APPLICATIONS OF FIXED POINT THEOREMS TO SO-
LUTIONS OF OPERATOR EQUATIONS IN BANACH
SPACES
Neeta Singh
Department of Mathematics, University of Allahabad, Allahabad (India).
E-mail:n_s32132@yahoo.com
ORCID:
Reception: 25/08/2022 Acceptance: 09/09/2022 Publication: 29/12/2022
Suggested citation:
Singh, N. (2022). Applications of Fixed Point Theorems to Solutions of Operator Equations in Banach Spaces. 3C TIC.
Cuadernos de desarrollo aplicados a las TIC,11 (2), 72-79. https://doi.org/10.17993/3ctic.2022.112.72-79
https://doi.org/10.17993/3ctic.2022.112.72-79
ABSTRACT
In this paper we use Browder’s and Gohde’s fixed point theorem, Kirk’s fixed point theorem and the
Sadovskii fixed point theorem to obtain solutions of operator equations in Banach spaces.
KEYWORDS
fixed point, Banach spaces
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
72
APPLICATIONS OF FIXED POINT THEOREMS TO SO-
LUTIONS OF OPERATOR EQUATIONS IN BANACH
SPACES
Neeta Singh
Department of Mathematics, University of Allahabad, Allahabad (India).
E-mail:n_s32132@yahoo.com
ORCID:
Reception: 25/08/2022 Acceptance: 09/09/2022 Publication: 29/12/2022
Suggested citation:
Singh, N. (2022). Applications of Fixed Point Theorems to Solutions of Operator Equations in Banach Spaces. 3C TIC.
Cuadernos de desarrollo aplicados a las TIC,11 (2), 72-79. https://doi.org/10.17993/3ctic.2022.112.72-79
https://doi.org/10.17993/3ctic.2022.112.72-79
ABSTRACT
In this paper we use Browder’s and Gohde’s fixed point theorem, Kirk’s fixed point theorem and the
Sadovskii fixed point theorem to obtain solutions of operator equations in Banach spaces.
KEYWORDS
fixed point, Banach spaces
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
73
1 INTRODUCTION
Perhaps the most famous fixed-point theorem is the Banach’s contraction principle which has several
applications. Motivated by this we have considered in this review article, applications of other well-
known fixed-point theorems in various kinds of Banach spaces. This article should be of interest to
mathematicians working in the fields of fixed-point theory and functional analysis.
In Section 1we apply the Browder’s and G
¨o
hde’s fixed point theorem for the existence of solutions
of operator equations involving asymptotically nonexpansive mappings in uniformly convex Banach
spaces. In Section 2we apply Kirk’s fixed point theorem for the existence of solutions of the operator
equation
x−Tx
=
f
in reflexive Banach spaces and in Section 3we apply the Sadovskii fixed point
theorem for existence of solutions of the operator equation x−Tx=fin arbitrary Banach spaces.
2 Application of Browder’s and G¨ohde’s fixed point theorem
Definition 1. [1] A mapping T from a metric space (
X, d
)into another metric space (
Y,ρ
)is said to
satisfy Lipschitz condition on X if there exists a constant L>0such that
ρ(T x, T y)≤Ld(x, y)
for all
x, y ∈X
. If
L
is the least number for which Lipschitz condition holds, then
L
is called Lipschitz
constant. If L=1, the mapping is said to be nonexpansive.
Definition 2. [2] Let K be a nonempty subset of a Banach space X. A mapping
T
:
K→K
is
said to be asymptotically nonexpansive if for each
n∈N
there exists a positive constant
kn≥
1with
limn→∞ kn=1such that
||Tnx−Tny|| ≤ kn||x−y||
for all x, y ∈K.
The Browder’s and G¨ohde’s fixed point theorem is as follows:
Theorem 1. [3] Let X be a uniformly convex Banach space and C a nonempty, closed, convex and
bounded subset of X. Then every nonexpansive mapping T:C→Chas a fixed point in C.
We now state the main theorem of Section 1.
Theorem 2. Let X be a uniformly convex Banach space and K a nonempty subset of X. Let
T
:
K→K
be an asymptotically nonexpansive mapping and fn∈K, then the operator equation
knx=Tnx+fn
where
n∈N
and
kn
is the Lipschitz constant of the iterates
Tn
, has a solution if and only if, for any
x1∈K, the sequence of iterates {xn}in K defined by
knxn+1 =Tnxn+fn
n∈Nis bounded.
Proof. For every n∈N, let Tfnbe defined to be a mapping from K into K by
Tfn(u)= 1
kn
[Tnu+fn].
Then un∈Kis a solution of
x=1
kn
[Tnx+fn]
https://doi.org/10.17993/3ctic.2022.112.72-79
if and only if
un
is a fixed point of
Tfn
. Since T is asymptotically nonexpansive it follows that
Tfn
is
nonexpansive for all n∈N.
||Tfn(x)−Tfn(y)|| =1
kn||Tn(x)−Tn(y)||≤||x−y||.
Suppose Tfnhas a fixed point un∈K. Then
||xn+1 −un|| =|| 1
kn
[Tnxn+fn]−un|| =||Tfn(xn)−Tfn(un)||≤||xn−un||,
Tfn
being nonexpasive. Since
{||xn−un||}
is non-increasing, hence
{xn}
is bounded. Conversely, suppose
that{xn}is bounded. Let d=diam({xn})and
Bd[x]={y∈K:||x−y|| ≤ d}
for each x∈K. Set
Cn=
i≥n
Bd[xi]⊂K.
Hence
Cn
is a nonempty, convex set for each
n∈N
. Now we claim that
Tfn
(
Cn
)
⊂Cn+1
. Let
y∈Bd
[
xn
]
which implies ||y−xn|| ≤ d. Since Tfnis nonexpansive, we get
||Tfn(y)−Tfn(xn)|| ≤ d
|| 1
kn
[Tn(y)+fn]−1
kn
[Tn(xn)+fn]|| ≤ d
or
|| 1
kn
[Tn(y)+fn]−xn+1|| ≤ d
or 1
kn
[Tn(y)+fn]∈Bd[xn+1]
giving
Tfn(y)∈Bd[xn+1]
proving that Tfn(Cn)⊂Cn+1.
Let
C
=
n∈NCn
. Since
Cn
increases with n, C is a closed, convex and bounded subset of K. We
can easily see that Tfnmaps C into C.
Tfn(C)=Tfn(
n∈N
Cn)⊆Tfn(
n∈N
Cn)=
n∈N
Tfn(Cn)⊆
n∈N
Cn+1 =C.
Applying the Browder’s and G
¨o
hde’s theorem to
Tfn
and C we get a fixed point of
Tfn
in
C
. Since
C⊂K, we obtain a fixed point of Tfnin K.
3 Application of Kirk’s Fixed Point Theorem
Let us recall some definitions and results that we shall require for the proof of the Main Theorem of
Section 2.
Definition 3. [2] Let (
X, ρ
)and (
M,d
)be metric spaces. A mapping
f
:
X→M
is said to be
nonexpansive if for each x, y ∈X,
d(f(x),f(y)) ≤ρ(x, y).
Definition 4. [1] A convex subset
K
of a Banach space
X
is said to have normal structure if each
bounded, convex subset Sof Kwith diam S>0contains a nondiametral point.
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
74
1 INTRODUCTION
Perhaps the most famous fixed-point theorem is the Banach’s contraction principle which has several
applications. Motivated by this we have considered in this review article, applications of other well-
known fixed-point theorems in various kinds of Banach spaces. This article should be of interest to
mathematicians working in the fields of fixed-point theory and functional analysis.
In Section 1we apply the Browder’s and G
¨o
hde’s fixed point theorem for the existence of solutions
of operator equations involving asymptotically nonexpansive mappings in uniformly convex Banach
spaces. In Section 2we apply Kirk’s fixed point theorem for the existence of solutions of the operator
equation
x−Tx
=
f
in reflexive Banach spaces and in Section 3we apply the Sadovskii fixed point
theorem for existence of solutions of the operator equation x−Tx=fin arbitrary Banach spaces.
2 Application of Browder’s and G¨ohde’s fixed point theorem
Definition 1. [1] A mapping T from a metric space (
X, d
)into another metric space (
Y,ρ
)is said to
satisfy Lipschitz condition on X if there exists a constant L>0such that
ρ(T x, T y)≤Ld(x, y)
for all
x, y ∈X
. If
L
is the least number for which Lipschitz condition holds, then
L
is called Lipschitz
constant. If L=1, the mapping is said to be nonexpansive.
Definition 2. [2] Let K be a nonempty subset of a Banach space X. A mapping
T
:
K→K
is
said to be asymptotically nonexpansive if for each
n∈N
there exists a positive constant
kn≥
1with
limn→∞ kn=1such that
||Tnx−Tny|| ≤ kn||x−y||
for all x, y ∈K.
The Browder’s and G¨ohde’s fixed point theorem is as follows:
Theorem 1. [3] Let X be a uniformly convex Banach space and C a nonempty, closed, convex and
bounded subset of X. Then every nonexpansive mapping T:C→Chas a fixed point in C.
We now state the main theorem of Section 1.
Theorem 2. Let X be a uniformly convex Banach space and K a nonempty subset of X. Let
T
:
K→K
be an asymptotically nonexpansive mapping and fn∈K, then the operator equation
knx=Tnx+fn
where
n∈N
and
kn
is the Lipschitz constant of the iterates
Tn
, has a solution if and only if, for any
x1∈K, the sequence of iterates {xn}in K defined by
knxn+1 =Tnxn+fn
n∈Nis bounded.
Proof. For every n∈N, let Tfnbe defined to be a mapping from K into K by
Tfn(u)= 1
kn
[Tnu+fn].
Then un∈Kis a solution of
x=1
kn
[Tnx+fn]
https://doi.org/10.17993/3ctic.2022.112.72-79
if and only if
un
is a fixed point of
Tfn
. Since T is asymptotically nonexpansive it follows that
Tfn
is
nonexpansive for all n∈N.
||Tfn(x)−Tfn(y)|| =1
kn||Tn(x)−Tn(y)||≤||x−y||.
Suppose Tfnhas a fixed point un∈K. Then
||xn+1 −un|| =|| 1
kn
[Tnxn+fn]−un|| =||Tfn(xn)−Tfn(un)||≤||xn−un||,
Tfn
being nonexpasive. Since
{||xn−un||}
is non-increasing, hence
{xn}
is bounded. Conversely, suppose
that{xn}is bounded. Let d=diam({xn})and
Bd[x]={y∈K:||x−y|| ≤ d}
for each x∈K. Set
Cn=
i≥n
Bd[xi]⊂K.
Hence
Cn
is a nonempty, convex set for each
n∈N
. Now we claim that
Tfn
(
Cn
)
⊂Cn+1
. Let
y∈Bd
[
xn
]
which implies ||y−xn|| ≤ d. Since Tfnis nonexpansive, we get
||Tfn(y)−Tfn(xn)|| ≤ d
|| 1
kn
[Tn(y)+fn]−1
kn
[Tn(xn)+fn]|| ≤ d
or
|| 1
kn
[Tn(y)+fn]−xn+1|| ≤ d
or 1
kn
[Tn(y)+fn]∈Bd[xn+1]
giving
Tfn(y)∈Bd[xn+1]
proving that Tfn(Cn)⊂Cn+1.
Let
C
=
n∈NCn
. Since
Cn
increases with n, C is a closed, convex and bounded subset of K. We
can easily see that Tfnmaps C into C.
Tfn(C)=Tfn(
n∈N
Cn)⊆Tfn(
n∈N
Cn)=
n∈N
Tfn(Cn)⊆
n∈N
Cn+1 =C.
Applying the Browder’s and G
¨o
hde’s theorem to
Tfn
and C we get a fixed point of
Tfn
in
C
. Since
C⊂K, we obtain a fixed point of Tfnin K.
3 Application of Kirk’s Fixed Point Theorem
Let us recall some definitions and results that we shall require for the proof of the Main Theorem of
Section 2.
Definition 3. [2] Let (
X, ρ
)and (
M,d
)be metric spaces. A mapping
f
:
X→M
is said to be
nonexpansive if for each x, y ∈X,
d(f(x),f(y)) ≤ρ(x, y).
Definition 4. [1] A convex subset
K
of a Banach space
X
is said to have normal structure if each
bounded, convex subset Sof Kwith diam S>0contains a nondiametral point.
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
75
The following theorem gives application of the Browder-Göhde-Kirk’s theorem for the existence of
solutions of the operator equation
x−Tx =f.
It is known that every uniformly convex Banach space is reflexive. We generalize the theorem below to
reflexive Banach spaces using Kirk’s fixed point theorem
Theorem 3. [3] Let
X
be a uniformly convex Banach space,
f
an element in
X
and
T
:
X→X
a
nonexpansive mapping, then the operator equation
x−Tx =f
has a solution
x
if and only if for any
x0∈X
, the sequence of Picard iterates
{xn}
in
X
defined by
xn+1 =Tx
n+f,n∈N0is bounded.
Definition 5. [1] A Banach space
X
is said to satisfy the Opial condition if whenever a sequence
{xn}in Xconverges weakly to x0∈X, then
lim
n→∞ inf ∥xn−x0∥<lim
n→∞ inf ∥xn−x∥
for all x∈X,x=x0.
Lemma 1. [3] Let
X
be a reflexive Banach space with the Opial condition. Then
X
has normal
structure.
Lemma 2. [3] A closed subspace of a reflexive Banach space is reflexive.
Now we state the Kirk’s fixed point theorem.
Theorem 4. [3] Let
X
be a Banach space and
C
a nonempty weakly compact, convex subset of
X
with normal structure, then every nonexpansive mapping T:C→Chas a fixed point.
We state the main theorem of Section 2.
Theorem 5. Let
X
be a reflexive Banach space satisfying Opial condition. Let
f∈X
and
T
:
X→X
be a nonexpansive mapping. Then the operator equation
x−Tx =f
has a solution
x
if and only if for any
x0∈X
, the sequence of Picard iterates
{xn}
in
X
defined by
xn+1 =Tx
n+f,n∈N0is bounded.
Proof. Let Tfbe the mapping from Xinto Xgiven by
Tf(u)=Tu+f.
Then uis a solution of
x−Tx =f
if and only if
u
is a fixed point of
Tf
. Clearly
Tf
is nonexpansive. Suppose
Tf
has a fixed point
u∈X
.
Then for all n∈N,
∥xn+1 −u∥≤∥xn−u∥.
Hence {xn}is bounded.
Conversely, suppose that {xn}is bounded. Let d=diam({xn})and
Bd[x]={y∈X:∥x−y∥≤d}
for each x∈X. Set Cn=i≥nBd[xi]. Then Cnis a nonempty convex set for each n, and
Tf(Cn)⊂Cn+1.
https://doi.org/10.17993/3ctic.2022.112.72-79
Let Cbe the closure of the union of Cnfor n∈N,
C=
n∈N
Cn.
Since
Cn
increases with
n
,
C
is a closed, convex and bounded subset of
X
. It is known that [1] bounded,
closed and convex subsets of reflexive Banach spaces are weakly compact, hence we get that
C
is weakly
compact.
Now since
Tf(C)=Tf(Cn)⊆Tf(Cn)=Tf(Cn)⊆Cn+1 =C,
we get that
Tf
maps
C
into itself. By Lemma 1
.
3
.
6,
C
is a reflexive Banach space. Now
X
satisfies
Opial condition and
C
being a closed subset of
X
, will also satisfy Opial condition. Hence by Lemma
1
.
3
.
5,
C
has normal structure. Finally, applying Kirk’s fixed point theorem we get that
Tf
has a fixed
point in Cwhich proves the theorem.
4 Application of Sadovskii Fixed Point Theorem
We recall some definitions
Definition 6. [1] Let (
M,ρ
)denote a complete metric space and let
B
denote the collection of
nonempty and bounded subsets of
M
. Define the Kuratowski measure of noncompactness
α
:
B→R+
by taking for A∈B,
α(A)=inf{ϵ>0Ais contained in the union of a finite number of sets in Beach
having diameter less than ϵ}.
If Mis a Banach space the function αhas the following properties for A, B ∈B
1.α(A)=0⇔Ais compact,
2.α(A+B)≤α(A)+α(B).
Definition 7. [2] Let
K
be a subset of a metric space
M
. A mapping
T
:
K→M
is said to be
condensing if Tis bounded and continuous and if
α(T(D)) <α(D)
for all bounded subsets Dof Mfor which α(D)>0.
We state the Sadovskii fixed point theorem.
Theorem 6. [2] Let
K
be a nonempty, bounded closed and convex subset of a Banach space and let
T:K→Kbe a condensing mapping, then Thas a fixed point.
The main result of section 3is the following:
Theorem 7. Let
X
be an arbitrary Banach space, let
f∈X
and
T
:
X→X
be a condensing mapping,
then the operator equation
x−Tx =f
has a solution if and only if for any
x0∈X
, the sequence of Picard iterates
{xn}
in
X
, defined by
xn+1 =Tx
n+f,n∈N0is bounded.
Proof. Let the mapping Tf:X→Xbe defined by
Tf(u)=Tu+f.
Then uis a solution of the operator equation
x−Tx =f
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
76
The following theorem gives application of the Browder-Göhde-Kirk’s theorem for the existence of
solutions of the operator equation
x−Tx =f.
It is known that every uniformly convex Banach space is reflexive. We generalize the theorem below to
reflexive Banach spaces using Kirk’s fixed point theorem
Theorem 3. [3] Let
X
be a uniformly convex Banach space,
f
an element in
X
and
T
:
X→X
a
nonexpansive mapping, then the operator equation
x−Tx =f
has a solution
x
if and only if for any
x0∈X
, the sequence of Picard iterates
{xn}
in
X
defined by
xn+1 =Tx
n+f,n∈N0is bounded.
Definition 5. [1] A Banach space
X
is said to satisfy the Opial condition if whenever a sequence
{xn}in Xconverges weakly to x0∈X, then
lim
n→∞ inf ∥xn−x0∥<lim
n→∞ inf ∥xn−x∥
for all x∈X,x=x0.
Lemma 1. [3] Let
X
be a reflexive Banach space with the Opial condition. Then
X
has normal
structure.
Lemma 2. [3] A closed subspace of a reflexive Banach space is reflexive.
Now we state the Kirk’s fixed point theorem.
Theorem 4. [3] Let
X
be a Banach space and
C
a nonempty weakly compact, convex subset of
X
with normal structure, then every nonexpansive mapping T:C→Chas a fixed point.
We state the main theorem of Section 2.
Theorem 5. Let
X
be a reflexive Banach space satisfying Opial condition. Let
f∈X
and
T
:
X→X
be a nonexpansive mapping. Then the operator equation
x−Tx =f
has a solution
x
if and only if for any
x0∈X
, the sequence of Picard iterates
{xn}
in
X
defined by
xn+1 =Tx
n+f,n∈N0is bounded.
Proof. Let Tfbe the mapping from Xinto Xgiven by
Tf(u)=Tu+f.
Then uis a solution of
x−Tx =f
if and only if
u
is a fixed point of
Tf
. Clearly
Tf
is nonexpansive. Suppose
Tf
has a fixed point
u∈X
.
Then for all n∈N,
∥xn+1 −u∥≤∥xn−u∥.
Hence {xn}is bounded.
Conversely, suppose that {xn}is bounded. Let d=diam({xn})and
Bd[x]={y∈X:∥x−y∥≤d}
for each x∈X. Set Cn=i≥nBd[xi]. Then Cnis a nonempty convex set for each n, and
Tf(Cn)⊂Cn+1.
https://doi.org/10.17993/3ctic.2022.112.72-79
Let Cbe the closure of the union of Cnfor n∈N,
C=
n∈N
Cn.
Since
Cn
increases with
n
,
C
is a closed, convex and bounded subset of
X
. It is known that [1] bounded,
closed and convex subsets of reflexive Banach spaces are weakly compact, hence we get that
C
is weakly
compact.
Now since
Tf(C)=Tf(Cn)⊆Tf(Cn)=Tf(Cn)⊆Cn+1 =C,
we get that
Tf
maps
C
into itself. By Lemma 1
.
3
.
6,
C
is a reflexive Banach space. Now
X
satisfies
Opial condition and
C
being a closed subset of
X
, will also satisfy Opial condition. Hence by Lemma
1
.
3
.
5,
C
has normal structure. Finally, applying Kirk’s fixed point theorem we get that
Tf
has a fixed
point in Cwhich proves the theorem.
4 Application of Sadovskii Fixed Point Theorem
We recall some definitions
Definition 6. [1] Let (
M,ρ
)denote a complete metric space and let
B
denote the collection of
nonempty and bounded subsets of
M
. Define the Kuratowski measure of noncompactness
α
:
B→R+
by taking for A∈B,
α(A)=inf{ϵ>0Ais contained in the union of a finite number of sets in Beach
having diameter less than ϵ}.
If Mis a Banach space the function αhas the following properties for A, B ∈B
1.α(A)=0⇔Ais compact,
2.α(A+B)≤α(A)+α(B).
Definition 7. [2] Let
K
be a subset of a metric space
M
. A mapping
T
:
K→M
is said to be
condensing if Tis bounded and continuous and if
α(T(D)) <α(D)
for all bounded subsets Dof Mfor which α(D)>0.
We state the Sadovskii fixed point theorem.
Theorem 6. [2] Let
K
be a nonempty, bounded closed and convex subset of a Banach space and let
T:K→Kbe a condensing mapping, then Thas a fixed point.
The main result of section 3is the following:
Theorem 7. Let
X
be an arbitrary Banach space, let
f∈X
and
T
:
X→X
be a condensing mapping,
then the operator equation
x−Tx =f
has a solution if and only if for any
x0∈X
, the sequence of Picard iterates
{xn}
in
X
, defined by
xn+1 =Tx
n+f,n∈N0is bounded.
Proof. Let the mapping Tf:X→Xbe defined by
Tf(u)=Tu+f.
Then uis a solution of the operator equation
x−Tx =f
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
77
if and only if uis a fixed point of Tf.
Since
T
is bounded and continuous,
Tf
is also bounded and continuous. Using the properties of the
Kuratowski measure of noncompactness, for all bounded subsets Dof X, we have
α(Tf(D)) = α(T(D)+{f})≤α(T(D)) + α({f}).
Since {f}is compact, {f}is compact, implying α({f})=0, giving
α(Tf(D)) ≤α(T(D)) <α(T(D)).
Since Tis condensing mapping and it follows that Tfis a condensing mapping.
Suppose
Tf
has a fixed point
u
in
X
. Then for all
n∈N
, since
Tf
is a continuous mapping being
condensing, we get
∥xn+1 −u∥=∥Tx
n+f−u∥=∥Tf(xn)−Tf(u)∥≤∥xn−u∥.
Hence {xn}is bounded.
Conversely, suppose that {xn}is bounded. Let d=diam({xn})and for each x∈X
Bd[x]={y∈X:∥x−y∥≤d}.
Set
Cn
=
i≥nBd
[
xi
], then
Cn
is a nonempty convex set for each
n
. Using that
T
is a continuous
mapping and the given Picard iteration, we have
y∈Bd[xn]⇒∥y−xn∥≤d
⇒∥Ty−Tx
n∥≤d
⇒∥Ty−[xn+1 −f]∥≤d
⇒∥(Ty+f)−xn+1∥≤d
⇒(Ty+f)∈Bd[xn+1].
Applying this, we get the following
Tf(Cn)=Tf(
i≥n
Bd[xi])
⊆
i≥n
Tf(Bd[xi])
=
i≥n{Tf(y): ∥y−xi∥≤d}
=
i≥n{(Ty+f): ∥y−xi∥≤d}
⊆
i≥n+1
Bd[xi]=Cn+1.
Let us define
C=
n∈N
Cn.
Since Cnincreases with n,
Cn⊂Cn+1 ⊂Cn+2 ⊂.......,
it follows that Cis a closed, convex and bounded subset of X. Now we have
Tf(C)=Tf
n∈N
Cn⊆Tf
n∈N
Cn=
n∈N
Tf(Cn)⊆
n∈N
Cn+1 =C
giving Tf:C→Csince Tfis continuous mapping.
Finally, applying the Sadovskii fixed point theorem to
Tf
and
C
, we obtain that
Tf
has a fixed point
in Cwhich proves the theorem.
https://doi.org/10.17993/3ctic.2022.112.72-79
REFERENCES
[1]
Goebel K. and Kirk W. A. (1990). Topics in metric fixed point theory. Cambridge Studies in
Advanced Mathematics, volume 28,Cambridge University Press.
[2]
Khamsi M. A. and Kirk W. A. (2001). An Introduction to Metric Spaces and Fixed Point
Theory. Pure and Applied Mathematics. John Wiley & Sons, Inc.
[3]
Agarwal R. P. ,O’Regan R. P. , and Sahu D. R. (2009). Fixed point theory for Lipschitzian-
type mappings with applications. Topological fixed point theory and its Applications, volume 6.
Springer.
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
78
if and only if uis a fixed point of Tf.
Since
T
is bounded and continuous,
Tf
is also bounded and continuous. Using the properties of the
Kuratowski measure of noncompactness, for all bounded subsets Dof X, we have
α(Tf(D)) = α(T(D)+{f})≤α(T(D)) + α({f}).
Since {f}is compact, {f}is compact, implying α({f})=0, giving
α(Tf(D)) ≤α(T(D)) <α(T(D)).
Since Tis condensing mapping and it follows that Tfis a condensing mapping.
Suppose
Tf
has a fixed point
u
in
X
. Then for all
n∈N
, since
Tf
is a continuous mapping being
condensing, we get
∥xn+1 −u∥=∥Tx
n+f−u∥=∥Tf(xn)−Tf(u)∥≤∥xn−u∥.
Hence {xn}is bounded.
Conversely, suppose that {xn}is bounded. Let d=diam({xn})and for each x∈X
Bd[x]={y∈X:∥x−y∥≤d}.
Set
Cn
=
i≥nBd
[
xi
], then
Cn
is a nonempty convex set for each
n
. Using that
T
is a continuous
mapping and the given Picard iteration, we have
y∈Bd[xn]⇒∥y−xn∥≤d
⇒∥Ty−Tx
n∥≤d
⇒∥Ty−[xn+1 −f]∥≤d
⇒∥(Ty+f)−xn+1∥≤d
⇒(Ty+f)∈Bd[xn+1].
Applying this, we get the following
Tf(Cn)=Tf(
i≥n
Bd[xi])
⊆
i≥n
Tf(Bd[xi])
=
i≥n{Tf(y): ∥y−xi∥≤d}
=
i≥n{(Ty+f): ∥y−xi∥≤d}
⊆
i≥n+1
Bd[xi]=Cn+1.
Let us define
C=
n∈N
Cn.
Since Cnincreases with n,
Cn⊂Cn+1 ⊂Cn+2 ⊂.......,
it follows that Cis a closed, convex and bounded subset of X. Now we have
Tf(C)=Tf
n∈N
Cn⊆Tf
n∈N
Cn=
n∈N
Tf(Cn)⊆
n∈N
Cn+1 =C
giving Tf:C→Csince Tfis continuous mapping.
Finally, applying the Sadovskii fixed point theorem to
Tf
and
C
, we obtain that
Tf
has a fixed point
in Cwhich proves the theorem.
https://doi.org/10.17993/3ctic.2022.112.72-79
REFERENCES
[1]
Goebel K. and Kirk W. A. (1990). Topics in metric fixed point theory. Cambridge Studies in
Advanced Mathematics, volume 28,Cambridge University Press.
[2]
Khamsi M. A. and Kirk W. A. (2001). An Introduction to Metric Spaces and Fixed Point
Theory. Pure and Applied Mathematics. John Wiley & Sons, Inc.
[3]
Agarwal R. P. ,O’Regan R. P. , and Sahu D. R. (2009). Fixed point theory for Lipschitzian-
type mappings with applications. Topological fixed point theory and its Applications, volume 6.
Springer.
https://doi.org/10.17993/3ctic.2022.112.72-79
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
79
