Journal of Applied Mathematics and Physics, 2014, 2, 170-175
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.25021
How to cite this paper: Khan, A.R. and Fukhar-ud-din, H. (2014) Common Fixed Point Iterations of Generalized Asymptoti-
cally Quasi-Nonexpansive Mappings in Hyperbolic Spaces. Journal of Applied Mathematics and Physics, 2, 170-175.
http://dx.doi.org/10.4236/jamp.2014.25021
Common Fixed Point Iterations of
Generalized Asymptotically
Quasi-Nonexpansive Mappings in
Hyperbolic Spaces
A. R. Khan, H. Fukhar-ud-din
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran,
Saudi Arabia
Received Dec emb er 2013
Abstract
We introduce a general iterative method for a finite family of generalized asymptotically quasi-
nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative
method includes multi-step iterative method of Khan et al. [1] as a special case. Our results are
new in hyperbolic spaces and generalize many known results in Banach spaces and CAT(0) sp aces,
simultaneousl y.
Keywords
Hyperbolic Space, General Iterative Method, Generali zed Asymptot ically Qu asi-Nonex pansi ve
Mapping, Common Fixed Point, S tron g Conve rgenc e
1. Introduction
Let
C
be a nonempty subset of a metric space
X
and
:TC C→
be a mapping. Throughout this paper, we
assume that
()FT
, the set of fixed points of
T
is nonempty and
{1,2, 3,,}.Ir= …
The mapping
T
is: 1)
asymptotically nonexpansive if there exists a sequence of real numbers
{ }
n
u
in
[ )
0,∞
with
lim 0
n
n
u
→∞
=
such
that
( )
( )
,1(, )
nn n
dTxTyud xy≤+
for all
,xy C∈
and
1n≥
2) asymptotically quasi -nonexpansive if there
exists a sequence of real numbers
{ }
n
u
in
[ )
0,∞
with
lim 0
n
n
u
→∞
=
such that
( )
( )
,1(, )
nn
dTxpu dxp≤+
for all
, ()x CpFT∈∈
and
1n≥
3) generalized asymptotically quasi-nonexpansive if there exist two se-
quences of real numbers
{ }
n
u
and
{ }
n
c
in
[ )
0,∞
with
lim0 lim
nn
nn
uc
→∞ →∞
= =
such that
( )
( )( )
,, ,
nnn
dTxpdxp udxpc≤+ +
for all
, ()x CpFT∈∈
and
1n≥
(iv) uniformly
L
-Lipschitzian if
there exists a constant
0L>
such that
( )
,(, )
nn
dTxTx Ldxy≤
for all
,xy C∈
and
1n≥
(v)
( )
L γ −−

A. R. Khan, H. Fukhar-ud-din
171
uniformly Lipschitzian if there are constants
0, 0L γ >>
such that
( )
,(, )
nn
dTxTx Ldxy
γ
≤
for all
,xy C∈
and
1n≥
and (vi) semi-compact if for any sequence
{ }
n
x
in
C
wit h
( )
lim ,0
nn
n
d x Tx
→∞
=
, there exists a
subsequence
{ }
i
n
x
of
{ }
n
x
such that
.
i
n
x cC→∈
Let
( )
,Xd
be a metric space. Suppose that there exists a family
F
of metric segments such that any two
points
,xy
in
X
are endpoints of a unique metric segment
[ ][ ]
,,xy Fxy∈
is an isometric image of the real
line interval
( )
0, ,d xy 

). We shall denote by
(1 )xy αα ⊕−
the unique point of
[ ]
,xy
which satisfies
( ,)(1)(,)and(,)( ,)for[0,1].dxzdxy dzydxyJ α αα =−= ∈=
Such metric spaces are usually called convex metric spaces [2] [3]. One can easily deduce that
01x yy⊕=
,
10x yx⊕=
and
(1 )x xx αα ⊕− =
from the definition of a convex metric space [2].
A convex metric space
X
is hyperbolic if
((1) ,(1))(,)(1) ( ,)dxy zwdxzdyw αααααα ⊕−⊕−≤+−
for all
,,,xyzw X∈
and
J α ∈
. For
zw=
, the hyperbolic inequality reduces to convex structure [3].
( ) ( ) ()() ()
1,, 1,.dxyzd xzdyz αα αα ⊕− ≤+−
(1.1)
A nonempty subset
C
of a convex metric space
X
is convex if
(1 )x yC αα ⊕− ∈
for all
,xy C∈
and
.J α ∈
Normed spaces and their subsets are linear hyperbolic spaces while
(0) CAT
spaces [4]-[6] qualify for the
criteria of nonlinear hyperbolic spaces [2] [7].
A convex metric space
X
is uniformly convex [7] if
()() () ()
1 11
,inf 1,:,,,,,0,
22
rdaxyd axrdayrdxyr
r
δε ε


= −⊕≤≤≥>




for any
,0a Xr∈>
and
0 ε >
.
From now onwards we assume that
X
is a uniformly convex hyperbolic space with the property that for
every
0, 0s ε ≥>
, there exists
(, )0s ηε >
depending on  and  such that
(,)(,) 0rs δεηε >>
for any
rs>
.
We now translate the iterative method (1.3) [1] from normed space setting to the more general setup of hy-
perbolic space as follows:
11
, ,1
nrn n
xCxU xn
+
∈= ≥
(1.2)
whe r e
( )
( )
( )
( )
( )
0
111 01
2 2212
1
the identity mapping
1
1
1
n
n
n nnn
n
n nnn
n
rnrn rrn rn
UI
U xaTUxax
Uxa TUxax
U xaTUxax
−
=
= ⊕−
= ⊕−
= ⊕−

and
{ }
:
i
Ti I∈
is a family of generalized asymptotically quasi-nonexpansive self-mappings of
C
, i.e. ,
( )
()()
,1 ,
n
iiini in
dTxpu dxpc≤+ +
for all
xC∈
and
( ){}
,,
i iin
pFTiI u∈∈
and
{ }
in
c
are sequences in
[0, )∞
with
1in n
u
∞
=
<∞
∑
and
1
in
n
c
∞
=
<∞
∑
for each
i
.
The purpose of this paper is to:
1) establish convergence of iterative method (1.2) to a common fixed point of a finite family of generalized
asymptotically quasi-nonexpansive mappings on a hyperbolic space(uniformly convex hyperbolic space).
Our work is a significant generalization of the corresponding results in Banach spaces and
(0)CAT
spaces.
In the sequel, we assume that
( )
.
i
iI
F FT
∈
= ≠∅

2. Convergence Theorems in Hyperbolic Space
Lemma 2.1. Let
C
be a nonempty, closed and convex subset of a hyperbolic space
X
. Then, for the sequence

A. R. Khan, H. Fukhar-ud-din
172
{ }
n
x
in (1.2), there are sequences
{ }
n
v
and
{ }
n
ξ
in
[ )
0,∞
satisfying
11
,
nn
nn
v ξ
∞∞
= =
<∞ <∞
∑∑
such that
1)
()()()
1
,1, ,
nnn n
dxpv dx p ξ
+
≤+ +
for all
pF∈
and all
1
n≥
2)
()()
11
,( ,),
nmn n
n
dxpMdx p ξ
∞
+=
≤+
∑
for all
pF∈
and
1
1, 1,0.nmM≥≥ >
Proof. (a) Let
pF∈
and
max
n in
iI
vu
∈
=
for all
1.n≥
Since
1in nu
∞
=<∞
∑
for each
i
, therefore
1n nv
∞
=<∞
∑
.
Now we have
()( )
( )
()()
( )
()()() ()()()()()
111 01111
1
111111 1
,1 ,1,,
1,1, 1,1,.
nn
nnnnnn nnnnn
nnnnn nnnnnn n
dUxpdaTUxaxpa dxpadTxp
adx paudx pcudx pcvdx pc
=⊕− ≤−+
 ≤− ++ +≤+ +≤++

Assume that
()()() ()
1
1
,1 ,1
kk
kn nnnnin
k
i
dUxpv dxpvc
−
=
≤+++ ∑
holds for some
1.k>
Consider
( )
( )
( )( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )()
( )
( )
( )( )
( )
( )
( )
( )( )( )( )
( )
( )
( )
( )
( )
( )( )( )
( )( )
11
1111 1
11 111
111 11
111 1
,1 ,1,,
1 ,1,
1,1 ,
1 ,11
nn
nkknnnnkkn n
kn knknknkn
nkn n
knkn knknkn
nkn n
knknkn knkn
k
n nn
knknkn kn
dUxpdaTUxaxpadxp adTUxp
adxp audUxp ac
adxp acaudUxp
adx pacavv
++
++++ +
++ +++
+++ ++
+++ +
=⊕−≤−+
≤−+ ++
≤−++ +
≤−++++
()( )
( )
( )
()()
( )
( )
( )( )
()()
( )
( )
( )()( )
1
1 11
11 11
1
1
1
1
1
1
1
,1
11 ,11 ,1
1 ,1
k
nn in
k kk
nn nnn nin
knknkn knkn
kk
n
k
i
k
i
k
i
nn in
dx pvc
a vdxpavcavdxpavc
vdx pvc
−
+ +−
=
+
=
+
=
++ +++
+

++

≤−+++++ ++
≤+ ++
∑
∑
∑
By mathematical induction, we have
( )
() () ()
1
1
,1, 1,1.
jj j
jn nnnnin
i
dUxpvdxpvcjr
−
=
≤+ ++≤≤
∑
(2.1)
Now, by (1.2) and (2.1), we obtain
( )
( )
( )
( )
( )
( )
()()
( )
( )
( )
()()
()()( )()()( )
( )()( )
11
1
1
1
12
1
1
1
, 1,
,1 ,
1,1,
1 1,11,
1 ,11
n
nrn rrn rn
n
rn rnrnn
rn
rnrnnrnrn n
rn
r
rr
rnrnnnninrn rnrnn
i
r
rr
rn nnrn nininr
i
dxpd aTUxaxp
adTUxpa dxp
audUxpcadxp
auvdx pvcacadx p
avdxpav cca
+−
−
−
−
−−
=
−
=
= ⊕−
≤ +−

≤++ +−


≤ ++++++−


≤++ ++−
∑
∑
()()
( )()( )
() () ( )
( )()
() ()()() ()
1
1
1
1
11
1
1
,
1 1,1
11
1 1,1
!
1,1 1,,
nn
rr r
rn rnnnrnnin
i
rr
r
k
rn rnnnrnnin
ki
r
r rr
nnninnnn
i
dx p
aavdxpav c
rrr k
aavdx pavc
k
vdx pvcvdx p ξ
−
=
−
−
= =
−
=

≤−++ ++



−… −+

=−+++ + 




≤+ ++≤+ +
∑
∑∑
∑
Where
( )
1
1
supsup 1,
rr
n nin
i
M MvMc ξ
−
=
==+=
∑
and
1n n
ξ
∞
=
<∞
∑
.
(b) We know that
1 exptt+≤
for
0t≥
. Thus, by part (a), we have

A. R. Khan, H. Fukhar-ud-din
173
() ()( )
()( )
() ()
()
()
( )
( )
( )
( )
()
11 1
11 1
1 2212
1 11
1
1
11 1
1
,1 ,
exp ,
exp ,
exp ,
exp,, ,
where
r
nmnm nmnm
nm nmnm
nmnmnmnm nm
nmnmnm
ini i
inin in
ini ni
ii i
dx pdxp
rdx p
rrdx p
rvdx pv
rvdx pM dx p
M
νξ
νξ
ν νξξ
ξ
ξξ
++−+−+−
+− +−+−
+−+−+−+− +−
+−+− +−
= =+=
∞∞ ∞
= ==
≤+ +
≤+
≤ +++
≤+
≤+=+
∑ ∑∑
∑∑ ∑

( )
1
exp .
i
i
rv
∞
=
=∑
Theorem 2.2. Let
C
be a nonempty, closed and convex subset of a complete hyperbolic space
X
. Then
the sequence
{ }
n
x
in (1.2) converges strongly to a point in
F
if and only if
liminf(,)0
nn
dxF
→∞
=
, where
( )
,inf( ,)
pF
d xFdxp
∈
=
.
Proof. We only prove the sufficiency. By Lemma 2.1 (a), we have
()()()
1
,1,for all and 1.
r
nnn n
dxpvdx ppFn ξ
+
≤++∈ ≥
Therefore,
( )
() () ( )
( )
( )
( )
11
, 111!,
rk
nnn n
k
dxFrrrkk vdxF ξ
+=
≤+−… −++
∑
As
1
,
n
n
v
∞
=
<∞
∑
so
()()
( )
( )
11
1 1!
rk
n
nk
rrrkk v
∞
= =
− …− +<∞
∑∑
. Now
1n n
ξ
∞
=
<∞
∑
in Lemma 2.1 (a), so
by Lemma 1.1 [1] and
liminf(,)0
nn
dxF
→∞
=
, we get that
lim(, )0
nn
dxF
→∞
=
. Let
0.
ε >
From the proof of
Lemma 2.1 (b), we have
()() () ( ) ( )
11
,, ,1,
nm nnmnni
in
dxxdxFdxFM dxFM ξ
++
∞
=
≤+≤++ ∑
(2.2)
Since
lim(, )0
nn
dxF
→∞
=
and
i
in
ξ
∞
=
<∞
∑
, therefore there exists a natural number
0
n
such that
( ) ( )
1
, 21
n
dxF M ε ≤+
and
1
2
i
in
M ξε
∞
=
<
∑
for all
0
nn≥
.
So for all integers
0
, 1,
n nm ≥≥
we obtain from (2.2) that
( )
() ()
11
1
1
,1 .
2
21
nmn
dx xMMM
M
εε
ε
+


≤++ =


+

Thus,
{ }
n
x
is a Cauchy sequence in
X
and so converges to
qX∈
. Finally, we show that
qF∈
. For
any
0 ε >
, there exists natural number
1
n
such that
()( )
, inf,3
npFn
dxFdx p
ε
∈
= <
and
( )
,2
n
dxq
ε
<
for all
1
nn≥
.
There must exist
*
pF∈
such that
( )
*
,2
n
dx p
ε
<
for all
1
nn≥
, in particular,
( )
*
1
,2
n
dx p
ε
<
and
( )
1,.
2
n
dx q
ε
<
Hence
() ()
( )
**
11
,, ,
nn
dpqdxpdx q ε ≤ +<
. Since
ε
is arbitrary, therefore
( )
*
, 0.dpq=
That is,
*.qp F
= ∈
Theorem 2.3. Let
C
be a nonempty, closed and convex subset of a complete convex metric space
X
, If
( )
lim ,0
nn in
d xTx
→∞
=
for the sequence
{ }
n
x
in (1.2),
iI∈
and one of the mappings is semi-compact, then
{ }
n
x
converges strongly to
pF∈
.
Proof. Let
l
T
be semi-compact for some
1.lr≤≤
Then there exists a subsequence
{ }
i
x
of
{ }
n
x
such
that
.
i
x pC →∈
Hence
( )()
,lim ,0.
j
li li
n
dpTpdx Tx
→∞
= =
Thus
pF∈
and so by Theorem 2.2,
{ }
n
x
converges strongly to a common fixed point of the family of
mappings.

A. R. Khan, H. Fukhar-ud-din
174
3. Results in a Uniformly Convex Hyperbolic Space
Lemma 3.1.Let
C
be a nonempty, closed and convex subset of a uniformly convex hyperbolic space
X
. Then,
for the sequence
{ }
n
x
in (1.2) with
[ ]
,1
in
a δδ ∈−
for some
1
0, 2 δ

∈

, we have
( )
( )lim,
nn
adx p
→∞
exists for all
pF∈
(b)
( )
lim, 0,
nn jn
d xTx
→∞ =
for each
jI∈
.
Proof. (a) Let
pF∈
and
max ,
ni Iin
vu
∈
=
for all
1 n≥
.By Lemma 1.1 [1] and Lemma 2.1 (a), it follows
that
( )
lim ,
nn
dx p
→∞
exists. Assume that
()
lim ,.
nn
dx pc
→∞
=
(3.1)
(b) The inequality (2.1) together with (3.1) gives that
( )
limsup, ,1.
njn n
dU xpcjr
→∞
≤ ≤≤
(3.2)
Note that
() ()
( )
( )
()
( )
( )
()
()( )
( )
( )()()
( )
()()
( )
()( )
()
( )
( )
()( ) ()
11
1
1
12 1
1 12 1
, ,1,
1,1,
11,1 ,
1,1,1 ,
n
nrnnrn rnrnn
rn
rnnnrnrn n
rn
n
rnnrnnrn rnrnn
rnr nrn
n
rnnrnnrn rnrnn rnr nrn
r
dxpdUxpdaTUxa xp
avdUxpcadxp
avdaTUxaxpaca dxp
avadTUxpadxpaca dxp
a
+−
−
−
−− −
− −− −
= =⊕−

≤++ +−

=+⊕−++ −

≤ ++−++−

≤
( )
( )
( )
( )
( )
( )
() ()
( )
()
( )
()
( ) ( ) () ()
()
()
( )( )
22
12 1
22
11
11
1
12
1,11 ,
11
1,11,
11 1.
nnnrnn n rnrnrn
rnnrnnrn
rn rn
rr
rj rj
innjn ninnn
ij ij
rr
rjrj rj
in nin n jnrnnrn
jn
ij ij
av dUxpaav dxp
aavca vc
av dUxpav dxp
a vca vcavc
−− −
−−
−−
=+=+
− −−
+
=+=+
++− +
++ ++

≤ ++−+


+++++…+ +
∏∏
∏∏
and therefore, we have
( )()()( )
( )
( )
11
1
,,
,,
1
jn nn rn
rj rjrj
njn njn
rj n
c
dx pdxpc
dxpdUxp c
v
δ
δδ
δ
+
− −−+
+
−

 
≤−+++++

 

+



Hence
( )
liminf ,
njnn
cdU xp
→∞
≤
,
1.jr≤≤
(3.3)
Using (3.2) and (3.3), we have
( )
lim ,
njn n
dU xpc
→∞
=
.
That is,
( )
( )
( )
1
li 1,m
n
njn jnjnn
jn
d aTUxaxpc
→∞ −
⊕− =
for
1.jr≤≤
This together with (3.1), (3.2) and Lemma 2.5 [8] gives that
( )
( )
1
lm ,0i
n
n jnn
jn
dTUx x
→∞ −
=
for
1.jr≤≤
(3.4)
If
1j=
, we have by (3.4),
( )
1
lm ,0i
n
n nn
dTx x
→∞
=
.
In case
{ }
2,3,4,,jr∈
, we observe that
( )
( )
( )()()
( ) ( )
( )()
( )
11
11211 2
,, ,1,0
nn
nnnjnnjnn
jnjnj njnjnjn
dxUxdxaTU xaxadTU xx
−−
−−−−−−
=⊕− ≤→
(3.5)

A. R. Khan, H. Fukhar-ud-din
175
Since
j
T
is
( )
L γ −−
uniformly Lipschitzian, therefore the inequality
()
()
()
( )
( )
( )
( )
()
( )
,
111 1
, ,,,,
n nnnn
jnnjnj nj nnnnjnn
jn jnjnjn
dTxxdTxTU xdTU xxLdxU xdTU xx
γ
−−− −
≤+ ≤+
together with (3.4) and (3.5) gives that
( )
lim , 0.
n
jnn n
dTx x
→∞ =
Hence,
( )
, 0as
n
jn n
dTx xn→→∞
for
1.jr≤≤
(3.6)
Note that
( )
( )
( )
( )
( )
( )
111
, ,1,0.
nn
nnnrn rnnrnnrnnrn
rn rn
dxxdxaTU xax adxTU x
+−−
=⊕− ≤→
Let us observe that:
( )
( )
()() ( )
( )
()
( )
( )
111 1
1 111
1
1 111
,, ,,,
, ,,,.
nnn n
njnnnnjnjnjnjnjn
nn
nnnjnnnjnn
dxTxdxxdxTxdTxTx dTxTx
dx xdxTxLdxxLdTx x
γ
γ
+++ +
+ +++
+
+ +++
≤+ ++
≤++ +
So by
( )
L γ −−
uniformly Lipschitzian property of
j
T
, (3.5) and (3.6), we get
( )
, 0,1lim
nn jn
d xTxjr
→∞
= ≤≤
.
Theorem 3.2. Under the hypotheses of Lemma 3.1, assume that, for some
1jr ≤≤
,
m
j
T
is semi-compact
for some positive integer m. Then
{}
n
x
in (1.2), converges strongly to a point in
F
.
Proof. Fix
jI∈
and suppose
m
j
T
is semi-compact for some
1.m≥
By Lemma 3.1 (b), we obtain
()() ()()
( )
( )
( )
1 122
,,,, ,
,1(, )0.
mmmmm
jnnjnj nj njnjnjnjnn
jn njn n
dTxxdT xTxdTx TxdTxTxdTxx
d TxxmLdTxx
γ
− −−
≤++++
≤ +−→

Since
{ }
n
x
is bounded and
m
j
T
is semi-compact,
{ }
n
x
has a convergent subsequence
{ }
i
n
x
such that
i
n
x qC→∈
. Hence, by Lemma 3.1 (b), we have
( )
( )
, lim,0,
jj
inn in
d qTqdxTxiI
→∞
== ∈
.
Thus
qF∈
and so by Theorem 2.2,
{ }
n
x
converges strongly to a common fixed point
q
of the family
{}
:
i
Ti I ∈
.
Acknowledgements
The author A. R. Khan is grateful to KACST for supporting the research project ARP-32-34. The author H.
Fukhar-ud-din acknowledges King Fahd University of Petroleum & Minerals for supporting research project
IN121037.
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