International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 2 (2016), 118-128
http://www.etamaths.com

AN IMPLICIT ALGORITHM FOR A FAMILY OF TOTAL ASYMPTOTICALLY

NONEXPANSIVE MAPPINGS IN CAT(0) SPACES

G. S. SALUJA∗

Abstract. In this paper, we establish some strong convergence theorems of an implicit algorithm

for a finite family of of total asymptotically nonexpansive mappings in the setting of CAT(0) spaces.

Our results extend and generalize several recent results from the current existing literatures (see, e.g.,
[2, 9, 14, 16, 17, 25, 29]).

1. Introduction and Preliminaries

A metric space (X,d) is said to be a length space if any two points of X are joined by a rectifiable
path (i.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 mapping c
from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and let d(c(t),c(t′)) = |t− t′| for
t,t′ ∈ [0, l]. In particular, c is an isometry, and d(x,y) = l. The image α of c is called a geodesic (or
metric) segment joining x and y. We say that 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 denoted by [x,y], called the segment joining x to y.

A geodesic triangle 4(x1,x2,x3) in a geodesic metric space (X,d) consists of three points in X (the
vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). A comparison
triangle for geodesic triangle 4(x1,x2,x3) in (X,d) is a triangle 4(x1,x2,x3) := 4(x1,x2,x3) in R2
such that dR2 (xi,xj) = d(xi,xj) for i,j ∈{1, 2, 3}. Such a triangle always exists (see [3]).

A geodesic metric space is said to be a CAT(0) space if all geodesic triangles of appropriate size
satisfy the following CAT(0) inequality.

Let 4 be a geodesic triangle in X, and let 4⊂ R2 be a comparison triangle for 4. Then 4 is said
to satisfy the CAT(0) inequality if for all x,y ∈4 and all comparison points x,y ∈4,

d(x,y) ≤ dR2 (x,y).(1.1)

Complete CAT(0) spaces are often called Hadamard spaces (see [12]). If x,y1,y2 are points of a
CAT(0) space and y0 is the mid-point of the segment [y1,y2] which we will denote by (y1⊕y2)/2, then
the CAT(0) inequality implies

d2
(
x,
y1 ⊕y2

2

)
≤

1

2
d2(x,y1) +

1

2
d2(x,y2) −

1

4
d2(y1,y2).(1.2)

2010 Mathematics Subject Classification. 54H25, 54E40.
Key words and phrases. total asymptotically nonexpansive mapping; strong convergence; implicit iteration process;

common fixed point; CAT(0) space.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

118



AN IMPLICIT ALGORITHM FOR A FAMILY OF TOTAL. . . . . . 119

The inequality (1.2) is the (CN) inequality of Bruhat and Tits [4]. The above inequality has been
extended in [6] as

d2(z,αx⊕ (1 −α)y) ≤ αd2(z,x) + (1 −α)d2(z,y)
−α(1 −α)d2(x,y)(1.3)

for any α ∈ [0, 1] and x,y,z ∈ X.

Let us recall that a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN)
inequality (see [[3], page 163]). Moreover, if X is a CAT(0) metric space and x,y ∈ X, then for any
α ∈ [0, 1], there exists a unique point αx⊕ (1 −α)y ∈ [x,y] such that

d(z,αx⊕ (1 −α)y) ≤ αd(z,x) + (1 −α)d(z,y),(1.4)
for any z ∈ X and [x,y] = {αx⊕ (1 −α)y : α ∈ [0, 1]}.

A subset C of a CAT(0) space X is convex if for any x,y ∈ C, we have [x,y] ⊂ C.

Let T be a self mapping on a nonempty subset C of X. Denote the set of fixed points of T by
F(T) = {x ∈ C : T(x) = x}. We say that T is:

(1) nonexpansive if d(Tx,Ty) ≤ d(x,y) for all x,y ∈ C;

(2) asymptotically nonexpansive ([10]) if there exists a sequence {rn}⊂ [0,∞) with limn→∞rn = 0
such that d(Tnx,Tny) ≤ (1 + rn)d(x,y) for all x,y ∈ C and n ≥ 1;

(3) uniformly L-Lipschitzian if there exists a constant L > 0 such that d(Tnx,Tny) ≤ Ld(x,y) for
all x,y ∈ C and n ≥ 1;

(4) semi-compact if for a sequence {xn} in C with limn→∞d(xn,Txn) = 0, there exists a subsequence
{xnk} of {xn} such that xnk → p ∈ C.
Remark 1.1. From the above definitions, it is clear that each nonexpansive mapping is an asymptot-
ically nonexpansive mapping with the constant sequence {kn} = {1}, ∀n ≥ 1 and an asymptotically
nonexpansive mapping is a uniformly L-Lipschitzian mapping with L = supn≥1{kn}.

Chang et al. [5] defined the concept of total asymptotically nonexpansive mapping as follows.

Definition 1.2. ([5] Definition 2.1) Let (X,d) be a metric space, K be its nonempty subset and let
T : K → K be a mapping. T is said to be a total asymptotically nonexpansive mapping if there exist
non-negative real sequences {µn}, {νn} with µn → 0, νn → 0 and a strictly increasing continuous
function ψ : [0,∞) → [0,∞) with ψ(0) = 0 such that

d(Tnx,Tny) ≤ d(x,y) + νnψ(d(x,y)) + µn
for all x,y ∈ K and n ≥ 1.
Remark 1.3. From the above definition, it is clear that each asymptotically nonexpansive mapping is
a total asymptotically nonexpansive mapping with µn = 0, νn = kn − 1 for all n ≥ 1, ψ(t) = t, t ≥ 0.

Recently, there are a lot of papers have appeared on the iterative approximation of fixed points
of asymptotically nonexpansive mappings, asymptotically quasi-nonexpansive mappings, asymptoti-
cally nonexpansive mappings in the intermediate sense and their generalizations through Ishikawa,
S-iteration, modified S-iteration, Noor iteration and implicit iterations in uniformly convex Banach
spaces, convex metric spaces and CAT(0) spaces (see, e.g., [1, 2, 5, 8, 9, 13, 14, 15, 16, 17, 19, 20, 21,
22, 23, 24]).

Let E be a Hilbert space, let K be a nonempty closed convex subset of E and let {Ti}: K → K {i =
1, 2, . . . ,N} be nonexpansive mappings. In 2001, Xu and Ori [29] introduced the following implicit
iteration process {xn} defined by

xn = αnxn−1 + (1 −αn)Tn(modN)xn for n ≥ 1,(1.5)



120 SALUJA

where x0 ∈ K is an initial point, {αn} is a real sequence in (0, 1) and proved a weak convergence of
the sequence {xn} defined by (1.5) to a common fixed point p ∈ F = ∩Ni=1F(Ti).

In 2003, Sun [27] introduced the following implicit iterative sequence {xn}

xn = αnxn−1 + (1 −αn)T
k(n)
i(n)

xn for n ≥ 1,(1.6)

for a finite family of asymptotically quasi-nonexpansive self-mappings on a bounded closed convex
subset K of a Hilbert space E with {αn} a real sequence in (0, 1) and an initial point x0 ∈ K, where
n = (k(n) − 1)N + i(n), 1(n) ∈ {1, 2, . . . ,N}, and proved a strong convergence of the sequence {xn}
defined (1.6) to a common fixed point p ∈ F = ∩Ni=1F(Ti). The result of Sun [27] generalized and
extended the corresponding main result of Wittmann [28] and Xu and Ori [29].

Inspired and motivated by [28, 29], we now define a modified implicit iteration process for a finite
family of total asymptotically nonexpansive mappings as below.

Modified implicit iterative process in CAT(0) space

Let C be a nonempty closed convex subset of a CAT(0) space X, and {T1,T2, . . . ,TN} be a fi-
nite family of N ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive self mappings on C. From an
arbitrary x0 ∈ C, we define the sequence {xn} by:

x1 = (1 −α1)x0 ⊕α1T1x1,
x2 = (1 −α2)x1 ⊕α2T2x2,

...

xN = (1 −αN )xN−1 ⊕αNTNxN,
xN+1 = (1 −αN+1)xN ⊕αN+1T21 xN+1,(1.7)

...

x2N = (1 −α2N )x2N ⊕α2NT2Nx2N,
x2N+1 = (1 −α2N+1)x2N+1 ⊕α2N+1T31 x2N+1,

...

where {αn} is an appropriate sequence in (0, 1).

The above iteration can be written in the following compact form:

xn = αnxn−1 ⊕ (1 −αn)T
k(n)
i(n)

xn, for n ≥ 1(1.8)

where n =
(
k(n) − 1

)
N + i(n), k(n) > 1 is a positive integer such that k(n) →∞ as n →∞.

Let X be a CAT(0) space. Then, the following inequality holds:

d(λx⊕ (1 −λ)z,λy ⊕ (1 −λ)w) ≤ λd(x,y) + (1 −λ)d(z,w),(1.9)

for all x,y,z,w ∈ X (see [6]).

Let {Ti : i ∈ I = {1, 2, . . . ,N}} be the set of N uniformly Li (i = 1, 2, . . . ,N)-Lipschitzian self
mappings of C. We show that (1.8) exists. Let x0 ∈ C and x1 = α1x0 ⊕ (1 − α1)T1x1. Define
W : C → C by W(x) = α1x0 ⊕ (1 −α1)T1x for x ∈ C. The existence of x1 is guaranteed if W has a
fixed point. For any x,y ∈ C, we have

d(Wx,Wy) ≤ (1 −α1)d(T1x,T1y) ≤ (1 −α1)L1 d(x,y)
≤ (1 −α1)Ld(x,y)(1.10)

where L = max{Li : i ∈ I}. Now, W is a contraction if (1−α1)L < 1 or L > 1/(1−α1). As α1 ∈ (0, 1),
therefore W is a contraction if 1 < L < 2. By the Banach contraction principle W has a unique fixed



AN IMPLICIT ALGORITHM FOR A FAMILY OF TOTAL. . . . . . 121

point. Thus, the existence of x1 is established. Thus, the implicit algorithm (1.8) is well defined.

The goal of this paper is to study strong convergence of iterative algorithm (1.8) for the class
of uniformly Li-Lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings (for
i = 1, 2, . . . ,N) in the setting of CAT(0) spaces. Our results extend, improve and generalize several
results from the current existing literature.

We need the following useful notion and lemmas for the development of our main results.

Let {Ti : i ∈ I} be the set of N self mappings of C. A mapping T : C → C is said to satisfy con-
dition (A) if there exists a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for all
r ∈ (0,∞) such that d(x.p) ≥ f(d(x,F(T ))) for x ∈ C where d(x,F(T)) = inf{d(x,p) : p ∈ F(T) 6= ∅}.
Condition (A) was introduced by Senter and Dotson [26].

Lemma 1.4. ([6]) Let X be a CAT(0) space.
(i) For x,y ∈ X and t ∈ [0, 1], there exists a unique point z ∈ [x,y] such that

d(x,z) = td(x,y) and d(y,z) = (1 − t) d(x,y). (A)
We use the notation (1 − t)x⊕ ty for the unique point z satisfying (A).
(ii) For x,y,z ∈ X and t ∈ [0, 1], we have

d((1 − t)x⊕ ty,z) ≤ (1 − t) d(x,z) + td(y,z).

Lemma 1.5. ([18]) Suppose that {an}, {bn} and {rn} be sequences of nonnegative numbers such that
an+1 ≤ (1 + bn)an + rn for all n ≥ 1. If

∑∞
n=1 bn < ∞ and

∑∞
n=1 rn < ∞, then limn→∞an exists.

2. Main Results

In this section, we establish strong convergence theorems using implicit iteration scheme (1.8) for
({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings (for i = 1, 2, . . . ,N) in the setting of
CAT(0) spaces.

Lemma 2.1. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let {Ti : i ∈
I} be N uniformly Li-Lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings
with F = ∩Ni=1F(Ti) 6= ∅. Suppose that the sequence {xn} is defined by the algorithm (1.8), where
{αn}⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). If the following conditions are satisfied:

(i)
∑∞
n=1 µi,n < ∞,

∑∞
n=1 νi,n < ∞ for i ∈ I;

(ii) there exists a constant M > 0 such that ψ(t) ≤ Mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈
I}, a ≥ 0.

Then limn→∞d(xn,p) and limn→∞d(xn,F) exist for p ∈F.

Proof. Let p ∈F. Then, from (1.8) and Lemma 1.4(ii), we have

d(xn,p) = d(αnxn−1 ⊕ (1 −αn)T
k(n)
i(n)

xn,p)

≤ αnd(xn−1,p) + (1 −αn)d(T
k(n)
i(n)

xn,p)

≤ αnd(xn−1,p) + (1 −αn)[d(xn,p) + νi,k(n)ψ(d(xn,p)) + µi,k(n)]
≤ αnd(xn−1,p) + (1 −αn)[d(xn,p) + Mνi,k(n) d(xn,p) + µi,k(n)]
= αnd(xn−1,p) + (1 −αn)[(1 + Mνi,k(n)) d(xn,p) + µi,k(n)](2.1)
≤ αnd(xn−1,p) + (1 −αn + Mνi,k(n)) d(xn,p) + (1 −αn)µi,k(n).

Since αn ∈ [δ, 1 − δ], the above inequality gives that

d(xn,p) ≤ d(xn−1,p) +
Mνi,k(n)

δ
d(xn,p) +

(1
δ
− 1
)
µi,k(n).(2.2)



122 SALUJA

On simplification, we get that

d(xn,p) ≤
( δ
δ −Mνi,k(n)

)
d(xn−1,p) +

(1
δ
− 1
)( δ

δ −Mνi,k(n)

)
µi,k(n)

=
(

1 +
Mνi,k(n)

δ −Mνi,k(n)

)
d(xn−1,p) +

(1
δ
− 1
)( δ

δ −Mνi,k(n)

)
µi,k(n)

= (1 + Ai,k(n))d(xn−1,p) + Bi,k(n)(2.3)

where Ai,k(n) =
Mνi,k(n)
δ−Mνi,k(n)

and Bi,k(n) =
(

1
δ
− 1
)(

δ
δ−Mνi,k(n)

)
µi,k(n). Since

∑∞
k(n)=1 νi,k(n) < ∞ for

i ∈ I therefore limk(n)→∞νi,k(n) = 0, and hence, there exists a natural number n1 such that νik(n) < δ/2
for k(n) ≥ n1/N + 1 or n > n1. Then, we have that

∑∞
k(n)=1 Ai,k(n) <

(
2M

δ(2−M)

)∑∞
k(n)=1 νik(n) < ∞.

Similarly,
∑∞
k(n)=1 wi,k(n) < ∞. Similarly,

∑∞
k(n)=1 Bi,k(n) < ∞.

Now, for any p ∈F, from (2.3), for k(n) ≥ n1/N + 1, we have

d(xn,F) ≤ (1 + Ai,k(n)) d(xn−1,F) + Bi,k(n),(2.4)

By Lemma 1.5, (2.3) and (2.4), we obtain limn→∞d(xn,p) and limn→∞d(xn,F) both exist. This
completes the proof. �

Theorem 2.2. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let
{Ti : i ∈ I} be N uniformly Li-Lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpan-
sive mappings with F = ∩Ni=1F(Ti) is nonempty and closed. Suppose that the sequence {xn} is defined
by the algorithm (1.8), where {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). If the following conditions are
satisfied:

(i)
∑∞
n=1 µi,n < ∞,

∑∞
n=1 νi,n < ∞ for i ∈ I;

(ii) there exists a constant M > 0 such that ψ(t) ≤ Mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈
I}, a ≥ 0.

Then the sequence {xn} converges strongly to a common fixed point of {Ti : i ∈ I} if and only if
lim infn→∞d(xn,F) = 0.

Proof. If xn → p as n → ∞, then limn→∞d(xn,p) = 0. Since 0 ≤ d(xn,F) ≤ d(xn,p), we have
lim infn→∞d(xn,F) = 0.

Conversely, suppose that lim infn→∞d(xn,F) = 0. By Lemma 1.5, we have that limn→∞d(xn,F)
exists. Further, by assumption lim infn→∞d(xn,F) = 0, we conclude that limn→∞d(xn,F) = 0.
Next, we show that {xn} is a Cauchy sequence.

Since x ≤ exp(x− 1) for x ≥ 1, therefore from (2.3), we have

d(xn+m,p) ≤ (1 + Ai,k(n))d(xn−1,p) + Bi,k(n)

≤
(
e
∑N

i=1

∑∞
k(n)=1

Ai,k(n)
)
d(xn,p) +

N∑
i=1

∞∑
k(n)=1

Bi,k(n)

< Rd(xn,p) + R

N∑
i=1

∞∑
k(n)=1

Bi,k(n)(2.5)

for all natural numbers m,n, where R =
(
e
∑N

i=1

∑∞
k(n)=1 Ai,k(n)

)
+ 1 < ∞. Since limn→∞d(xn,F) = 0,

without loss of generality, we may assume that a subsequence {xnk} of {xn} and a sequence {pnk}⊂F



AN IMPLICIT ALGORITHM FOR A FAMILY OF TOTAL. . . . . . 123

such that d(xnk,pnk ) → 0 as k →∞. Then for any ε > 0, there exists kε ∈ N such that

d(xnk,pnk ) <
ε

4R
and

N∑
i=1

∞∑
j=nkε

Bi,j <
ε

4R
(2.6)

for k ≥ kε.

Hence, for any m ∈ N and for n ≥ nkε , by (2.5) we have

d(xn+m,xn) ≤ d(xn+m,pnk ) + d(xn,pnk )

≤ Rd(xn,pnk ) + R
N∑
i=1

∞∑
j=nkε

Bi,j

+Rd(xn,pnk ) + R

N∑
i=1

∞∑
j=nkε

Bi,j(2.7)

= 2Rd(xn,pnk ) + 2R
N∑
i=1

∞∑
j=nkε

Bi,j

< 2R.
ε

4R
+ 2R.

ε

4R
= ε.

This implies that {xn} is a Cauchy sequence in C. By the completeness of C, we can assume that
limn→∞xn = q. We will prove that q is a common fixed point of {Ti : i ∈ I}, that is, we will show
that q ∈F. Since C is closed, therefore q ∈ C. Next, we show that q ∈F. Since limn→∞d(xn,F) = 0,
gives that d(q,F) = 0. Since F is closed, q ∈F. Thus q is a common fixed point of {Ti : i ∈ I}. This
completes the proof. �

Theorem 2.3. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let
{Ti : i ∈ I} be N uniformly Li-Lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpan-
sive mappings with F = ∩Ni=1F(Ti) 6= ∅. Suppose that the sequence {xn} defined by the algorithm
(1.8), where {αn}⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). If the following conditions are satisfied:

(i)
∑∞
n=1 µi,n < ∞,

∑∞
n=1 νi,n < ∞ for i ∈ I;

(ii) there exists a constant M > 0 such that ψ(t) ≤ Mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈
I}, a ≥ 0.

Then lim infn→∞d(xn,F) = lim supn→∞d(xn,F) = 0 if {xn} converges to a unique point in F.

Proof. Let p ∈F. Since {xn} converges to p, limn→∞d(xn,p) = 0. So, for a given ε > 0, there exists
n0 ∈ N such that

d(xn,p) < ε for n ≥ n0.
Taking the infimum over p ∈F, we obtain that

d(xn,F) < ε for n ≥ n0.

This means that limn→∞d(xn,F) = 0. Thus we obtain that

lim inf
n→∞

d(xn,F) = lim sup
n→∞

d(xn,F) = 0.

This completes the proof. �

As shown in the preceding proof, the property needed to assure that p ∈ F is exactly the fol-
lowing one. Given any sequence {un} of real numbers there is a subsequence {unj} of {un} such
that limj→∞unj = lim infn→∞un. In general, if {umj} is a convergent subsequence of {un}, then
lim infn→∞un ≤ limj→∞umj . This immediately gives the following result.



124 SALUJA

Corollary 2.4. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let
{Ti : i ∈ I} be N uniformly Li-Lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive
mappings with F = ∩Ni=1F(Ti) 6= ∅. Suppose that the sequence {xn} defined by the algorithm (1.8),
where {αn}⊂ [δ, 1 −δ] for some δ ∈ (0, 1/2). If the following conditions are satisfied:

(i)
∑∞
n=1 µi,n < ∞,

∑∞
n=1 νi,n < ∞ for i ∈ I;

(ii) there exists a constant M > 0 such that ψ(t) ≤ Mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈
I}, a ≥ 0.

Then {xn} converges strongly to a common fixed point of {Ti : i ∈ I} if and only if there exists
some subsequence {xnj} of {xn} which converges to p ∈F.

Corollary 2.5. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let {Ti : i ∈
I} be N asymptotically nonexpansive mappings of C with {ki,n}⊂ [1,∞) such that

∑∞
n=1(ki,n−1) < ∞

for all i ∈ I. Suppose that F = ∩Ni=1F(Ti) is nonempty and closed. Starting from arbitrary x0 ∈ C,
define the sequence {xn} by the algorithm (1.8), where {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). Then
{xn} converges strongly to a common fixed point of {Ti : i ∈ I} if and only if lim infn→∞d(xn,F) = 0.

Proof. Follows from Theorem 2.2 with µi,n = 0, νi,n = (ki,n−1) for all i ∈ I and ψ(t) = t, t ≥ 0. This
completes the proof. �

Lemma 2.6. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let {Ti : i ∈
I} be N uniformly Li-Lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings
with F = ∩Ni=1F(Ti) 6= ∅. Suppose that the sequence {xn} is defined by the algorithm (1.8), where
{αn}⊂ [δ, 1 −δ] for some δ ∈ (0, 1/2). If the following conditions are satisfied:

(i)
∑∞
n=1 µi,n < ∞,

∑∞
n=1 νi,n < ∞ for i ∈ I;

(ii) there exists a constant M > 0 such that ψ(t) ≤ Mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈
I}, a ≥ 0.

Then limn→∞d(xn,Tlxn) = 0 for each l ∈ I.

Proof. Let L = max{Li : i ∈ I}. Note that {xn} is bounded as limn→∞d(xn,p) exists by Lemma 2.1.
So, there exists R′ > 0 and x0 ∈ X such that xn ∈ B′R(x0) = {x : d(x,x0) < R

′} for n ≥ 1. Denote
d(xn−1,T

k(n)
i(n)

) by ρn.

We claim that limn→∞ρn = 0.
For any p ∈F, apply (1.3) to (1.8), we have

d2(xn,p) = d
2(αnxn−1 ⊕ (1 −αn)T

k(n)
i(n)

xn,p)

≤ αnd2(xn−1,p) + (1 −αn)d2(T
k(n)
i(n)

xn,p)

−αn(1 −αn)d2(xn−1,T
k(n)
i(n)

xn)

≤ αnd2(xn−1,p) + (1 −αn)[d(xn,p) + νi,k(n)ψ(d(xn,p)) + µi,k(n)]2

−αn(1 −αn)d2(xn−1,T
k(n)
i(n)

xn)(2.8)

≤ αnd2(xn−1,p) + (1 −αn)[d(xn,p) + Mνi,k(n)d(xn,p) + µi,k(n)]2

−αn(1 −αn)d2(xn−1,T
k(n)
i(n)

xn)

= αnd
2(xn−1,p) + (1 −αn)[(1 + Mνi,k(n))d(xn,p) + µi,k(n)]2

−αn(1 −αn)d2(xn−1,T
k(n)
i(n)

xn).



AN IMPLICIT ALGORITHM FOR A FAMILY OF TOTAL. . . . . . 125

Now, using (2.3), we get

αn(1 −αn)ρ2n ≤ αnd
2(xn−1,p) −d2(xn,p) + (1 −αn)[(1 + Mνi,k(n))

×{(1 + Ai,k(n)) d(xn−1,p) + Bi,k(n)} + µi,k(n)]2

= αnd
2(xn−1,p) −d2(xn,p) + (1 −αn)[(1 + Mνi,k(n))(1 + Ai,k(n)) ×

d(xn−1,p) + (1 + Mνi,k(n))Bi,k(n) + µi,k(n)]
2(2.9)

= αnd
2(xn−1,p) −d2(xn,p) + (1 −αn)[(1 + fi,k(n))d(xn−1,p) ×

+gi,k(n)]
2

where fi,k(n) = Mνi,k(n) + Ai,k(n) + MAi,k(n)νi,k(n) and gi,k(n) = (1 + Mνi,k(n))Bi,k(n) + µi,k(n). Since∑∞
k(n)=1 µi,k(n) < ∞,

∑∞
k(n)=1 νi,k(n) < ∞ and

∑∞
k(n)=1 Bi,k(n) < ∞, it follows that

∑∞
k(n)=1 fi,k(n) <

∞ and
∑∞
k(n)=1 gi,k(n) < ∞. Again, note that

αn(1 −αn)ρ2n ≤ αnd
2(xn−1,p) −d2(xn,p) + (1 −αn)[d(xn−1,p) + li,k(n)]2

= d2(xn−1,p) −d2(xn,p) + (1 −αn)qi,k(n),(2.10)

where li,k(n) = fi,k(n)d(xn−1,p) + gi,k(n) and qi,k(n) = l
2
i,k(n)

+ 2li,k(n)d(xn−1,p). Since {d(xn−1,p)}
is convergent,

∑∞
k(n)=1 fi,k(n) < ∞ and

∑∞
k(n)=1 gi,k(n) < ∞, it follows that

∑∞
k(n)=1 li,k(n) < ∞ and∑∞

k(n)=1 qi,k(n) < ∞. This implies that

ρ2n ≤
1

αn(1 −αn)
[d2(xn−1,p) −d2(xn,p)] +

qi,k(n)

αn

≤
1

δ2
[d2(xn−1,p) −d2(xn,p)] +

qi,k(n)

δ
.(2.11)

Since
∑∞
k(n)=1 qi,k(n) < ∞, {d(xn,p)} is convergent and δ > 0, therefore on taking limit as n →∞ in

(2.11), we get

lim
n→∞

ρn = 0.(2.12)

Further,

d(xn,xn−1) ≤ (1 −αn)d
(
T
k(n)
i(n)

xn,xn−1
)

= (1 −αn)ρn ≤ (1 −δ)ρn,(2.13)

which implies that limn→∞d(xn,xn−1) = 0.

For a fixed j ∈ I, we have d(xn+j,xn) ≤ d(xn+j,xn+j−1) + · · · + d(xn,xn−1), and hence

lim
n→∞

d(xn+j,xn) = 0 for j ∈ I.(2.14)

For n > N, n = (n−N)(modN). Also, n = (k(n)−1)N+i(n). Hence, n−N = ((k(n)−1)−1)N+i(n) =
(k(n−N))N + i(n−N). That is, k(n−N) = k(n) − 1 and i(n−N) = i(n).

Therefore, we have

d(xn−1,Tnxn) ≤ d
(
xn−1,T

k(n)
i(n)

xn
)

+ d
(
T
k(n)
i(n)

xn,Tnxn
)

≤ ρn + Ld
(
T
k(n)−1
i(n)

xn,xn
)

≤ ρn + L2 d(xn,xn−N ) + Ld
(
T
k(n−N)
i(n−N) xn−N,x(n−N)−1

)
+Ld(x(n−N)−1,xn)(2.15)

≤ ρn + L2 d(xn,xn−N ) + Lρn−N
+Ld(x(n−N)−1,xn).

Using (2.12) and (2.14) in (2.15), we get

lim
n→∞

d(xn−1,Tnxn) = 0.(2.16)



126 SALUJA

Since

d(xn,Tnxn) ≤ d(xn,xn−1) + d(xn−1,Tnxn),(2.17)

using (2.13) and (2.16) in (2.17), we have

lim
n→∞

d(xn,Tnxn) = 0.(2.18)

Hence, for all l ∈ I, we have

d(xn,Tn+lxn) ≤ d(xn,xn+l) + d(xn+l,Tn+lxn+l)
+d(Tn+lxn+l,Tn+lxn)

≤ (1 + L) d(xn,xn+l) + d(xn+l,Tn+lxn+l).(2.19)

Using (2.14) and (2.18) in (2.19), we obtain

lim
n→∞

d(xn,Tn+lxn) = 0, ∀ l ∈ I.(2.20)

Thus, limn→∞d(xn,Tlxn) = 0 for l ∈ I. This completes the proof. �

As an application of Theorem 2.2, we establish some strong convergence results as follows.

Theorem 2.7. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let
{Ti : i ∈ I} be N uniformly Li-Lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpan-
sive mappings with F = ∩Ni=1F(Ti) 6= ∅ and there exists one member T in {Ti : i ∈ I} which is either
semicompact or satisfies condition (A). Suppose that the sequence {xn} is defined by the algorithm
(1.8), where {αn}⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). If the following conditions are satisfied:

(i)
∑∞
n=1 µi,n < ∞,

∑∞
n=1 νi,n < ∞ for i ∈ I;

(ii) there exists a constant M > 0 such that ψ(t) ≤ Mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈
I}, a ≥ 0.

Then {xn} converges strongly to a common fixed point of {Ti : i ∈ I}.

Proof. By Lemma 2.1, we see that

lim
n→∞

d(xn,x
∗) and lim

n→∞
d(xn,F) exist.

Let one of T ′is, say, Ts, s ∈ I is either semicompact or satisfies condition (A). If Ts is semicompact,
then there exists a subsequence {xnj} of {xn} such that xnj → z ∈ C as j → ∞. Now, Lemma 2.6
guarantees that limn→∞d(xnj,Tsxnj ) = 0 for s ∈ I and so d(z,Tsz) = 0 for s ∈ I. This implies
that z ∈ F. Therefore, lim infn→∞d(xn,F) = 0. If Ts satisfies condition (A), then we also have
lim infn→∞d(xn,F) = 0. Now, Theorem 2.2 implies that {xn} converges strongly to a point in F.
This completes the proof. �

Theorem 2.8. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let
{Ti : i ∈ I} be N ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings. Suppose that
F = ∩Ni=1F(Ti) 6= ∅ (Ti, i = 1, 2, . . . ,N, need not to be continuous). Starting from arbitrary x0 ∈ C,
define the sequence {xn} by the algorithm (1.8), where {αn}⊂ [δ, 1−δ] for some δ ∈ (0, 1/2). Assume
that (i′) limn→∞d(xn,xn+1) = 0 if the sequence {zn} in C satisfies (ii′) limn→∞d(zn,zn+1) = 0, then
lim infn→∞d(zn,F) = 0 or lim supn→∞d(zn,F) = 0. If the following conditions are satisfied:

(i)
∑∞
n=1 µi,n < ∞,

∑∞
n=1 νi,n < ∞ for all i ∈ I;

(ii) there exists a constant M > 0 such that ψ(t) ≤ Mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈
I}, a ≥ 0.

Then {xn} converges to a unique point in F.



AN IMPLICIT ALGORITHM FOR A FAMILY OF TOTAL. . . . . . 127

Proof. By hypothesis (i′) and (ii′), we have that

lim inf
n→∞

d(xn,F) = 0 or lim sup
n→∞

d(xn,F) = 0.

Therefore, we obtain from Theorem 2.2 that the sequence {xn} converges to a unique point in F.
This completes the proof. �

Finally, we obtain the following result from Theorem 2.7 as corollary.

Corollary 2.9. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let
{Ti : i ∈ I} be N asymptotically nonexpansive mappings of C with {hi,n} ⊂ [1,∞) for i ∈ I such
that

∑∞
n=1(hn − 1) < ∞, where hn = max{hi,n : i ∈ I}. Suppose that F = ∩

N
i=1F(Ti) 6= ∅ and there

exists one member T in {Ti : i ∈ I} which is either semicompact or satisfies condition (A). From
an arbitrary x0 ∈ C, define the sequence {xn} by algorithm (1.8), where {αn} ⊂ [δ, 1 − δ] for some
δ ∈ (0, 1/2).. Then {xn} converges strongly to a common fixed point of {Ti : i ∈ I}.

Remark 2.10. Our results extend, generalize and improve several corresponding approximation results
from the current existing literature to the case of implicit iteration process and more general class of
nonexpansive and asymptotically nonexpansive mappings considered in this paper (see, e.g., [2, 7, 14,
16, 17, 29] and many others).

Remark 2.11. Our results also extend the corresponding results [25] to the case of finite family of
mappings and implicit iteration process considered in this paper.

Example 2.12. ([11], Example 3.1) Let R be the real line with the usual norm ‖.‖ and C = [−1, 1].
Define a mapping T : C → C by

T(x) =

{
−2 sinx

2
, if x ∈ [0, 1],

2 sinx
2
, if x ∈ [−1, 0).

Then T is an asymptotically nonexpansive mapping with constant sequence {kn} = {1} for n ≥ 1
and uniformly L-Lipschtzian mapping with L = supn≥1{kn} and hence it is a total asymptotically
nonexpansive mapping by remark 1.3. Also the fixed point of T , that is, F(T) = {0}.

3. Conclusion

In this paper, we establish some strong convergence theorems using implicit algorithm (1.8) for a
finite family of ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings which is more general
than the class of nonexpansive and asymptotically nonexpansive mappings in the framework of CAT(0)
spaces.

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Department of Mathematics, Govt. Nagarjuna P.G. College of Science, Raipur - 492010 (C.G.), India

∗Corresponding author: saluja1963@gmail.com