Approximations of Fixed Point of Nonexpansive Mappings in Banach
Spaces
ANANTACHAI PADCHAROEN, NAKNIMIT AKKASRIWORN
Department of Mathematics, Faculty of Science and Technology
Rambhai Barni Rajabhat University, Chanthaburi 22000, THAILAND
anantachai.p@rbru.ac.th, naknimit.a@rbru.ac.th
Abstract: -In this paper, we used condition (E)to study approximation on the Banach space, demonstrating the
convergence theorem as well as an example that supports the main theorem. Iterative fixed point approximation
for nonlinear operators is a novel area of investigation. As a result, the literature contains a number of iterative
techniques for overcoming such impediments and improving the rate of convergence.
Key-Words: Approximations of Fixed Point, Nonexpansive Mappings, Banach Spaces
Received: May 17, 2022. Revised: June 18, 2022. Accepted: July 21, 2022. Published: August 30, 2022.
1 Introduction
Iterative fixed point approximation for nonlinear op-
erators is a novel area of investigation (see, for exam-
ple, [1, 2, 3, 4, 5] and others). Using a Picard itera-
tive technique, the Banach contraction principle de-
termines the unique fixed point of a contraction map-
ping. On the other hand, the Picard iterative technique
does not necessarily converge to the fixed point of a
nonexpansive mapping.
Let Xbe a Banach space, whereas ∅ =C⊆X, and
T:C→C. If v=Tv, an element v∈Cis regarded
to as a fixed point for T. The set of all fixed points of
the map Tis denoted by F(T).The set of all natural
numbers shall be denoted by Nthroughout the work.
When Tis nonexpansive, that is, for all η, δ ∈C,
∥Tη−Tδǁ≤ ∥η−δ∥.
If Xis uniformly convex and Cis convex closed
bounded, then F(T)is nonempty. In 2008, Suzuki [6]
presented a new class of nonlinear mappings that is
just a generalization of the nonexpansive mappings
class. A mapping T:C→Cis said to obey the
condition (C)(or Suzuki mapping) if for all η, δ ∈C,
1
2∥η−Tη∥ ≤ ∥η−δ∥ ⇒ ∥Tη−Tδ∥ ≤ ∥η−δ∥.
García-Falset et al. [7] extended condition (C)to the
following general formulations in 2011. A mapping
T:C→Cis said to satisfy condition (Eµ)if there
exists some µ≥1such that
∥Tη−δ∥ ≤ µ∥Tδ−δ∥+∥η−δ∥for all η, δ ∈C.
A mapping Tis said to satisfy condition (Eµ)(or
García-Falset mapping) when it does so for some µ≥
1.García-Falset et al. demonstrated that every Suzuki
mapping meets condition (E)with µ= 3.It is also
worth noting that the class of Garcia-Falset mappings
includes many other classes of generalized nonexpan-
sive mappings (see [8] for details).
Agarwal iteration process introduced in [9], also
called S-iteration process, is defined as:
η0∈C,
ζn= (1 −ιn)ηn+ιnTηn,
ηn+1 = (1 −τn)Tηn+τnTζn,
(1)
where {ιn},{τn}are sequences in [0,1].
Suantai and Phuengrattana [10] also introduced
one another three-step iteration process known as
SP - iteration process, defined as:
η0∈C,
ζn= (1 −ιn)ηn+ιnTηn,
ϑn= (1 −τn)ζn+τnTζn,
ηn+1 = (1 −σn)ϑn+σnTϑn,
(2)
where {ιn},{τn},{σn}are sequences in [0,1].
Thakur et. al. [11] used a new iteration process, de-
fined as:
η0∈C,
ζn= (1 −ιn)ηn+ιnTηn,
ϑn=T((1 −τn)ηn+τnζn)
ηn+1 =Tϑn,
(3)
where {ιn},{τn}are sequences in [0,1].
Hussain et al. [12] also introduced the K-iteration
process, a three-step iteration procedure defined as:
η0∈C,
ζn= (1 −ιn)ηn+ιnTηn,
ϑn=T((1 −τn)Tηn+τnTζn),
ηn+1 =Tϑn,
(4)
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DOI: 10.37394/23206.2022.21.71
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Ullah et al. [13] also created the K∗-iteration process,
a three-step iteration procedure defined as:
η0∈C,
ζn= (1 −ιn)ηn+ιnTηn,
ϑn=T((1 −τn)ζn+τnTζn),
ηn+1 =Tϑn,
(5)
where {ιn},{τn}are sequences in [0,1].
Ullah et al. [14] also described the AK-iteration
process, which consists of three steps:
η0∈C,
ζn=T((1 −ιn)ηn+ιnTηn),
ϑn=T((1 −τn)ζn+τnTζn),
ηn+1 =Tϑn,
(6)
where {ιn},{τn}are sequences in [0,1].
In this paper, we present a novel iteration process
designated as the SP ∗-iteration process, which is for-
mally defined:
η0∈C,
ζn=T((1 −ιn)ηn+ιnTηn),
ϑn=T((1 −τn)ζn+τnTζn),
ηn+1 =T((1 −σn)ϑn+σnTϑn),
(7)
where {ιn},{τn},{σn}are sequences in [0,1] such
that 0< a ≤ιn, τn, σn≤b < 1for all n≥1.
2 Preliminaries
Let Cbe a nonempty closed convex subset of a Banach
space X,and let {ηn}be a bounded sequence in X.For
ρ∈X,we set r(ρ, {ηn}) = lim supn→∞ ∥ηn−ρ∥.
The asymptotic radius of {ηn}relative to Cis given
by r(C,{ηn}) = inf{r(ρ, {ηn}) : ρ∈C}and the
asymptotic center of {ηn}relative to Cis the set
A(C,{ηn}) = {ρ∈C:r(ρ, {ηn}) = r(C,{ηn})}.
It is known that, in a uniformly convex Banach space,
A(C,{ηn})consists of exactly one point.
We say that a Banach space Xhas Opial property
[15] if and only if for all {ηn}in Cwhich weakly con-
verges to ρ∈Xand
Lemma 2.1. [7] Let Tbe a mapping on a subset Cof
a Banach space Xhaving the Opial property. Assume
that Tsatisfies the condition (E).If {ηn}converges
weakly to ωand limn→∞ ∥Tηn−ηn∥= 0,then ω∈
F(T).
Lemma 2.2. [7] Let Tbe a mapping on a subset Cof
a Banach space X. If Tsatisfies condition (E),then
for all v∈F(T)and ρ∈C,we have
∥Tρ−v∥ ≤ ∥ρ−v∥.
Lemma 2.3. [7] Let Tbe a mapping on a subset Cof
a Banach space X. If Tsatisfies condition (C),then
Talso satisfies condition (Eµ)with µ= 3.
Lemma 2.4. [16] Let Xbe a uniformly convex Ba-
nach space and 0< a ≤σn≤b < 1for all n≥1.
If {ηn}and {δn}are two sequences in Xsuch that
lim supn→∞ ∥ηn∥ ≤ λ, lim sup n→ ∞∥δn∥ ≤ λ,
and limn→∞ ∥σnηn+ (1 −σn)δn∥=λfor some
λ≥0,then limn→∞ ∥ηn−δn∥= 0.
3 Main results
Lemma 3.1. Let Cbe a nonempty closed convex sub-
set of X, which is a uniformly convex Banach space,
and T:C→Cbe a mapping obeying condition (E)
with F(T)=∅. For all n≥1, η0∈C. Autho-
rize the sequence {ηn}to be produced by (7), then
the limn→∞ ∥ηn−v∥exists for any v∈F(T).
Proof. Let v∈F(T)and ηn, ζn, ϑn∈C. Because T
is a mapping satisfying condition (E), we obtain
∥Tηn−v∥ ≤ µ∥Tv−v∥+∥ηn−v∥,
∥Tζn−v∥ ≤ µ∥Tv−v∥+∥ζn−v∥,
∥Tϑn−v∥ ≤ µ∥Tv−v∥+∥ϑn−v∥.
(8)
So,
∥ζn−v∥
=∥T((1 −ιn)ηn+ιnTηn)−v∥
≤ ∥(1 −ιn)ηn+ιnTηn−v∥
≤(1 −ιn)∥ηn−v∥+ιn∥Tηn−v∥
≤(1 −ιn)∥ηn−v∥+ιn[µ∥Tv−v∥+∥ηn−v∥]
= (1 −ιn)∥ηn−v∥+ιn∥ηn−v∥
=∥ηn−v∥.(9)
Using (9), we obtain
∥ϑn−v∥
=∥T((1 −τn)ζn+τnTζn)−v∥
≤ ∥(1 −τn)ζn+τnTζn−v∥
≤(1 −τn)∥ζn−v∥+τn∥Tζn−v∥
≤(1 −τn)∥ζn−v∥+τn[µ∥Tv−v∥+∥ζn−v∥]
=∥ζn−v∥
≤ ∥ηn−v∥.(10)
Similarly, using (10), we obtain
∥ηn+1 −v∥
=∥T((1 −σn)ϑn+σnTϑn)−v∥
≤ ∥(1 −σn)ϑn+σnTϑn−v∥
≤(1 −σn)∥ϑn−v∥+σn∥Tϑn−v∥
≤(1 −σn)∥ϑn−v∥+σn[µ∥Tv−v∥+∥ϑn−v∥]
=∥ϑn−v∥
≤ ∥ηn−v∥.(11)
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This means that for all v∈F(T),{∥ηn−v∥}
is bounded and non-increasing. As a
result,limn→∞ ∥ηn−v∥exists, as necessary.
Theorem 3.2. Let Cbe a nonempty closed convex
subset of X, which is a uniformly convex Banach
space, and T:C→Cbe a mapping obeying
condition (E).For all n≥1, η0∈C. Autho-
rize the sequence {ηn}to be produced by (7), where
{ιn},{τn},{σn}are sequences in [0,1] such that
0< a ≤ιn, τn, σn≤b < 1. Then F(T)=∅if
and only if limn→∞ ∥Tηn−ηn∥= 0.
Proof. Suppoes F(T)=∅and let v∈F(T).Then,
by Lemma 3.1, limn→∞ ∥ηn−v∥exists and {ηn}is
bounded. Put
lim
n→∞ ∥ηn−v∥=λ≥0.(12)
From (9) and (12), we obtain
lim sup
n→∞
∥ζn−v∥ ≤ lim sup
n→∞
∥ηn−v∥=λ. (13)
From (8), we obtain
lim sup
n→∞
∥Tηn−v∥
≤lim sup
n→∞
µ∥Tv−v∥+lim sup
n→∞
∥ηn−v∥
=lim sup
n→∞
∥ηn−v∥=λ. (14)
From (10) and (12), we obtain
lim sup
n→∞
∥ϑn−v∥ ≤ lim sup
n→∞
∥ηn−v∥=λ. (15)
From (8), we obtain
lim sup
n→∞
∥Tϑn−v∥
≤lim sup
n→∞
µ∥Tv−v∥+lim sup
n→∞
∥ϑn−v∥
=lim sup
n→∞
∥ηn−v∥=λ. (16)
From (11), we obtain
∥ηn+1 −v∥ ≤ ∥ϑn−v∥
Therefore,
λ≤lim inf
n→∞ ∥ϑn−v∥.(17)
Using (15) and (17), we have
λ=lim
n→∞ ∥ϑn−v∥.(18)
From (10), we obtain
∥ϑn−v∥ ≤ ∥ζn−v∥.
So,
λ≤lim inf
n→∞ ∥ζn−v∥.(19)
From (13) and (19), we have
λ=lim
n→∞ ∥ζn−v∥.(20)
Using (20), (12) and (14), we obtain
λ=lim
n→∞ ∥ζn−v∥
=lim
n→∞ ∥T((1 −ιn)ηn+ιnTηn)−v∥
≤lim
n→∞ ∥(1 −ιn)ηn+ιnTηn−v∥
≤(1 −ιn)lim
n→∞ ∥ηn−v∥+ιnlim
n→∞ ∥Tηn−v∥
=λ.
Therefore,
λ=lim
n→∞ ∥T((1 −ιn)ηn+ιnTηn)−v∥.(21)
Applying Lemma 2.4, we have
lim
n→∞ ∥Tηn−ηn∥= 0.(22)
Conversely, suppose that {ηn}is bounded and
limn→∞ ∥Tηn−ηn∥= 0.Let A(C,{ηn}).By (8),
we have
r(Tv, {ηn}) = lim sup
n→∞
∥ηn−Tv∥
≤lim sup
n→∞
(µ∥Tηn−ηn∥+∥ηn−v∥)
≤lim sup
n→∞
∥ηn−v∥
=r(v, {ηn}).
This means that Tv∈A(C,{ηn}).Because Xis uni-
formly convex, is a singleton set and hence we have
Tv=v. Thus, F(T)=∅.
Theorem 3.3. Let Cbe a nonempty closed convex
subset of X, which is a uniformly convex Banach
space with the Opial property, and T:C→Cbe a
mapping obeying condition (E). For all n≥1, η0∈
C. Authorize the sequence {ηn}to be produced by
(7), where {ιn},{τn},{σn}are sequences in [0,1]
such that 0< a ≤ιn, τn, σn≤b < 1. Then {ηn}
converges weakly to a fixed point of T.
Proof. Let F(T)=∅implies that {ηn}is bounded
and limn→∞ ∥Tηn−ηn∥= 0.Because Xis uni-
formly convex, it is reflexive. According to Eberlin’s
theorem, there exists a subsequence of {ηnj}of {ηn}
which thus converges weakly to some ω∈X. Be-
cause Cis closed and convex, Mazur’s theorem states
that ω∈C. And ω∈F(T), that according Lemma
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2.1. We also show that {ηn}weakly converges to
ω. If this is not the circumstance, then there must be
a subsequence {ηnk}of {ηn}such that {ηnk}con-
verges weakly to ϖ∈Cand ω=ϖ. By Lemma 2.1,
ϖ∈F(T).Because limn→∞ ∥ηn−v∥exists for ev-
ery v∈F(T). By Lemma 3.1 and Opial property, we
have
lim
n→∞ ∥ηn−ω∥=lim
j→∞ ∥xnj−ω∥ ≤ lim
j→∞ ∥ηnj−ϖ∥
=lim
n→∞ ∥ηn−ϖ∥=lim
k→∞
∥ηnk−ϖ∥
<lim
k→∞
∥ηnk−ω∥=lim
n→∞ ∥ηn−ω∥,
which is a contradiction. So ω=ϖ. This means that
{ηn}converges weakly to a fixed point of T.
Next we prove the strong convergence theorem.
Theorem 3.4. Let Cbe a nonempty closed convex
subset of X, which is a uniformly convex Banach
space, and T:C→Cbe a mapping obeying
condition (E). For all n≥1, η0∈C. Autho-
rize the sequence {ηn}to be produced by (7), where
{ιn},{τn},{σn}are sequences in [0,1] such that
0< a ≤ιn, τn, σn≤b < 1. Then {ηn}converges
strongly to a fixed point of T.
Proof. We know that F(T)=∅according to Lemma
2.1, and that limn→∞ ∥Tηn−ηn∥= 0 owing to The-
orem 3.2. Because Cis compact, there exists a sub-
sequence {ηnk}of {ηn}such that {ηnk}converges
strongly to vfor some v∈C. Because Tsatisfies
condition (E),we have
∥ηnk−Tv∥ ≤ µ∥Tηnk−v∥+∥ηnk−v∥,for all n≥1.
Taking k→ ∞,we obtain Tv=v i.e., v ∈F(T).
By Lemma 3.1, limn→∞ ∥ηn−v∥exists for every v∈
F(T)and so {ηn}converge strongly to v.
Senter and Dotson [19] suggested the concept of
a mapping obeying the condition (I), which will be
defined as: If there exists a nondecreasing function
f: [0,∞)→[0,∞)with f(0) = 0 and f(u)>0for
all u > 0such that ∥ρ−Tρ∥ ≥ f(dist(F(T)) for all
ρ∈C, where dist(ρ, F(T)) = infv∈F(T)∥ρ−v∥, and
T:C→C.
We now use condition (I)to show the strong con-
vergence theorem.
Theorem 3.5. Let Cbe a nonempty closed convex
subset of X, which is a uniformly convex Banach
space, and T:C→Cbe a mapping obeying
condition (E). For all n≥1, η0∈C. Autho-
rize the sequence {ηn}to be produced by (7), where
{ιn},{τn},{σn}are sequences in [0,1] such that
0< a ≤ιn, τn, σn≤b < 1with F(T)=∅.If T
satisfies condition (I),then {ηn}converges strongly
to a fixed point of T.
Proof. By Lemma 3.1, limn→∞ ∥ηn−v∥exists for
every v∈F(T)and limn→∞ d(ηn,F(T)) exists. As-
sume that limn→∞ ∥ηn−v∥for some u≥0. If u= 0
then the result follows. Suppose u > 0. From the hy-
pothesis and condition (I),
f(dist(ηn,F(T))) ≤ |Tηn−ηn∥.(23)
Because F(T)=∅, by Theorem 3.2, we have
limn→∞ ∥Tηn−ηn∥= 0.So (23) implies that
lim
n→∞ f(dist(ηn,F(T))) = 0.(24)
Because fis nondecreasing function, as a result of
(24), we have limn→∞ dist(ηn,F(T)) = 0.Thus, we
have a subsequence {ηnk}of {ηn}and a sequence
{ϑn}such that
∥ηnk−ϑk∥ ≤ 1
2kfor all k∈N.
So, using (11), we get
∥ηnk+1 −ϑk∥ ≤ ∥ϑnk−ϑk∥ ≤ 1
2k.
Hence
∥ϑnk+1 −ϑk∥ ≤ ∥ϑnk+1 −ηnk+1 ∥+∥ηnk+1 −ϑk∥
≤1
2k+1 +1
2k
<1
2k−1→0as k→ ∞.
This shows that {ϑn}is Cauchy sequence in F(T)and
so it converges to a point v. Because F(T)is closed,
v∈F(T)and then {ηnk}converges strongly to v.
Because limn→∞ ∥ηn−v∥exists, we have that ηn→
v∈F(T).
4 Numerical Example
Let E= (−∞,∞)with usual norm and C= [1,10].
Define Ton Csatisfying condition (E)with µ= 3 as
follow:
Tη=2η+ 5
3.
We will show that
|Tη−δ| ≤ 3|Tδ−δ|+|η−δ|for all η, δ ∈C.
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In fact,
|Tη−δ| ≤ |δ−Tδ|+|Tδ−Tη|
≤ |δ−Tδ|+
2δ+ 5
3−2η+ 5
3
=|δ−Tδ|+2
3|η−δ|
≤3|Tδ−δ|+|η−δ|.
Now, we conclude that Tsatisfies condition (E). Us-
ing the initial value η1= 8.5and letting the stop-
ping criteria |ηn−5|<10−6,reckoning the iter-
ative values of K∗-iteration process, AK-iteration
process and SP ∗-iteration process for choose ιn=
8n
9n+1 , τn=9n
10n+1 ,and σn=7n
8n+1 as show in Table
1 and Figure 1.
Table 1: Comparative sequence
Iter. K∗AK SP ∗
1 8.500000 8.500000 8.500000
2 5.829630 5.553086 5.446939
3 5.189445 5.084198 5.053214
4 5.042681 5.012646 5.006173
5 5.009549 5.001886 5.000706
6 5.002127 5.000280 5.000080
7 5.000473 5.000041 5.000009
8 5.000105 5.000006 5.000001
9 5.000023 5.000006 5.000000
10 5.000005 5.000000 5.000000
11 5.000001 5.000000 5.000000
12 5.000000 5.000000 5.000000
Figure 1: The plotting of comparative sequence in Ta-
ble 1
5 Conclusion
In this study, we proposed a new modified fixed point
algorithms to approximate the solution of fixed points
problem of a nonexpansive mapping in the framework
of Banach space. We performed convergence analy-
sis of the proposed algorithm and hence proved some
convergence theorems. Also, we provided some il-
lustrative numerical examples to show the efficiency
of the proposed algorithm.
6 Acknowledgments
This project was supported by the Research and De-
velopment Institute, Rambhai Barni Rajabhat Univer-
sity (Grant no.2220/2565).
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.71
Anantachai Padcharoen, Naknimit Akkasriworn
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Volume 21, 2022
