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Volume 2011, Article ID 173621, 11 pagesdoi:10.1155/2011/173621 Research Article Strong Convergence of an Implicit Algorithm in CAT0 Spaces 1 Department of Mathematics, The Islamia Unive

Volume 2011, Article ID 173621, 11 pages

doi:10.1155/2011/173621

Research Article

Strong Convergence of an Implicit Algorithm in CAT(0) Spaces

1 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan

2 Department of Mathematics, Taibah University, Madinah Munawarah 30002, Saudi Arabia

3 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,

Dhahran 31261, Saudi Arabia

Correspondence should be addressed to Abdul Rahim Khan,arahim@kfupm.edu.sa

Received 23 November 2010; Accepted 23 December 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq 2011 Hafiz Fukhar-ud-din et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We establish strong convergence of an implicit algorithm to a common fixed point of a finite family

of generalized asymptotically quasi-nonexpansive maps in CAT0 spaces Our work improves and extends several recent results from the current literature

1 Introduction

A metric spaceX, d is said to be a length space if any two points of X are joined by a rectifiable

pathi.e., a path of finite length, and the distance between any two points of X is taken to

be the infimum of the lengths of all rectifiable paths joining them In this case, d is said to be

a length metric otherwise known as an inner metric or intrinsic metric In case no rectifiable

path joins two points of the space, the distance between them is taken to be∞

A geodesic path joining x ∈ X to y ∈ X or, more briefly, a geodesic from x to y is a map

c from a closed interval 0, l ⊂ R to X such that c0  x, cl  y, and dct, ct  |t − t|

for all t, t∈ 0, l In particular, c is an isometry, and dx, y  l The image α of c is called a

geodesicor metric segment joining x and y We say X is i a geodesic space if any two points

of X are joined by a geodesic and ii uniquely geodesic if there is exactly one geodesic joining

x and y for each x, y ∈ X, which we will denote by x, y, called the segment joining x to y.

A geodesic triangle Δx1 , x2, x3 in a geodesic metric space X, d consists of three points

in X the vertices of Δ and a geodesic segment between each pair of vertices the edges of Δ.

A comparison triangle for geodesic triangle Δx1 , x2, x3 in X, d is a triangle Δx1, x2, x3 :

Δx1 , x2, x3 in R 2 such that dR2x i , x j   dx i , x j  for i, j ∈ {1, 2, 3} Such a triangle always

existssee 1

A geodesic metric space is said to be a CAT0 space if all geodesic triangles of appropriate size satisfy the following CAT0 comparison axiom

LetΔ be a geodesic triangle in X, and let Δ ⊂ R2be a comparison triangle forΔ Then

Δ is said to satisfy the CAT0 inequality if for all x, y ∈ Δ and all comparison points x, y ∈ Δ,

d

x, y

≤ dx, y

Complete CAT0 spaces are often called Hadamard spaces see 2 If x, y1 , y2are points of

a CAT0 space and y0 is the midpoint of the segment y1 , y2, which we will denote by

y1 ⊕ y2/2, then the CAT0 inequality implies

d2



x, y1⊕ y2

2



≤ 1

2d2

x, y1



1

2d2

x, y2



−1

4d2

y1, y2



. 1.2

The inequality1.2 is the CN inequality of Bruhat and Titz 3 The above inequality has been extended in4 as

d2

z, αx ⊕ 1 − αy≤ αd2z, x  1 − αd2

z, y

− α1 − αd2

x, y

, 1.3

for any α ∈ 0, 1 and x, y, z ∈ X.

Let us recall that a geodesic metric space is a CAT0 space if and only if it satisfies the (CN)

inequalitysee 1, page 163 Moreover, if X is a CAT0 metric space and x, y ∈ X, then for

any α ∈ 0, 1, there exists a unique point αx ⊕ 1 − αy ∈ x, y such that

d

z, αx ⊕ 1 − αy≤ αdz, x  1 − αdz, y

for any z ∈ X and x, y  {αx ⊕ 1 − αy : α ∈ 0, 1}.

A subset C of a CAT0 space X is convex if for any x, y ∈ C, we have x, y ⊂ C Let T be a selfmap on a nonempty subset C of X Denote the set of fixed points of T

by FT  {x ∈ C : Tx  x} We say T is: i asymptotically nonexpansive if there is a

sequence{u n } ⊂ 0, ∞ with lim n → ∞ u n  0 such that dT n x, T n y ≤ 1  u n dx, y for all

x, y ∈ C and n ≥ 1, ii asymptotically quasi-nonexpansive if FT /  φ and there is a sequence {u n } ⊂ 0, ∞ with lim n → ∞ u n  0 such that dT n x, p ≤ 1  u n dx, p for all x ∈ C, p ∈ FT and n ≥ 1, iii generalized asymptotically quasi-nonexpansive 5 if FT / ∅ and there exist

two sequences of real numbers {u n } and {c n} with limn → ∞ u n  0  limn → ∞ c n such that

dT n x, p ≤ dx, p  1  u n dx, p  c n for all x ∈ C, p ∈ FT and n ≥ 1, iv uniformly

L-Lipschitzian if for some L > 0, dT n x, T n y ≤ Ldx, y for all x, y ∈ C and n ≥ 1, and v

semicompact if for any bounded sequence{x n } in C with dx n , Tx n  → 0 as n → ∞, there is

a convergent subsequence of{x n}

Denote the indexing set{1, 2, 3, , N} by I Let {T i : i ∈ I} be the set of N selfmaps

of C Throughout the paper, it is supposed that F  N

i1 FT i  / φ We say condition A is satisfied if there exists a nondecreasing function f : 0, ∞ → 0, ∞ with f0  0, fr > 0 for all r ∈ 0, ∞ and at least one T ∈ {T i : i ∈ I} such that dx, Tx ≥ fdx, F for all x ∈ C where dx, F  inf{dx, p : p ∈ F}.

If in definition iii, c n  0 for all n ≥ 1, then T becomes asymptotically

quasi-nonexpansive, and hence the class of generalized asymptotically quasi-nonexpansive maps includes the class of asymptotically quasi-nonexpansive maps

Let{x n } be a sequence in a metric space X, d, and let C be a subset of X We say

that{x n } is: vi of monotone typeA with respect to C if for each p ∈ C, there exist two

sequences{r n } and {s n} of nonnegative real numbers such that∞n1 r n < ∞,∞

n1 s n < ∞

and dx n1 , p ≤ 1  r n dx n , p  s n,vii of monotone typeB with respect to C if there exist

sequences{r n } and {s n} of nonnegative real numbers such that∞

n1 r n < ∞,∞

n1 s n < ∞

and dx n1 , C ≤ 1  r n dx n , C  s nalso see 6

From the above definitions, it is clear that sequence of monotone typeA is a sequence

of monotone typeB but the converse is not true, in general

Recently, numerous papers have appeared on the iterative approximation of fixed points of asymptotically nonexpansiveasymptotically quasi-nonexpansive maps through Mann, Ishikawa, and implicit iterates in uniformly convex Banach spaces, convex metric spaces and CAT0 spaces see, e.g., 5,7 16

Using the concept of convexity in CAT0 spaces, a generalization of Sun’s implicit algorithm15 is given by

x0∈ C,

x1 α1 x0⊕ 1 − α1T1 x1,

x2 α2 x1⊕ 1 − α2T2 x2,

x N  α N x N−1 ⊕ 1 − α N T N x N ,

x N1  α N1 x N ⊕ 1 − α N1 T2

1x N1 ,

x 2N  α2N x 2N−1 ⊕ 1 − α2N T2

x 2N1  α2N1 x 2N ⊕ 1 − α2N1T3

1x 2N1 ,

.,

1.5

where 0≤ α n≤ 1

Starting from arbitrary x0, the above process in the compact form is written as

x n  α n x n−1 ⊕ 1 − α n T kn

where n  k − 1N  i, i  in ∈ I and k  kn ≥ 1 is a positive integer such that kn → ∞

as n → ∞.

In a normed space, algorithm1.6 can be written as

x0∈ C, x n  α n x n−1  1 − α n T kn

i n x n , n ≥ 1, 1.7

where n  k − 1N  i, i  in ∈ I and k  kn ≥ 1 is a positive integer such that kn → ∞

as n → ∞.

The algorithms1.6-1.7 exist as follows

Let X be a CAT0 space Then, the following inequality holds:

d

λx ⊕ 1 − λz, λy ⊕ 1 − λw≤ λdx, y

 1 − λdz, w, 1.8

for all x, y, z, w ∈ X see 17

Let{T i : i ∈ I} be the set of N uniformly L-Lipschitzian selfmaps of C We show that

1.6 exists Let x0 ∈ C and x1  α1 x0⊕1−α1T1x1 Define S : C → C by: Sx  α1x0⊕1−α1T1x

for all x ∈ C The existence of x1 is guaranteed if S has a fixed point For any x, y ∈ C, we

have

d

Sx, Sy

≤ 1 − α1dT1x, T1y

≤ 1 − α1Lx − y. 1.9

Now, S is a contraction if 1 − α1L < 1 or L < 1/1 − α1 As α1 ∈ 0, 1, therefore S

is a contraction even if L > 1 By the Banach contraction principle, S has a unique fixed point Thus, the existence of x1 is established Similarly, we can establish the existence of

x2, x3, x4, Thus, the implicit algorithm 1.6 is well defined Similarly, we can prove that

1.7 exists

For implicit iterates, Xu and Ori16 proved the following theorem

Theorem XO see 16, Theorem 2 Let {Ti : i ∈ I} be nonexpansive selfmaps on a closed convex

subset C of a Hilbert space with F /  φ, let x0 ∈ C, and let {α n } be a sequence in 0, 1 such that

converges weakly to a point in F.

They posed the question: what conditions on the maps {T i : i ∈ I} and or the

parameters{α n} are sufficient to guarantee strong convergence of the sequence in Theorem XO?

The aim of this paper is to study strong convergence of iterative algorithm1.6 for

the class of uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive

selfmaps on a CAT0 space Thus, we provide a positive answer to Xu and Ori’s question for the general class of maps which contains asymptotically quasi-nonexpansive, asymptotically nonexpansive, quasi-nonexpansive, and nonexpansive maps in the setup of CAT0 spaces

It is worth mentioning that if an implicit iteration algorithm without an error term converges, then the method of proof generally carries over easily to algorithm with bounded error terms Thus, our results also hold if we add bounded error terms to the implicit iteration scheme considered Our results constitute generalizations of several important known results

We need the following useful lemma for the development of our convergence results

Lemma 1.1 see 14, Lemma 1.1 Let {rn } and {s n } be two nonnegative sequences of real

numbers, satisfying the following condition:

r n1 ≤ 1  s n r n ∀n ≥ n0 for some n0≥ 1. 1.10

If∞

n1 s n < ∞, then lim n → ∞ r n exists.

2 Convergence in CAT(0) Spaces

We establish some convergence results for the algorithm1.6 to a common fixed point of a

finite family of uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive

selfmaps in the general class of CAT0 spaces The following result extends Theorem XO; our methods of proofs are based on the ideas developed in15

Theorem 2.1 Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset of

X Let {T i : i ∈ I} be N uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive

selfmaps of C with {u in }, {c in } ⊂ 0, ∞ such that∞

n1 u in < ∞ and∞

n1 c in < ∞ for all i ∈ I Suppose that F is closed Starting from arbitrary x0 ∈ C, define the sequence {x n } by the algorithm

1.6, where {α n } ⊂ δ, 1−δ for some δ ∈ 0, 1/2 Then, {x n } is of monotone type(A) and monotone

type(B) with respect to F Moreover, {x n } converges strongly to a common fixed point of the maps {T i : i ∈ I} if and only if lim inf n → ∞ dx n , F  0.

Proof First, we show that {x n } is of monotone type(A) and monotone type(B) with respect to F Let

p ∈ F Then, from 1.6, we obtain that

d

x n , p

 d α n x n−1 ⊕ 1 − α n T kn

i n x n , p

≤ α n d

x n−1 , p

 1 − α n d T i kn n x n , p

≤ α n d

x n−1 , p

 1 − α nd

x n , p

 u ikn d

x n , p

≤ α n d

x n−1 , p



d

x n , p

 1 − α n c ikn

2.1

Since α n ∈ δ, 1 − δ, the above inequlaity gives that

d

x n , p

≤ dx n−1 , p

δ d



x n , p



 1

δ− 1



On simplification, we have that

d

x n , p

≤ δ

δ − u ikn d

x n−1 , p



 1

δ − 1



δ

δ − u ikn c ikn 2.3

Let 1 v ikn  δ/δ − u ikn   1  u ikn /δ − u ikn  and γ ikn  1/δ − 11  v ikn c ikn Since∞

natural number n1 such that u ikn < δ/2 for kn ≥ n1/N  1 or n > n1 Then, we have that

∞

Now, from2.3, for kn ≥ n1 /N  1, we get that

d

x n , p



d

x n−1 , p



d x n−1 , F   γ ikn 2.5

These inequalities, respectively, prove that {x n} is a sequence of monotone typeA and monotone typeB with respect to F

Next, we prove that {x n} converges strongly to a common fixed point of the maps

{T i : i ∈ I} if and only if lim inf n → ∞ dx n , F  0.

If x n → p ∈ F, then lim n → ∞ dx n , p  0 Since 0 ≤ dx n , F ≤ dx n , p, we have

lim infn → ∞ dx n , F  0.

Conversely, suppose that lim infn → ∞ dx n , F  0 Applying Lemma 1.1to2.5, we have that limn → ∞ dx n , F exists Further, by assumption lim inf n → ∞ dx n , F  0, we conclude

that limn → ∞ dx n , F  0 Next, we show that {x n} is a Cauchy sequence

Since x ≤ expx − 1 for x ≥ 1, therefore from 2.4, we have

d

x nm , p

≤ exp

⎛

⎝N

i1

∞



k n1

⎞

⎠dx n , p

N

i1

∞



k n1

γ ikn

< Md

x n , p

N

i1

∞



k n1

γ ikn ,

2.6

for the natural numbers m, n, where M  exp{N

i1

∞

dx n , F < /4M andN

i1

∞

jn γ ij ≤ /4 for all n ≥ n0 So, we can find p∗ ∈ F such that

dx n0 , p∗ ≤ /4M Hence, for all n ≥ n0 and m ≥ 1, we have that

d x nm , x n  ≤ dx nm , p∗

 dx n , p∗

< Md

x n0 , p∗

N

i1

∞



jn0

γ ij  Mdx n0 , p∗

N

i1

∞



jn0

γ ij

 2

⎛

⎝Mdx n0 , p∗

N

i1

∞



jn0

γ ij  Mdx n0 , p∗⎞⎠ ≤ 2M

4M  4



 .

2.7

This proves that{x n} is a Cauchy sequence Let limn → ∞ x n  z Since C is closed, therefore

z ∈ C Next, we show that z ∈ F Now, the following two inequalities:

d

z, p

≤ dz, x n   dx n , p

∀p ∈ F, n ≥ 1,

d z, x n  ≤ dz, p

 dx n , p

∀p ∈ F, n ≥ 1 2.8

give that

−dz, x n  ≤ dz, F − dx n , F  ≤ dz, x n , n ≥ 1. 2.9 That is,

|dz, F − dx n , F | ≤ dz, x n , n ≥ 1. 2.10

As limn → ∞ x n  z and lim n → ∞ dx n , F  0, we conclude that z ∈ F.

We deduce some results fromTheorem 2.1as follows

Corollary 2.2 Let X, d be a complete CAT0 space, and let C be a nonempty closed convex

subset of X Let {T i : i ∈ I} be N uniformly L-Lipschitzian and generalized asymptotically

quasi-nonexpansive selfmaps of C with {u in }, {c in } ⊂ 0, ∞ such that∞

n1 u in < ∞ and∞

n1 c in < ∞ for all i ∈ I Suppose that F is closed Starting from arbitaray x0∈ C, define the sequence {x n } by the

algorithm1.6, where {α n } ⊂ δ, 1 − δ for some δ ∈ 0, 1/2 Then, {x n } converges strongly to a

common fixed point of the maps {T i : i ∈ I} if and only if there exists some subsequence {x n j } of {x n}

which converges to p ∈ F.

Corollary 2.3 Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset

of X Let {T i : i ∈ I} be N uniformly L-Lipschitzian and asymptotically quasi-nonexpansive selfmaps

of C with {u in } ⊂ 0, ∞ such that∞n1 u in < ∞ for all i ∈ I Starting from arbitaray x0∈ C, define

the sequence {x n } by the algorithm 1.6, where {α n } ⊂ δ, 1 − δ for some δ ∈ 0, 1/2 Then, {x n}

is of monotone type(A) and monotone type(B) with respect to F Moreover, {x n } converges strongly to

a common fixed point of the maps {T i : i ∈ I} if and only if lim inf n → ∞ dx n , F  0.

Proof Follows fromTheorem 2.1with c in  0 for all n ≥ 1.

Corollary 2.4 Let X be a Banach space, and let C be a nonempty closed convex subset of X Let

{T i : i ∈ I} be N asymptotically quasi-nonexpansive self-maps of C with {u in } ⊂ 0, ∞ such that

∞

n1 u in < ∞ for all i ∈ I Starting from arbitaray x0∈ C, define the sequence {x n } by the algorithm

1.7, where {α n } ⊂ δ, 1−δ for some δ ∈ 0, 1/2 Then, {x n } is of monotone type(A) and monotone

type(B) with respect to F Moreover, {x n } converges strongly to a common fixed point of the maps {T i : i ∈ I} if and only if lim inf n → ∞ dx n , F  0.

Proof Take λx ⊕ 1 − λy  λx  1 − λy inCorollary 2.3

The lemma to follow establishes an approximate sequence, and as a consequence of that, we find another strong convergence theorem for1.6

Lemma 2.5 Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset of

X Let {T i : i ∈ I} be N uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive

selfmaps of C with {u in }, {c in } ⊂ 0, ∞ such that∞n1 u in < ∞ and∞

n1 c in < ∞ for all i ∈ I Suppose that F is closed Let {α n } ⊂ δ, 1 − δ for some δ ∈ 0, 1/2 From arbitaray x0 ∈ C, define

the sequence {x n } by 1.6 Then, lim n → ∞ dx n , T l x n   0 for all i ∈ I.

Proof Note that {x n} is bounded as limn → ∞ dx n , p exists proved inTheorem 2.1 So, there

exists R > 0 and x0 ∈ X such that x n ∈ B R x0  {x : dx, x0 < R} for all n ≥ 1 Denote

dx n−1 , T in kn x n  by σ n

We claim that limn → ∞ σ n 0

For any p ∈ F, apply 1.3 to 1.6 and get

d2

x n , p

 d2

α n x n−1 ⊕ 1 − α n T kn

i n x n , p

≤ α n d2

x n−1 , p

 1 − α n1 u ikn



d

x n , p

2

− α n 1 − α n d2

T i kn n x n , x n−1

2.11

further, using2.4, we obtain

2δ3σ n2≤ α n d2x n−1 , x∗ − d2x n , x∗

 1 − α n1 u ikn





d x n−1 , x∗ 1 u ikn



γ ikn  c ikn

2

,

2.12

which implies that

2δ3σ n2≤ α n d2

x n−1 , p

 1 − α n d2

x n−1 , p



M − d2

x n , p

, 2.13

for some consant M > 0 This gives that

2δ3σ n2≤ d2

x n−1 , p

− d2

x n , p

where σ ikn  u ikn  v ikn  γ ikn  c ikn

For m ≥ 1, we have that

2δ3

m



n1

σ n2≤ d2

x0, p

− d2

x m , p

 m

k n1

≤ d2

x0, p

 m

k n1

2.15

When m → ∞, we have that∞

n < ∞ as∞

Hence,

lim

d x n , x n−1  ≤ 1 − α n d T i kn n x n , x n−1

 1 − α n σ n ≤ 1 − δσ n

2.17

implies that limn → ∞ dx n , x n−1  0

For a fixed j ∈ I, we have dx nj , x n  ≤ dx nj , x nj−1   · · ·  dx n , x n−1, and hence

lim

x nj , x n



 0 ∀j ∈ I. 2.18

For n > N, n  n − Nmod N Also, n  kn − 1N  in Hence, n − N  kn − 1 −

1N  in  kn − NN  in − N

That is, kn − N  kn − 1 and in − N  in.

Therefore, we have

d x n−1 , T n x n  ≤ d x n−1 , T i kn n x n  d T i kn n x n , Tx n

≤ σ n  Ld T i kn−1 n x n , x n

≤ σ n  L2d x n , x n−N   Ld T i kn−N n−N x n−N , x n−N−1  Ldx n−N−1 , x n



 σ n  L2d x n , x n−N   Lσ n−N  Ldx n−N−1 , x n



,

2.19

which together with2.16 and 2.18 yields that limn → ∞ dx n−1 , Tx n  0

Since

d x n , Tx n  ≤ dx n , x n−1   dx n−1 , T n x n , 2.20

we have

lim

Hence, for all l ∈ I,

d x n , T nl x n  ≤ dx n , x nl   dx nl , T nl x nl   dT nl x nl , T nl x n

≤ 1  Ldx n , x nl   dx nl , T nl x nl , 2.22

together with2.18 and 2.21 implies that

lim

Thus, limn → ∞ dx n , T l x n   0 for all l ∈ I.

Theorem 2.6 Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset of

X Let {T i : i ∈ I} be N-uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive

selfmaps of C with {u in }, {c in } ⊂ 0, ∞ such that∞n1 u in < ∞ and∞

n1 c in < ∞ for all i ∈ I Suppose that F is closed, and there exists one member T in {T i : i ∈ I} which is either semicompact or

satisfies condition (A) Let {α n } ⊂ δ, 1 − δ for some δ ∈ 0, 1/2 From arbitaray x0 ∈ C, define the

sequence {x n } by algorithm 1.6 Then, {x n } converges strongly to a common fixed point of the maps

in {T i : i ∈ I}.

Proof Without loss of generality, we may assume that T1 is either semicompact or satisfies conditionA If T1 is semicompact, then there exists a subsequence{x nj } of {x n} such that

x n j → x∗∈ C as j → ∞ Now,Lemma 2.5guarantees that limn → ∞ dx n j , T l x n j   0 for all l ∈ I and so dx∗, T l x∗  0 for all l ∈ I This implies that x∗∈ F Therefore, lim inf n → ∞ dx n , F  0.

If T1 satisfies conditionA, then we also have lim inf n → ∞ dx n , F  0 Now, Theorem 2.1

gaurantees that{x n } converges strongly to a point in F.

Finally, we state two corollaries to the above theorem

Corollary 2.7 Let X, d be a complete CAT0 space and let C be a nonempty closed convex subset

of X Let {T i : i ∈ I} be N uniformly L-Lipschizian and asymptotically quasi-nonexpansive selfmaps

of C with {u in } ⊂ 0, ∞ such that∞n1 u in < ∞ for all i ∈ I Suppose that there exists one member

T in {T i : i ∈ I} which is either semicompact or satisfies condition (A) From arbitaray x0 ∈ C, define

the sequence {x n } by algorithm 1.6, where {α n } ⊂ δ, 1 − δ for some δ ∈ 0, 1/2 Then, {x n}

converges strongly to a common fixed point of the maps in {T i : i ∈ I}.

Corollary 2.8 Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset

of X Let {T i : i ∈ I} be N asymptotically nonexpansive selfmaps of C with {u in } ⊂ 0, ∞ such that

∞

n1 u in < ∞ for all i ∈ I Suppose that there exists one member T in {T i : i ∈ I} which is either

semicompact or satisfies condition (A) From arbitrary x0∈ C, define the sequence {x n } by algorithm

1.6, where {α n } ⊂ δ, 1 − δ for some δ ∈ 0, 1 Then, {x n } converges strongly to a common fixed

point of the maps in {T i : i ∈ I}.

Remark 2.9 The corresponding approximation results for a finite family of asymptotically

quasi-nonexpansive maps on: i uniformly convex Banach spaces 5, 14, 15, ii convex metric spaces13, iii CAT0 spaces 12 are immediate consequences of our results

Remark 2.10 Various algorithms and their strong convergence play an important role in

finding a common element of the set of fixedcommon fixed point for different classes of mappings and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces; for details we refer to18–20

Acknowledgments

The author A R Khan gratefully acknowledges King Fahd University of Petroleum and Minerals and SABIC for supporting research project no SB100012

References

1 M R Bridson and A Haefliger, Metric Spaces of Non-Positive Curvature, vol 319 of Grundlehren der

Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.

finding a common element of the set of fixedcommon fixed point for different classes of mappings and the set of solutions of an equilibrium problem in the... n   for all i ∈ I.

Proof Note that {x n} is bounded... as∞

Hence,

lim

d x n , x n−1