EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 348-357
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Common Fixed Points of Two Multivalued
Asymptotically Nonexpansive Mappings

Safeer Hussain Khan1,∗, Hira Iqbal2, Mujahid Abbas3,4

1Department of Mathematics, Statistics and Physics, Qatar University, Doha, 2713,Qatar
2 Department of Sciences and Humanities, National University of Computer and Emerging
Sciences, Lahore Campus, Pakistan
3Department of Mathematics,Government College University Lahore 54000, Pakistan
4Department of Mathematics, King Abdulaziz University, P. O. Box. 80203, Jeddah 21589,
Saudi Arabia

Abstract. In this paper, we construct a modified Ishikawa iterative process to approximate com-
mon fixed points of two multivalued asymptotically nonexpansive mappings and prove some con-
vergence theorems in uniformly convex hyperbolic spaces.

2010 Mathematics Subject Classifications: AMS 47H10, 47H09
Key Words and Phrases: Asymptotically nonexpansive mapping, multivalued mapping, com-
mon fixed point, Ishikawa iteration process

1. Introduction

Let D be a nonempty subset of a metric space (X, d). A mapping T : D → D is called
asymptotically nonexpansive if for any x, y ∈ D, there exists a sequence {kn} with kn ≥ 1
and lim

n→∞
kn = 1 such that

d(Tnx, Tny) ≤ knd(x, y).

Let P (D) represent the set of all nonempty subsets ofD, Cl(D) denote the set all nonempty
closed subsets of D and CB(D) denote the set all nonempty closed and bounded subsets
of D . For A,B ∈ CB(D) and x, y ∈ D, define

d(x,A) = inf
a∈A

d(x, a).

Then H is known as the generalized Pompeiu-Hausdorff distance induced by d if

H(A,B) = max{sup
a∈A

d(a,B), sup
b∈B

d(A, b)}.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3371

Email addresses: safeer@qu.edu.qa (S. H. Khan),
hira.iqbal@nu.edu.pk (H. Iqbal), abbas.mujahid@gmail.com (M. Abbas)

http://www.ejpam.com 348 c© 2019 EJPAM All rights reserved.



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 349

We say that a multivalued mapping T : D → P (D) has a fixed point x if x ∈ Tx. It
is obvious that the theory of mutlivalued mappings is more complicated than that of the
single valued mappings. Many different techniques have been employed to approximate
fixed points of multivalued mappings. Rus [7] introduced the concept of generalized orbits
in multivalued mappings. Khamsi and Kirk extended this concept of generalized orbits
for the iterates of a multivalued mapping in [3].

Definition 1. Let D be a nonempty subset of X and T : D → P (D) be a multivalued
mapping. We call O(x, T ) = {xn} a generalized orbit of x if for any x ∈ D and n ≥ 0,
the sequence {xn} is defined by x0 = x and xn+1 ∈ T (xn) where n ∈ N ∪ {0}.

Recently, in 2017, Khamsi and Khan [2], introduced the concept of a multivalued
asymptotically nonexpansive mapping.

Definition 2. A mapping T : D → P (D) is called a multivalued asymptotically nonex-
pansive mapping if there exists a sequence {kn} with kn ≥ 1 and limn→∞ kn = 1 such that
for any x, y ∈ D, and any generalized orbit O(x, T ) = {xn} of x, there exists a generalized
orbit O(y, T ) = {yn} of y such that

d(xn+h, yh) ≤ khd(xn, y),

where n, h ∈ N.

They established the existence of a fixed point for a multivalued asymptotically non-
expansive mapping in hyperbolic metric spaces. They also proved the convergence of a
modified Mann iterative process.

Motivated by this, we construct a modified Ishikawa iterative process for two multival-
ued asymptotically nonexpansive mappings and then prove some convergence theorems in
uniformly convex hyperbolic metric spaces.

Next we recall the concept of hyperbolic metric spaces. Let (X, d) be a metric space
and x, y be any two points in X. Then the unique metric segment [x, y] is an isometric
image of the real line interval [0, d(x, y)]. A point z in [x, y] is denoted as βx ⊕ (1 − β)y
which satisfies

d(x, z) = (1− β)d(x, y) and d(z, y) = βd(x, y)

where β ∈ [0, 1]. Metric spaces with the class of metric segments are usually called convex
metric spaces [5]. Further, if

d(αu⊕ (1− α)x, αv ⊕ (1− α)y) ≤ αd(u, v) + (1− α)d(x, y)

is satisfied for any x, y, u, v ∈ X and α ∈ [0, 1], then X is said to be a hyperbolic metric
space [6]. The following definition of uniformly convex hyperbolic metric space can be
found in [1].

Definition 3. A hyperbolic metric space (X, d) is called uniformly convex if for any
x, y, w ∈ X, for every r > 0, and for each ε > 0

δ(r, ε) = inf

{
1− 1

r
d

(
1

2
x⊕ 1

2
y, w

)
;



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 350

d(x,w) ≤ r, d(y, w) ≤ r, d(x, y) ≥ rε
}
> 0.

Following is an important result in a uniformly convex hyperbolic metric space which
will be used later.

Theorem 1. [4] Let (X, d) be a uniformly convex complete hyperbolic metric space. Let
c > 0 and z ∈ X. Assume that {xn} and {yn} are two sequences in X such that

lim sup
n→∞

d(xn, z) ≤ c, lim sup
n→∞

d(yn, z) ≤ c,

and, lim
n→∞

d(αxn ⊕ (1− α)yn, z) = c,

then we have
lim
n→∞

d(xn, yn) = 0.

The following definition is needed in the sequel.

Definition 4. A multivalued mapping T : D → P (D) is said to be H-continuous if for
any sequence {xn} which converges to x in D, we have

lim
n→∞

d(yn, T (x)) = 0

for any sequence {yn} such that yn ∈ Txn for any n ∈ N.

Khamsi and Khan [2] have shown that every multivalued asymptotically nonexpansive
mappings is H-continuous.

In this paper, we construct a modified Ishikawa iterative process which extends the
Mann Type iterative process considered by Khamsi and Khan [2]. We use this iterative
process to approximate common fixed points for two multivalued asymptotically nonex-
pansive mappings and prove some convergence theorems in uniformly convex hyperbolic
spaces.

2. Main Results

Let T1 and T2 be two multivalued asymptotically nonexpansive mappings such that

there exist sequences of positive numbers {k(1)m } and {k(2)m } with k
(i)
m ∈ [1,∞) and

∑
(k

(i)
m −

1) <∞. Let
km = max{k(1)m , k(2)m },

then
∑

(km − 1) <∞. So, we take {km} for both T1 and T2.
Now, we construct modified Ishikawa iterative process as follows:

Let O(x1, T1) = {x́1n} be a generalized orbit of x1 associated with T1 and O(x1, T2) =
{x1n} be a generalized orbit of x1 associated with T2. Fix 0 < α, β < 1 and set

x2 = αx1 ⊕ (1− α)ý11,



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 351

y1 = βx1 ⊕ (1− β)x11.

Again, let O(x2, T1) = {x́2n} be a generalized orbit of x2 associated with T1 and O(x2, T2) =
{x2n} be a generalized orbit of x2 associated with T2 where

d(x1n+h, x
2
h) ≤ khd(x1n, x

2),

and d( ´x1n+h, x́
2
h) ≤ khd(x́1n, x

2).

Hence, using induction for any m ≥ 1 we have a sequence {xm} in D and the orbits
O(xm, T1) = {x́mn } and O(xm, T2) = {xmn } such that

xm+1 = αxm ⊕ (1− α)ýmm,

ym = βxm ⊕ (1− β)xmm (1)

and

d(xm−1n+h , x
m
h ) ≤ khd(xm−1n , xm),

and d( ´xm−1n+h , x́
m
h ) ≤ khd( ´xm−1n , xm).

Throughout this section, we denote F = F (T1) ∩ F (T2).

Lemma 1. Let (X, d) be a complete uniformly convex hyperbolic metric space. Let D
be a nonempty bounded, closed and convex subset of X. Let T1 and T2 be multivalued
asymptotically nonexpansive mappings with km ∈ [1,∞) and

∑∞
m=1(km − 1) <∞. Define

the sequence as in (1). If

lim
m→∞

d(xm, xmm) = 0 = lim
m→∞

d(xm, x́mm),

then
lim
m→∞

d(xm, xm1 ) = 0 = lim
m→∞

d(xm, x́m1 ).

Proof. Let
d(xm, x́mm) = am, and d(xm, xmm) = bm.

Since xm+1 = αxm ⊕ (1− α)ýmm, we have

d(xm+1, xm) ≤ (1− α)d(xm, ýmm)

≤ d(xm, ýmm)

≤ d(xm, x́mm) + d(x́mm, ý
m
m)

≤ am + d(x́mm, x́
m
m+m) + d(x́mm+m, ý

m
m)

≤ am + kmd(x́mm, x
m) + kmd(x́mm, y

m)

≤ am + kmam + km(βd(x́mm, x
m)



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 352

+(1− β)d(x́mm, x
m
m))

≤ am + kmam + kmβam + km(1− β)d(x́mm, x
m)

+km(1− β)d(xmm, x
m)

≤ am + kmam + kmβam

+km(1− β)am + km(1− β)bm

≤ (1 + 2km)am + kmbm.

Taking limm→∞ in the above inequality, we get

lim
m→∞

d(xm+1, xm) ≤ lim
m→∞

(1 + 2km)am + lim
m→∞

kmbm = 0.

That is
lim
m→∞

d(xm+1, xm) = 0 (2)

Moreover, from (2)

d(xm+1, ´xm+1
1 ) ≤ d(xm+1, ´xm+1

m+1) + d( ´xm+1
m+1,

´xm+1
1 ),

≤ am+1 + k1d(xm+1, ´xm+1
m ),

≤ am+1 + k1[d(xm, xm+1) + d(xm, x́mm) + d(x́mm,
´xm+1
m )],

≤ am+1 + k1[d(xm, xm+1) + d(xm, x́mm) + kmd(xm, xm+1)],

= am+1 + k1[am + (1 + km)d(xm, xm+1)].

Then

lim
m→∞

d(xm+1, ´xm+1
1 ) ≤ lim

m→∞
am+1 + k1 lim

m→∞
[am + (1 + km)d(xm, xm+1)]

= 0.

Hence
lim
m→∞

d(xm, x́m1 ) = 0.

Similarly

d(xm+1, xm+1
1 ) ≤ d(xm+1, xm+1

m+1) + d(xm+1
m+1, x

m+1
1 )

≤ bm+1 + k1d(xm+1, xm+1
m )

≤ bm+1 + k1[d(xm, xm+1) + d(xm, xmm) + d(xmm, x
m+1
m )]

≤ bm+1 + k1[d(xm, xm+1) + d(xm, xmm) + kmd(xm, xm+1)]

= bm+1 + k1[bm + (1 + km)d(xm, xm+1)].

Consequently
lim
m→∞

d(xm, xm1 ) = 0.



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 353

Theorem 2. Let (X, d) be a complete uniformly convex hyperbolic metric space. Let D
be a nonempty bounded, closed and convex subset of X. Let T1 and T2 be multivalued
asymptotically nonexpansive mappings. Let km be the Lipschitz sequence associated with
T1 and T2 such that km ∈ [1,∞] and

∑∞
m=1(km−1) <∞. Let F 6= ∅ and T1p = T2p = {p}

for p ∈ F. Fix x1 ∈ D and α ∈ (0,∞). Suppose xm is defined as in (1). Then,

lim
m→∞

d(xm, xm1 ) = 0, and lim
m→∞

d(xm, x́m1 ) = 0.

That is,
lim
m→∞

d(xm, T1x
m) = 0, and lim

m→∞
d(xm, T2x

m) = 0.

Proof. Let p ∈ D be such that p ∈ F and T1p = T2p = {p}. Then

d( ´xmn+h, p) ≤ khd(xmn , p)

and d(xmn+h, p) ≤ khd(xmn , p).

Now

d(xm+1, p) ≤ αd(xm, p) + (1− α)d(ýmm, p)

≤ αd(xm, p) + (1− α)kmd(ym, p) (3)

and

d(ym, p) ≤ βd(xm, p) + (1− β)d(x́mm, p)

≤ β d(xm, p) + (1− β)kmd(xm, p). (4)

Inequalities (3) and (4) imply,

d(xm+1, p) ≤ αd(xm, p) + (1− α)km[β + (1− β)km]d(xm, p)

= [α+ β(1− α)km + (1− β)(1− α)k2m]d(xm, p)

= Vmd(xm, p),

where Vm = α+ β(1− α)km + (1− β)(1− α)k2m.
This implies,

d(xm+1, p)− d(xm, p) ≤ (Vm − 1)d(xm, p) ≤ (Vm − 1)δ(D)

for any m ∈ N, where δ(D) = supx,y∈D {d(x, y)} is the diameter of D. Therefore,

d(xm+h, p)− d(xm, p) ≤
m+(h−1)∑
i=m

(Vi − 1)δ(D).

Since
∑∞

m=1(km − 1) <∞ so
∑∞

m=1(Vm − 1) <∞. Let h→∞,

lim sup
n→∞

d(xn, p)− d(xm, p) ≤ δ(D)
∞∑
i=m

(Vi − 1),



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 354

for any m ∈ N. Now letting m→∞,

lim sup
n→∞

d(xn, p) ≤ lim inf
m→∞

d(xm, p).

This implies {d(xn, p)} is convergent. Let

c = lim
n→∞

d(xn, p). (5)

If c = 0, then we have nothing to prove. So we take c > 0. Since d(xnn, p) ≤ knd(xn, p),

lim sup
n→∞

d(xnn, p) ≤ c. (6)

Also,
lim sup
n→∞

d(yn, p) ≤ c. (7)

We know that if limn→∞ kn = 1 then limn→∞ k
2
n = 1, therefore,

lim sup
n→∞

d(ýnn, p) ≤ lim sup
n→∞

knd(yn, p)

≤ lim sup
n→∞

kn[βd(xn, p) + (1− β)d(xnn, p)]

≤ lim sup
n→∞

kn[βd(xn, p) + (1− β)knd(xn, p)]

= c,

and
c = lim

n→∞
d(xn+1, p) = lim

n→∞
d(αxn ⊕ (1− α)ýnn, p). (8)

Then, from Theorem 1 , (5), (7) and (8), we get

lim
n→∞

d(xn, ýnn) = 0. (9)

Next

d(xn, p) ≤ d(xn, ýnn) + d(ýnn, p)

≤ d(xn, ýnn) + knd(p, yn)

implies
lim inf
n→∞

d(xn, p) ≤ lim inf
n→∞

(d(xn, ýnn) + knd(p, yn)).

Using (5) and (9), we get
c ≤ lim inf

n→∞
d(yn, p). (10)

Then (7) and (10) imply
lim
n→∞

d(yn, p) = c. (11)



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 355

That is,
c = lim

n→∞
d(yn, p) = lim

n→∞
d(βxn ⊕ (1− β)xnn). (12)

From Theorem 1, (5), (6) and (12), we get,

d(xn, xnn) = 0. (13)

Also,

d(xn, x́nn) ≤ d(xn, ýnn) + d(ýnn, x́
n
n)

≤ d(xn, ýnn) + knd(xn, yn)

≤ d(xn, ýnn) + kn(1− α)d(xn, xnn).

Hence, (9) and (13) give
lim
n→∞

d(xn, x́nn) = 0. (14)

Now using Lemma 1, (13) and (14), we get our desired results.

We now give some convergence results.

Theorem 3. Let D be a compact and convex subset of a uniformly convex hyperbolic
space. Let T1, T2 and xn be as in Theorem 2. If F 6= ∅ with T1p = T2p = {p} for p ∈ F
then there is a subsequence of {xn} which converges to a common fixed point of T1 and T2.

Proof. Since D is compact so there exists a subsequence {xnk} of {xn} such that {xnk}
converges to some z ∈ D. From Theorem 2 we know that

lim
m→∞

d(xn, xn1 ) = 0, and lim
n→∞

d(xn, x́n1 ) = 0.

Applying H-continuity of T1 and T2, we have

lim
n→∞

d(Tz, xnk
1 ) = lim

n→∞
d(T1z,

´xnk
1 ) = 0

where xnk
1 ∈ T1xnk and ´xnk

1 ∈ T2xnk . Thus

d(z, T1z) ≤ d(T1z,
´xnk
1 ) + d(z, xnk) + d( ´xnk

1 , xnk).

Therefore, as n→∞
d(z, T1z) = 0.

Similarly,

d(z, T2z) ≤ d(T2z,
´xnk
1 ) + d(z, xnk) + d( ´xnk

1 , xnk).

Thus, as n→∞,
d(z, T2z) = 0.

Hence, z is a common fixed point of T1 and T2.



S. H. Khan, H. Iqbal, M. Abbas / Eur. J. Pure Appl. Math, 12 (2) (2019), 348-357 356

Theorem 4. Let D be a nonempty, closed, convex and bounded subset of a complete
uniformly convex hyperbolic space (X, d). Let T1, T2 and xn be as in Theorem 2. If F 6= ∅
with T1p = T2p = {p} for p ∈ F, then {xn} converges to a common fixed point of T1 and
T2 if and only if lim infn→∞ d(xn, F ) = 0 where d(xn, F ) = inf{d(xn, p) : p ∈ F}.

Proof. The necessity of the conditions is obvious. Conversely, suppose that lim infn→∞ d(xn, F ) =
0. Since,

d(xn+1, p) ≤ Vnd(xn, p)

⇒ d(xn+1, F ) ≤ Vnd(xn, F ).

Thus, limn→∞(xn, F ) exists.
Since lim infn→∞ d(xn, F ) = 0, limn→∞(xn, F ) = 0. Next, from 1 + x ≤ ex for all

x ≥ 0, we obtain

d(xn+k, p) ≤ Vn+(k−1)d(xn+(k−1), p)

= (1 + (Vn+(k−1) − 1))d(xn+(k−1), p)

≤ eVn+(k−1)−1d(xn+(k−1), p)

≤ eVn+(k−1)−1eVn+(k−2)−1d(xn+(k−2), p)

≤: : :

≤: : :

≤ e
∑n+(k−1)

i=n (Vi−1)d(xn, p).

We know that
∑

n(Vn − 1) <∞,so there exists some W such that,

d(xn+k, p) ≤Wd(xn, p)

for all p ∈ F and n ∈ N. Since limn→∞ d(xn, F ) = 0, ∃ n0 such that

d(xn0 , F ) <
ε

W + 1
.

Thus there must exist p∗ such that

d(xn0 , p∗) <
ε

W + 1
.

Hence

d(xn0+k, xn0) ≤ d(xn0+k, p∗) + d(p∗, xn0)

≤Wd(p∗, xn0) + d(p∗, xn0)

< W

(
ε

W + 1

)
+

ε

W + 1

= ε.



REFERENCES 357

This shows that {xn} is Cauchy in D. Since D is closed, xn converges to some z in D.
Next, we show that z ∈ F. Let d(z, xn) < ε

4(k1+1) and let z∗ ∈ F. Then d(xn, z∗) <
ε

4(k1+1)

and d(z∗, T z) <
ε

2(k1+1) so that

d(z, z∗) < d(z, xn) + d(xn, z∗) <
ε

2(k1 + 1)
.

Finally,

d(z, T1z) ≤ d(z, xn) + d(xn, z∗) + d(z∗, T1z)

≤ d(z, xn) + d(xn, z∗) + k1d(z∗, z)

<
ε

4(k1 + 1)
+

ε

4(k1 + 1)
+

ε

2(k1 + 1)

< ε.

Therefore z ∈ T1z. Similarly z ∈ T2z.
Hence T1 and T2 have a common fixed point.

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