FIXED POINT THEOREMS FOR SUZUKI NONEXPAN-
SIVE MAPPINGS IN BANACH SPACES
John Sebastian
Department of Mathematics, Central University of Kerala, Kasaragod, India.
E-mail: john.sebastian@cukerala.ac.in
ORCID: 0000-0002-6759-9228
Shaini Pulickakunnel
Department of Mathematics, Central University of Kerala, Kasaragod, India.
E-mail: shainipv@cukerala.ac.in
ORCID: 0000-0001-9958-9211
Reception: 05/08/2022 Acceptance: 20/08/2022 Publication: 29/12/2022
Suggested citation:
Sebastian, J. and Pulickakunnel, S. (2022). Fixed point theorems for suzuki nonexpansive mappings in banach spaces. 3C
TIC. Cuadernos de desarrollo aplicados a las TIC,11 (2), 15-24. https://doi.org/10.17993/3ctic.2022.112.15-24
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
15
ABSTRACT
In this paper, we investigate the existence of fixed points for Suzuki nonexpansive mappings in the setting
of Banach spaces using the asymptotic center technique. We also establish the convergence of regular
approximate fixed point sequence to the fixed points of Suzuki nonexpansive mappings. Examples are
also given to illustrate the results. Our theorems generalize several results in the literature.
KEYWORDS
Banach space, Nonexpansive mapping, Suzuki nonexpansive mapping, Fixed point, Approximate fixed
point.
https://doi.org/10.17993/3ctic.2022.112.15-24
1 INTRODUCTION
Fixed point results for nonexpansive mappings in Banach spaces are of great importance in the
development of fixed point theory and are widely used to solve problems in diverse fields such as
differential equations, game theory, engineering, medicine and many more (see [3,14,16]. The possibility
of using the theory in a wide range of applications has attracted many researchers and has consequently
resulted in a rapid growth of research in this field. Several authors have introduced extensions of
nonexpansive mappings such as generalized nonexpansive mappings, relatively nonexpansive mappings,
α
- nonexpansive mappings, etc. (see [1, 6, 15]) and proved fixed point results in Banach spaces and
various other spaces as well. In 2008, Suzuki introduced a new condition called condition (C) [17]. The
mapping which satisfies condition (C) is now known as Suzuki nonexpansive mapping. Suzuki proved
that all nonexpansive mappings satisfy the condition (C). Unlike nonexpansive mappings, the Suzuki
nonexpansive mappings need not be always continuous. We can find a couple of examples for mappings
which are not continuous but satisfying condition (C) in [17].
There are several techniques for finding the fixed points of nonexpansive mappings. One of the
most widely used techniques, introduced by Edelstein in 1972 [5], uses the concept of asymptotic radius
and asymptotic center of a sequence relative to a set
K
. Many researchers have used the properties
and geometric behavior of the asymptotic center of sequences under consideration, to prove several
fixed point results for nonexpansive mappings (see [5, 9]). For any sequence, asymptotic center can
be considered as the intersection of some closed balls [7]. Therefore, the asymptotic center is always
closed. But it need not be nonempty. Researchers proved that if a set
K
is nonempty, weakly compact
and convex, then the asymptotic center of any sequence in
K
has the same properties as
K
[7]. These
results boosted the usefulness of asymptotic center technique as a tool to find fixed points for Suzuki
nonexpansive mapping. Dhompongsa [4] used this technique in Suzuki nonexpansive mapping and
proved that if
K
is a Banach space and
T
is a self mapping of
K
satisfying condition (C), then for any
bounded approximate fixed point sequence in
K
, the asymptotic center relative to
K
is invariant under
T
. Another equally important method to find fixed points in Banach spaces is the Chebyshev center
technique, which uses the concept of Chebyshev radius and Chebyshev center to analyze the geometric
structure of a set. A good amount of research work is reported in literature which makes use of these
two techniques to find fixed points (see [4,5, 7–10,13].
In this paper, we primarily focus on the asymptotic center technique and show that it is possi-
ble to derive an interesting relation between the asymptotic radius and Chebyshev radius under certain
conditions. In [8], Kirk proved a fixed point result for nonexpansive mapping in reflexive Banach spaces
having normal structure. We extended this result for Suzuki nonexpansive mapping in weakly compact
Banach spaces. Also, we investigated some sufficient conditions for the existence of fixed points for
Suzuki nonexpansive mapping in a closed, bounded and convex subset of a Banach space. Apart from
these, we also developed certain sufficient conditions for the convergence of regular approximate fixed
point sequences to the fixed points of Suzuki nonexpansive mappings.
1.1 PRELIMINARIES
Definition 1. [7] A mapping
T
on a subset
K
of a Banach space
X
is called a nonexpansive mapping
if ||Tx−Ty|| ≤ ||x−y|| for all x, y ∈K.
Definition 2. [17] Let
T
be a mapping on a subset
K
of a Banach space
X
. Then
T
is said to satisfy
condition (C) if for all x, y ∈K,
1
2||x−Tx|| ≤ ||x−y|| =⇒ ||Tx−Ty||≤||x−y||.
The mapping satisfying condition (C) is called Suzuki nonexpansive mapping.
Definition 3. [10] Let
T
:
K→K
be any mapping. A sequence
{xn}
in
K
is called approximate fixed
point sequence if ||Tx
n−xn|| → 0as n→∞.
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
16
ABSTRACT
In this paper, we investigate the existence of fixed points for Suzuki nonexpansive mappings in the setting
of Banach spaces using the asymptotic center technique. We also establish the convergence of regular
approximate fixed point sequence to the fixed points of Suzuki nonexpansive mappings. Examples are
also given to illustrate the results. Our theorems generalize several results in the literature.
KEYWORDS
Banach space, Nonexpansive mapping, Suzuki nonexpansive mapping, Fixed point, Approximate fixed
point.
https://doi.org/10.17993/3ctic.2022.112.15-24
1 INTRODUCTION
Fixed point results for nonexpansive mappings in Banach spaces are of great importance in the
development of fixed point theory and are widely used to solve problems in diverse fields such as
differential equations, game theory, engineering, medicine and many more (see [3,14,16]. The possibility
of using the theory in a wide range of applications has attracted many researchers and has consequently
resulted in a rapid growth of research in this field. Several authors have introduced extensions of
nonexpansive mappings such as generalized nonexpansive mappings, relatively nonexpansive mappings,
α
- nonexpansive mappings, etc. (see [1, 6, 15]) and proved fixed point results in Banach spaces and
various other spaces as well. In 2008, Suzuki introduced a new condition called condition (C) [17]. The
mapping which satisfies condition (C) is now known as Suzuki nonexpansive mapping. Suzuki proved
that all nonexpansive mappings satisfy the condition (C). Unlike nonexpansive mappings, the Suzuki
nonexpansive mappings need not be always continuous. We can find a couple of examples for mappings
which are not continuous but satisfying condition (C) in [17].
There are several techniques for finding the fixed points of nonexpansive mappings. One of the
most widely used techniques, introduced by Edelstein in 1972 [5], uses the concept of asymptotic radius
and asymptotic center of a sequence relative to a set
K
. Many researchers have used the properties
and geometric behavior of the asymptotic center of sequences under consideration, to prove several
fixed point results for nonexpansive mappings (see [5, 9]). For any sequence, asymptotic center can
be considered as the intersection of some closed balls [7]. Therefore, the asymptotic center is always
closed. But it need not be nonempty. Researchers proved that if a set
K
is nonempty, weakly compact
and convex, then the asymptotic center of any sequence in
K
has the same properties as
K
[7]. These
results boosted the usefulness of asymptotic center technique as a tool to find fixed points for Suzuki
nonexpansive mapping. Dhompongsa [4] used this technique in Suzuki nonexpansive mapping and
proved that if
K
is a Banach space and
T
is a self mapping of
K
satisfying condition (C), then for any
bounded approximate fixed point sequence in
K
, the asymptotic center relative to
K
is invariant under
T
. Another equally important method to find fixed points in Banach spaces is the Chebyshev center
technique, which uses the concept of Chebyshev radius and Chebyshev center to analyze the geometric
structure of a set. A good amount of research work is reported in literature which makes use of these
two techniques to find fixed points (see [4,5, 7–10,13].
In this paper, we primarily focus on the asymptotic center technique and show that it is possi-
ble to derive an interesting relation between the asymptotic radius and Chebyshev radius under certain
conditions. In [8], Kirk proved a fixed point result for nonexpansive mapping in reflexive Banach spaces
having normal structure. We extended this result for Suzuki nonexpansive mapping in weakly compact
Banach spaces. Also, we investigated some sufficient conditions for the existence of fixed points for
Suzuki nonexpansive mapping in a closed, bounded and convex subset of a Banach space. Apart from
these, we also developed certain sufficient conditions for the convergence of regular approximate fixed
point sequences to the fixed points of Suzuki nonexpansive mappings.
1.1 PRELIMINARIES
Definition 1. [7] A mapping
T
on a subset
K
of a Banach space
X
is called a nonexpansive mapping
if ||Tx−Ty|| ≤ ||x−y|| for all x, y ∈K.
Definition 2. [17] Let
T
be a mapping on a subset
K
of a Banach space
X
. Then
T
is said to satisfy
condition (C) if for all x, y ∈K,
1
2||x−Tx|| ≤ ||x−y|| =⇒ ||Tx−Ty||≤||x−y||.
The mapping satisfying condition (C) is called Suzuki nonexpansive mapping.
Definition 3. [10] Let
T
:
K→K
be any mapping. A sequence
{xn}
in
K
is called approximate fixed
point sequence if ||Tx
n−xn|| → 0as n→∞.
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
17
Definition 4. [4] Let
K
be a nonempty closed and convex subset of a Banach space
X
and
{xn}
,a
bounded sequence in X. For x∈X, the asymptotic radius of {xn}at xis defined as
r(x, {xn}) = lim sup{||xn−x||}.
The asymptotic radius and asymptotic center of {xn}relative to Kare defined as follows:
r≡r(K, {xn})=inf{r(x, {xn}):x∈K}
A≡A(K, {xn})={x∈K:r(x, {xn})=r}.
Definition 5. [9] A bounded sequence is said to be regular if each of its subsequence has the same
asymptotic radius.
Definition 6. [9] A bounded sequence is said to be uniform if each of its subsequence has the same
asymptotic center.
Definition 7. [7] For any subset
K
of
X
, the radius of
K
relative to
x
, Chebyshev radius of
K
,
Chebyshev center of Kand the diameter of Kare defined as follows:
For any x∈X,rx(K) = sup{||x−y|| :y∈K}
r(K) = inf{rx(K):x∈K}
C(K)={x∈K:rx(K)=r(K)}
diam(K)=sup{rx(K):x∈K}.
Definition 8. A nonempty, closed, convex subset
D
of a given set
K
is said to be a minimal invariant
set for a mapping
T
:
K→K
if
T
(
D
)
⊆D
and
D
has no nonempty, closed and convex proper subsets
which are T-invariant.
Definition 9. [7, 8] A convex subset
K
of
X
is said to have normal structure if each bounded, convex
subset Sof Kwith diamS > 0contains a nondiametral point.
Definition 10. [2] A convex set
K
of
X
is said to have asymptotic normal structure if, given any
bounded convex subset
S
of
K
which contains more than one point and given any decreasing net of
nonempty subsets
{sα
;
α∈A}
of
S
, the asymptotic center of
{sα
;
α∈A}
in
S
is a proper subset of
S
.
Definition 11. [11] Let X be a Banach space.
X
is said to have Opial property if for each weakly
convergent sequence {xn}in Xwith weak limit z and for all x∈Xwith x=z,
lim sup ||xn−z|| <lim sup ||xn−x||.
Lemma 1. [17, Lemma 6] Let
T
be a mapping on a bounded convex subset
K
of a Banach space
X
.
Assume that Tsatisfies condition (C). Define a sequence {xn}in Kby x1∈Kand
xn+1 =λT xn+ (1 −λ)xn
for n∈N, where λis a real number belonging to [1
2,1). Then
lim
n→∞ ||Tx
n−xn|| =0
holds.
Lemma 2. [7, Lemma 9.1] Let
{xn}
be a sequence in a Banach space
X
and
K
a nonempty subset of
X.
(a) If Kis weakly compact, then A(K, {xn})=ϕ.
(b) If Kis convex, then A(K, {xn})is convex.
Lemma 3. [4, Lemma 3.1] Let
K
be a subset of a Banach space
X
, and
T
:
K→K
be a mapping
satisfying condition (C). Suppose
{xn}
is a bounded approximate fixed point sequence for
T
. Then
A(K, {xn})is invariant under T.
https://doi.org/10.17993/3ctic.2022.112.15-24
Proposition 1. [9, Preposition 1] Every bounded sequence has a regular subsequence.
Theorem 1. [7, Theorem 3.2] Suppose
K
is a nonempty, weakly compact, convex subset of a Banach
space. Then for any mapping
T
:
K→K
there exists a closed convex subset of
K
which is minimal
T-invariant.
Theorem 2. [12, Theorem 1] A convex subset
K
of
X
has normal structure if and only if
K
has
asymptotic normal structure.
Remark 1. [7] It is clear that if for any sequence
{xn}
in
K
and
x∈X
,
r
(
x, {xn}
)=0if and only if
lim
n→∞ xn=x.
Remark 2. [9] If
{xnk}
is a subset of
{xn}
, then
r
(
K, {xnk}
)
≤r
(
K, {xn}
)and if
r
(
K, {xnk}
)=
r(K, {xn}), then A(K, {xn})⊆A(K, {xnk}).
2 RESULTS
Theorem 3. Let
K
be a weakly compact convex subset of a Banach space
X
and
T
:
K→K
satisfies
condition (C). Assume that
K
is minimal
T
-invariant and
{xn}
is an approximate fixed point sequence
in K. Then
(i) A(K, {xn})=K
(ii) r(K, {xn})=r(K).
Proof. Let Kbe a weakly compact and convex subset of a Banach space X.
Suppose diam(K)=0. Then there is nothing to prove.
Now, suppose diam(K)>0.
Let {xn}be any bounded approximate fixed point sequence in K. Then A(K, {xn})is closed.
Also by Lemma 2, A(K, {xn})is nonempty and convex. Thus A(K, {xn})is weakly compact.
By Lemma 3, A(K, {xn})is T-invariant and by the minimality of K, we have A(K, {xn})=K.
Since
K
is weakly compact, there exist a subsequence
{xnk}
of
{xn}
and
z∈K
such that
xnk→z
weakly.
Clearly {xnk}is an approximate fixed point sequence in K.
Since A(K, {xnk})=K,lim sup ||xnk−x|| =r(K, {xnk})for all x∈K.
Also, we have for any x∈K,lim sup ||xnk−x|| ≤ sup{||x−y|| :y∈K}.
Therefore,
r(K, {xnk})≤rx(K)=⇒r(K, {xnk})≤r(K).(1)
Now, for any y∈K,xnk−y→z−yweakly and hence we have
||z−y|| ≤ lim sup ||xnk−y|| =r(K, {xnk}).
Hence for all y∈K,
||z−y|| ≤ r(K, {xnk})=⇒rz(K)≤r(K, {xnk})
=⇒r(K)≤r(K, {xnk}).(2)
Thus from (1) and (2) we get
r(K, {xnk})=r(K).(3)
We know that for any subsequence {xnk}of {xn},
r(K, {xnk})≤r(K, {xn}).(4)
Since r(K, {xnk})=r(K), if x∈C(K), then for all y∈K,||x−y|| ≤ r(K, {xnk}).
Therefore, for all xn,
||xn−x|| ≤ r(K, {xnk})=⇒r(K, {xn})≤r(K, {xnk}).(5)
Hence from (4) and (5), r(K, {xn})=r(K, {xnk}).Thus from (3) we get
r(K, {xn})=r(K).
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
18
Definition 4. [4] Let
K
be a nonempty closed and convex subset of a Banach space
X
and
{xn}
,a
bounded sequence in X. For x∈X, the asymptotic radius of {xn}at xis defined as
r(x, {xn}) = lim sup{||xn−x||}.
The asymptotic radius and asymptotic center of {xn}relative to Kare defined as follows:
r≡r(K, {xn})=inf{r(x, {xn}):x∈K}
A≡A(K, {xn})={x∈K:r(x, {xn})=r}.
Definition 5. [9] A bounded sequence is said to be regular if each of its subsequence has the same
asymptotic radius.
Definition 6. [9] A bounded sequence is said to be uniform if each of its subsequence has the same
asymptotic center.
Definition 7. [7] For any subset
K
of
X
, the radius of
K
relative to
x
, Chebyshev radius of
K
,
Chebyshev center of Kand the diameter of Kare defined as follows:
For any x∈X,rx(K) = sup{||x−y|| :y∈K}
r(K) = inf{rx(K):x∈K}
C(K)={x∈K:rx(K)=r(K)}
diam(K)=sup{rx(K):x∈K}.
Definition 8. A nonempty, closed, convex subset
D
of a given set
K
is said to be a minimal invariant
set for a mapping
T
:
K→K
if
T
(
D
)
⊆D
and
D
has no nonempty, closed and convex proper subsets
which are T-invariant.
Definition 9. [7, 8] A convex subset
K
of
X
is said to have normal structure if each bounded, convex
subset Sof Kwith diamS > 0contains a nondiametral point.
Definition 10. [2] A convex set
K
of
X
is said to have asymptotic normal structure if, given any
bounded convex subset
S
of
K
which contains more than one point and given any decreasing net of
nonempty subsets
{sα
;
α∈A}
of
S
, the asymptotic center of
{sα
;
α∈A}
in
S
is a proper subset of
S
.
Definition 11. [11] Let X be a Banach space.
X
is said to have Opial property if for each weakly
convergent sequence {xn}in Xwith weak limit z and for all x∈Xwith x=z,
lim sup ||xn−z|| <lim sup ||xn−x||.
Lemma 1. [17, Lemma 6] Let
T
be a mapping on a bounded convex subset
K
of a Banach space
X
.
Assume that Tsatisfies condition (C). Define a sequence {xn}in Kby x1∈Kand
xn+1 =λT xn+ (1 −λ)xn
for n∈N, where λis a real number belonging to [1
2,1). Then
lim
n→∞ ||Tx
n−xn|| =0
holds.
Lemma 2. [7, Lemma 9.1] Let
{xn}
be a sequence in a Banach space
X
and
K
a nonempty subset of
X.
(a) If Kis weakly compact, then A(K, {xn})=ϕ.
(b) If Kis convex, then A(K, {xn})is convex.
Lemma 3. [4, Lemma 3.1] Let
K
be a subset of a Banach space
X
, and
T
:
K→K
be a mapping
satisfying condition (C). Suppose
{xn}
is a bounded approximate fixed point sequence for
T
. Then
A(K, {xn})is invariant under T.
https://doi.org/10.17993/3ctic.2022.112.15-24
Proposition 1. [9, Preposition 1] Every bounded sequence has a regular subsequence.
Theorem 1. [7, Theorem 3.2] Suppose
K
is a nonempty, weakly compact, convex subset of a Banach
space. Then for any mapping
T
:
K→K
there exists a closed convex subset of
K
which is minimal
T-invariant.
Theorem 2. [12, Theorem 1] A convex subset
K
of
X
has normal structure if and only if
K
has
asymptotic normal structure.
Remark 1. [7] It is clear that if for any sequence
{xn}
in
K
and
x∈X
,
r
(
x, {xn}
)=0if and only if
lim
n→∞ xn=x.
Remark 2. [9] If
{xnk}
is a subset of
{xn}
, then
r
(
K, {xnk}
)
≤r
(
K, {xn}
)and if
r
(
K, {xnk}
)=
r(K, {xn}), then A(K, {xn})⊆A(K, {xnk}).
2 RESULTS
Theorem 3. Let
K
be a weakly compact convex subset of a Banach space
X
and
T
:
K→K
satisfies
condition (C). Assume that
K
is minimal
T
-invariant and
{xn}
is an approximate fixed point sequence
in K. Then
(i) A(K, {xn})=K
(ii) r(K, {xn})=r(K).
Proof. Let Kbe a weakly compact and convex subset of a Banach space X.
Suppose diam(K)=0. Then there is nothing to prove.
Now, suppose diam(K)>0.
Let {xn}be any bounded approximate fixed point sequence in K. Then A(K, {xn})is closed.
Also by Lemma 2, A(K, {xn})is nonempty and convex. Thus A(K, {xn})is weakly compact.
By Lemma 3, A(K, {xn})is T-invariant and by the minimality of K, we have A(K, {xn})=K.
Since
K
is weakly compact, there exist a subsequence
{xnk}
of
{xn}
and
z∈K
such that
xnk→z
weakly.
Clearly {xnk}is an approximate fixed point sequence in K.
Since A(K, {xnk})=K,lim sup ||xnk−x|| =r(K, {xnk})for all x∈K.
Also, we have for any x∈K,lim sup ||xnk−x|| ≤ sup{||x−y|| :y∈K}.
Therefore,
r(K, {xnk})≤rx(K)=⇒r(K, {xnk})≤r(K).(1)
Now, for any y∈K,xnk−y→z−yweakly and hence we have
||z−y|| ≤ lim sup ||xnk−y|| =r(K, {xnk}).
Hence for all y∈K,
||z−y|| ≤ r(K, {xnk})=⇒rz(K)≤r(K, {xnk})
=⇒r(K)≤r(K, {xnk}).(2)
Thus from (1) and (2) we get
r(K, {xnk})=r(K).(3)
We know that for any subsequence {xnk}of {xn},
r(K, {xnk})≤r(K, {xn}).(4)
Since r(K, {xnk})=r(K), if x∈C(K), then for all y∈K,||x−y|| ≤ r(K, {xnk}).
Therefore, for all xn,
||xn−x|| ≤ r(K, {xnk})=⇒r(K, {xn})≤r(K, {xnk}).(5)
Hence from (4) and (5), r(K, {xn})=r(K, {xnk}).Thus from (3) we get
r(K, {xn})=r(K).
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
19
Corollary 1. Let
K
be a weakly compact convex subset of a Banach space
X
and
T
:
K→K
satisfies
condition (C). If
K
is minimal
T
-invariant, then every approximate fixed point sequence in
K
are
regular and uniform.
Proof. Let
{xn}
be an approximate fixed point sequence in
K
. Then every subsequence
{xnk}
of
{xn}is also an approximate fixed point sequence.
Hence by (ii) in Theorem 3, r(K, {xn})=r(K, {xnk}).
Therefore, {xn}is regular.
By (i) in Theorem 3, A(K, {xn})=A(K, {xnk})=K.
Hence {xn}is uniform.
Corollary 2. [10, Preposition 6.3] Let
K
be a weakly compact convex subset of a Banach space
X
,
and
T
:
K→K
be a nonexpansive mapping. Assume that
K
is minimal for
T
, that is, no closed convex
bounded proper subset of
K
is invariant for
T
. If
{xn}
is an approximate fixed point sequence in
K
,
then A(K, {xn})=K.
Proof. Since every nonexpansive mapping satisfies condition (C), by above theorem,
A
(
K, {xn}
)=
K
.
Theorem 4. Let
K
be a nonempty, weakly compact and convex subset of a Banach space
X
and suppose
Khas normal structure. Then every mapping T:K→Ksatisfying condition (C) has a fixed point.
Proof. By Theorem 1, we can consider Kas closed, convex minimal T- invariant subset.
Suppose diamK > 0.
Consider an approximate fixed point sequence {xn}⊆K.
By (i) in Theorem 3, A(K, {xn})=K.
Now, define Wn:= {xm:m≥n},n∈N.
Clearly, {Wn,n∈N}is a decreasing chain of nonempty bounded subsets of K.
We can easily prove that Asymptotic center of {Wn,n ∈N}=A(K, {xn})=K.
Since Khas normal structure, by Theorem 2, Khas asymptotic normal structure.
Thus A(K, {xn})=K, which is a contradiction.
Therefore, diamK =0and hence Khas only one element x(say).
Thus T(x)=x.
We obtain the result of Kirk [8] and Theorem 4.1 in [7] as corollaries of our result.
Corollary 3. [8, Theorem] Let
K
be a nonempty, bounded, closed and convex subset of a reflexive
Banach space
X
, and suppose that
K
has normal structure. If
T
:
K→K
is nonexpansive, then
T
has
a fixed point.
Proof. Since a bounded, closed and convex subset of a reflexive Banach space is weakly compact and
every nonexpansive mapping is Suzuki nonexpansive, by the above theorem, Thas a fixed point.
Corollary 4. [7, Theorem 4.1] Let
K
be a nonempty, weakly compact, convex subset of a Banach
space, and suppose
K
has normal structure. Then every nonexpansive mapping
T
:
K→K
has a fixed
point.
Proof. Since every nonexpansive mapping is Suzuki nonexpansive, by the above theorem,
T
has a
fixed point.
The following theorem gives sufficient conditions for the existence of fixed points for Suzuki nonexpansive
mapping in Banach spaces.
Theorem 5. Let
K
be a closed, bounded and convex subset of a Banach space
X
and
T
:
K→K
satisfies condition (C). If T(K)is contained in a compact subset of K, then Thas a fixed point in K.
https://doi.org/10.17993/3ctic.2022.112.15-24
Proof. Let {xn}be an approximate fixed point sequence in K.
Therefore, ||Tx
n−xn|| → 0.
Consider the sequence {Tx
n}in T(K).
Since
T
(
K
)is a subset of a compact set of
K
, there exist a subsequence
{Tx
nk}
of
{Tx
n}
and
z∈K
such that Tx
nk→z.
Therefore, lim
k→∞||xnk−z|| = lim
k→∞ ||xnk−Tx
nk|| =0.
Hence xnk→z.
Clearly, {xnk}is a bounded approximate fixed point sequence and A(K, {xnk})={z}.
Also by Lemma 3, A(K, {xnk})is T-invariant and hence T(z)=z.
The following is an example to illustrate this theorem.
Example 1. In the space l2consider the closed unit ball
K={x=(x1,x
2, ...)∈l2:||x||2≤1}.
Define T:K→Kas
T(x)=(1
3,0,0, ..), if x = (1,0,0, ...)
0, otherwise.
For all x, y = (1,0,0, ...)in K,||Tx−Ty||2=0≤ ||x−y||2.
If x= (1,0,0, ...)then 1
2||x−Tx||2=1
2||(2
3,0,0, ...)||2=1
3.
Therefore, if 1
2||x−Tx||2≤ ||x−y||2for any y∈K, then 1
3≤ ||x−y||2.
Thus ||Tx−Ty||2=1
3≤ ||x−y||2.
Hence Tsatisfies condition (C).
T(K)=(0,0,0, ...),(1
3,0,0, ...)is a subset of a compact set of K.
Thus all conditions in the above theorem are satisfied.
Since T(0) = 0,Thas a fixed point.
Theorem 6. Let
X
be a Banach space with Opial property and
K
be a closed, bounded and convex
subset of
X
. Let
T
:
K→K
satisfies condition (C) and if
T
(
K
)is contained in a weakly compact
subset of K, then Thas a fixed point in K.
Proof. Let {xn}be an approximate fixed point sequence in K. Then we have, ||Tx
n−xn|| → 0.
Since
1
2||Tx
n−xn|| ≤ ||Tx
n−xn||
and
T
satisfies condition (C), we have
||T2xn−Tx
n||≤||Tx
n−xn||
for all n∈N.
Therefore, lim
n→∞||T2xn−Tx
n|| ≤ lim
n→∞ ||Tx
n−xn|| =0.
Hence {T(xn)}is an approximate fixed point sequence.
Since
T
(
K
)is a subset of a weakly compact set of
K
, there exist a subsequence
{Tx
nk}
of
{Tx
n}
and
z∈Ksuch that Tx
nk→zweakly as k→∞.
By Opial property, for any x=z,lim sup ||Tx
nk−z|| <lim sup ||Tx
nk−x||.
Therefore, for any
x
=
z,r
(
z,{Tx
nk}
)
<r
(
x, {Tx
nk}
)=
⇒r
(
K, {Tx
nk}
)=
r
(
z,{Tx
nk}
)and
A(K, {Tx
nk})={z}.
By Lemma 3, A(K, {xnk})is T-invariant and hence T(z)=z.
Corollary 5. [17, Theorem 4] Let
T
be a mapping on a convex subset
K
of a Banach space
X
. Assume
that Tsatisfies condition (C). Assume also that either of the following holds:
(i) Kis compact;
(ii) Kis weakly compact and Xhas the Opial property.
Then Thas a fixed point.
Proof. Suppose Kis compact and convex. Therefore, Kis closed, bounded and convex.
Also, since T:K→K, we have T(K)⊆K. Thus T(K)is a subset of a compact set.
Hence by Theorem 5, Thas a fixed point.
Now suppose Kis weakly compact, convex and Xhas the Opial property.
Therefore, Kis closed, bounded and convex. Also, since T:K→K, we have T(K)⊆K.
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
20
Corollary 1. Let
K
be a weakly compact convex subset of a Banach space
X
and
T
:
K→K
satisfies
condition (C). If
K
is minimal
T
-invariant, then every approximate fixed point sequence in
K
are
regular and uniform.
Proof. Let
{xn}
be an approximate fixed point sequence in
K
. Then every subsequence
{xnk}
of
{xn}is also an approximate fixed point sequence.
Hence by (ii) in Theorem 3, r(K, {xn})=r(K, {xnk}).
Therefore, {xn}is regular.
By (i) in Theorem 3, A(K, {xn})=A(K, {xnk})=K.
Hence {xn}is uniform.
Corollary 2. [10, Preposition 6.3] Let
K
be a weakly compact convex subset of a Banach space
X
,
and
T
:
K→K
be a nonexpansive mapping. Assume that
K
is minimal for
T
, that is, no closed convex
bounded proper subset of
K
is invariant for
T
. If
{xn}
is an approximate fixed point sequence in
K
,
then A(K, {xn})=K.
Proof. Since every nonexpansive mapping satisfies condition (C), by above theorem,
A
(
K, {xn}
)=
K
.
Theorem 4. Let
K
be a nonempty, weakly compact and convex subset of a Banach space
X
and suppose
Khas normal structure. Then every mapping T:K→Ksatisfying condition (C) has a fixed point.
Proof. By Theorem 1, we can consider Kas closed, convex minimal T- invariant subset.
Suppose diamK > 0.
Consider an approximate fixed point sequence {xn}⊆K.
By (i) in Theorem 3, A(K, {xn})=K.
Now, define Wn:= {xm:m≥n},n∈N.
Clearly, {Wn,n∈N}is a decreasing chain of nonempty bounded subsets of K.
We can easily prove that Asymptotic center of {Wn,n ∈N}=A(K, {xn})=K.
Since Khas normal structure, by Theorem 2, Khas asymptotic normal structure.
Thus A(K, {xn})=K, which is a contradiction.
Therefore, diamK =0and hence Khas only one element x(say).
Thus T(x)=x.
We obtain the result of Kirk [8] and Theorem 4.1 in [7] as corollaries of our result.
Corollary 3. [8, Theorem] Let
K
be a nonempty, bounded, closed and convex subset of a reflexive
Banach space
X
, and suppose that
K
has normal structure. If
T
:
K→K
is nonexpansive, then
T
has
a fixed point.
Proof. Since a bounded, closed and convex subset of a reflexive Banach space is weakly compact and
every nonexpansive mapping is Suzuki nonexpansive, by the above theorem, Thas a fixed point.
Corollary 4. [7, Theorem 4.1] Let
K
be a nonempty, weakly compact, convex subset of a Banach
space, and suppose
K
has normal structure. Then every nonexpansive mapping
T
:
K→K
has a fixed
point.
Proof. Since every nonexpansive mapping is Suzuki nonexpansive, by the above theorem,
T
has a
fixed point.
The following theorem gives sufficient conditions for the existence of fixed points for Suzuki nonexpansive
mapping in Banach spaces.
Theorem 5. Let
K
be a closed, bounded and convex subset of a Banach space
X
and
T
:
K→K
satisfies condition (C). If T(K)is contained in a compact subset of K, then Thas a fixed point in K.
https://doi.org/10.17993/3ctic.2022.112.15-24
Proof. Let {xn}be an approximate fixed point sequence in K.
Therefore, ||Tx
n−xn|| → 0.
Consider the sequence {Tx
n}in T(K).
Since
T
(
K
)is a subset of a compact set of
K
, there exist a subsequence
{Tx
nk}
of
{Tx
n}
and
z∈K
such that Tx
nk→z.
Therefore, lim
k→∞||xnk−z|| = lim
k→∞ ||xnk−Tx
nk|| =0.
Hence xnk→z.
Clearly, {xnk}is a bounded approximate fixed point sequence and A(K, {xnk})={z}.
Also by Lemma 3, A(K, {xnk})is T-invariant and hence T(z)=z.
The following is an example to illustrate this theorem.
Example 1. In the space l2consider the closed unit ball
K={x=(x1,x
2, ...)∈l2:||x||2≤1}.
Define T:K→Kas
T(x)=(1
3,0,0, ..), if x = (1,0,0, ...)
0, otherwise.
For all x, y = (1,0,0, ...)in K,||Tx−Ty||2=0≤ ||x−y||2.
If x= (1,0,0, ...)then 1
2||x−Tx||2=1
2||(2
3,0,0, ...)||2=1
3.
Therefore, if 1
2||x−Tx||2≤ ||x−y||2for any y∈K, then 1
3≤ ||x−y||2.
Thus ||Tx−Ty||2=1
3≤ ||x−y||2.
Hence Tsatisfies condition (C).
T(K)=(0,0,0, ...),(1
3,0,0, ...)is a subset of a compact set of K.
Thus all conditions in the above theorem are satisfied.
Since T(0) = 0,Thas a fixed point.
Theorem 6. Let
X
be a Banach space with Opial property and
K
be a closed, bounded and convex
subset of
X
. Let
T
:
K→K
satisfies condition (C) and if
T
(
K
)is contained in a weakly compact
subset of K, then Thas a fixed point in K.
Proof. Let {xn}be an approximate fixed point sequence in K. Then we have, ||Tx
n−xn|| → 0.
Since
1
2||Tx
n−xn|| ≤ ||Tx
n−xn||
and
T
satisfies condition (C), we have
||T2xn−Tx
n||≤||Tx
n−xn||
for all n∈N.
Therefore, lim
n→∞||T2xn−Tx
n|| ≤ lim
n→∞ ||Tx
n−xn|| =0.
Hence {T(xn)}is an approximate fixed point sequence.
Since
T
(
K
)is a subset of a weakly compact set of
K
, there exist a subsequence
{Tx
nk}
of
{Tx
n}
and
z∈Ksuch that Tx
nk→zweakly as k→∞.
By Opial property, for any x=z,lim sup ||Tx
nk−z|| <lim sup ||Tx
nk−x||.
Therefore, for any
x
=
z,r
(
z,{Tx
nk}
)
<r
(
x, {Tx
nk}
)=
⇒r
(
K, {Tx
nk}
)=
r
(
z,{Tx
nk}
)and
A(K, {Tx
nk})={z}.
By Lemma 3, A(K, {xnk})is T-invariant and hence T(z)=z.
Corollary 5. [17, Theorem 4] Let
T
be a mapping on a convex subset
K
of a Banach space
X
. Assume
that Tsatisfies condition (C). Assume also that either of the following holds:
(i) Kis compact;
(ii) Kis weakly compact and Xhas the Opial property.
Then Thas a fixed point.
Proof. Suppose Kis compact and convex. Therefore, Kis closed, bounded and convex.
Also, since T:K→K, we have T(K)⊆K. Thus T(K)is a subset of a compact set.
Hence by Theorem 5, Thas a fixed point.
Now suppose Kis weakly compact, convex and Xhas the Opial property.
Therefore, Kis closed, bounded and convex. Also, since T:K→K, we have T(K)⊆K.
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
21
Thus T(K)is a subset of a weakly compact set.
Hence by Theorem 6, Thas a fixed point.
Theorem 7. Let
K
be a compact subset of a Banach space
X
. Let
T
:
K→K
satisfies condition (C).
Let
{xn}
be a regular approximate fixed point sequence in
K
. Then
{xn}
converge strongly to a fixed
point of T.
Proof. Let {xn}be a regular approximate fixed point sequence in K.
Since Kis compact, there exist a subsequence {xnk}of {xn}and z∈Ksuch that xnk→z.
Therefore, A(K, {xnk})={z}.
Since
{xnk}
is an approximate fixed point sequence, by Lemma 3,
A
(
K, {xnk}
)is
T
-invariant. Hence
T(z)=z.
Since {xn}is regular, r(K, {xn})=r(K, {xnk})=0.
Since Kis compact and {xn}is an approximate fixed point sequence, A(K, {xn})is nonempty.
We know that if {xn}is regular, then for any subsequence {xnk},
A(K, {xn})⊆A(K, {xnk}).
Therefore, A(K, {xn})={z}. Hence xn→z.
The following example shows that even if
T
has fixed points, if
{xn}
is not regular, then
{xn}
need not
converge to a fixed point.
Example 2. Consider the compact set K=[−1,1] in Rand define T:K→Kas T(x)=x.
Clearly Tis a nonexpansive mapping and hence satisfies condition (C).
Consider xn=(−1)nf or all n ∈N
For all n∈N,||xn−Tx
n|| =0. Thus {xn}is an approximate fixed point sequence.
But {xn}does not converge to a fixed point.
This will not contradict the above theorem because {xn}is not regular.
Asymptotic radius of {xn}=r(K, {xn})=1and A(K, {xn})={0}.
Consider the subsequence {x2n}={1,1,1, ...}. Clearly x2n→1.
Therefore, A(K, {x2n})={1}and r(K, {x2n})=0.
Hence {xn}is not regular.
Theorem 8. Let
K
be a subset of a Banach space
X
and
T
:
K→K
satisfies condition (C). Suppose
T
(
K
)is contained in a compact subset of
K
and let
{xn}
be a regular approximate fixed point sequence
in Kwith nonempty asymptotic center. Then {xn}converge strongly to a fixed point of T.
Proof. Let
{xn}
be a regular approximate fixed point sequence in
K
with nonempty asymptotic
center.
Consider the sequence
{Tx
n}
in
T
(
K
). Since
T
(
K
)is compact, there exist a subsequence
{Tx
nk}
of
{Tx
n}and z∈T(K)such that Tx
nk→z.
Therefore, lim
n→∞||xnk−z|| ≤ lim
n→∞ ||xnk−Tx
nk|| =0.
Thus xnk→zand hence A(K, {xnk})={z}.
Since {xnk}is an approximate fixed point sequence, by Lemma 3, A(K, {xnk})is T-invariant.
Hence T(z)=z.
Since {xn}is regular, r(K, {xn})=r(K, {xnk})=0.
Also if {xn}is regular, then for any subsequence {xnk}, we have A(K, {xn})⊆A(K, {xnk}).
Thus we have A(K, {xn})={z}, which implies that xn→z.
3 CONCLUSIONS
In this paper, we have used the asymptotic center technique to establish the existence of fixed points
for Suzuki nonexpansive mappings in Banach spaces. We have shown that under certain condition, the
asymptotic radius and Chebyshev radius are equal. Using this result, we have established that every
https://doi.org/10.17993/3ctic.2022.112.15-24
approximate fixed point sequence is regular as well as uniform. The convergence of regular approximate
fixed point sequences to a fixed points of the Suzuki nonexpansive mapping is also established. A couple
of examples are given to illustrate the results.
ACKNOWLEDGMENT
The first author is highly grateful to University Grant Commission, India, for providing financial
support in the form of Junior Research fellowship.
REFERENCES
[1]
Aoyama, K., and Kohsaka, F. (2011). Fixed point theorem for
α
-nonexpansive mappings in
Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 74(13), 4387-4391.
[2]
Bogin, J. (1976). A generalization of a fixed point theorem of Goebel, Kirk and Shimi. Canadian
Mathematical Bulletin, 19(1), 7–12.
[3]
Byrne, C. (2003). A unified treatment of some iterative algorithms in signal processing and image
reconstruction. Inverse problems, 20(1), 103.
[4]
Dhompongsa, S.,Inthakon, W. and Kaewkhao, A. (2009). Edelstein’s method and fixed
point theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and
Applications, 350(1), 12–17.
[5]
Edelstein, M. (1972). The construction of an asymptotic center with a fixed point property.
Bulletin of the American Mathematical Society, 78(2), 206–208.
[6]
Eldred, A.,Kirk, W., and Veeramani, P. (2005). Proximal normal structure and relatively
nonexpansive mappings. Studia Mathematica, 3(171), 283-293.
[7]
Goebel, K. and Kirk, W. A. (1990). Topics in metric fixed point theory. Cambridge university
press.
[8]
Kirk, W.A. (1965). A fixed point theorem for mappings which do not increase distances. The
American Mathematical Monthly, 72(9), 1004–1006.
[9]
Kirk, W.A. and Massa, S. (1990). Remarks on asymptotic and Chebyshev centers. Houston
Journal of Mathematics, 16, 364–375.
[10]
Kirk, W. A., and Sims, B. (2002). Handbook of metric fixed point theory. Australian Mathema-
tical Society GAZETTE, 29(2).
[11]
Lami Dozo, E. (1973). Multivalued nonexpansive mappings and Opial’s condition. Proceedings
of the American Mathematical Society, 38(2), 286-92.
[12]
Lim, T.C. (1974). Characterization of normal structure. Proceedings of the American Mathematical
Society, 43(2), 313-319.
[13]
Lim, T.C. (1980). On asymptotic centers and fixed points of nonexpansive mappings. Canadian
Journal of Mathematics, 32(2), 421-430.
[14]
Lopez, G.,Martin, V., and Xu, H. K. (2009). Perturbation techniques for nonexpansive
mappings with applications. Nonlinear Analysis: Real World Applications, 10(4), 2369-2383.
[15]
Park, J. Y., and Jeong, J. U. (1994). Weak convergence to a fixed point of the sequence of
Mann type iterates. Journal of Mathematical Analysis and Applications, 184(1), 75-81.
https://doi.org/10.17993/3ctic.2022.112.15-24
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
22
Thus T(K)is a subset of a weakly compact set.
Hence by Theorem 6, Thas a fixed point.
Theorem 7. Let
K
be a compact subset of a Banach space
X
. Let
T
:
K→K
satisfies condition (C).
Let
{xn}
be a regular approximate fixed point sequence in
K
. Then
{xn}
converge strongly to a fixed
point of T.
Proof. Let {xn}be a regular approximate fixed point sequence in K.
Since Kis compact, there exist a subsequence {xnk}of {xn}and z∈Ksuch that xnk→z.
Therefore, A(K, {xnk})={z}.
Since
{xnk}
is an approximate fixed point sequence, by Lemma 3,
A
(
K, {xnk}
)is
T
-invariant. Hence
T(z)=z.
Since {xn}is regular, r(K, {xn})=r(K, {xnk})=0.
Since Kis compact and {xn}is an approximate fixed point sequence, A(K, {xn})is nonempty.
We know that if {xn}is regular, then for any subsequence {xnk},
A(K, {xn})⊆A(K, {xnk}).
Therefore, A(K, {xn})={z}. Hence xn→z.
The following example shows that even if
T
has fixed points, if
{xn}
is not regular, then
{xn}
need not
converge to a fixed point.
Example 2. Consider the compact set K=[−1,1] in Rand define T:K→Kas T(x)=x.
Clearly Tis a nonexpansive mapping and hence satisfies condition (C).
Consider xn=(−1)nf or all n ∈N
For all n∈N,||xn−Tx
n|| =0. Thus {xn}is an approximate fixed point sequence.
But {xn}does not converge to a fixed point.
This will not contradict the above theorem because {xn}is not regular.
Asymptotic radius of {xn}=r(K, {xn})=1and A(K, {xn})={0}.
Consider the subsequence {x2n}={1,1,1, ...}. Clearly x2n→1.
Therefore, A(K, {x2n})={1}and r(K, {x2n})=0.
Hence {xn}is not regular.
Theorem 8. Let
K
be a subset of a Banach space
X
and
T
:
K→K
satisfies condition (C). Suppose
T
(
K
)is contained in a compact subset of
K
and let
{xn}
be a regular approximate fixed point sequence
in Kwith nonempty asymptotic center. Then {xn}converge strongly to a fixed point of T.
Proof. Let
{xn}
be a regular approximate fixed point sequence in
K
with nonempty asymptotic
center.
Consider the sequence
{Tx
n}
in
T
(
K
). Since
T
(
K
)is compact, there exist a subsequence
{Tx
nk}
of
{Tx
n}and z∈T(K)such that Tx
nk→z.
Therefore, lim
n→∞||xnk−z|| ≤ lim
n→∞ ||xnk−Tx
nk|| =0.
Thus xnk→zand hence A(K, {xnk})={z}.
Since {xnk}is an approximate fixed point sequence, by Lemma 3, A(K, {xnk})is T-invariant.
Hence T(z)=z.
Since {xn}is regular, r(K, {xn})=r(K, {xnk})=0.
Also if {xn}is regular, then for any subsequence {xnk}, we have A(K, {xn})⊆A(K, {xnk}).
Thus we have A(K, {xn})={z}, which implies that xn→z.
3 CONCLUSIONS
In this paper, we have used the asymptotic center technique to establish the existence of fixed points
for Suzuki nonexpansive mappings in Banach spaces. We have shown that under certain condition, the
asymptotic radius and Chebyshev radius are equal. Using this result, we have established that every
https://doi.org/10.17993/3ctic.2022.112.15-24
approximate fixed point sequence is regular as well as uniform. The convergence of regular approximate
fixed point sequences to a fixed points of the Suzuki nonexpansive mapping is also established. A couple
of examples are given to illustrate the results.
ACKNOWLEDGMENT
The first author is highly grateful to University Grant Commission, India, for providing financial
support in the form of Junior Research fellowship.
REFERENCES
[1]
Aoyama, K., and Kohsaka, F. (2011). Fixed point theorem for
α
-nonexpansive mappings in
Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 74(13), 4387-4391.
[2]
Bogin, J. (1976). A generalization of a fixed point theorem of Goebel, Kirk and Shimi. Canadian
Mathematical Bulletin, 19(1), 7–12.
[3]
Byrne, C. (2003). A unified treatment of some iterative algorithms in signal processing and image
reconstruction. Inverse problems, 20(1), 103.
[4]
Dhompongsa, S.,Inthakon, W. and Kaewkhao, A. (2009). Edelstein’s method and fixed
point theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and
Applications, 350(1), 12–17.
[5]
Edelstein, M. (1972). The construction of an asymptotic center with a fixed point property.
Bulletin of the American Mathematical Society, 78(2), 206–208.
[6]
Eldred, A.,Kirk, W., and Veeramani, P. (2005). Proximal normal structure and relatively
nonexpansive mappings. Studia Mathematica, 3(171), 283-293.
[7]
Goebel, K. and Kirk, W. A. (1990). Topics in metric fixed point theory. Cambridge university
press.
[8]
Kirk, W.A. (1965). A fixed point theorem for mappings which do not increase distances. The
American Mathematical Monthly, 72(9), 1004–1006.
[9]
Kirk, W.A. and Massa, S. (1990). Remarks on asymptotic and Chebyshev centers. Houston
Journal of Mathematics, 16, 364–375.
[10]
Kirk, W. A., and Sims, B. (2002). Handbook of metric fixed point theory. Australian Mathema-
tical Society GAZETTE, 29(2).
[11]
Lami Dozo, E. (1973). Multivalued nonexpansive mappings and Opial’s condition. Proceedings
of the American Mathematical Society, 38(2), 286-92.
[12]
Lim, T.C. (1974). Characterization of normal structure. Proceedings of the American Mathematical
Society, 43(2), 313-319.
[13]
Lim, T.C. (1980). On asymptotic centers and fixed points of nonexpansive mappings. Canadian
Journal of Mathematics, 32(2), 421-430.
[14]
Lopez, G.,Martin, V., and Xu, H. K. (2009). Perturbation techniques for nonexpansive
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