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
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