CUBO A Mathematical Journal
Vol.14, No¯ 03, (143–166). October 2012

Weak and strong convergence theorems of a multistep
iteration to a common fixed point of a family of nonself
asymptotically nonexpansive mappings in banach spaces

Shrabani Banerjee and Binayak S.Choudhury

Department of Mathematics,

Bengal Engineering and Science University,

Shibpur, Howrah-711103, India.

email: banerjee.shrabani@yahoo.com, binayak12@yahoo.co.in

ABSTRACT

In this paper we have defined a multistep iterative scheme with errors involving a fam-

ily of asymptotically nonexpansive nonself mappings in Banach spaces. A retraction

has been used in the construction of the iteration. We prove here weak and strong

convergences of the iteration to common fixed points of the family of asymptotically

nonexpansive nonself mappings. We have used several concepts of Banach space geom-

etry. Our results improve and extend some recent results.

RESUMEN

En este art́ıculo definimos un esquema de multi paso iterativo con errores que involucran

una familia de aplicaciones no expansivas y no auto asintóticamente en espacios de

Banach. Una retracción se ha usado en la construcción de la iteración. Probamos

convergencias débiles y fuertes de las iteraciones a puntos fijos clásicos de la familia

de aplicaciones no expansivas no auto asintóticamente. Hemos usado varios conceptos

de geometŕıa en espacios de Banach. Nuestro resultado mejora y extiende algunos

resultados recientes.

Keywords and Phrases: Modified multistep iterative process with errors; nonself asymptotically

nonexpansive mapping; retraction; Opial’s condition; uniformly convex Banach space; common

fixed point; Kadec-klee property; Condition (B); weak and strong convergence.

2010 AMS Mathematics Subject Classification: 47H10



144 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

1 Introduction

Let K be a nonempty subset of real normed space E. A self mapping T : K → K is called nonex-
pansive if

‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ K

A self mapping T : K → K is called asymptotically nonexpansive if there exists a sequence {kn} ⊂
[1, ∞) with limn→∞ kn = 1 such that

‖Tnx − Tny‖ ≤ kn‖x − y‖ for all x, y ∈ K and n ≥ 1. (1.1)

T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that

‖Tnx − Tny‖ ≤ L‖x − y‖ for all x, y ∈ K and n ≥ 1. (1.2)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk[7] in 1972

as a generalization of the class of nonexpansive self mappings. They proved that if T is a selfmap

on K where K is a nonempty closed convex subset of a real uniformly convex Banach space, then T

has a fixed point. Fixed point iterative processes for asymptotically nonexpansive self-mappings on

convex subsets of Banach spaces have been studied extensively by many authors. Since T remains a

self- mapping of a nonempty closed convex subset K of a Banach space E, the well known Mann[11]

and Ishikawa[8] iterative processes are well defined. If however the domain K of T is a proper subset

of E (and it is the case of several applications) and T maps K into E, then the iteration processes

of Mann and Ishikawa and their modifications fail to be well defined. To overcome this problem

Chidume et al.[2] introduced the concept of nonself asymptotically nonexpansive mappings in 2003

as a generalization of asymptotically nonexpansive self mappings. A subset K of E is said to be a

retract of E if there exists a continuous mapping P : E → K such that Px = x for all x ∈ K. Every
closed convex subset of a uniformly convex Banach space is a retract. A mapping P : E → E is said
to be a retraction if P2 = P. It follows that if a map P is a retraction then Py = y for all y in the

range of P. The nonself asymptotically nonexpansive mapping is defined as follows:

Definition 1.1. ([2]) Let E be a real normed linear space, K be a nonempty subset of E and P : E → K
be the nonexpansive retraction of E onto K. Let T : K → E be a non-self mapping. T is said to
be a non-self asymptotically nonexpansive mapping if there exists a sequence {kn} ⊂ [1, ∞) with
kn → 1 as n → ∞ such that the following inequality holds:

‖T(PT)n−1x − T(PT)n−1y‖ ≤ kn‖x − y‖, for all x, y ∈ K and n ≥ 1. (1.3)

T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that

‖T(PT)n−1x − T(PT)n−1y‖ ≤ L‖x − y‖, for all x, y ∈ K and n ≥ 1. (1.4)

If T is a self map, then P becomes the identity map so that (1.3) and (1.4) coinside with (1.1) and

(1.2) respectively.



CUBO
14, 3 (2012)

Weak and strong convergence theorems of a multistep ... 145

Chidume et al.[2] introduced and studied the weak and strong convergences of the following

iterative process

{
x1 ∈ K

xn+1 = P((1 − αn)xn + αnT(PT)
n−1xn)

(1.5)

where {αn} is a appropriate real sequence in [0, 1]. If T is a self map, then P becomes the identity

map so that (1.5) reduces to the Mann-type iteration scheme[11]. Then Wang[19] used a similar

scheme to prove the weak and strong convergence theorems for a pair of non-self asymptotically

nonexpansive mappings which is given by






x1 ∈ K

xn+1 = P((1 − αn)xn + αnT1(PT1)
n−1yn)

yn = P((1 − βn)xn + βnT2(PT2)
n−1xn), n ≥ 1.

(1.6)

If T is a self map, then P becomes the identity map so that (1.6) reduces to the Ishikawa-like

iteration scheme without errors [9] involving two asymptotically nonexpansive self mappings. After

that Chidume and Bashir ali [3] introduced a new iteration process for approximating common

fixed points for finite families of nonself asymptotically nonexpansive mappings which is defined

as follows:






x1 ∈ K

xn+1 = P[(1 − α1n)xn + α1nT1(PT1)
n−1yn+r−2]

yn+r−2 = P[(1 − α2n)xn + α2nT2(PT2)
n−1yn+r−3]

·

·

·

·

yn = P[(1 − αmn)xn + αmnTm(PTm)
n−1xn]

(1.7)

Very recently Yang [20] introduced and studied a modified multistep iteration for a finite family

of nonself asymptotically nonexpensive mappings and discuss their convergences which is defined

as follows.



146 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

For a given x1 ∈ K and n ≥ 1, compute the iterative sequences {xn}, {yn}, ....., {yn+r−2} defined by






yn = P[(1 − anr)xn + anrTr(PTr)
n−1xn]

yn+1 = P[(1 − an(r−1) − bn(r−1))xn + an(r−1)Tr−1(PTr−1)
n−1yn

+bn(r−1)Tr−1(PTr−1)
n−1xn]

yn+2 = P[(1 − an(r−2) − bn(r−2))xn + an(r−2)Tr−2(PTr−2)
n−1yn+1

+bn(r−2)Tr−2(PTr−2)
n−1yn]

.

.

.

.

yn+r−2 = P[(1 − an2 − bn2)xn + an2T2(PT2)
n−1yn+r−3

+bn2T2(PT2)
n−1yn+r−4]

xn+1 = P[(1 − an1 − bn1)xn + an1T1(PT1)
n−1yn+r−2

+bn1T1(PT1)
n−1yn+r−3]

(1.8)

where {ani}, {bni}, {1 − ani − bni} are appropriate real sequences in [0, 1] for i ∈ I where I =

{1, 2, ....., r}. Motivated by these facts we have introduced and studied a new type of multistep

iterative process with errors which is defined as follows:

Let E be a normed space, K be a nonempty convex subset of E which is also a nonexpansive retract

of E. Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings
with sequences {kin} ⊂ [1, ∞) with limn→∞ k

i
n = 1 for i ∈ I. Then for a given x1 ∈ K and n ≥ 1,

compute the iterative sequences {xn}, {yn}, ....., {yn+r−2} defined by





yn = P[(1 − a
1
nr − bnr)xn + a

1
nrTr(PTr)

n−1xn + bnrunr]

yn+1 = P[(1 − a
1
n(r−1)

− a2
n(r−1)

− bn(r−1))xn + a
1
n(r−1)

Tr−1(PTr−1)
n−1yn

+a2
n(r−1)

Tr−1(PTr−1)
n−1xn + bn(r−1)un(r−1)]

yn+2 = P[(1 − a
1
n(r−2)

− a2
n(r−2)

− a3
n(r−2)

− bn(r−2))xn + a
1
n(r−2)

Tr−2(PTr−2)
n−1yn+1

+a2
n(r−2)

Tr−2(PTr−2)
n−1yn + a

3
n(r−2)

Tr−2(PTr−2)
n−1xn + bn(r−2)un(r−2)]

.

.

.

.

yn+r−2 = P[(1 − a
1
n2 − a

2
n2 − ..... − a

r−1
n2

− bn2)xn + a
1
n2T2(PT2)

n−1yn+r−3

+a2n2T2(PT2)
n−1yn+r−4 + ..... + a

r−1
n2

T2(PT2)
n−1xn + bn2un2]

xn+1 = P[(1 − a
1
n1 − a

2
n1 − ..... − a

r
n1 − bn1)xn + a

1
n1T1(PT1)

n−1yn+r−2

+a2n1T1(PT1)
n−1yn+r−3 + ..... + a

r
n1T1(PT1)

n−1xn + bn1un1]

(1.9)

where {aknj}, {bnj}, {1 −
∑r−j+1

k=1
aknj − bnj} are appropriate real sequences in [0, 1] for j ∈ I and

k ∈ {1, ..., r − j + 1} and {unj} are bounded sequences in K for j ∈ I. The iterative sequence



CUBO
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Weak and strong convergence theorems of a multistep ... 147

(1.9) is called the new modified multistep iteration for a finite family of nonself asymptotically

nonexpansive mappings.

The iterative sequence (1.9) can be written as in the compact form

yn+r−j = P[(1 −

r−j+1∑

k=1

aknj − bnj)xn +

r−j∑

k=1

aknjTj(PTj)
n−1yn+r−j−k +

a
r−j+1
nj

Tj(PTj)
n−1xn + bnjunj]

where j ∈ I and xn+1 = yn+r−1.

As an illustration, for r = 3, (1.9) reduces to the new modified three-step iteration with errors:





yn = P[(1 − a
1
n3 − bn3)xn + a

1
n3T3(PT3)

n−1xn + bn3un3]

yn+1 = P[(1 − a
1
n2 − a

2
n2 − bn2)xn + a

1
n2T2(PT2)

n−1yn

+a2n2T2(PT2)
n−1xn + bn2un2]

xn+1 = P[(1 − a
1
n1 − a

2
n1 − a

3
n1 − bn1)xn + a

1
n1T1(PT1)

n−1yn+1

+a2n1T1(PT1)
n−1yn + a

3
n1T1(PT1)

n−1xn + bn1un1]

(1.10)

where {aknj}, {bnj}, {1 −
∑3−j+1

k=1
aknj − bnj} are appropriate real sequences in [0, 1] for j ∈ {1, 2, 3}

and k ∈ {1, ..., 3 − j + 1} and {unj} are bounded sequences in K for j ∈ {1, 2, 3}.

For aknj = 0 for all j ∈ {1, 2, ...., r−2} and k ∈ {3, 4, ..., r−j+1} and bnj = 0 for all j ∈ I, (1.9) reduces

to the iteration (1.8). Again if aknj = 0 for all j ∈ {1, 2, ...., r − 2, r − 1} and k = {2, 3, 4, ..., r − j + 1}

and bnj = 0 for all j ∈ I, then (1.9) reduces to the iteration (1.7).

Next we recall the following definitions and results.

Let E be a real normed linear space. The modulus of convexity of E is a function δE : (0, 2] → [0, 1]
defined by

δE(ǫ) = inf{1 − ‖
1

2
(x + y)‖ : ‖x‖ = 1, ‖y‖ = 1, ǫ = ‖x − y‖} .

E is called uniformly convex if and only if δE(ǫ) > 0 for all ǫ ∈ (0, 2]. The norm of E is said to

be Frèchet differentiable if for each x ∈ E with ‖x‖ = 1 the limit limt→0
‖x+ty‖−‖x‖

t
exists and is

attained uniformly for y with ‖y‖ = 1 and in this case it has been shown that in [18] that

< h, J(x) > +
1

2
‖x‖2 ≤

1

2
‖x + h‖2 ≤< h, J(x) > +

1

2
‖x‖2 + b(‖h‖) (1.11)

for all x, h ∈ E where J is the Frèchet derivative of the functional ‖.‖2 at x ∈ E, < ., . > is the

pairing between E and E⋆ and b is a function defined on [0, ∞) such that limt→0
b(t)

t
= 0. A

Banach space E is said to satisfy Opial’s condition [12] if xn ⇀ x and x 6= y imply

lim sup
n→∞

‖xn − x‖ < lim sup
n→∞

‖xn − y‖ .

A Banach space E is said to satisfy Kadec-Klee property, if for every sequence {xn} ∈ E, xn ⇀ x
and ‖xn‖ → ‖x‖ together imply that xn → x as n → ∞. There are uniformly convex Banach
spaces which have neither a Frèchet differentiable norm nor satisfy Opial’s property but their dual

does have the Kadec-Klee property (see [6],[10]).



148 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

Lemma 1.1. ([18], Lemma1) Let {an}, {bn} and {δn} be sequences of nonnegative real numbers

satisfying the inequality

an+1 ≤ (1 + δn)an + bn, ∀n ≥ 1.

If
∑

∞

n=1
δn < ∞ and

∑
∞

n=1
bn < ∞, then

(i) limn→∞ an exists,

(ii) limn→∞ an = 0 whenever lim infn→∞ an = 0.

Lemma 1.2. ([21], Theorem2) Let p > 1 and r > 0 be two fixed real numbers. Then a Banach space

E is uniformly convex if and only if there exists a continuous strictly increasing convex function

g : [0, ∞) → [0, ∞) with g(0) = 0 such that

‖λx + (1 − λ)y‖p ≤ λ‖x‖p + (1 − λ)‖y‖p − ωp(λ)g(‖x − y‖)

for all x, y ∈ Br(0) = {x ∈ E : ‖x‖ ≤ r} and λ ∈ [0, 1] where ωp(λ) = λ
p(1 − λ) + λ(1 − λ)p.

Lemma 1.3. ([5], Lemma1.4) Let E be a uniformly convex Banach space and Br = {x ∈ E : ‖x‖ ≤ r},

r > 0. Then there exist a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞),
g(0) = 0 such that

‖λx + βy + γz‖2 ≤ λ‖x‖2 + β‖y‖2 + γ‖z‖2 − λβg(‖x − y‖)

for all x, y, z ∈ Br and all λ, β, γ ∈ [0, 1] with λ + β + γ = 1.

By using Lemma 1.2 and Lemma 1.3 we can easily prove the following Lemma:

Lemma 1.4. Let E be a uniformly convex Banach space and Br = {x ∈ E : ‖x‖ ≤ r}, r > 0. Then

there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞), g(0) = 0
such that

‖λ1x1 + λ2x2 + ... + λnxn‖
2 ≤ λ1‖x1‖

2
+ λ2‖x2‖

2
+ ... + λn‖xn‖

2
− λ1λ2g(‖x1 − x2‖)

for all xi ∈ Br and all λi ∈ [0, 1] for all i = 1, 2, ..., n with
∑n

i=1
λi = 1.

Proof: The Lemma is true for n = 2 since for n = 2 using Lemma 1.2 we get

‖λ1x1 + λ2x2‖
2 ≤ λ1‖x1‖

2 + λ2‖x2‖
2 − ω2(λ1)g(‖x1 − x2‖)

where ω2(λ1) = λ
2
1(1 − λ1) + λ1(1 − λ1)

2 = λ21λ2 + λ1λ
2
2 = λ1λ2(λ1 + λ2) = λ1λ2. Also by Lemma

1.3 we see that this Lemma is true for n = 3. Now let the Lemma is true for n = m. Now

‖λ1x1 + λ2x2 + ... + λmxm + λm+1xm+1‖
2

= ‖λ1x1 + λ2x2 + ... + λm−1xm−1 +

(1 − λ1 − λ2 − .... − λm−1)(
λm

1 − λ1 − λ2 − .... − λm−1
xm +

λm+1

1 − λ1 − λ2 − .... − λm−1
xm+1)‖

2.



CUBO
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Weak and strong convergence theorems of a multistep ... 149

By using the above inequality,

‖λ1x1 + λ2x2 + ... + λmxm + λm+1xm+1‖
2

= ‖λ1x1 + λ2x2 + ... + λm−1xm−1 +

(1 − λ1 − λ2 − .... − λm−1)(
λm

1 − λ1 − λ2 − .... − λm−1
xm +

λm+1

1 − λ1 − λ2 − .... − λm−1
xm+1)‖

2

≤ λ1‖x1‖
2 + λ2‖x2‖

2 + ... + λm−1‖xm−1‖
2 +

(1 − λ1 − λ2 − .... − λm−1)‖
λm

1 − λ1 − λ2 − .... − λm−1
xm +

λm+1

1 − λ1 − λ2 − .... − λm−1
xm+1‖

2

−λ1λ2g(‖x1 − x2‖) .

Since λ1 + λ2 + ... + λm + λm+1 = 1, so

λm

1 − λ1 − λ2 − .... − λm−1
+

λm+1

1 − λ1 − λ2 − .... − λm−1
=

λm + λm+1

1 − λ1 − λ2 − .... − λm−1
= 1 .

Then from Lemma 1.2 we get that

‖
λm

1 − λ1 − λ2 − .... − λm−1
xm +

λm+1

1 − λ1 − λ2 − .... − λm−1
xm+1‖

2

≤
λm

1 − λ1 − λ2 − .... − λm−1
‖xm‖

2
+

λm+1

1 − λ1 − λ2 − .... − λm−1
‖xm+1‖

2

−ω2(
λm

1 − λ1 − λ2 − .... − λm−1
)g(‖x1 − x2‖) .

Now,

ω2(
λm

1 − λ1 − λ2 − .... − λm−1
)

= (
λm

1 − λ1 − λ2 − .... − λm−1
)2(1 −

λm

1 − λ1 − λ2 − .... − λm−1
)

+
λm

1 − λ1 − λ2 − .... − λm−1
(1 −

λm

1 − λ1 − λ2 − .... − λm−1
)2

=
λm

1 − λ1 − λ2 − .... − λm−1
.

λm+1

1 − λ1 − λ2 − .... − λm−1

(
λm

1 − λ1 − λ2 − .... − λm−1
+

λm+1

1 − λ1 − λ2 − .... − λm−1
)

=
λmλm+1

(1 − λ1 − λ2 − .... − λm−1)2
≥ 0 .

Therefore from above we have

‖
λm

1 − λ1 − λ2 − .... − λm−1
xm +

λm+1

1 − λ1 − λ2 − .... − λm−1
xm+1‖

2

≤
λm

1 − λ1 − λ2 − .... − λm−1
‖xm‖

2 +
λm+1

1 − λ1 − λ2 − .... − λm−1
‖xm+1‖

2 .



150 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

So finally we get that

‖λ1x1 + λ2x2 + ... + λmxm + λm+1xm+1‖
2

= ‖λ1x1 + λ2x2 + ... + λm−1xm−1 +

(1 − λ1 − λ2 − .... − λm−1)(
λm

1 − λ1 − λ2 − .... − λm−1
xm +

λm+1

1 − λ1 − λ2 − .... − λm−1
xm+1)‖

2

≤ λ1‖x1‖
2 + λ2‖x2‖

2 + ... + λm−1‖xm−1‖
2 +

(1 − λ1 − λ2 − .... − λm−1)(
λm

1 − λ1 − λ2 − .... − λm−1
‖xm‖

2
+

λm+1

1 − λ1 − λ2 − .... − λm−1
‖xm+1‖

2
) − λ1λ2g(‖x1 − x2‖)

= λ1‖x1‖
2 + λ2‖x2‖

2 + ... + λm−1‖xm−1‖
2 + λm‖xm‖

2 + λm+1‖xm+1‖
2 − λ1λ2g(‖x1 − x2‖) .

Hence the Lemma is true for n = m + 1. Thus, by induction, the Lemma is true for all n ≥ 2.

This completes the proof of the Lemma.

Lemma 1.5. ([2], Theorem3.4) Let E be a real uniformly Banach space and K be a nonempty

closed convex subset of E and T : K → E be asymptotically nonexpansive mapping with a sequence
{kn} ⊂ [1, ∞) with limn→∞ kn = 1. Then I − T is demiclosed at zero, i.e. if {xn} is a sequence
in K which converges weakly to x and if the sequence {xn − Txn} converges strongly to zero, then

x − Tx = 0.

Lemma 1.6. ([10], Theorem2) Let E be a real reflexive Banach space such that E⋆ has the Kadec-

Klee property. Let {xn} be a bounded sequence in E and x
⋆, y⋆ ∈ ww(xn)(weak w-limit set of

{xn}). Suppose limn→∞ ‖txn + (1 − t)x
⋆ − y⋆‖ exists for all t ∈ [0, 1]. Then x⋆ = y⋆.

Lemma 1.7. ([1]) Let E be a uniformly convex Banach space K be a nonempty bounded closed

convex subset of E. Then there exists a strictly increasing continuous convex function φ : [0, ∞) →
[0, ∞) with φ(0) = 0 such that for any Lipschitzian mapping T : K → E with the Lipschitz constant
L ≥ 1 and for any x, y ∈ K and t ∈ [0, 1] the following inequality holds:

‖T(tx + (1 − t)y) − (tTx + (1 − t)Ty)‖ ≤ Lφ−1(‖x − y‖ − L−1‖Tx − Ty‖)

The purpose of this paper is to introduce a new modified multi step iteration with errors

for approximating common fixed points for finite families of nonself asymptotically nonexpansive

mappings. We prove some strong and weak convergence theorems in real uniformly convex Banach

spaces. More precisely we prove convergence theorems in a uniformly convex Banach space which

satisfy Opial’s condition or have Frèchet differentiable norm or whose duals have the Kadec-Klee

property. Our results generalize some recent results.

2 Main Results

We begin this section with the following lemmas.



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Weak and strong convergence theorems of a multistep ... 151

Lemma 2.1. Let E be a real normed space and K be a nonempty subset of E which is also a

nonexpansive retract of E. Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself asymptotically
nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with

∑
∞

n=1
(kin − 1) < ∞ for i ∈ I. Let {xn}

be defined by (1.9) with
∑

∞

n=1
bni < ∞ for i ∈ I. If F =

⋂r
i=1

F(Ti) 6= ∅, then limn→∞ ‖xn − q‖

exists for all q ∈ F.

Proof: Let q ∈ F. For each n ≥ 1, let kn = max {k
1
n, k

2
n, .........., k

r
n} so that {kn} ⊂

[1, ∞) with
∑

∞

n=1
(kn − 1) < ∞. Since {uni} are bounded sequences in K for i ∈ I, let M =

sup n≥1,i=1,2,...,r‖uni − q‖. From (1.9) we get

‖yn − q‖ = ‖P((1 − a
1
nr − bnr)xn + a

1
nrTr(PTr)

n−1xn + bnrunr) − Pq‖

≤ ‖(1 − a1nr − bnr)(xn − q) + a
1
nr(Tr(PTr)

n−1xn − q) + bnr(unr − q)‖

≤ (1 − a1nr − bnr)‖xn − q‖ + a
1
nr‖Tr(PTr)

n−1xn − q‖ + bnr‖unr − q‖

≤ (1 − a1nr)‖xn − q‖ + a
1
nrkn‖xn − q‖ + bnrM

≤ kn‖xn − q‖ + bnrM

= kn‖xn − q‖ + σ
1
n (2.1)

where σ1n = bnrM. By the given condition we get that
∑

∞

n=1
σ1n < ∞. Also from (1.9) and (2.1)

we have

‖yn+1 − q‖ = ‖P((1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))xn + a

1
n(r−1)Tr−1(PTr−1)

n−1yn +

a2n(r−1)Tr−1(PTr−1)
n−1xn + bn(r−1)un(r−1)) − Pq‖

≤ ‖(1 − a1n(r−1) − a
2
n(r−1) − bn(r−1))(xn − q) + a

1
n(r−1)(Tr−1(PTr−1)

n−1yn − q)

+a2n(r−1)(Tr−1(PTr−1)
n−1xn − q) + bn(r−1)(un(r−1) − q)‖

≤ (1 − a1n(r−1) − a
2
n(r−1) − bn(r−1))‖xn − q‖ + a

1
n(r−1)‖Tr−1(PTr−1)

n−1yn − q‖

+a2n(r−1)‖Tr−1(PTr−1)
n−1xn − q‖ + bn(r−1)‖un(r−1) − q‖

≤ (1 − a1n(r−1) − a
2
n(r−1))‖xn − q‖ + a

1
n(r−1)kn‖yn − q‖

+a2n(r−1)kn‖xn − q‖ + bn(r−1)M

≤ (1 − a1n(r−1) − a
2
n(r−1))‖xn − q‖ + a

1
n(r−1)kn[kn‖xn − q‖ + bnrM]

+a2n(r−1)kn‖xn − q‖ + bn(r−1)M

≤ [1 + a1n(r−1)(k
2
n − 1) + a

2
n(r−1)(kn − 1)]‖xn − q‖ + a

1
n(r−1)knbnrM +

bn(r−1)M

≤ k2n‖xn − q‖ + knbnrM + bn(r−1)M

= k2n‖xn − q‖ + σ
2
n (2.2)

where σ2n = knbnrM + bn(r−1)M. By the given condition we get that
∑

∞

n=1
σ2n < ∞. Also from



152 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

(1.9) and (2.2) we have

‖yn+2 − q‖ =

‖P((1 − a1n(r−2) − a
2
n(r−2) − a

3
n(r−2) − bn(r−2))xn + a

1
n(r−2)Tr−2(PTr−2)

n−1yn+1

+a2n(r−2)Tr−2(PTr−2)
n−1yn + a

3
n(r−2)Tr−2(PTr−2)

n−1xn + bn(r−2)un(r−2)) − Pq‖

≤ ‖(1 − a1n(r−2) − a
2
n(r−2) − a

3
n(r−2) − bn(r−2))(xn − q) + a

1
n(r−2)(Tr−2(PTr−2)

n−1yn+1 − q)

+a2n(r−2)(Tr−2(PTr−2)
n−1yn − q) + a

3
n(r−2)(Tr−2(PTr−2)

n−1xn − q)

+bn(r−2)(un(r−2) − q)‖

≤ (1 − a1n(r−2) − a
2
n(r−2) − a

3
n(r−2) − bn(r−2))‖xn − q‖ + a

1
n(r−2)‖Tr−2(PTr−2)

n−1yn+1 − q‖

+a2n(r−2)‖Tr−2(PTr−2)
n−1yn − q‖ + a

3
n(r−2)‖Tr−2(PTr−2)

n−1xn − q‖

+bn(r−2)‖un(r−2) − q‖

≤ (1 − a1n(r−2) − a
2
n(r−2) − a

3
n(r−2))‖xn − q‖ + a

1
n(r−2)kn‖yn+1 − q‖

+a2n(r−2)kn‖yn − q‖ + a
3
n(r−2)kn‖xn − q‖ + bn(r−2)M

≤ (1 − a1n(r−2) − a
2
n(r−2) − a

3
n(r−2))‖xn − q‖ + a

1
n(r−2)kn[k

2
n‖xn − q‖

+knbnrM + bn(r−1)M] + a
2
n(r−2)kn[kn‖xn − q‖ + bnrM]

+a3n(r−2)kn‖xn − q‖ + bn(r−2)M

≤ k3n‖xn − q‖ + σ
3
n (2.3)

where σ3n = k
2
nbnrM + knbnrM + knbn(r−1)M + bn(r−2)M. By the given condition we get that∑

∞

n=1
σ3n < ∞. In general after (j + 1) steps we get

‖yn+j − q‖ ≤ k
j+1
n ‖xn − q‖ + σ

j+1
n (2.4)

for j = 0, 1, ..., r − 2 and {σ
j+1
n } is a nonnegative real sequence such that

∑
∞

n=1
σ
j+1
n < ∞ for

j = 0, 1, ..., r − 2. Therefore it follows from (1.9) and (2.4) that

‖xn+1 − q‖ ≤ k
r
n‖xn − q‖ + σ

r
n = [1 + (k

r
n − 1)]‖xn − q‖ + σ

r
n (2.5)

where {σrn} is a nonnegative real sequence such that
∑

∞

n=1
σrn < ∞. Since 0 ≤ t

r −1 ≤ rtr−1(t−1)

for all t ≥ 1, so 0 ≤ krn − 1 ≤ rk
r−1
n (kn − 1). Since

∑
∞

n=1
(kn − 1) < ∞ so {kn} is bounded,

kn ∈ [1, M
′] for some M′ > 0. So

∑
∞

n=1
(krn−1) < ∞. Thus by Lemma 1.1 we get limn→∞ ‖xn−q‖

exists for all q ∈ F. ♦

Lemma 2.2. Let E be a uniformly convex Banach space and K be a nonempty closed convex subset

of E which is also a nonexpansive retract of E. Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself
asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with

∑
∞

n=1
(kin − 1) < ∞ for

i ∈ I. Let {xn} be defined by (1.9) with
∑

∞

n=1
bni < ∞ for i ∈ I. If F =

⋂r
i=1

F(Ti) 6= ∅, then the

following results hold

(1) If lim infn→∞ a
1
nk > 0, for all k < r and 0 < lim infn→∞ a

1
nr ≤ lim supn→∞(a

1
nr + bnr) < 1

then limn→∞ ‖Tr(PTr)
n−1xn − xn‖ = 0.



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Weak and strong convergence theorems of a multistep ... 153

(2) If 0 < lim infn→∞ a
1
n1 ≤ lim supn→∞(

∑r
k=1

akn1+bn1) < 1 then limn→∞ ‖T1(PT1)
n−1yn+r−2−

xn‖ = 0.

(3) If lim infn→∞ a
1
nk > 0, for all k < j and 0 < lim infn→∞ a

1
nj ≤ lim supn→∞(

∑r−j+1
m=1

amnj +

bnj) < 1, then limn→∞ ‖Tj(PTj)
n−1yn+r−j−1 − xn‖ = 0 for j = 2, 3, ....., r − 1.

Proof: Let q ∈ F. By Lemma 2.1 we have that limn→∞ ‖xn − q‖ exists for all q ∈ F. So

{xn −q} is bounded in K. Since {kn} and {σ
j+1
n } are bounded so it follows from (2.4) that {yn+j −q}

are bounded for j = 0, 1, ..., r − 2. Since Tj is a nonself asymptotically nonexpansive mapping, we

have

‖Tj(PTj)
n−1yn+r−j−1 − q‖ ≤ k

j
n‖yn+r−j−1 − q‖

for j = 1, ..., r−1. Therefore the sequences {Tj(PTj)
n−1yn+r−j−1 −q} are bounded for j = 1, ..., r−1.

Therefore there exists D > 0 such that K ⊆ BD. From (1.9) and Lemma 1.3 we get

‖yn − q‖
2 = ‖P((1 − a1nr − bnr)xn + a

1
nrTr(PTr)

n−1xn + bnrunr) − Pq‖
2

≤ ‖(1 − a1nr − bnr)(xn − q) + a
1
nr(Tr(PTr)

n−1xn − q) + bnr(unr − q)‖
2

≤ (1 − a1nr − bnr)‖xn − q‖
2 + a1nr‖Tr(PTr)

n−1xn − q‖
2 + bnr‖unr − q‖

2 −

(1 − a1nr − bnr)a
1
nrg1(‖Tr(PTr)

n−1xn − xn‖)

≤ (1 − a1nr)‖xn − q‖
2 + a1nrk

2
n‖xn − q‖

2 + bnrM
2 −

(1 − a1nr − bnr)a
1
nrg1(‖Tr(PTr)

n−1xn − xn‖)

≤ k2n‖xn − q‖
2
+ µ1n − a

1
nr(1 − a

1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖) (2.6)



154 Shrabani Banerjee and Binayak S.Choudhury CUBO
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where µ1n = bnrM
2 so that

∑
∞

n=1
µ1n < ∞. From (1.9) and (2.6) and from Lemma 1.4 we get

‖yn+1 − q‖
2 = ‖P[(1 − a1n(r−1) − a

2
n(r−1) − bn(r−1))xn + a

1
n(r−1)Tr−1(PTr−1)

n−1yn +

a2n(r−1)Tr−1(PTr−1)
n−1xn + bn(r−1)un(r−1)] − Pq‖

2

≤ ‖(1 − a1n(r−1) − a
2
n(r−1) − bn(r−1))(xn − q) + a

1
n(r−1)(Tr−1(PTr−1)

n−1yn − q)

+a2n(r−1)(Tr−1(PTr−1)
n−1xn − q) + bn(r−1)(un(r−1) − q)‖

2

≤ (1 − a1n(r−1) − a
2
n(r−1) − bn(r−1))‖xn − q‖

2 + a1n(r−1)‖Tr−1(PTr−1)
n−1yn − q‖

2

+a2n(r−1)‖Tr−1(PTr−1)
n−1xn − q‖

2 + bn(r−1)‖un(r−1) − q‖
2

−a1n(r−1)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖)

≤ (1 − a1n(r−1) − a
2
n(r−1) − bn(r−1))‖xn − q‖

2 + a1n(r−1)k
2
n‖yn − q‖

2

+a2n(r−1)k
2
n‖xn − q‖

2 + bn(r−1)M
2

−a1n(r−1)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖)

≤ (1 − a1n(r−1) − a
2
n(r−1) − bn(r−1))‖xn − q‖

2
+ a1n(r−1)k

2
n[k

2
n‖xn − q‖

2
+ µ1n

−a1nr(1 − a
1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖)] + a
2
n(r−1)k

2
n‖xn − q‖

2

+bn(r−1)M
2

−a1n(r−1)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖)

≤ k4n‖xn − q‖
2 + k2nµ

1
n + bn(r−1)M

2

−a1n(r−1)a
1
nr(1 − a

1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖)

−a1n(r−1)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖)

≤ k4n‖xn − q‖
2
+ µ2n

−a1n(r−1)a
1
nr(1 − a

1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖)

−a1n(r−1)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖) (2.7)

where µ2n = k
2
nµ

1
n + bn(r−1)M

2, so that
∑

∞

n=1
µ2n < ∞. Proceeding in this way we have

‖yn+j − q‖
2 ≤ k2(j+1)n ‖xn − q‖

2 + µ(j+1)n −

(

r∏

i=r−j

a1ni)(1 − a
1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖) −

(

r−1∏

i=r−j

a1ni)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖) −

.... − a1n(r−j)(1 −

j+1∑

k=1

akn(r−j) − bn(r−j))gj+1(‖Tr−j(PTr−j)
n−1yn+j−1 − xn‖)

(2.8)

for j = 1, 2, ..., r − 2 and {µ
(j+1)
n } is a nonnegative real sequence such that

∑
∞

n=1
µ
(j+1)
n < ∞. Thus



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Weak and strong convergence theorems of a multistep ... 155

we get

‖xn+1 − q‖
2 ≤ k2rn ‖xn − q‖

2 + µrn −

(

r∏

i=1

a1ni)(1 − a
1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖) −

(

r−1∏

i=1

a1ni)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(||Tr−1(PTr−1)

n−1yn − xn||)

−.... − a1n1a
1
n2(1 −

r−1∑

k=1

akn2 − bn2)gr−1(‖T2(PT2)
n−1yn+r−3 − xn‖)

−a1n1(1 −

r∑

k=1

akn1 − bn1)gr(‖T1(PT1)
n−1yn+r−2 − xn‖) (2.9)

where {µrn} is a nonnegative real sequence such that
∑

∞

n=1
µrn < ∞. Since {kn} is bounded so

there exists M1 > 0 such that kn ∈ [1, M1] for all n ≥ 1. Hence k
2r
n − 1 ≤ 2rk

2r−1
n (kn − 1) ≤

2rM2r−1
1

(kn − 1) holds for all n ≥ 1. So
∑

∞

n=1
(kn − 1) < ∞ implies that

∑
∞

n=1
(k2rn − 1) < ∞.

Therefore from (2.9) we get

‖xn+1 − q‖
2 ≤ ‖xn − q‖

2
+ (k2rn − 1)‖xn − q‖

2
+ µrn −

(

r∏

i=1

a1ni)(1 − a
1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖) −

(

r−1∏

i=1

a1ni)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖)

−.... − a1n1a
1
n2(1 −

r−1∑

k=1

akn2 − bn2)gr−1(‖T2(PT2)
n−1yn+r−3 − xn‖)

−a1n1(1 −

r∑

k=1

akn1 − bn1)gr(‖T1(PT1)
n−1yn+r−2 − xn‖)

≤ ‖xn − q‖
2 + 2rM2r−11 (kn − 1)D

2 + µrn −

(

r∏

i=1

a1ni)(1 − a
1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖) −

(

r−1∏

i=1

a1ni)(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))g2(‖Tr−1(PTr−1)

n−1yn − xn‖)

−.... − a1n1a
1
n2(1 −

r−1∑

k=1

akn2 − bn2)gr−1(‖T2(PT2)
n−1yn+r−3 − xn‖)

−a1n1(1 −

r∑

k=1

akn1 − bn1)gr(‖T1(PT1)
n−1yn+r−2 − xn‖) . (2.10)



156 Shrabani Banerjee and Binayak S.Choudhury CUBO
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From (2.10) we get

(

r∏

i=1

a1ni)(1 − a
1
nr − bnr)g1(‖Tr(PTr)

n−1xn − xn‖)

≤ ‖xn − q‖
2 − ‖xn+1 − q‖

2 + 2rM2r−1
1

(kn − 1)D
2 + µrn (2.11)

and

(

j∏

i=1

a1ni)(1 −

r−j+1∑

k=1

aknj − bnj)gr−j+1(‖Tj(PTj)
n−1yn+r−j−1 − xn‖)

≤ ‖xn − q‖
2 − ‖xn+1 − q‖

2 + 2rM2r−11 (kn − 1)D
2 + µrn (2.12)

for j = 1, 2, ..., r−1. If lim infn→∞ a
1
ni > 0, for all i < r and 0 < lim infn→∞ a

1
nr ≤ lim supn→∞(a

1
nr+

bnr), then there exists a positive integer n0 and η, η
′ ∈ (0, 1) such that 0 < η < a1ni(i ∈

I), a1nr + bnr < η
′ < 1, for all n ≥ n0. Thus from (2.11) we get

ηr(1 − η′)g1(‖Tr(PTr)
n−1xn − xn‖) ≤ ‖xn − q‖

2 − ‖xn+1 − q‖
2 + 2rM2r−11 (kn − 1)D

2

+µrn, for all n ≥ n0 .

This implies that

∞∑

n=n0

g1(‖Tr(PTr)
n−1xn − xn‖) ≤

1

ηr(1 − η′)
(‖xn0 − q‖

2 + 2rM2r−1
1

D2
∞∑

n=n0

(kn − 1)

+

∞∑

n=n0

µrn) < ∞

which further implies that limn→∞ g1(‖Tr(PTr)
n−1xn − xn‖) = 0. Since g1 is strictly increasing

and continuous with g1(0) = 0, so limn→∞ ‖Tr(PTr)
n−1xn − xn‖ = 0. Similarly from (2.12)

using the fact that gr−j+1 is strictly increasing and continuous with gr−j+1(0) = 0 we get

limn→∞ ‖Tj(PTj)
n−1yn+r−j−1 − xn‖ = 0 for j = 1, 2, ....., r − 1. ♦

Lemma 2.3. Let E be a uniformly convex Banach space and K be a nonempty closed convex subset

of E which is also a nonexpansive retract of E. Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself
asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with

∑
∞

n=1
(kin − 1) < ∞ for

i ∈ I. Let {xn} be defined by (1.9) with
∑

∞

n=1
bni < ∞ for i ∈ I. If F =

⋂r
i=1

F(Ti) 6= ∅ and

(1) 0 < lim infn→∞ a
1
nr ≤ lim supn→∞(a

1
nr + bnr) < 1

(2) 0 < lim infn→∞ a
1
nj ≤ lim supn→∞(

∑r−j+1
m=1

amnj + bnj) < 1,

then limn→∞ ‖xn − Tixn‖ = 0 for i ∈ I.

Proof: By Lemma 2.2 we get

lim
n→∞

‖Tr(PTr)
n−1xn − xn‖ = 0 and

lim
n→∞

‖Tj(PTj)
n−1yn+r−j−1 − xn‖ = 0 for j = 1, 2, ....., r − 1. (2.13)



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Weak and strong convergence theorems of a multistep ... 157

Since P is nonexpansive mapping so from (1.9) together with (2.13) we have

‖yn − xn‖ ≤ a
1
nr‖Tr(PTr)

n−1xn − xn‖ + bnr‖unr − xn‖

≤ ‖Tr(PTr)
n−1xn − xn‖ + bnr‖unr − xn‖ → 0 as n → ∞. (2.14)

Since Tr−1 is nonself asymptotically nonexpansive mapping, from (2.13) and (2.14) we get

‖Tr−1(PTr−1)
n−1xn − xn‖ ≤ ‖Tr−1(PTr−1)

n−1xn − Tr−1(PTr−1)
n−1yn‖ +

‖Tr−1(PTr−1)
n−1yn − xn‖

≤ kn‖yn − xn‖ + ‖Tr−1(PTr−1)
n−1yn − xn‖

→ 0 as n → ∞. (2.15)

Again from (1.9), (2.13) and (2.15) it follows that

‖yn+1 − xn‖ ≤ a
1
n(r−1)‖Tr−1(PTr−1)

n−1yn − xn‖ + a
2
n(r−1)‖Tr−1(PTr−1)

n−1xn − xn‖

+bn(r−1)‖un(r−1) − xn‖ → 0 as n → ∞. (2.16)

From (2.16) and (2.13) we have that

‖Tr−2(PTr−2)
n−1xn − xn‖ ≤ ‖Tr−2(PTr−2)

n−1xn − Tr−2(PTr−2)
n−1yn+1‖ +

‖Tr−2(PTr−2)
n−1yn+1 − xn‖

≤ kn‖yn+1 − xn‖ + ‖Tr−2(PTr−2)
n−1yn+1 − xn‖

→ 0 as n → ∞. (2.17)

Also from (2.17) and (2.14) we have that

‖Tr−2(PTr−2)
n−1yn − xn‖ ≤ ‖Tr−2(PTr−2)

n−1yn − Tr−2(PTr−2)
n−1xn‖ +

‖Tr−2(PTr−2)
n−1xn − xn‖

≤ kn‖yn − xn‖ + ‖Tr−2(PTr−2)
n−1xn − xn‖

→ 0 as n → ∞ . (2.18)

Continuing in this way we have that

lim
n→∞

‖Tj(PTj)
n−1yn+r−j−2 − xn‖ = 0 for j = 1, 2, ....., r − 2. (2.19)

Again from (1.9), (2.13), (2.17) and (2.18) it follows that

‖yn+2 − xn‖ ≤ a
1
n(r−2)‖Tr−2(PTr−2)

n−1yn+1 − xn‖ + a
2
n(r−2)‖Tr−2(PTr−2)

n−1yn − xn‖

+a3n(r−2)‖Tr−2(PTr−2)
n−1xn − xn‖ + bn(r−2)‖un(r−2) − xn‖

→ 0 as n → ∞. (2.20)



158 Shrabani Banerjee and Binayak S.Choudhury CUBO
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From (2.20) and (2.13) we have that

‖Tr−3(PTr−3)
n−1xn − xn‖ ≤ ‖Tr−3(PTr−3)

n−1xn − Tr−3(PTr−3)
n−1yn+2‖ +

‖Tr−3(PTr−3)
n−1yn+2 − xn‖

≤ kn‖yn+2 − xn‖ + ‖Tr−3(PTr−3)
n−1yn+2 − xn‖

→ 0 as n → ∞. (2.21)

Thus from (2.21) and (2.14) it follows that

‖Tr−3(PTr−3)
n−1yn − xn‖ ≤ ‖Tr−3(PTr−3)

n−1yn − Tr−3(PTr−3)
n−1xn‖ +

‖Tr−3(PTr−3)
n−1xn − xn‖

≤ kn‖yn − xn‖ + ‖Tr−3(PTr−3)
n−1xn − xn‖

→ 0 as n → ∞ .

Continuing in this way we have that

lim
n→∞

‖Tj(PTj)
n−1yn+r−j−3 − xn‖ = 0 for j = 1, 2, ....., r − 3.

Continuing in this way after a finite steps we have that

lim
n→∞

‖Ti(PTi)
n−1xn − xn‖ = 0, for i ∈ I, and

lim
n→∞

‖Tj(PTj)
n−1yn+r−j−k − xn‖ = 0 for j = 1, 2, ....., r − k. (2.22)

From (1.9), (2.22) we have that

‖xn+1 − xn‖ = ‖P[(1 −
r∑

k=1

akn1 − bn1)xn +

r−1∑

k=1

akn1T1(PT1)
n−1yn+r−1−k +

arn1T1(PT1)
n−1xn + bn1un1] − xn‖

≤
r−1∑

k=1

akn1‖T1(PT1)
n−1yn+r−1−k − xn‖ + a

r
n1‖T1(PT1)

n−1xn − xn‖

+bn1‖un1 − xn‖ → 0 as n → ∞. (2.23)

Since every nonself asymptotically nonexpansive mapping uniformly L-Lipschitzian, so from (2.22)

and (2.23) we get

‖Ti(PTi)
n−2xn − xn‖ ≤ ‖Ti(PTi)

n−2xn − Ti(PTi)
n−2xn−1‖ +

‖Ti(PTi)
n−2xn−1 − xn−1‖ + ‖xn−1 − xn‖

≤ (1 + L)‖xn − xn−1‖ + ‖Ti(PTi)
n−2xn−1 − xn−1‖

→ 0 as n → ∞. (2.24)

Now from (2.22) and (2.24) it follows that

‖xn − Tixn‖ ≤ ‖xn − Ti(PTi)
n−1xn‖ + ‖Ti(PTi)

n−1xn − Tixn‖

≤ ‖xn − Ti(PTi)
n−1xn‖ + L‖Ti(PTi)

n−2xn − xn‖

→ 0 as n → ∞ .



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Weak and strong convergence theorems of a multistep ... 159

Thus we have that limn→∞ ‖xn − Tixn‖ = 0 for i ∈ I.♦

Lemma 2.4. Let E be a uniformly convex Banach space and K be a nonempty closed convex

subset of E which is also a nonexpansive retract of E which has a Frèchet differentiable norm.

Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself asymptotically nonexpansive mappings with
sequences {kin} ⊂ [1, ∞) with

∑
∞

n=1
(kin − 1) < ∞ for i ∈ I. Let {xn} be defined by (1.9) with∑

∞

n=1
bni < ∞ for i ∈ I. If F =

⋂r
i=1

F(Ti) 6= ∅ and

(1) 0 < lim infn→∞ a
1
nr ≤ lim supn→∞(a

1
nr + bnr) < 1

(2) 0 < lim infn→∞ a
1
nj ≤ lim supn→∞(

∑r−j+1
m=1

amnj + bnj) < 1,

then for any p1, p2 ∈ F, limn→∞ < xn, J(p1 − p2) > exists. In particular

limn→∞ < p − q, J(p1 − p2) >= 0 for all p, q ∈ ww(xn).

Proof: Since E has Frèchet differentiable norm, taking x = p1 − p2 with p1 6= p2 and

h = t(xn − p1) in the inequality (1.11) we get that

t < xn − p1, J(p1 − p2) > +
1

2
‖p1 − p2‖

2 ≤
1

2
‖txn + (1 − t)p1 − p2‖

2 ≤

t < xn − p1, J(p1 − p2) > +
1

2
‖p1 − p2‖

2 + b(t‖xn − p1‖) . (2.25)

Again p1 ∈ F, so by Lemma 2.1 we have that limn→∞ ‖xn − p1‖ exists. Let sup{‖xn − p1‖ : n ∈

N} ≤ M′ for some M′ > 0. Thus from (2.25) we get

1

2
‖p1 − p2‖

2 + lim sup
n→∞

t < xn − p1, J(p1 − p2) >≤
1

2
lim

n→∞
‖txn + (1 − t)p1 − p2‖

2

≤
1

2
‖p1 − p2‖

2 + b(tM′) + lim inf
n→∞

t < xn − p1, J(p1 − p2) >

⇒ lim sup
n→∞

< xn − p1, J(p1 − p2) >≤ lim inf
n→∞

< xn − p1, J(p1 − p2) > +
b(tM′)

tM′
M′

⇒ lim
n→∞

< xn − p1, J(p1 − p2) > exists as t → 0.

In particular limn→∞ < p − q, J(p1 − p2) >= 0 for all p, q ∈ ww(xn).

Lemma 2.5. Let E be a uniformly convex Banach space and K be a nonempty closed convex subset

of E which is also a nonexpansive retract of E. Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself
asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with

∑
∞

n=1
(kin − 1) < ∞ for

i ∈ I. Let {xn} be defined by (1.9) with
∑

∞

n=1
bni < ∞ for i ∈ I. If F =

⋂r
i=1

F(Ti) 6= ∅ and

(1) 0 < lim infn→∞ a
1
nr ≤ lim supn→∞(a

1
nr + bnr) < 1

(2) 0 < lim infn→∞ a
1
nj ≤ lim supn→∞(

∑r−j+1
m=1

amnj + bnj) < 1,

then limn→∞ ‖txn + (1 − t)p1 − p2‖ exists for all p1, p2 ∈ F.



160 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

Proof: Let dn(t) = ‖txn + (1 − t)p1 − p2‖ for all t ∈ [0, 1] and p1, p2 ∈ F. Then

limn→∞ dn(0) = ‖p1 − p2‖ exists and limn→∞ dn(1) = ‖xn − p2‖ exists by Lemma 2.1. De-

fine Qn : K → E by

Qnx = P[(1 − a
1
n1 − a

2
n1 − ..... − a

r
n1 − bn1)x + a

1
n1T1(PT1)

n−1xr−2 +

a2n1T1(PT1)
n−1xr−3 + ..... + a

r
n1T1(PT1)

n−1x + bn1un1]

xr−2 = P[(1 − a
1
n2 − a

2
n2 − ..... − a

r−1
n2 − bn2)x + a

1
n2T2(PT2)

n−1xr−3 +

a2n2T2(PT2)
n−1xr−4 + ..... + a

r−1
n2 T2(PT2)

n−1x + bn2un2]

.

.

.

.

x2 = P[(1 − a
1
n(r−2) − a

2
n(r−2) − a

3
n(r−2) − bn(r−2))x + a

1
n(r−2)Tr−2(PTr−2)

n−1x1

+a2n(r−2)Tr−2(PTr−2)
n−1x0 + a

3
n(r−2)Tr−2(PTr−2)

n−1x + bn(r−2)un(r−2)]

x1 = P[(1 − a
1
n(r−1) − a

2
n(r−1) − bn(r−1))x + a

1
n(r−1)Tr−1(PTr−1)

n−1x0 +

a2n(r−1)Tr−1(PTr−1)
n−1x + bn(r−1)un(r−1)]

x0 = P[(1 − a
1
nr − bnr)x + a

1
nrTr(PTr)

n−1x + bnrunr]

for all x ∈ K. Thus for all x, z ∈ K

‖x0 − z0‖ ≤ (1 − a
1
nr − bnr)‖x − z‖ + a

1
nr‖Tr(PTr)

n−1x − Tr(PTr)
n−1z‖

≤ (1 − a1nr − bnr)‖x − z‖ + a
1
nrkn‖x − z‖

≤ kn‖x − z‖ .

Proceeding in this way we get

‖Qnx − Qnz‖ ≤ k
r
n‖x − z‖ = [1 + (k

r
n − 1)]‖x − z‖ .

Set

Sn,m = Qn+m−1Qn+m−2........Qn, m ≥ 1 and

bn,m = ‖Sn,m(txn + (1 − t)p1) − (txn+m + (1 − t)p1)‖.

Then

‖Sn,mx − Sn,my‖ ≤ (
n+m−1∏

j=n

krj )‖x − y‖ = Hnmr‖x − y‖

where Hnmr = (
∏n+m−1

j=n
krj ) for n ≥ 1, Sn,mxn = xn+m and Sn,mp = p for all p ∈ F. From the

facts
∑

∞

n=1
(kn−1) < ∞ and 0 ≤ tr−1 ≤ rtr−1(t−1) for all t ≥ 1 we have that

∑
∞

n=1
(krn−1) < ∞



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Weak and strong convergence theorems of a multistep ... 161

which in turn implies that Hnmr → 1 as n, m → ∞. Also we have that Sn,m is Lipschitzian with
the lipschitz constant Hnmr. By Lemma 1.7 we have

bn,m ≤ Hnmrφ
−1(‖xn − p1‖ − H

−1
nmr‖Sn,mxn − Sn,mp1‖)

= Hnmrφ
−1(‖xn − p1‖ − H

−1
nmr‖xn+m − p1‖) .

Now,

dn+m(t) = ‖txn+m + (1 − t)p1 − p2‖

≤ bn,m + ‖Sn,m(txn + (1 − t)p1) − p2‖

= bn,m + ‖Sn,m(txn + (1 − t)p1) − Sn,mp2‖

≤ bn,m + Hnmr‖txn + (1 − t)p1 − p2‖

= bn,m + Hnmrdn(t) .

It then follows from Lemma 2.1 that the sequence {bn,m} converges uniformly to 0 as n → ∞ for
all m ≥ 1. Thus from above we get

lim sup
n→∞

dn(t) ≤ φ
−1(0) + lim inf

n→∞
dn(t) = lim inf

n→∞
dn(t) .

This completes the proof.

Theorem 1. Let E be a uniformly convex Banach space and K be a nonempty closed convex subset

of E which is also a nonexpansive retract of E. Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself
asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with

∑
∞

n=1
(kin − 1) < ∞ for

i ∈ I. Let {xn} be defined by (1.9) with
∑

∞

n=1
bni < ∞ for i ∈ I. Let F =

⋂r
i=1

F(Ti) 6= ∅ and

(1) 0 < lim infn→∞ a
1
nr ≤ lim supn→∞(a

1
nr + bnr) < 1

(2) 0 < lim infn→∞ a
1
nj ≤ lim supn→∞(

∑r−j+1
m=1

amnj + bnj) < 1.

Assume that any one of the following conditions holds:

(1)E satisfies Opial’s property

(2)E has a Frechet differentiable norm

(3)E⋆ has the Kadec-Klee property

then {xn} converges weakly to some common fixed point of {Ti}, i ∈ I.

Proof: Since F 6= ∅, so let q ∈ F. Then by Lemma 2.1 limn→∞ ‖xn − q‖ exists and so {xn}

is bounded. Since E be a uniformly convex Banach space so {xn} has a subsequence {xnj} which is

weakly convergent to p ∈ K (say). From Lemma 2.3 we get limn→∞ ‖xn − Tixn‖ = 0 for i ∈ I. By

Lemma 1.5 we have Ti is demiclosed at 0 so p ∈ F(Ti) for all i ∈ I. Then p ∈ F. If possible let {xn}

has another subsequence {xnk} which converges weakly to another point q ∈ K. Then by similar



162 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

argument as above we have that q ∈ F(T). Let (1) hold. Then by Opial’s property we have

‖xn − p‖ = lim sup
j→∞

‖xnj − p‖

< lim sup
j→∞

‖xnj − q‖ = lim
n→∞

‖xn − q‖

= lim sup
k→∞

‖xnk − q‖

< lim sup
k→∞

‖xnk − p‖ = lim
n→∞

‖xn − p‖

a contradiction. So p = q.

Let (2) hold. Then from Lemma 2.4 we get limn→∞ < p − q, J(p1 − p2) >= 0 for all p, q ∈

ww(xn) and p1, p2 ∈ F. Since p, q ∈ F also so from above we get < p − q, J(p − q) >= 0, that is,

‖p − q‖2 = 0 which implies that p = q.

Let (3) hold. Then from Lemma 2.5 we get limn→∞ ‖txn + (1 − t)p − q‖ exists, so by Lemma 1.6

we have that p = q. So {xn} converges weakly to some common fixed point of {Ti}, i ∈ I. This

completes the proof of the Theorem. ♦

Condition(A)[14] A mapping T : K → E with nonempty fixed point set F(T) in K satisfies Condition
(A) if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(m) > 0 for all
m ∈ (0, ∞) such that

f(d(x, F(T))) ≤ ‖x − Tx‖ for all x ∈ K .

A finite family of mappings Ti : K → E, for all i = 1, 2, 3, ..., r with nonempty fixed point set
F =

⋂r
i=1 F(Ti) 6= ∅ satisfies

(i) Condition(A)[4] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and
f(m) > 0 for all m ∈ (0, ∞) such that

f(d(x, F)) ≤
1

r
(

r∑

i=1

‖x − Tix‖) for all x ∈ K

(ii) Condition(B)[4] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and
f(m) > 0 for all m ∈ (0, ∞) such that

f(d(x, F)) ≤ max
1≤i≤r

{‖x − Tix‖} for all x ∈ K

(iii) Condition(C)[4] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and
f(m) > 0 for all m ∈ (0, ∞) such that at least one of the Ti’s satisfies condition(A).
Clearly if Ti = T, for all i = 1, 2, 3, ..., r, then Condition(A) reduces to Condition(A). Also

Condition(B) reduces to Condition(A) if all but one of Ti’s are identities. Also it contains Condition(A).

Furthermore Condition(C) and Condition(B) are equivalent. Tan and Xu [18] pointed out that the

Condition(A) is weaker than the compactness of K. It is well known that every continuous and

demicompact mapping must satisfy Condition(A) [14]. Since every completely continuous map-

ping is continuous and demicompact so it must satisfy Condition(A). Also Condition(B) contains



CUBO
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Weak and strong convergence theorems of a multistep ... 163

Condition(A) therefore to study the strong convergence of the iterative sequence {xn} be defined

by (1.9) we use Condition(B) instead of the complete continuity of the mappings {T1, T2, ...., Tr}

and Condition(A).

Theorem 2. Let E be a uniformly convex Banach space and K be a nonempty closed convex subset

of E which is also a nonexpansive retract of E. Let Ti : K → E(i ∈ I = {1, 2, ..., r}) be given nonself
asymptotically nonexpansive mappings with sequences {kin} ⊂ [1, ∞) with

∑
∞

n=1
(kin − 1) < ∞ for

i ∈ I. Let {xn} be defined by (1.9) with
∑

∞

n=1
bni < ∞ for i ∈ I. Let F =

⋂r
i=1

F(Ti) 6= ∅ and

(1) 0 < lim infn→∞ a
1
nr ≤ lim supn→∞(a

1
nr + bnr) < 1

(2) 0 < lim infn→∞ a
1
nj ≤ lim supn→∞(

∑r−j+1
m=1

amnj + bnj) < 1.

If the family of mappings {T1, T2, ...., Tr} satisfy Condition(B), then {xn} converges strongly to some

common fixed point of {T1, T2, ...., Tr}.

Proof: Let q ∈ F then by Lemma 2.1 limn→∞ ‖xn − q‖ exists. Let limn→∞ ‖xn − q‖ = a,

for some a ≥ 0. Let a > 0. Now from (2.5) we get

‖xn+1 − q‖ ≤ [1 + (k
r
n − 1)]‖xn − q‖ + σ

r
n

= [1 + δn]‖xn − q‖ + σ
r
n (2.26)

where {σrn} is a nonnegative real sequence such that
∑

∞

n=1
σrn < ∞ and δn = k

r
n − 1 such that∑

∞

n=1 δn < ∞. So

d(xn+1, F) ≤ (1 + δn)d(xn, F) + σ
r
n .

By Lemma 1.1 we have that limn→∞ d(xn, F) exists. By Condition (B) and Lemma 2.3 we get,

lim
n→∞

f(d(xn, F)) = 0.

Since f : [0, ∞) → [0, ∞) is a nondecreasing function with f(0) = 0 so we have
limn→∞ d(xn, F) = 0. Since limn→∞ ‖xn −q‖ exists, it follows that {‖xn −q‖} is bounded, so there

exists M′′ > 0 such that ‖xn − q‖ ≤ M
′′. From (2.26) we get

‖xn+1 − q‖ ≤ ‖xn − q‖ + δnM
′′ + σrn

= ‖xn − q‖ + θn

where θn = δnM
′′ + σrn. Now

∑
∞

n=1
θn < ∞. Now for any m > 1 we have that

‖xn+m − q‖ ≤ ‖xn+m−1 − q‖ + θn+m−1

≤ ‖xn+m−2 − q‖ + θn+m−2 + θn+m−1

· · · · · · · · · · · · · · · · · · · · · · · · · · ·

≤ ‖xn − q‖ +
n+m−1∑

k=n

θk .



164 Shrabani Banerjee and Binayak S.Choudhury CUBO
14, 3 (2012)

Since
∑

∞

n=1
θn < ∞ and limn→∞ d(xn, F) = 0, there exists N1 ∈ N such that for all n ≥ N1 we

have d(xn, F) <
ǫ
3
and

∑
∞

n=N1
θn <

ǫ
6
. Therefore there exists x ∈ F such that ||xN1 − x|| <

ǫ
3
.

Therefore we have

||xn+m − xn|| ≤ ||xn+m − x|| + ||xn − x||

< ||xN1 − x|| +

n+m−1∑

k=N1

θk + ||xN1 − x|| +

n−1∑

k=N1

θk

<
ǫ

3
+

ǫ

6
+

ǫ

3
+

ǫ

6
= ǫ (2.27)

, .nonumber (2.28)

Hence {xn} is a Cauchy sequence. Since E is complete so xn → p ∈ E, so for given ǫ > 0 there
exists n1 ∈ N such that for all n ≥ n1, ‖xn − p‖ ≤

ǫ
2(1+k1)

. Again since limn→∞ d(xn, F) = 0, so

for given ǫ > 0 there exists n2 ∈ N such that for all n ≥ n2(≥ n1), d(xn, F) <
ǫ

2(1+k1)
. so there

exists p ∈ F such that ‖xn2 − p‖ ≤
ǫ

2(1+k1)
. Therefore

‖p − Tip‖ = ‖p − xn2 + xn2 − p + p − Tip‖

≤ ‖p − xn2‖ + ‖xn2 − p‖ + ‖p − Tip‖

= ‖p − xn2‖ + ‖xn2 − p‖ + ‖Tip − Tip‖

≤ ‖p − xn2‖ + ‖xn2 − p‖ + k1‖p − p‖

≤ (1 + k1)‖p − xn2‖ + (1 + k1)‖xn2 − p‖

≤ (1 + k1)
ǫ

2(1 + k1)
+ (1 + k1)

ǫ

2(1 + k1)
= ǫ .

Since ǫ is arbitrary so we have Tip = p for all i ∈ I. So p ∈ F(Ti) for all i ∈ I. Thus p ∈ F. Hence

{xn} converges strongly to some common fixed point of {T1, T2, ...., Tr}.

Remark 2.6. Theorem 1 and Theorem 2 extends and generalize Theorem 2.1 and Theorem 2.5 of

[20].

ACKNOWLEDGEMENT

This work is supported by Council of Scientific and Industrial Research(CSIR), Government of

India .

Received: December 2011. Revised: September 2012.

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