x1, x2 · · · , xn−1).

Thus we have proved that (x1, x2, · · · , xn) is a common n-tupled fixed point of S and T .
Uniqueness

Let (x1
∗
, x2

∗
, · · · , xn

∗
)∈ Xn be second common n-tupled fixed point of S and T . From

condition (1) of Theorem 1, we can write

d(x1, x1
∗
) = d(S(x1, x2, · · · , xn), T (x1

∗
, x2

∗
, · · · , xn

∗
))

≤ α1

d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

n



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 340

+α2

d(x1, S(x1, x2, · · · , xn))d(x1
∗
, T (x1

∗
, x2

∗
, · · · , xn

∗
))

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α3

d(x1
∗
, S(x1, x2, · · · , xn))d(x1, T (x1

∗
, x2

∗
, · · · , xn

∗
))

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α4

d(S(x1, x2, · · · , xn), T (x1
∗
, x2

∗
, · · · , xn

∗
))d(x1, x1

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α5

d(S(x1, x2, · · · , xn), T (x1
∗
, x2

∗
, · · · , xn

∗
))d(x2, x2

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α6

d(x1
∗
, T (x1

∗
, x2

∗
, · · · , xn

∗
))d(x2, x2

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α7

d(x1
∗
, S(x1, x2, · · · , xn))d(x1, x1

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α8

d(x1
∗
, S(x1, x2, · · · , xn))d(x2, x2

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α9

d(x1
∗
, T (x1

∗
, x2

∗
, · · · , xn

∗
))d(xn, xn

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α10

d(x1
∗
, S(x1, x2, · · · , xn))d(xn, xn

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

= α1

d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

n

+α2

d(x1, x1)d(x1
∗
, x1

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α3

d(x1
∗
, x1)d(x1, x1

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α4

d(x1, x1
∗
)d(x1, x1

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α5

d(x1, x1
∗
)d(x2, x2

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α6

d(x1
∗
, x1

∗
)d(x2, x2

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α7

d(x1
∗
, x1)d(x1, x1

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α8

d(x1
∗
, x1)d(x2, x2

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 341

+α9

d(x1
∗
, x1

∗
)d(xn, xn

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

+α10

d(x1
∗
, x1)d(xn, xn

∗
)

1 + d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

≤ α1

d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

n
+ α3d(x

1, x1
∗
)

+α4d(x
1, x1

∗
) + α5d(x

1, x1
∗
) + α7d(x

1, x1
∗
) + α8d(x

1, x1
∗
) + α10d(x

1, x1
∗
)

which implies that

(1−
α1

n
− α3 − α4 − α5 − α7 − α8 − α10)d(x

1, x1
∗
) ≤

α1

d(x2, x2
∗
) + d(x3, x3

∗
) + · · ·+ d(xn, xn

∗
)

n

d(x1, x1
∗
) ≤

α1[d(x
2, x2

∗
) + d(x3, x3

∗
) + · · ·+ d(xn, xn

∗
)]

(n− α1 − nα3 − nα4 − nα5 − nα7 − nα8 − nα10)
(C1)

similarly,

d(x2, x2
∗
) ≤

α1[d(x
1, x1

∗
) + d(x3, x3

∗
) + · · ·+ d(xn, xn

∗
)]

(n− α1 − nα3 − nα4 − nα5 − nα7 − nα8 − nα10)
. (C2)

Proceeding n-times, one can write

d(xn, xn
∗
) ≤

α1[d(x
1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn−1, xn−1

∗
)]

(n− α1 − nα3 − nα4 − nα5 − nα7 − nα8 − nα10)
. (Cn)

Adding, C1, C2, · · · , and Cn, we get

d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
) ≤

(n− 1)α1[d(x
1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

(n− α1 − nα3 − nα4 − nα5 − nα7 − nα8 − nα10)
[

1−
(n− 1)α1[d(x

1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

(n− α1 − nα3 − nα4 − nα5 − nα7 − nα8 − nα10)

]

≤ 0

n(1− α1 − α3 − α4 − α5 − α7 − α8 − α10)

(n− α1 − nα3 − nα4 − nα5 − nα7 − nα8 − nα10)
[d(x1, x1

∗
)+d(x2, x2

∗
)+· · ·+d(xn, xn

∗
)] ≤ 0

since α1 + α3 + α4 + α5 + α7 + α8 + α10 < 1.
Therefore

n(1− α1 − α3 − α4 − α5 − α7 − α8 − α10)

(n− α1 − nα3 − nα4 − nα5 − nα7 − nα8 − nα10)
> 0.

Hence
[d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)] ≤ 0

which implies that x1 = x1
∗
, x2 = x2

∗
, · · · , xn = xn

∗
.

So (x1, x2, · · · , xn) = (x1
∗
, x2

∗
, · · · , xn

∗
).

Thus, S and T have unique common n-fixed point.

Theorem 1 yields the following corollary.



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 342

Corollary 1. Let (X, d) be a complete b-metric space with parameter s ≥1 and let the
mapping T : Xn → X satisfy:

d(T (x1, x2, · · · , xn), T (y1, y2, · · · , yn)) ≤ α1

d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

n

+α2

d(x1, T (x1, x2, · · · , xn))d(y1, T (y1, y2, · · · , yn))

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α3

d(y1, T (x1, x2, · · · , xn))d(x1, T (y1, y2, · · · , yn))

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α4

d(T (x1, x2, · · · , xn), T (y1, y2, · · · , yn))d(x1, y1)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α5

d(T (x1, x2, · · · , xn), T (y1, y2, · · · , yn))d(x2, y2)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α6

d(y1, T (y1, y2, · · · , yn))d(x2, y2)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α7

d(y1, T (x1, x2, · · · , xn))d(x1, y1)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α8

d(y1, T (x1, x2, · · · , xn))d(x2, y2)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α9

d(y1, T (y1, y2, · · · , yn))d(xn, yn)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

+α10

d(y1, T (x1, x2, · · · , xn))d(xn, yn)

1 + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

for all x1, x2, x3,· · · , xn and y1, y2, y3, · · · , yn ∈ X and αi ≥ 0, i = 1, 2, · · · , 10 with the
conditions sα1 + α2 + α4 + α5 + α6 + α9 < 1 and α1 + α3 + α4 + α5 + α7 + α8 + α10 < 1.
Then T has unique common n-tupled fixed point in X.

Proof. Proof is very easy if we take S = T in Theorem 1.

Theorem 2. Let (X, d) be a complete b metric space with parameter s ≥ 1 and let the
mappings S, T : Xn −→ X satisfy:

d(S(x1, x2, · · · , xn), T (y1, y2, · · · , yn)) ≤ α1

d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

n

+β
d(x1, S(x1, x2, · · · , xn))d(y1, T (y1, y2, · · · , yn))

1 + s[d(x1, T (y1, · · · , yn)) + d(y1, S(x1, · · · , xn)) + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)]



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 343

+γ
d(x1, S(x1, x2, · · · , xn))d(x1, T (y1, y2, · · · , yn))

1 + s[d(x1, T (y1, · · · , yn)) + d(y1, S(x1, · · · , xn)) + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)]
.

(2)

For all x1, x2, x3,· · · , xn and y1, y2, y3, · · · , yn ∈ X and α, β, γ are non-negative real
numbers with s(α+ β + γ) < 1. Then S and T have unique common n-tupled fixed point.

Proof. Taking n arbitrary points x10, x
2
0, x

3
0,· · · , x

n
0 , in X, define

x12k+1 = S(x12k, x
2
2k, x

3
2k, · · · , x

n
2k),

x22k+1 = S(x22k, x
1
2k, x

3
2k, · · · , x

n
2k),

x32k+1 = S(x32k, x
2
2k, x

1
2k, · · · , x

n
2k),

...
xn2k+1 = S(xn2k, x

n−1
2k , xn−2

2k , · · · , x22k, x
1
2k),

and
x12k+2 = T (x12k+1, x

2
2k+1, x

3
2k+1, · · · , x

n
2k+1),

x22k+2 = T (x22k+1, x
1
2k+1, x

3
2k+1, · · · , x

n
2k+1),

x32k+2 = T (x32k+1, x
2
2k+1, x

1
2k+1, · · · , x

n
2k+1),

...
xn2k+2 = T (xn2k+1, x

n−1
2k+1

, xn−2
2k+1

, · · · , x22k+1, x
1
2k+1) for k=0,1,2,· · · .

For the sake of simplicity, we take
λ = d(x12k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1).

Consider

d(x12k+1, x
1
2k+2) = d(S(x12k, x

2
2k, · · · , x

n
2k), T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1)).

Then by using condition (2) of Theorem 2, we have

d(x12k+1, x
1
2k+2) ≤ α1

d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)

n

+β
d(x12k, S(x

1
2k, x

2
2k, · · · , x

n
2k))d(x

1
2k+1, T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + s[d(x1
2k, T (x

1
2k+1

, · · · , xn
2k+1

)) + d(x1
2k+1

, S(x1
2k, · · · , x

n
2k) + λ]

+γ
d(x12k, S(x

1
2k, x

2
2k, · · · , x

n
2k))d(x

1
2k, T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + s[d(x1
2k, T (x

1
2k+1

, · · · , xn
2k+1

)) + d(x1
2k+1

, S(x1
2k, · · · , x

n
2k)) + λ]

= α1

d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)

n

+β
d(x12k, x

1
2k+1)d(x

1
2k+1, x

1
2k+2)

1 + s[d(x1
2k, x

1
2k+2

) + d(x1
2k+1

, x1
2k+1

) + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)]

+γ
d(x12k, x

1
2k+1)d(x

1
2k, x

1
2k+2)

1 + s[d(x1
2k, x

1
2k+2

) + d(x1
2k+1

, x1
2k+1

) + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)]



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 344

= α1

d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)

n

+β
d(x12k, x

1
2k+1)d(x

1
2k+1, x

1
2k+2)

1 + s[d(x1
2k+1

, x1
2k+2

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)]

+γ
d(x12k, x

1
2k+1)d(x

1
2k, x

1
2k+2)

1 + s[d(x1
2k, x

1
2k+2

) + d(x1
2k, x

1
2k+1

) + d(x2
2k, x

2
2k+1

) + · · ·+ d(xn
2k, x

n
2k+1

)]

≤ α
d(x12k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)

n
+ βd(x12k, x

1
2k+1) + γd(x12k, x

1
2k+1)

which implies that

d(x12k+1, x
1
2k+2) ≤

α+ nβ + nγ

n
d(x12k, x

1
2k+1)

+
α

n
[d(x22k, x

2
2k+1) + d(x32k, x

3
2k+1) + · · ·+ d(xn2k, x

n
2k+1)]. (D1)

Similarly, we can prove

d(x22k+1, x
2
2k+2) ≤

α+ nβ + nγ

n
d(x22k, x

2
2k+1)

+
α

n
[d(x12k, x

1
2k+1) + d(x32k, x

3
2k+1) + · · ·+ d(xn2k, x

n
2k+1)]. (D2)

Proceeding similarly, one can write

d(xn2k+1, x
n
2k+2) ≤

α+ nβ + nγ

n
d(xn2k, x

n
2k+1)

+
α

n
[d(x12k, x

1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn−1

2k , xn−1
2k+1

)]. (Dn)

Adding equations (D1), (D2), · · · , and (Dn), we get

[d(x12k+1, x
1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)] ≤

(α+ β + γ)[d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)].

Also

d(x12k+2, x
1
2k+3) ≤

α+ nβ + nγ

n
d(x12k+1, x

1
2k+2)

+
α

n
[d(x22k+1, x

2
2k+2) + d(x32k+1, x

3
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]. (E1)

d(x22k+2, x
2
2k+3) ≤

α+ nβ + nγ

n
d(x22k+1, x

2
2k+2)

+
α

n
[d(x12k+1, x

1
2k+2) + d(x32k+1, x

3
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]. (E2)

...

d(xn2k+2, x
n
2k+3) ≤

α+ nβ + nγ

n
d(xn2k+1, x

n
2k+2)



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 345

+
α

n
[d(x12k+1, x

1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn−1

2k+1
, xn−1

2k+2
)]. (En)

Adding, (E1), (E2) · · · , and (En), we get

[d(x12k+2, x
1
2k+3) + d(x22k+2, x

2
2k+3) + · · ·+ d(xn2k+2, x

n
2k+3)]

≤ (α+ β + γ)[d(x12k+1, x
1
2k+2) + d(x22k+1, x

2
2k+2) + · · ·+ d(xn2k+1, x

n
2k+2)]

≤ (α+ β + γ)2[d(x12k, x
1
2k+1) + d(x22k, x

2
2k+1) + · · ·+ d(xn2k, x

n
2k+1)].

Therefore we have the following
d(x1n, x

1
n+1) + d(x2n, x

2
n+1) + · · ·+ d(xnn, x

n
n+1)

≤ (α+ β + γ)[d(x1n−1, x
1
n) + d(x2n−1, x

2
n) + · · ·+ d(xnn−1, x

n
n)]

≤ (α+ β + γ)2[d(x1n−2, x
1
n−1) + d(x2n−2, x

2
n−1) + · · ·+ d(xnn−2, x

n
n−1)]

≤ · · · ≤ (α+ β + γ)n[d(x10, x
1
1) + d(x20, x

2
1) + · · ·+ d(xn0 , x

n
1 )]

where h = α+ β + γ < 1.
Now if we set d(x1n, x

1
n+1) + d(x2n, x

2
n+1) + · · ·+ d(xnn, x

n
n+1) = ψn.

Then ψn ≤ hψn−1 ≤ h2ψn−2 ≤ · · · ≤ hnψ0.

So for m > n, we have

[d(x1n, x
1
m) + d(x2n, x

2
m) + · · ·+ d(xnn, x

n
m)] ≤

s[d(x1n, x
1
n+1) + d(x2n, x

2
n+1) + · · ·+ d(xnn, x

n
n+1)]

+s2[d(x1n+1, x
1
n+2) + d(x2n+1, x

2
n+2) + · · ·+ d(xnn+1, x

n
n+2)]

+ · · ·+ sm−n[d(x1m−1, x
1
m) + d(x2m−1, x

2
m) + · · ·+ d(xnm−1, x

n
m)]

≤ shnψ0 + s2hn+1ψ0 + · · · sm−nhm−1ψ0

< shn[1 + sh+ (sh)2 + · · · ]ψ0

= shn

1−sh
−→ 0 as n −→ ∞.

This shows that {x1n}, {x
1
n}, · · · , {x

1
n} are Cauchy sequences in X. As X is complete

b-metric space, so there exists x1, x2, x3,· · · , xn ∈ X such that x1n−→ x1, x2n−→ x2 ,· · · ,
xnn−→ xn as n−→∞.

Now we will prove that x1 = S(x1, x2, · · · , xn), x2 = S(x2, x3, x4, · · · , xn, x1) ,· · · ,
xn = S(xn, x1, x2 · · · , xn−1). On contrary suppose that x1 6= S(x1, x2, · · · , xn), x2 6=
S(x2, x3, x4, · · · , xn, x1), · · · , xn 6= S(xn, x1, x2 · · · , xn−1).
Then

d(x1, S(x1, x2, · · · , xn)) = l1 > 0, d(x2, S(x2, x3, x4, · · · , xn, x1)) = l2 > 0,· · · , d(xn, S(xn, x1, x2 · · · , xn−1)) =
l3 > 0.
For the sake of simplicity, we take again

λ = d(x1, x12k+1) + d(x2, x22k+1) + · · ·+ d(xn, xn2k+1).
Now consider the following and using condition (2) of Theorem 2, we get

l1 = d(x1, S(x1, x2, · · · , xn)) ≤



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 346

s[d(x1, x12k+2) + d(x12k+2, S(x
1, x2, · · · , xn))]

= sd(x1, x12k+2) + sd(S(x1, x2, · · · , xn), x12k+2))

= sd(x1, x12k+2) + sd(S(x1, x2, · · · , xn), T (x12k+1, x
2
2k+1, · · · , x

n
2k+1))

≤ sd(x1, x12k+2) + sα
d(x1, x12k+1) + d(x2, x22k+1) + · · ·+ d(xn, xn2k+1)

n

+sβ
d(x1, S(x1, x2, · · · , xn))d(x12k+1, T (x

1
2k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + s[d(x1, T (x1
2k+1

, x2
2k+1

, · · · , xn
2k+1

)) + d(x1
2k+1

, S(x1, x2, · · · , xn)) + λ]

+sγ
d(x1, S(x1, x2, · · · , xn))d(x1, T (x12k+1, x

2
2k+1, · · · , x

n
2k+1))

1 + s[d(x1, T (x1
2k+1

, x2
2k+1

, · · · , xn
2k+1

)) + d(x1
2k+1

, S(x1, x2, · · · , xn)) + λ]

= sα
d(x1, x12k+1) + d(x2, x22k+1) + · · ·+ d(xn, xn2k+1)

n

+sβ
d(x1, S(x1, x2, · · · , xn))d(x12k+1, x

1
2k+2)

1 + s[d(x1, x1
2k+2

) + d(x1
2k+1

, S(x1, · · · , xn)) + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)]

+sγ
d(x1, S(x1, x2, · · · , xn))d(x1, x12k+2)

1 + s[d(x1, x1
2k+2

) + d(x1
2k+1

, S(x1, · · · , xn)) + d(x1, x1
2k+1

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)]

= sα
d(x1, x12k+1) + d(x2, x22k+1) + · · ·+ d(xn, xn2k+1)

n

+sβ
d(x1, S(x1, x2, · · · , xn))d(x12k+1, x

1
2k+2)

1 + s[d(x1
2k+1

, S(x1, x2, · · · , xn)) + d(x1
2k+1

, x1
2k+2

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)]

+sγ
d(x1, S(x1, x2, · · · , xn))d(x1, x12k+2)

1 + s[d(x1
2k+1

, S(x1, x2, · · · , xn)) + d(x1
2k+1

, x1
2k+2

) + d(x2, x2
2k+1

) + · · ·+ d(xn, xn
2k+1

)]
.

Taking limit k → ∞ we get l1 ≤ 0
so d(x1, S(x1, x2, · · · , xn)) = 0 ⇒ x1 = S(x1, x2, · · · , xn).
Similarly we can prove that
x2 = S(x2, x3, x4, · · · , xn, x1) ,· · · , xn = S(xn, x1, x2 · · · , xn−1).
Also we can prove that
x1 = T (x1, x2, · · · , xn), x2 = T (x2, x3, · · · , xn, x1) ,· · · , xn = T (xn, x1, x2 · · · , xn−1).

Thus we have proved that (x1, x2, · · · , xn) is a common n-fixed point of S and T .
Uniqueness:

Let (x1
∗
, x2

∗
, · · · , xn

∗
)∈Xn be another common n-fixed point of S and T . Using condition

(2) of Theorem 2 here, we get

d(x1, x1
∗
) = d(S(x1, x2, · · · , xn), T (x1

∗
, x2

∗
, · · · , xn

∗
))

≤ α
d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

n



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 347

+β
d(x1, S(x1, x2, · · · , xn))d(x1

∗
, T (x1

∗
, x2

∗
, · · · , xn

∗
))

1 + s[d(x1, T (x1
∗
, · · · , xn

∗
)) + d(x1

∗
, S(x1, · · · , xn)) + d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

+γ
d(x1, S(x1, x2, · · · , xn))d(x1, T (x1

∗
, x2

∗
, · · · , xn

∗
))

1 + s[d(x1, T (x1
∗
, · · · , xn

∗
)) + d(x1

∗
, S(x1, · · · , xn)) + d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

= α
d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

n

+β
d(x1, x1)d(x1

∗
, x1

∗
)

1 + s[d(x1, x1
∗
) + d(x1

∗
, x1) + d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

+γ
d(x1, x1)d(x1, x1

∗
)

1 + s[d(x1, x1
∗
) + d(x1

∗
, x1) + d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

= α
d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

n

+β
d(x1, x1)d(x1

∗
, x1

∗
)

1 + s[3d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

+γ
d(x1, x1)d(x1, x1

∗
)

1 + s[3d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

d(x1, x1
∗
) ≤ α

d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)

n
.

Which implies that

d(x1, x1
∗
)[1−

α

n
] ≤ α

d(x2, x2
∗
) + d(x3, x3

∗
) + · · ·+ d(xn, xn

∗
)

n

d(x1, x1
∗
) ≤

α

n− α
[d(x2, x2

∗
) + d(x3, x3

∗
) + · · ·+ d(xn, xn

∗
)]. (F1)

Similarly, we can prove that

d(x2, x2
∗
) ≤

α

n− α
[d(x1, x1

∗
) + d(x3, x3

∗
) + · · ·+ d(xn, xn

∗
)] (F2)

...

d(xn, xn
∗
) ≤

α

n− α
[d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn−1, xn−1

∗
)]. (Fn)

Adding, (F1), (F2), · · · , and (Fn), we get

d(x1, x1
∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
) ≤

(n− 1)α

n− α
[d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)]

which implies that

(1−
(n− 1)α

n− α
)[d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)] ≤ 0



S. Hussain, M. Sarwar and Y. Li / Eur. J. Pure Appl. Math, 11 (1) (2018), 331-351 348

so
(1− α)[d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
)] ≤ 0.

Since 0 < α < 1.
Therefore
d(x1, x1

∗
) + d(x2, x2

∗
) + · · ·+ d(xn, xn

∗
) = 0.

Thus
(x1, x2, · · · , xn) = (x1

∗
, x2

∗
, · · · , xn

∗
).

Hence S and T have unique common n-tupled fixed point.

Corollary 2. Let (X, d) be a complete b metric space with parameter s ≥ 1 and let the
mapping T : Xn −→ X satisfy:

d(S(x1, x2, · · · , xn), T (y1, y2, · · · , yn)) ≤ α1

d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)

n

+β
d(x1, T (x1, x2, · · · , xn))d(y1, T (y1, y2, · · · , yn))

1 + s[d(x1, T (y1, · · · , yn)) + d(y1, T (x1, · · · , xn)) + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)]

+γ
d(x1, T (x1, x2, · · · , xn))d(x1, T (y1, y2, · · · , yn))

1 + s[d(x1, T (y1, · · · , yn)) + d(y1, T (x1, · · · , xn)) + d(x1, y1) + d(x2, y2) + · · ·+ d(xn, yn)]
.

For all x1, x2, x3,· · · , xn and y1, y2, y3,· · · , yn ∈ X and α, β, γ are non-negative real
numbers with s(α+ β + γ) < 1. Then T has unique common n-tupled fixed point.

Remarks:

• If we put n = 2 and α9 = 0, α10 = 0 in Theorem 1, then we get coupled fixed point
result of Sarwar et al. [15].

• If we put n = 2 and αi = 0, i = 4, 5, · · · , 10 in Theorem 1, then we get the result of
Malhotra and Bansal [16].

Example 3. Suppose S=[0, 1] and a b-metric d : X×X → R defined by d(x, y) = 2
3
(x−y)2

for each x, y ∈ X.
Then (X, d) is b metric space having parameter s = 2. If we define S, T : Xn → X by

S(x1, x2, x3, · · · , xn) = x1+x2+x3+···+xn

n
, T (x1, x2, x3, · · · , xn) = x1+x2+x3+···+xn

n+1
for each

x1, x2, x3,· · · , xn ∈ Xn. Then it can be proved simply that the maps S and T satisfy the
contraction in Theorem 1 with α1 = 1

25
, α2 = 2

25
, α3 = 3

25
, α4 = 4

25
, α5 = 1

50
, α6 = 3

50
, α7 =

7
50
, α8 =

9
50
, α9 =

1
75
, α10 =

2
75
.

Clearly (0, 0, 0, · · · , 0) is a unique common n-tupled fixed point of S and T .

3. Conclusion

The derived results generalized the results of [16] and [15] in the setting of b metric
spaces.



REFERENCES 349

Authors contributions All authors contributed equally to the writing of this manuscript.
All authors read and approved the final version.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11571378).
The authors are grateful to the editor and anonymous reviewers for their careful reviews,
valuable comments and remarks to improve this manuscript.

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