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