Convergence analysis of viscosity implicit rules of nonexpansive mappings in Banach spaces

Abstract

In this paper, the study of implicit viscosity approximation methods for nonexpansive mappings in Banach spaces is explored. A new iterative sequence is introduced for the class of nonexpansive mappings in Banach spaces. Suitable conditions are imposed on the control parameters to prove a strong convergence theorem. Moreover, the strong convegence of the newly introduced sequence to a fixed point of a nonexpansive mapping is obtained which also solves the variational inequality problem. These results improve and extend some recent corresponding results announced.

Full text

Nonlinear Functional Analysis and Applications
Vol. 24, No. 4 (2019), pp. 691-713
ISSN: 1229-1595(print), 2466-0973(online)
http://nfaa.kyungnam.ac.kr/journal-nfaa
Copyright c
2019 Kyungnam University Press
KUP
ress
CONVERGENCE ANALYSIS OF VISCOSITY IMPLICIT
RULES OF NONEXPANSIVE MAPPINGS
IN BANACH SPACES
Mathew O. Aibinu1and Jong Kyu Kim2
1School of Mathematics, Statistics and Computer Science
University of KwaZulu-Natal, Durban, South Africa
Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)
e-mails: moaibinu@yahoo.com, 216040407@stu.ukzn.ac.za
2Department of Mathematics Education, Kyungnam University
Changwon, Gyeongnam, 51767, Korea
e-mail: jongkyuk@kyungnam.ac.kr
Abstract. In this paper, the study of implicit viscosity approximation methods for non-
expansive mappings in Banach spaces is explored. A new iterative sequence is introduced
for the class of nonexpansive mappings in Banach spaces. Suitable conditions are imposed
on the control parameters to prove a strong convergence theorem. Moreover, the strong
convergence of the newly introduced sequence to a fixed point of a nonexpansive mapping is
obtained which also solves the variational inequality problem. These results are improvement
and extension of some recent corresponding results announced.
1. Introduction
Following the idea of Attouch [3], the viscosity approximation method for
nonexpansive mappings in Hilbert spaces was introduced in 2000 by Moudafi
[10].
Let Hbe a real Hilbert space with inner product h., .iand norm k.k, K be
a nonempty, closed and convex subset of H. Let G:K→Kbe a contraction
(i.e., kG(u)−G(v)k ≤ cku−vkfor all u, v ∈Kand for some c∈[0,1)), and
0Received January 15, 2019. Revised March 22, 2019.
02010 Mathematics Subject Classification: 47H06, 47J05, 47J25, 47H10.
0Keywords: Viscosity, implicit rule, generalized contraction, nonexpansive.
0Corresponding author: M. O. Aibinu(moaibinu@yahoo.com).

692 M. O. Aibinu and J. K. Kim
let T:K→Kbe a nonexpansive mapping (i.e., kT u −T vk ≤ ku−vkfor all
u, v ∈K). The set of fixed points of Twill be denoted by F(T).Recently, Xu
et al. [16] proposed the implicit midpoint procedure:
xn+1 =λnG(xn) + (1 −λn)Txn+xn+1
2, n ∈N,(1.1)
where {λn}∞
n=1 ⊂[0,1].Under certain conditions imposed on the control pa-
rameter, it was established that the implicit midpoint procedure (1.1) con-
verges to a fixed point pof Twhich also solves the variational inequality:
h(I−G)p, x −pi ≥ 0,∀x∈F(T).(1.2)
Ke and Ma [5] introduced generalized viscosity implicit rules which extend
the results of Xu et al. [16]. The generalized viscosity implicit procedures are
given by
xn+1 =λnG(xn) + (1 −λn)T(δnxn+ (1 −δn)xn+1), n ∈N,(1.3)
and
yn+1 =λnG(yn) + βnyn+γnT(δnyn+ (1 −δn)yn+1), n ∈N,(1.4)
where {λn}∞
n=1 ,{βn}∞
n=1 ,{γn}∞
n=1 ⊂[0,1] with λn+βn+γn= 1.Suitable
conditions were imposed on the control parameters to show that the sequence
{xn}∞
n=1 converges strongly to a fixed point pof the nonexpansive mapping
T, which is also the unique solution of the variational inequality (1.2). In
other words, pis the unique fixed point of the contraction PF(T)G, that is,
PF(T)G(p) = p. Replacement of strict contractions in (1.4) by the generalized
contractions and extension to uniformly smooth Banach spaces was considered
by Yan et al. [17]. Under certain conditions on imposed on the parameters
which are involved, the sequence {xn}∞
n=1 converges strongly to a fixed point
pof the nonexpansive mapping T, which is also the unique solution of the
variational inequality
h(I−G)p, J(x−p)i ≥ 0,∀x∈F(T),(1.5)
where Jis the normalized duality mapping.
Inspired by the previous works in this direction, we propose a new implicit
iterative algorithm. Precisely, for a nonempty closed convex subset Kof a
uniformly smooth Banach space Eand for real sequences λi
n∞
n=13
i=1 ⊂
[0,1] and {δn}∞
n=1 ⊂(0,1),the implicit iterative scheme is defined from an
arbitrary x1∈Kby
xn+1 =λ1
nG1(xn) + λ2
nxn+λ3
nT((1 −δn)G2(xn) + δnxn+1),(1.6)
where T:K→Kis a nonexpansive mapping and Gi:K→Kis a generalized
contraction mapping for each i= 1,2.

Convergence analysis of viscosity implicit rules of nonexpansive mappings 693
2. Preliminaries
Let Ebe a real Banach space with dual E∗and denotes the norm on Eby
k.k.The normalized duality mapping J:E→2E∗is defined as
J(x) = {f∈E∗:hx, f i=kxkkfk,kxk=kfk} ,
where h·,·i is the duality pairing between Eand E∗.Let BEdenotes the unit
ball of E. The modulus of convexity of Eis defined as
δE() = inf 1−kx+yk
2:x, y ∈BE,kx−yk ≥ ,0≤≤2.
Eis uniformly convex if and only if δE()>0 for every ∈(0,2]. Eis said to
be smooth (or G´ateaux differentiable) if the limit
lim
t→0+
kx+tyk−kxk
t
exists for each x, y ∈BE. E is said to have uniformly Gˆateaux differentiable
norm if for each y∈BE,the limit is attained uniformly for x∈BEand
uniformly smooth if it is smooth and the limit is attained uniformly for each
x, y ∈BE.Recall that if Eis smooth, then Jis single-valued and onto if
Eis reflexive. Furthermore, the normalized duality mapping Jis uniformly
continuous on bounded subsets of Efrom the strong topology of Eto the
weak-star topology of E∗if Eis a Banach space with a uniformly Gˆateaux
differentiable norm.
Let Tbe a self-mapping of K. T :K→Kis said to be L-Lipschitzian if
there exists a constant L > 0,such that for all u, v ∈K,
kT u −T vk ≤ Lku−vk.
Let (X, d) be a metric space and Ka subset of X. A mapping G:K→Kis
said to be a Meir-Keeler contraction if for each  > 0 there exists δ=δ()>0
such that for each u, v ∈K, with ≤d(u, v)<  +δ, we have
d(G(u), G(v)) < .
Let Nbe the set of all positive integers and R+the set of all positive real
numbers. A mapping ψ:R+→R+is said to be an L-function if ψ(0) =
0, ψ(t)>0 for all t > 0 and for every s > 0,there exists u>ssuch that
ψ(t)≤sfor each t∈[s, u].A mapping G:E→Eis called a (ψ, L)-contraction
if ψ:R+→R+is an L-function and
d(G(x), G(y)) < ψ(d(x, y)),
for all x, y ∈E, x 6=y.
The following interesting results about the Meir-Keeler contraction are well
known.

694 M. O. Aibinu and J. K. Kim
Proposition 2.1. ([9]) Let (X, d)be a complete metric space and let Gbe a
Meir-Keeler contraction on X. Then Ghas a unique fixed point in X.
Remark 2.2. If Kis a nonempty closed (convex) subset of a complete metric
space (X, d),then the conclusion of Proposition 2.1 is still true.
Proposition 2.3. ([13]) Let Ebe a Banach space, Ka convex subset of E
and G:K→Ka Meir-Keeler contraction. Then for all  > 0,there exists a
c∈(0,1) such that
kG(u)−G(v)k ≤ cku−vk(2.1)
for all u, v ∈Kwith ku−vk ≥ .
Proposition 2.4. ([8]) Let Kbe a nonempty convex subset of a Banach space
E, T :K→Ka nonexpansive mapping and G:K→Ka Meir-Keeler
contraction. Then T G and GT :K→Kare Meir-Keeler contractions.
The following lemmas are also needed in the sequel.
Lemma 2.5. ([11]) Let Kbe a nonempty closed and convex subset of a uni-
formly smooth Banach space E. Let T:K→Kbe a nonexpansive mapping
such that F(T)6=∅and G:K→Kbe a generalized contraction mapping.
Then {xt}defined by
xt=tG(xt) + (1 −t)T xt
for t∈(0,1),converges strongly to p∈F(T),which solves the variational
inequality:
hG(p)−p, J(z−p)i ≤ 0,∀z∈F(T).
Lemma 2.6. ([11]) Let Kbe a nonempty closed and convex subset of a uni-
formly smooth Banach space E. Let T:K→Kbe a nonexpansive mapping
such that F(T)6=∅and G:K→Kbe a generalized contraction mapping.
Assume that {xt}defined by
xt=tG(xt) + (1 −t)T xt
for t∈(0,1),converges strongly to p∈F(T)as t→0.Suppose that {xn}is a
bounded sequence such that kxn−T xnk → 0as n→ ∞.Then
lim sup
n→∞
hG(p)−p, J(xn−p)i ≤ 0.
Lemma 2.7. ([12]) Let {un}∞
n=1 and {vn}∞
n=1 be bounded sequences in a
Banach space Eand {tn}∞
n=1 be a sequence in [0,1] with 0<lim inf
n→∞
tn≤
lim sup
n→∞
tn<1.Suppose that for all n≥0,
un+1 = (1 −tn)un+tnvn
and
lim sup
n→∞
(kun+1 −unk−kvn+1 −vnk)≤0.

Convergence analysis of viscosity implicit rules of nonexpansive mappings 695
Then lim
n→∞ kun−vnk= 0.
Lemma 2.8. ([15]) Let {an}be a sequence of nonnegative real numbers sat-
isfying the following relations:
an+1 ≤(1 −αn)an+αnσn+γn, n ∈N,
where
(i) {α}n⊂(0,1),
∞
X
n=1
αn=∞;
(ii) lim sup
n→∞
σn≤0;
(iii) γn≥0,
∞
X
n=1
γn<∞.
Then, an→0as n→ ∞.
In this paper, the generalized contraction mappings refer to Meir-Keeler
contractions or (ψ, L)-contractions. It is assumed from the definition of (ψ, L)-
contraction that L-function is continuous, strictly increasing and lim
t→∞
φ(t) =
∞,where φ(t) = t−ψ(t) for all t∈R+.Whenever there is no confusion, φ(t)
and ψ(t) will be written as φ t and ψ t, respectively.
3. Main results
Assumption 3.1. Let Kbe a nonempty closed convex subset of a uniformly
smooth Banach space E. Let Gi:K→Kbe generalized contraction mappings
and Ta nonexpansive self-mapping defined on Kwith F(T)6=∅, for each
i= 1,2.The real sequences λi
n∞
n=13
i=1 ⊂[0,1] and {δn}∞
n=1 ⊂(0,1) are
assumed to satisfy the following conditions:
(i)
3
X
i=1
λi
n= 1;
(ii) lim
n→∞(1 −λ2
n−λ3
nδn) = 0,
∞
X
n=1
(1 −λ2
n−λ3
nδn) = ∞;
(iii) 0 <lim inf
n→∞
λ2
n≤lim sup
n→∞
λ2
n<1;
(iv) lim
n→∞
λ3
n= 0,
∞
X
n=1
λ3
n(1 −δn)<∞;
(v) 0 <  ≤δn≤δn+1 ≤δ < 1,∀n∈N.
The convergence of the iterative scheme (1.6) is being considered under the
conditions (i)-(v) of Assumption 3.1 stated above.

696 M. O. Aibinu and J. K. Kim
First, it is observed that for all ω∈K, the mapping defined by
u7→ Tω(u) : = λ1
nG1(ω) + λ2
nω+λ3
nT((1 −δn)G2(ω) + δnu),(3.1)
for all u∈K, where λi
n∞
n=13
i=1 ⊂[0,1],{δn}∞
n=1 ⊂(0,1),is a contraction
with the contractive constant δ∈(0,1).
Indeed, for all u, v ∈K,
kTω(u)−Tω(v)k=λ3
nkT((1 −δn)G2(ω) + δnu)−T((1 −δn)G2(ω) + δnv)k
≤λ3
nk(1 −δn)G2(ω) + δnu−(1 −δn)G2(ω)−δnvk
≤λ3
nδnku−vk
≤δnku−vk
≤δku−vk.(3.2)
Therefore, Tωis a contraction. Thus, (1.6) is well defined since every contrac-
tion in a Banach space has a fixed point.
The proof of the following lemmas which are useful in establishing our main
result are given as below.
Lemma 3.2. Let Kbe a nonempty closed convex subset of a uniformly smooth
Banach space E. Let Gi:K→Kbe a generalized contraction mapping and
Ta nonexpansive self-mapping defined on Kwith F(T)6=∅for each i= 1,2.
For an arbitrary x1∈K, define the iterative sequence {xn}∞
n=1 by (1.6). Then
the sequence {xn}∞
n=1 is bounded under the conditions (i)-(v) of Assumption
3.1.
Proof. It is shown that the sequence {xn}∞
n=1 is bounded. Let ψ= max {ψ1, ψ2}
and G= max {kG1(p)−pk,kG2(p)−pk} .For p∈F(T),
kxn+1 −pk=kλ1
nG1(xn) + λ2
nxn+λ3
nT((1 −δn)G2(xn) + δnxn+1)−pk
≤λ1
nkG1(xn)−pk+λ2
nkxn−pk
+λ3
nkT((1 −δn)G2(xn) + δnxn+1)−pk
≤λ1
nkG1(xn)−G1(p)k+λ1
nkG1(p)−pk+λ2
nkxn−pk
+λ3
nk(1 −δn)G2(xn) + δnxn+1 −pk
=λ1
nkG1(xn)−G1(p)k+λ1
nkG1(p)−pk+λ2
nkxn−pk
+λ3
nk(1 −δn)(G2(xn)−p) + δn(xn+1 −p)k
≤λ1
nkG1(xn)−G1(p)k+λ1
nkG1(p)−pk+λ2
nkxn−pk
+λ3
n(1 −δn)kG2(xn)−G2(p)k+λ3
n(1 −δn)kG2(p)−pk
+λ3
nδnkxn+1 −pk

Convergence analysis of viscosity implicit rules of nonexpansive mappings 697
≤λ1
nψ1kxn−pk+λ1
nkG1(p)−pk+λ2
nkxn−pk
+λ3
n(1 −δn)ψ2kxn−pk+λ3
n(1 −δn)kG2(p)−pk
+λ3
nδnkxn+1 −pk
≤λ1
nψ+λ2
n+λ3
n(1 −δn)ψkxn−pk
+λ1
n+λ3
n(1 −δn)G+λ3
nδnkxn+1 −pk
=ψ+λ2
n(1 −ψ)−λ3
nδnψkxn−pk
+1−λ2
n−λ3
nδnG+λ3
nδnkxn+1 −pk.
Therefore, we have
kxn+1 −pk ≤ ψ+λ2
n(1 −ψ)−λ3
nδnψ
1−λ3
nδn
kxn−pk+1−λ2
n−λ3
nδn
1−λ3
nδn
G
=1−(1 −λ2
n−λ3
nδn)φ
1−λ3
nδnkxn−pk+(1 −λ2
n−λ3
nδn)φ
1−λ3
nδn
φ−1G
≤max kxn−pk, φ−1G.(3.3)
Then by induction,
kxn+1 −pk ≤ max kx1−pk, φ−1G.
This shows that the sequence {xn}∞
n=1 is bounded and hence {{Gi(xn)}∞
n=1}2
i=1
and {T((1 −δn)G2(xn) + δnxn+1)}∞
n=1 are bounded. Certainly, for p∈F(T),
kG1(xn)k≤kG1(xn)−G1(p)k+kG1(p)k
≤ψ1kxn−pk+kG1(p)k
≤max ψ1kx1−pk, ψ1φ−1G+kG1(p)k(by induction).
Similarly,
kG2(xn)k ≤ max ψ1kx1−pk, ψ1φ−1G+kG2(p)k.
Furthermore,
kT((1 −δn)G2(xn) + δnxn+1)k
=kT((1 −δn)G2(xn) + δnxn+1)−p+pk
≤ kT((1 −δn)G2(xn) + δnxn+1)−T pk+kpk
≤ k(1 −δn)G2(xn) + δnxn+1 −pk+kpk
≤(1 −δn)kG2(xn)−pk+δnkxn+1 −pk+||p||
≤(1 −δn)kG2(xn)−G2(p)k+(1−δn)kG2(p)−pk+δnkxn+1 −pk+||p||
≤(1 −δn)ψ2kxn−pk+δnkxn+1 −pk+ (1 −δn)kG2(p)−pk+||p||
≤(1 −)ψ2kxn−pk+δkxn+1 −pk+ (1 −)kG2(p)−pk+||p||.

698 M. O. Aibinu and J. K. Kim
Therefore, we have
kT((1 −δn)G2(xn) + δnxn+1)k
≤max (1 + δ−)ψkx1−pk,(1 + δ−)ψφ−1G
+ (1 −)|G2(p)−pk+||p|| (by induction).

Lemma 3.3. Let Kbe a nonempty closed convex subset of a uniformly smooth
Banach space E. Let G:K→Kbe a generalized contraction mapping and
Ta nonexpansive self-mapping defined on Kwith F(T)6=∅.Suppose that
{δn}∞
n=1 is a real sequence in (0,1) and {xn}∞
n=1 ⊂K. Set
vn= (1 −δn)G(xn) + δnxn+1 .
Then, we have
kT vn+1 −T vnk ≤ (1 −δn+1)ψkxn+1 −xnk+ (δn+1 −δn)kxn+1 −G(xn)k
+δn+1kxn+2 −xn+1k.
Proof.
kT vn+1 −T vnk
=kT((1 −δn+1)G(xn+1) + δn+1xn+2 )−T((1 −δn)G(xn) + δnxn+1)k
≤ k(1 −δn+1)G(xn+1) + δn+1xn+2 −(1 −δn)G(xn)−δnxn+1 k
=k(1 −δn+1)G(xn+1)−(1 −δn+1)G(xn)
+ (1 −δn+1)G(xn)−(1 −δn)G(xn)
+δn+1xn+2 −δn+1xn+1 +δn+1xn+1 −δnxn+1 k
=k(1 −δn+1)(G(xn+1)−G(xn)) −(δn+1 −δn)G(xn)
+δn+1(xn+2 −xn+1)+(δn+1 −δn)xn+1k
=k(1 −δn+1)(G(xn+1)−G(xn)) + (δn+1 −δn)(xn+1 −G(xn))
+δn+1(xn+2 −xn+1)k
≤(1 −δn+1)kG(xn+1)−G(xn)k+ (δn+1 −δn)kxn+1 −G(xn)k
+δn+1kxn+2 −xn+1k
≤(1 −δn+1)ψkxn+1 −xnk+ (δn+1 −δn)kxn+1 −G(xn)k
+δn+1kxn+2 −xn+1k.(3.4)

Theorem 3.4. Let Kbe a nonempty closed convex subset of a uniformly
smooth Banach space E. Let Gi:K→Kbe generalized contraction mapping
and Ta nonexpansive self-mapping defined on Kwith F(T)6=∅, for each

Convergence analysis of viscosity implicit rules of nonexpansive mappings 699
i= 1,2.Assume that the conditions (i)−(v)of Assumption 3.1 are satisfied.
Then the iterative sequence {xn}∞
n=1 which is defined from an arbitrary x1∈K
by (1.6), converges strongly to a fixed point pof T, which solves the variational
inequality
h(I−G1)p, J(x−p)i ≥ 0,∀x∈F(T).(3.5)
Proof. Set un=xn+1−λ2
nxn
1−λ2
nand vn= (1 −δn)G2(xn) + δnxn+1 .Then it could
be obtained that,
un+1 −un=xn+2 −λ2
n+1xn+1
1−λ2
n+1
−xn+1 −λ2
nxn
1−λ2
n
=λ1
n+1G1(xn+1) + λ3
n+1T(yn+1)
1−λ2
n+1
−λ1
nG1(xn) + λ3
nT(yn)
1−λ2
n
=λ1
n+1
1−λ2
n+1
(G1(xn+1)−G1(xn)) + λ1
n+1
1−λ2
n+1
−λ1
n
1−λ2
nG1(xn)
+λ3
n+1
1−λ2
n+1
(T(yn+1)−T(yn)) + λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
nT(yn)
=λ1
n+1
1−λ2
n+1
(G1(xn+1)−G1(xn)) −λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
nG1(xn)
+λ3
n+1
1−λ2
n+1
(T(yn+1)−T(yn)) + λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
nT(yn)
=λ1
n+1
1−λ2
n+1
(G1(xn+1)−G1(xn))
+λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
n(T(yn)−G1(xn))
+λ3
n+1
1−λ2
n+1
(T(yn+1)−T(yn)).
Let
M1
n= sup
n
{kT(yn)−G1(xn)k} ,
M2
n= sup
n
{kxn−G1(xn)k} ,
M3
n= sup
n
{kxn+1 −G2(xn)k}
and M= max M1
n, M2
n, M3
n.Put ψ= max {ψ1, ψ2}.Then, it can be
obtained from (3.4) that

700 M. O. Aibinu and J. K. Kim
kun+1 −unk ≤ λ1
n+1
1−λ2
n+1
kG1(xn+1)−G1(xn)k
+
λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
n
kT(yn)−G1(xn)k
+λ3
n+1
1−λ2
n+1
kT(yn+1)−T(yn)k
≤λ1
n+1
1−λ2
n+1
ψ1kxn+1 −xnk
+
λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
n
kT(yn)−G1(xn)k
+λ3
n+1
1−λ2
n+1
[(1 −δn+1)ψ2kxn+1 −xnk
+(δn+1 −δn)kxn+1 −G2(xn)k+δn+1kxn+2 −xn+1k]
≤λ1
n+1ψ+λ3
n+1(1 −δn+1)ψ
1−λ2
n+1
kxn+1 −xnk
+
λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
n
+λ3
n+1(δn+1 −δn)
1−λ2
n+1 M
+λ3
n+1δn+1
1−λ2
n+1
kxn+2 −xn+1k.(3.6)
Next is to evaluate kxn+1 −xnk.
xn+2 −xn+1 =λ1
n+1G1(xn+1) + λ2
n+1xn+1 +λ3
n+1T yn+1
−λ1
nG1(xn) + λ2
nxn+λ3
nT yn
=λ1
n+1(G1(xn+1)−G1(xn)) + λ2
n+1(xn+1 −xn)
+λ3
n+1(T yn+1 −T yn)+(λ1
n+1 −λ1
n)G1(xn)
+(λ2
n+1 −λ2
n)xn+ (λ3
n+1 −λ3
n)T yn
=λ1
n+1(G1(xn+1)−G1(xn)) + λ2
n+1(xn+1 −xn)
+λ3
n+1(T yn+1 −T yn)
+((λ2
n−λ2
n+1)+(λ3
n−λ3
n+1))G1(xn)
+(λ2
n+1 −λ2
n)xn+ (λ3
n+1 −λ3
n)T yn
=λ1
n+1(G1(xn+1)−G1(xn)) + λ2
n+1(xn+1 −xn)
+λ3
n+1(T yn+1 −T yn)+(λ2
n+1 −λ2
n)(xn−G1(xn))
+(λ3
n+1 −λ3
n)(T yn−G1(xn)).

Convergence analysis of viscosity implicit rules of nonexpansive mappings 701
Then, from (3.4)) it leads to
kxn+2 −xn+1k ≤ λ1
n+1ψkxn+1 −xnk+λ2
n+1kxn+1 −xnk
+λ3
n+1kT yn+1 −T ynk
+|λ2
n+1 −λ2
n|kxn−G1(xn)k
+|λ3
n+1 −λ3
n|kT yn−G1(xn)k
≤λ1
n+1ψkxn+1 −xnk+λ2
n+1kxn+1 −xnk
+λ3
n+1[(1 −δn+1)ψkxn+1 −xnk
+(δn+1 −δn)kxn+1 −G2(xn)k+δn+1kxn+2 −xn+1k]
+|λ2
n+1 −λ2
n|kxn−G1(xn)k
+|λ3
n+1 −λ3
n|kT yn−G1(xn)k
=λ2
n+1 + (λ1
n+1 +λ3
n+1)ψ−λ3
n+1δn+1ψkxn+1 −xnk
+λ3
n+1δn+1kxn+2 −xn+1k
+|λ2
n+1 −λ2
n|+|λ3
n+1 −λ3
n|+λ3
n+1(δn+1 −δn)M
=λ2
n+1 + (1 −λ2
n+1)ψ−λ3
n+1δn+1ψkxn+1 −xnk
+λ3
n+1δn+1kxn+2 −xn+1k
+|λ2
n+1 −λ2
n|+|λ3
n+1 −λ3
n|+λ3
n+1(δn+1 −δn)M
=ψ+λ2
n+1(1 −ψ)−λ3
n+1δn+1ψkxn+1 −xnk
+λ3
n+1δn+1kxn+2 −xn+1k
+|λ2
n+1 −λ2
n|+|λ3
n+1 −λ3
n|+λ3
n+1(δn+1 −δn)M
=λ2
n+1(1 −ψ) + (1 −λ3
n+1δn+1)ψkxn+1 −xnk
+λ3
n+1δn+1kxn+2 −xn+1k
+|λ2
n+1 −λ2
n|+|λ3
n+1 −λ3
n|+λ3
n+1(δn+1 −δn)M.
Putting dn=|λ2
n+1 −λ2
n|+|λ3
n+1 −λ3
n|+λ3
n+1(δn+1 −δn),it could be ob-
tained that,
kxn+2 −xn+1k ≤ λ2
n+1(1 −ψ) + (1 −λ3
n+1δn+1)ψ
1−λ3
n+1δn+1
kxn+1 −xnk
+dnM
1−λ3
n+1δn+1
.(3.7)
Let Sn=
λ3
n+1
1−λ2
n+1
−λ3
n
1−λ2
n+λ3
n+1(δn+1 −δn)
1−λ2
n+1
and substitute (3.7) into (3.6) to
obtain

702 M. O. Aibinu and J. K. Kim
kun+1 −unk
≤[λ1
n+1ψ+λ3
n+1(1 −δn+1)ψ
1−λ2
n+1
+λ3
n+1δn+1
1−λ2
n+1
×λ2
n+1(1 −ψ) + (1 −λ3
n+1δn+1)ψ
1−λ3
n+1δn+1
]kxn+1 −xnk
+SnM+λ3
n+1δn+1
1−λ2
n+1
×dnM
1−λ3
n+1δn+1
= [λ1
n+1ψ+λ3
n+1(1 −δn+1)ψ−λ3
n+1δn+1(λ1
n+1ψ+λ3
n+1(1 −δn+1)ψ)
[1 −λ2
n+1][1 −λ3
n+1δn+1]
+λ3
n+1δn+1(λ2
n+1(1 −ψ) + (1 −λ3
n+1δn+1)ψ)
[1 −λ2
n+1][1 −λ3
n+1δn+1]]kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M
= [λ1
n+1ψ+λ3
n+1(1 −δn+1)ψ−λ3
n+1δn+1(λ1
n+1ψ+λ3
n+1ψ−λ3
n+1δn+1ψ)
[1 −λ2
n+1][1 −λ3
n+1δn+1]
+λ3
n+1δn+1(λ2
n+1 −λ2
n+1ψ+ψ−λ3
n+1δn+1ψ)
[1 −λ2
n+1][1 −λ3
n+1δn+1]]kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M
=hλ1
n+1ψ+λ3
n+1(1 −δn+1)ψ−λ3
n+1δn+1((1 −λ2
n+1)ψ−λ3
n+1δn+1ψ)
[1 −λ2
n+1][1 −λ3
n+1δn+1]
+λ3
n+1δn+1(λ2
n+1 + (1 −λ2
n+1)ψ−λ3
n+1δn+1ψ)
[1 −λ2
n+1][1 −λ3
n+1δn+1]ikxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M
=λ1
n+1ψ+λ3
n+1(1 −δn+1)ψ+λ3
n+1δn+1λ2
n+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M
=(1 −λ2
n+1)ψ−λ3
n+1δn+1ψ+λ3
n+1δn+1λ2
n+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M

Convergence analysis of viscosity implicit rules of nonexpansive mappings 703
=1−(1 −λ2
n+1)(1 −ψ)−λ3
n+1δn+1(1 −ψ)
[1 −λ2
n+1][1 −λ3
n+1δn+1]kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M
=1−(1 −λ2
n+1)φ−λ3
n+1δn+1φ
[1 −λ2
n+1][1 −λ3
n+1δn+1]kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M
=1−(1 −λ2
n+1 −λ3
n+1δn+1)φ
[1 −λ2
n+1][1 −λ3
n+1δn+1]kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M
≤1−(1 −λ2
n+1 −λ3
n+1δn+1)φ
1−λ2
n+1 kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
[1 −λ2
n+1][1 −λ3
n+1δn+1]M.
It then follows that
kun+1 −unk−kxn+1 −xnk≤−(1 −λ2
n+1 −λ3
n+1δn+1)φ
1−λ2
n+1
kxn+1 −xnk
+Sn+dnλ3
n+1δn+1
(1 −λ2
n+1)(1 −λ3
n+1δn+1)M,
and thus,
lim sup
n→∞
(kun+1 −unk−kxn+1 −xnk)≤0.(3.8)
Invoking Lemma 2.7 gives
lim
n→∞ kun−xnk= 0.(3.9)
Consequently,
kxn+1 −xnk=k(1 −λ2
n)un+λ2
nxn−xnk
=k(1 −λ2
n)un−(1 −λ2
n)xnk
=k(1 −λ2
n)(un−xn)k
≤(1 −λ2
n)kun−xnk → 0 as n→ ∞.(3.10)

704 M. O. Aibinu and J. K. Kim
Next is to show that lim
n→∞ kxn−T(xn)k= 0.From (1.6), we could have that
kxn−T xnk ≤ kxn−xn+1k+kxn+1 −T(xn)k
≤ kxn+1 −xnk+kλ1
nG1(xn) + λ2
nxn+λ3
nT(vn)−T(xn)k
≤ kxn+1 −xnk+λ1
nkG1(xn)−T(xn)k+λ2
nkxn−T(xn)k
+λ3
nkT(vn)−T(xn)k
≤ kxn+1 −xnk+λ1
nkG1(xn)−T(xn)k+λ2
nkxn−T(xn)k
+λ3
nkvn−xnk
≤ kxn+1 −xnk+λ1
nkG1(xn)−T(xn)k+λ2
nkxn−T(xn)k
+λ3
nk(1 −δn)G2(xn) + δnxn+1 −xnk
≤ kxn+1 −xnk+λ1
nkG1(xn)−T(xn)k+λ2
nkxn−T(xn)k
+λ3
n(1 −δn)kxn−G2(xn)k+λ3
nδnkxn+1 −xnk
= (1 + λ3
nδn)kxn+1 −xnk+ (λ1
n+λ3
n(1 −δn))M
+λ2
nkxn−T(xn)k
= (1 + λ3
nδn)kxn+1 −xnk+ (1 −λ3
nδn−λ2
n)M
+λ2
nkxn−T(xn)k.
From 0 <lim inf
n→∞
λ2
n≤lim sup
n→∞
λ2
n<1,let 0 < η ≤λ2
n<1.Then
kxn−T xnk ≤ 1 + λ3
nδn
1−λ2
n
kxn+1 −xnk+1−λ2
n−λ3
nδn
1−λ2
n
M
≤1 + λ3
nδn
1−ηkxn+1 −xnk+1−λ2
n−λ3
nδn
1−ηM, (3.11)
which goes to zero as n→ ∞ by (3.10) and condition (ii) of Assumption 3.1.
Let a net {xt}be defined by xt=tG1(xt) + (1 −t)T xtfor t∈(0,1).It is
known by Lemma 2.5 that {xt}converges strongly to p∈F(T),which solves
the variational inequality:
hG1(p)−p, J(x−p)i ≤ 0,∀x∈F(T),
which is equivalent to
h(I−G1)p, J(x−p)i ≥ 0,∀x∈F(T).
It is claimed that
lim sup
n→∞
hG1(p)−p, J(xn+1 −p)i ≤ 0,(3.12)

Convergence analysis of viscosity implicit rules of nonexpansive mappings 705
where p∈F(T) is the unique fixed point of the generalized contraction
PF(T)G1(p) (Proposition 2.4), that is, p=PF(T)G1(p).
By (3.11), lim
n→∞ kxn−T xnk= 0.So it follows from Lemma 2.6 that
lim sup
n→∞
hG1(p)−p, J(xn−p)i ≤ 0.
Due to the norm-to-weak∗uniform continuity on bounded sets of the duality
map and the fact that kxn+1 −xnk → 0 as n→ ∞ by (3.10), we obtain that,
lim sup
n→∞
hG1(p)−p, J(xn+1 −p)i
= lim sup
n→∞
hG1(p)−p, J(xn+1 −xn+xn−p)i
= lim sup
n→∞
hG1(p)−p, J(xn−p)i ≤ 0.(3.13)
Lastly, it is established that xn→p∈F(T) as n→ ∞.Suppose that the
sequence {xn}∞
n=1 does not converge strongly to p∈F(T).Then there exists
 > 0 and a subsequence {xnk}∞
k=1 of {xn}∞
n=1 such that kxnk−pk ≥ , for all
k∈N.Therefore, for this , there exists ci∈(0,1
2) such that
kGi(xnk)−Gi(p)k ≤ cikxnk−pk, i = 1,2.
Let c= max {c1, c2}.Then,
||xnk+1 −p||2=λ1
nkG1(xnk)−p, J(xnk+1 −p)
+λ2
nkxnk−p, J(xnk+1 −p)
+λ3
nkT(ynk)−p, J(xnk+1 −p)
=λ1
nkG1(xnk)−G1(p), J(xnk+1 −p)
+λ1
nG1(p)−p, J(xnk+1 −p)
+λ2
nkxnk−p, J(xnk+1 −p)
+λ3
nkT(ynk)−p, J(xnk+1 −p)
≤cλ1
nkkxnk−pk kxnk+1 −pk
+λ1
nG1(p)−p, J(xnk+1 −p)
+λ2
nkkxnk−pk kxnk+1 −pk
+λ3
nk||(1 −δnk)G2(xnk) + δnkxnk+1 −p|| ||xnk+1 −p||
≤cλ1
nkkxnk−pk kxnk+1 −pk

706 M. O. Aibinu and J. K. Kim
+λ1
nG1(p)−p, J(xnk+1 −p)
+λ2
nkkxnk−pk kxnk+1 −pk
+λ3
nk(1 −δnk)||G2(xnk)−p|| ||xnk+1 −p||
+λ3
nkδnkkxnk+1 −pk2
≤cλ1
nkkxnk−pk kxnk+1 −pk+λ1
nG1(p)−p, J(xnk+1 −p)
+λ2
nkkxnk−pk kxnk+1 −pk+cλ3
nk(1 −δnk)||xnk−p|| ||xnk+1 −p||
+λ3
nk(1 −δnk)||G2(p)−p|| ||xnk+1 −p|| +λ3
nkδnkkxnk+1 −pk2
=cλ1
nk+λ2
nk+cλ3
nk(1 −δnk)kxnk−pk kxnk+1 −pk
+λ1
nG1(p)−p, J(xnk+1 −p)
+λ3
nk(1 −δnk)||G2(p)−p|| ||xnk+1 −p|| +λ3
nkδnkkxnk+1 −pk2
≤1
2cλ1
nk+λ2
nk+cλ3
nk(1 −δnk)kxnk−pk2+kxnk+1 −pk2
+λ1
nG1(p)−p, xnk+1 −p+λ3
nkδnkkxnk+1 −pk2
+1
2λ3
nk(1 −δnk)kG2(p)−pk2+kxnk+1 −pk2
=1
2c(λ1
nk+λ3
nk(1 −δnk)) + λ2
nkkxnk−pk2
+λ1
nG1(p)−p, J(xnk+1 −p)
+1
2c(λ1
nk+λ3
nk(1 −δnk)) + λ2
nk+ 2λ3
nkδnk+λ3
nk(1 −δnk)kxnk+1 −pk2
+1
2λ3
nk(1 −δnk)kG2(p)−pk2
=1
2c(λ1
nk+λ3
nk(1 −δnk)) + λ2
nkkxnk−pk2
+λ1
nG1(p)−p, J(xnk+1 −p)
+1
2c(λ1
nk+λ3
nk(1 −δnk)) + λ2
nk+λ3
nk(1 + δnk)kxnk+1 −pk2
+1
2λ3
nk(1 −δnk)kG2(p)−pk2
=1
2c(1 −λ2
nk−λ3
nkδnk) + λ2
nkkxnk−pk2
+λ1
nG1(p)−p, J(xnk+1 −p)
+1
2c(1 −λ2
nk−λ3
nkδnk) + λ2
nk+λ3
nk(1 + δnk)kxnk+1 −pk2
+1
2λ3
nk(1 −δnk)kG2(p)−pk2.(3.14)

Convergence analysis of viscosity implicit rules of nonexpansive mappings 707
Observe that
2−c(1 −λ2
nk−λ3
nkδnk)−λ2
nk−λ3
nk(1 + δnk)
= 2 −c+cλ2
nk+cλ3
nkδnk−λ2
nk−λ3
nk−λ3
nkδnk
= 2 −c−(1 −c)λ2
nk−(1 −c)λ3
nkδnk−λ3
nk
= 1 −c−(1 −c)λ2
nk−(1 −c)λ3
nkδnk+ 1 −λ3
nk
= 1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk(3.15)
and
λ1
nk= 1 −λ2
nk−λ3
nk
≤1−λ2
nk−λ3
nkδnk(since δnk∈(0,1)).(3.16)
Simplifying (3.14) by 2 gives
||xnk+1 −p||2
≤c(1 −λ2
nk−λ3
nkδnk) + λ2
nk
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk
kxnk−pk2
+λ1
n
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nkG1(p)−p, J(xnk+1 −p)
+λ3
nk(1 −δnk)
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk
kG2(p)−pk2
= 1−(1 −2c)(1 −λ2
nk−λ3
nkδnk) + λ1
nk
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk!kxnk−pk2
+λ1
nk
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nkG1(p)−p, J(xnk+1 −p)
+λ3
nk(1 −δnk)
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk
kG2(p)−pk2
≤ 1−(1 −2c)(1 −λ2
nk−λ3
nkδnk)
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk!kxnk−pk2
+(1 −2c)(1 −λ2
nk−λ3
nkδnk)
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk
1
1−2cG1(p)−p, J(xnk+1 −p)
+λ3
nk(1 −δnk)
1 + (1 −c)1−λ2
nk−λ3
nkδnk−λ3
nk
kG2(p)−pk2(By (3.16)).

708 M. O. Aibinu and J. K. Kim
By taking αn= (1 −2c)(1 −λ2
nk−λ3
nkδnk), σn=G1(p)−p, J(xnk+1 −p)
and γn=λ3
nk(1 −δnk) in Lemma 2.8, it shows that xnk→pas k→ ∞,which
is a contradiction. Hence, {xn}∞
n=1 converges strongly to p∈F(T).
The next result shows that under suitable conditions, the implicit iterative
sequences (1.4) and (1.6) converge to the same fixed point.
Theorem 3.5. Let Kbe a nonempty closed convex subset of a uniformly
smooth Banach space E. Let Gi:K→Kbe a c-contraction mapping and
Tbe a nonexpansive self-mapping defined on Kwith F(T)6=∅for each i=
1,2.Let λi
n∞
n=13
i=1 ⊂[0,1] and {δn}∞
n=1 ⊂(0,1) be real sequences such
that
3
X
i=1
λi
n= 1.Suppose that Gin (1.4) is the same as G1in (1.6) and
lim
n→∞
λ3
n
(1 −λ2
n−λ3
nδn)= 0.Then {xn}∞
n=1 defined by (1.6) converges to pif
and only if {yn}∞
n=1 defined by (1.4) converges to p.
Proof. Let c= max {c1, c2}.
kxn+1 −yn+1k
=||λ1
nG1(xn) + λ2
nxn+λ3
nT((1 −δn)G2(xn) + δnxn+1)
−λ1
nG(yn) + λ2
nyn+λ3
nT(δnyn+ (1 −δn)yn+1)||
=kλ1
n(G1(xn)−G1(yn)) + λ2
n(xn−yn)
+λ3
n(T((1 −δn)G2(xn) + δnxn+1)−T(δnyn+ (1 −δn)yn+1 )) k
≤λ1
n||G1(xn)−G1(yn)|| +λ2
nkxn−ynk
+λ3
nkT((1 −δn)G2(xn) + δnxn+1)−T(δnyn+ (1 −δn)yn+1 )k
≤λ1
nc1||xn−yn|| +λ2
nkxn−ynk
+λ3
nk(1 −δn)(G2(xn)−yn+1) + δn(xn+1 −yn)k
≤λ1
nc1||xn−yn|| +λ2
nkxn−ynk
+λ3
n(1 −δn)kG2(xn)−G2(yn) + G2(yn)−yn+1k
+λ3
nδnkxn+1 −yn+1 +yn+1 −ynk
≤λ1
nc1||xn−yn|| +λ2
nkxn−ynk+λ3
n(1 −δn)c2kxn−ynk
+λ3
n(1 −δn)kyn+1 −G2(yn)k+λ3
nδnkxn+1 −yn+1k+λ3
nδnkyn+1 −ynk
=λ1
nc+λ3
n(1 −δn)c+λ2
n||xn−yn|| +λ3
nδnkxn+1 −yn+1k
+λ3
n(1 −δn)kyn+1 −G2(yn)k+λ3
nδnkyn+1 −ynk.

Convergence analysis of viscosity implicit rules of nonexpansive mappings 709
Since {yn}∞
n=1 and {G2(yn)}∞
n=1 are bounded [5], let
M2= max sup
n
kyn+1 −G2(yn)k,sup
n
kyn+1 −ynk.
Then
kxn+1 −yn+1k
≤λ1
nc+λ3
n(1 −δn)c+λ2
n
1−λ3
nδn
||xn−yn|| +λ3
n
1−λ3
nδn
M2
=1−(1 −λ2
n−λ3
nδn)(1 −c)
1−λ3
nδn||xn−yn|| +λ3
n
1−λ3
nδn
M2
=1−(1 −λ2
n−λ3
nδn)(1 −c)
1−λ3
nδn||xn−yn|| +λ3
n
1−λ3
nδn
M2
= (1 −βn)||xn−yn|| +λ3
n
(1 −λ2
n−λ3
nδn)(1 −c)βnM2,(3.17)
where βn=(1−λ2
n−λ3
nδn)(1−c)
1−λ3
nδn.From the given condition, it follows that
lim sup
n→∞
λ3
n
(1 −λ2
n−λ3
nδn)≤0.
Apply Lemma 2.8 with γn= 0 to (3.17) to get that ||xn−yn|| → 0 as n→ ∞.
Furthermore, suppose ||yn−p|| → 0 as n→ ∞,it follows that,
||xn−p|| =||xn−yn+yn−p||
≤ ||xn−yn|| +||yn−p||
=||yn−p||
→0 (as n→ ∞).
Similary, suppose ||xn−p|| → 0 as n→ ∞,it follows that,
||yn−p|| =||yn−xn+xn−p||
≤ ||yn−xn|| +||xn−p||
=||xn−p||
→0 (as n→ ∞).

Corollary 3.6. ([17]) Let Ebe a uniformly smooth Banach space and K
a nonempty closed convex subset of E. Let T:K→Kbe a nonexpansive
mapping with F(T)6=∅and G:K→Ka generalized contraction mapping.
Pick any x0∈K. Let {xn}∞
n=1 be a sequence generated by
xn+1 =anG(xn) + bnxn+cnT(snxn+ (1 −sn)xn+1),(3.18)

710 M. O. Aibinu and J. K. Kim
where {an}∞
n=1 ,{bn}∞
n=1 and {cn}∞
n=1 are three sequences in [0,1] satisfying
the following conditions:
(i) an+bn+cn= 1;
(ii)
∞
X
n=1
an=∞,lim
n→∞
an= 0;
(iii)
∞
X
n=1
|bn+1 −bn|<∞and 0<lim inf
n→∞
bn≤lim sup
n→∞
bn<1;
(iv) 0 <  ≤sn≤sn+1 <1for all n∈N.
Then {xn}∞
n=1 converges strongly to a fixed point pof the nonexpansive map-
ping T, which is also the solution of the variational inequality (1.5).
Proof. Observe that λ1
n=an, λ2
n=bnand λ3
n=cn,by comparing (1.6) and
(3.18). Taking G1=G, δn= 1−snand G2to be the identity mapping of Kin
(1.6), we obtain (3.18). Hence, the conclusion follows from Theorem 3.4. 
Corollary 3.7. Let Kbe a nonempty closed convex subset of a uniformly
smooth Banach space E. Let Tbe a nonexpansive self-mapping defined on
Kwith F(T)6=∅.Assume that the real sequences {λn}∞
n=1 ⊂(0,1) and
{δn}∞
n=1 ⊂(0,1) satisfy the conditions:
(i) lim
n→∞
λn= 0;
(ii)
∞
X
n=1
λn=∞;
(iii)
∞
X
n=1
|λn+1 −λn|<∞;
(iv) 0 <  ≤δn≤δn+1 <1for all n∈N.
Then the iterative sequence {xn}∞
n=1 which is defined from an arbitrary x1∈K
by
xn+1 =λnxn+ (1 −λn)T((1 −δn)xn+δnxn+1) (3.19)
converges strongly to a fixed point pof Twhich solves the variational inequality
(1.5).
Proof. The result follows from Theorem 3.4 by simply taking G1=G2to be
the identity mappings of Kin (1.6). 
Corollary 3.8. ([1]) Let Ebe a uniformly smooth Banach space and Ka
nonempty closed convex subset of E. Let T:K→Kbe a nonexpansive
mapping with F(T)6=∅and G:K→Kan α-contraction. Suppose that the
real sequences {an} ⊂ (0,1),{bn} ⊂ [0,1) and {cn} ⊂ (0,1) are such that
an+bn+cn= 1,for all n∈Nand satisfy the fol lowing conditions:
(i) lim
n→∞
an= 0;

Convergence analysis of viscosity implicit rules of nonexpansive mappings 711
(ii)
∞
X
n=1
an=∞;
(iii) 0 <lim inf
n→∞
bn≤lim sup
n→∞
bn<1;
(iv) lim
n→∞ |bn+1 −bn|= 0.
For an arbitrary x1∈K, define the iterative sequence {xn}by
xn+1 =anG(xn) + bnxn+cnTxn+xn+1
2, n ∈N.(3.20)
Then the sequence {xn}converges in norm to a fixed point pof T, where pis
the unique solution in F(T)to the variational inequality (1.5).
Proof. It is known that a generalized contraction is more broad that an α-
contraction. Comparing (1.6) and (3.20), it is noted that λ1
n=an, λ2
n=bn
and λ3
n=cn.Taking G2to be the identity mappings of Kand δn= 2 for all
n∈Nin (1.6), it reduces to (3.20) with G1=G. Therefore, the desire result
follows from Theorem 3.4. 
Corollary 3.9. Let Kbe a nonempty closed convex subset of a uniformly
smooth Banach space E. Let Tbe a nonexpansive self-mapping defined on K
with F(T)6=∅.Assume that the real sequence {λn}∞
n=1 ⊂(0,1) satisfies the
following conditions:
(i) lim
n→∞
λn= 0;
(ii)
∞
X
n=1
λn=∞;
(iii)
∞
X
n=1
|λn+1 −λn|<∞.
Then the iterative sequence {xn}∞
n=1 which is defined from an arbitrary x1∈K
by
xn+1 =λnxn+ (1 −λn)T(xn+xn+1
2) (3.21)
converges strongly to a fixed point pof Twhich solves the variational inequality
(1.5).
Proof. The result follows from Theorem 3.4 by simply taking G1=G2to be
the identity mappings of Kand δn= 2 for all n∈N.Therfore, this improves
and extend the results of Alghamdi et al. [2]. 
Acknowledgments The first author acknowledges with thanks the bursary
and financial support from Department of Science and Technology and Na-
tional Research Foundation, Republic of South Africa Center of Excellence in

712 M. O. Aibinu and J. K. Kim
Mathematical and Statistical Sciences (DST-NRF CoE-MaSS) Doctoral Bur-
sary. Opinions expressed and conclusions arrived at are those of the authors
and are not necessarily to be attributed to the CoE-MaSS. And the second
author was supported by the Basic Science Research Program through the
National Research Foundation(NRF) Grant funded by Ministry of Education
of the republic of Korea (2018R1D1A1B07045427).
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