Common Fixed Point Iterations of Generalized Asymptotically Quasi-Nonexpansive Mappings in Hyperbolic Spaces

doi:10.4236/jamp.2014.25021

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

Email: [email protected], [email protected]

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

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

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

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

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

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

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

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.

jj

j

jn nnnnin

i

dUxpvdxpvcjr

−

=

≤+ ++≤≤

∑

(2.1)

Now, by (1.2) and (2.1), we obtain

( )

()()

( )

()()

()()( )()()( )

( )()( )

11

1

12

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

11

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

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

1 2212

1 11

1

11 1

1

,1 ,

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

ν νξξ

ξ

ξξ

++−+−+−

+− +−+−

+−+−+−+− +−

+−+− +−

= =+=

∞∞ ∞

= ==

≤+ +

≤+

≤ +++

≤+

≤+=+

∑ ∑∑

∑∑ ∑



( )

1

exp .

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

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 .

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

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

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

+++ +

+ +++

+

+ +++

≤+ ++

≤++ +

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.

References

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Asymptotically Quasi-Nonexpansive Mappings in Banach Space. Journal of Mathematical Analysis and Applications.

341, 1-11. http://dx.doi.org/10.1016/j.jmaa.2007.06.051

[2] Menger, K. (1928) Untersuchungenüberallgemeine Metrik. Mathematische Annalen, 100, 75-163.

http://dx.doi.org/10.1007/BF01448840

[3] Takahashi, W. (1970) A Convexity in Metric Spaces and Nonexpansive Mappings. Kodai. Math Sem. Rep., 22,

142-149. http://dx.doi.org/10.2996/kmj/1138846111

[4] Bridson, M. and Haefliger, A. (1999) Metric Spaces of Non-Positive Curvature. Springer-Verlag, Berlin, Heidelberg,

New York. http://dx.doi.org/10.1007/978-3-662-12494 -9

[5] Fukhar-ud-din, H. (2013) Strong Convergence of an Ishikawa-type Algorithm inCAT (0) Spaces. Fixed Point Theory

and Applications, 2013, 207.

[6] Khan, A.R., Khamsi, M.A. and Fukhar-ud-din, H. (2011) Strong Convergence of a General Iteration Scheme in CAT(0)

Spaces, Nonlinear Anal. 74, 783-791. http://dx.doi.org/10.1016/j.na.2010.09.029

[7] Goebel, K. and Reich, S. (1984) Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Series of

Monographs and Textbooks in Pure and Applied Mathematics, Dekker, New York.

[8] Khan, A.R., Fukhar-ud-din, H. and Khan, M.A.A. (2012) An Implicit Algorithm for Two Finite Families of Nonex-

pansive Maps in Hyperbolic Spaces. Fixed Point Theory and Applications, 2012, 54.

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