THE LERAY-SCHAUDER PRINCIPLE IN GEODESIC SPA-
CES
Sreya Valiya Valappil
Department of Mathematics, Central University of kerala, Kasaragod, India.
E-mail:sreya.v.v@cukerala.ac.in
ORCID:0000-0003-1089-3036
Shaini Pulickakunnel
Department of Mathematics, Central University of kerala, Kasaragod, India.
E-mail:shainipv@cukerala.ac.in
ORCID:0000-0001-9958-9211
Reception: 16/09/2022 Acceptance: 01/10/2022 Publication: 29/12/2022
Suggested citation:
Valappil, S. V. and Pulickakunnel, S. (2022). The Leray-Schauder Principle in Geodesic Spaces. 3C TIC. Cuadernos de
desarrollo aplicados a las TIC,11 (2), 99-106 https://doi.org/10.17993/3ctic.2022.112.99-106
https://doi.org/10.17993/3ctic.2022.112.99-106
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
99
ABSTRACT
The essential maps introduced by Granas in 1976 is a best tool for proving continuation results for
compact maps. Several authors modified this idea to different scenarios. This is a review article, here,
we consider the continuation results based on essential maps and the Leray-Schauder principle in the
setting of uniquely geodesic spaces [20].
KEYWORDS
Γ−uniquely geodesic spaces, Essential maps, Fixed point, Leray Schauder principle.
https://doi.org/10.17993/3ctic.2022.112.99-106
1 INTRODUCTION
This paper is based on the notion of essential maps defined by Granas [9] in 1976. Many studies and
extensions of this concept in variety of settings have been done by several authors [2,8,16
–
18]. Essential
map techniques are one of the best tools to prove continuation results for compact maps [1].
Compact maps play a vital role in proving the existence and uniqueness of solutions of differential and
integral equations. The classical Schauder fixed point theorem proved by Juliusz Schauder in 1930 in
the setting of Banach spaces as a generalization of the celebrated Brouwer’s fixed point theorem and
the famous Leray-Schauder principle are milestones in the theory of fixed points and both of these
incredible results are based on compact maps. For many years, researchers were trying to extend the
concepts in normed spaces to more general spaces. In that direction, several authors have extended very
many important fixed point theorems to various general spaces [3, 15]. Most importantly, the geodesic
spaces grasped the attention of many fixed point theorists after the publication of the papers [11, 12]
due to Kirk. A version of Schauder fixed point theorem has been proved by Niculescu and Roventa in
CAT
(0) spaces [15] by assuming the compactness of convex hull of finite number of elements. Followed
by this, in [3], Ariza, Li and Lopez have proved the Schauder Fixed point theorem in the setting of
Γ
−
uniquely geodesic spaces, which includes Busemann spaces, Linear spaces,
CAT
(
κ
)spaces with
diameter less than Dk/2, Hyperbolic spaces [19] etc.
Granas introduced essential maps in order to prove continuation results for compact maps in Banach
spaces [9]. In his paper, he has proved topological transversality principle and the Leray Schauder
principle, using essential maps techniques. The Leray Schauder principle was first proved for compact
mappings in Banach spaces and this has been broadly used to obtain fixed points of variety of mappings
under different settings and many interesting contributions can be found in the literature [4, 6,7,13, 14].
In [9], Granas has also proved several fixed point results for compact maps using essential mappings
and homotopical methods. Followed by the results established by Granas, Agarwal and O’Regan [2]
have extended the concept of essential maps to a large class of mappings and established several fixed
point theorems in 2000. The concept of essential maps has been further extended to
d−
essential maps,
and d−L-essential maps [18].
In this review article, we consider the continuation results based on essential maps, and the Leray-
Schauder principle in the setting of uniquely geodesic spaces.
2 Preliminaries
We recall some basic definitions and results used in this paper.
Definition 1. [5] Let (
X, ρ
)be a metric space. A geodesic segment (or geodesic)from
x∈X
to
y∈X
is
a map
γ
from a closed interval [0
,l
]
⊆R
to
X
such that
γ
(0) =
u
,
γ
(
l
)=
v
and
ρ
(
γ
(
t1
)
,γ
(
t2
)) =
|t1−t2|
for all t1,t2in [0,l].
(
X, ρ
)is said to be a geodesic space if every two points in
X
are joined by a geodesic and (
X, ρ
)is
uniquely geodesic if there is exactly one geodesic joining uto v, for all u, v ∈X.
We denote the set of all geodesic segments in
X
by Ω. For
A⊆X
, closure and boundary of
A
in
X
is
denoted by Aand ∂A respectively.
Definition 2. [3] Let (
X, ρ
)be a metric space and Γ
⊆
Ωbe a family of geodesic segments. We say
that (
X, ρ
)is a Γ
−
uniquely geodesic space if for every
x, y ∈X
, there exists a unique geodesic in Γ
passes through xand y. We denote a unique geodesic segment in Γjoining xand yby γx,y.
Remark 1. [3] Let (
X, ρ
)be a Γ
−
uniquely geodesic space. Then, the family Γinduces a unique mapping
Γ
:
X2×
[0
,
1]
→X
such that
Γ
(
x, y, τ
)
∈γx,y
and the following properties hold for each
u, v ∈X
:
https://doi.org/10.17993/3ctic.2022.112.99-106
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
100
ABSTRACT
The essential maps introduced by Granas in 1976 is a best tool for proving continuation results for
compact maps. Several authors modified this idea to different scenarios. This is a review article, here,
we consider the continuation results based on essential maps and the Leray-Schauder principle in the
setting of uniquely geodesic spaces [20].
KEYWORDS
Γ−uniquely geodesic spaces, Essential maps, Fixed point, Leray Schauder principle.
https://doi.org/10.17993/3ctic.2022.112.99-106
1 INTRODUCTION
This paper is based on the notion of essential maps defined by Granas [9] in 1976. Many studies and
extensions of this concept in variety of settings have been done by several authors [2,8,16
–
18]. Essential
map techniques are one of the best tools to prove continuation results for compact maps [1].
Compact maps play a vital role in proving the existence and uniqueness of solutions of differential and
integral equations. The classical Schauder fixed point theorem proved by Juliusz Schauder in 1930 in
the setting of Banach spaces as a generalization of the celebrated Brouwer’s fixed point theorem and
the famous Leray-Schauder principle are milestones in the theory of fixed points and both of these
incredible results are based on compact maps. For many years, researchers were trying to extend the
concepts in normed spaces to more general spaces. In that direction, several authors have extended very
many important fixed point theorems to various general spaces [3, 15]. Most importantly, the geodesic
spaces grasped the attention of many fixed point theorists after the publication of the papers [11, 12]
due to Kirk. A version of Schauder fixed point theorem has been proved by Niculescu and Roventa in
CAT
(0) spaces [15] by assuming the compactness of convex hull of finite number of elements. Followed
by this, in [3], Ariza, Li and Lopez have proved the Schauder Fixed point theorem in the setting of
Γ
−
uniquely geodesic spaces, which includes Busemann spaces, Linear spaces,
CAT
(
κ
)spaces with
diameter less than Dk/2, Hyperbolic spaces [19] etc.
Granas introduced essential maps in order to prove continuation results for compact maps in Banach
spaces [9]. In his paper, he has proved topological transversality principle and the Leray Schauder
principle, using essential maps techniques. The Leray Schauder principle was first proved for compact
mappings in Banach spaces and this has been broadly used to obtain fixed points of variety of mappings
under different settings and many interesting contributions can be found in the literature [4, 6,7,13, 14].
In [9], Granas has also proved several fixed point results for compact maps using essential mappings
and homotopical methods. Followed by the results established by Granas, Agarwal and O’Regan [2]
have extended the concept of essential maps to a large class of mappings and established several fixed
point theorems in 2000. The concept of essential maps has been further extended to
d−
essential maps,
and d−L-essential maps [18].
In this review article, we consider the continuation results based on essential maps, and the Leray-
Schauder principle in the setting of uniquely geodesic spaces.
2 Preliminaries
We recall some basic definitions and results used in this paper.
Definition 1. [5] Let (
X, ρ
)be a metric space. A geodesic segment (or geodesic)from
x∈X
to
y∈X
is
a map
γ
from a closed interval [0
,l
]
⊆R
to
X
such that
γ
(0) =
u
,
γ
(
l
)=
v
and
ρ
(
γ
(
t1
)
,γ
(
t2
)) =
|t1−t2|
for all t1,t2in [0,l].
(
X, ρ
)is said to be a geodesic space if every two points in
X
are joined by a geodesic and (
X, ρ
)is
uniquely geodesic if there is exactly one geodesic joining uto v, for all u, v ∈X.
We denote the set of all geodesic segments in
X
by Ω. For
A⊆X
, closure and boundary of
A
in
X
is
denoted by Aand ∂A respectively.
Definition 2. [3] Let (
X, ρ
)be a metric space and Γ
⊆
Ωbe a family of geodesic segments. We say
that (
X, ρ
)is a Γ
−
uniquely geodesic space if for every
x, y ∈X
, there exists a unique geodesic in Γ
passes through xand y. We denote a unique geodesic segment in Γjoining xand yby γx,y.
Remark 1. [3] Let (
X, ρ
)be a Γ
−
uniquely geodesic space. Then, the family Γinduces a unique mapping
Γ
:
X2×
[0
,
1]
→X
such that
Γ
(
x, y, τ
)
∈γx,y
and the following properties hold for each
u, v ∈X
:
https://doi.org/10.17993/3ctic.2022.112.99-106
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
101
1. ρ(Γ(x, y, τ),Γ(u,v,s)) = |τ−s|ρ(u, v)for all τ,s ∈[0,1]
2. Γ(u, v, 0) = uand Γ(u, v, 1) = v
Also, it is enough to consider
Γ
(
u,v,τ
)=
γu,v
(
τρ
(
u, v
)), i.e., it is a point on
γ
at a distance
τρ
(
u, v
)
from uand we denote it by (1 −τ)u⊕τv.
Definition 3. [1] Let
X
and
Y
be two metric spaces. A map
ζ
:
X→Y
is called compact if
ζ
(
X
)is
contained in a compact subset of Y.
Definition 4. [1] Let
T
be a closed convex subset of a Banach space
X
and
S
be a closed subset of
T
.
Denote set of all continuous, compact maps
ζ
:
S→T
by
K
(
S, T
)and set of all maps
ζ∈K
(
S, T
)
with x=ζ(x)by K∂S(S, T )for x∈∂S.
A map
ζ∈K∂S
(
S, T
)is essential in
K∂S
(
S, T
)if for every map
ξ∈K∂S
(
S, T
)with
ζ|∂S
=
ξ|∂S
there
exists
x∈intS
with
x
=
ξ
(
x
). Otherwise
ζ
is inessential in
K∂S
(
S, T
), that is, there exists a fixed point
free ξ∈K∂S(S, T )with ζ|∂S =ξ|∂S.
Definition 5. [1] Two maps
ζ,ξ ∈K∂S
(
S, T
)are homotopic in
K∂S
(
S, T
), written
ζ≃ξ
in
K∂S
(
S, T
),
if there exists a continuous, compact mapping
H
:
S×
[0
,
1]
→T
such that
Ht
(
x
) :=
H
(
·,τ
):
S→T
belongs to K∂S(S, T )for each τ∈[0,1] with H0=ζand H1=ξ.
Definition 6. [3] AΓ−uniquely geodesic space is said to have property (Q) if
lim
ε→0sup{ρ((1 −τ)x⊕τy, (1 −τ)x⊕τz):τ∈[0,1], x, y, z ∈X, ρ(y, z)≤ε}=0.
Definition 7. [3] Let
S
be a nonempty subset of a Γ
−
uniquely geodesic space. We say that
S
is
Γ−convex if γx,y ⊆Sfor all x, y ∈S
Remark 2. [3] Let (
X, ∥.∥
)be a normed linear space, Γ
L
the family of linear segments and let
ρ
denote
the metric induced by the norm ∥.∥. Then (X, ρ)is a ΓL-uniquely geodesic space with property (Q).
Definition 8. [3] A metric space (
X, ρ
)is a hyperbolic space (in the sense of Reich-Shafir [19]) if
X
is Γ−uniquely geodesic and the following inequality holds
ρ(1
2x⊕1
2y, 1
2x⊕1
2z)≤1
2ρ(y, z).
Many authors have developed various extensions, and modifications to Schauder’s fixed point theorem
[10,15]. In [3], Schauder-type fixed point theorem in the setting of geodesic spaces with the property
(Q) is proved as follows:
Theorem 1. [3] Let (
X, ρ
)be a Γ
−
uniquely geodesic space with property (Q), and all balls are
Γ-convex. Let
K
be a nonempty, closed, Γ
−
convex subset of (
X, ρ
). Then, any continuous mapping
T:K→Kwith compact range T(K)has at least one fixed point in K.
Throughout this paper, we denote a closed Γ
−
convex subset of the geodesic space
X
by
T
and interior
of
S
by
int S
, where
S
is a closed subset of
T
. We denote the set of all continuous, compact maps
ζ:S→Tby K(S, T )and set of all maps ζ∈K(S, T )with x=ζ(x)by K∂S(S, T )for x∈∂S.
3 RESULTS
Theorem 2. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q) and
ζ,ξ ∈K∂S
(
S, T
).
Suppose that for all (u, τ)∈∂S ×[0,1],
u= (1 −τ)ζ(u)⊕τξ(u)
i.e., geodesic segment joining ζ(u)and ξ(u)does not contain u. Then ζ≃ξin K∂S(S, T ).
https://doi.org/10.17993/3ctic.2022.112.99-106
Proof. Let H(u, τ)=Ht(u)=Γ(ζ(u),ξ(u),τ) = (1 −τ)u⊕τξu. Clearly, His continuous.
We show that H:S×[0,1] →Tis a compact map.
Let
{xi}
be a sequence in
S
. Since
ζ,ξ
:
S→T
are compact maps,
ζ
(
xi
)
→u
and
ξ
(
xi
)
→v
as
i→∞
for some subsequence
S
of natural numbers and
u, v ∈T
. Let
τ∈
[0
,
1] be such that
τ
is the limit of
some sequence τi∈[0,1]. Now using Remark 1,
ρH(xi,τ
i),(1 −τ)u⊕τv=ρ
Γ
(ζ(xi),ξ(xi),τ
i),(1 −τ)u⊕τv
≤ρ(1 −τi)ζ(xi)⊕τiξ(xi),(1 −τ)ζ(xi)⊕τξ(xi)
+ρ(1 −τ)ζ(xi)⊕τξ(xi),(1 −τ)u⊕τv
≤|τi−τ|ρ(ζ(xi),ξ(xi))
+ρ((1 −τ)ζ(xi)⊕τξ(xi),(1 −τ)ζ(xi)⊕τv)
+ρ((1 −τ)ζ(xi)⊕τv, (1 −τ)u⊕τv).(1)
Since τi→τ, we get
|τi−τ|ρ(ζ(xi),ξ(xi)) →0.(2)
Since ξ(xi)→v, we have ρ(ξ(xi),v)≤ϵfor every ϵ>0. Thus using property (Q), we get,
ρ(1 −τ)ζ(xi)⊕τξ(xi),(1 −τ)ζ(xi)⊕τv→0.(3)
Similarly,
ρ(1 −τ)ζ(xi)⊕τv, (1 −τ)u⊕τv→0.(4)
Using (2), (3) and (4) in (1), we have
{H(xi,τ
i)}→(1 −τ)u⊕τv, for ti∈[0,1].(5)
Since Tis Γ−convex, (1 −τ)u⊕τv ∈T. Hence His compact.
But it is given that
u
= (1
−τ
)
ζ
(
u
)
⊕τξ
(
u
)for (
u, τ
)
∈∂S ×
[0
,
1]. Hence
Hτ
(
u
)
∈K∂S
(
S, T
),
H(u, 0) = ζ(u)and H(u, 1) = ξ(u). Therefore ζ≃ξin K∂S(S, T ).
Following result in [1] is a corollary to our theorem.
Corollary 1. [1, Theorem 6.1] Let
X
be a Banach space,
T
a closed, convex subset of
X
,
S
a closed
subset of Tand ζ,ξ ∈K∂S(S, T ). Suppose that for all (u, λ)∈∂S ×[0,1],
u= (1 −λ)ζ(u)+λξ(u).
Then ζ≃ξin K∂S(S, T ).
Proof. Using Remark 2,
X
is a Γ
L
-uniquely geodesic space. Hence by Theorem 2, the result follows.
Next result is a characterization of inessential maps in K∂S(S, T )in Γ−uniquely geodesic spaces.
Theorem 3. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q) and let
ζ∈K∂S
(
S, T
).
Then
ζ
is inessential in
K∂S
(
S, T
)if and only if there exists
ξ∈K∂S
(
S, T
)with
ξ
(
u
)
=
u
for all
u∈S
and ζ≃ξin K∂S(S, T ).
Proof. Assume that
ζ
is inessential in
K∂S
(
S, T
). Hence there exists a map
ξ∈K∂S
(
S, T
)such that
ξ
(
u
)
=
u
for all
u∈S
and
ζ|∂S
=
ξ|∂S
by definition. Suppose there exists (
u, τ
)
∈∂S ×
[0
,
1] such that
u
= (1
−τ
)
ζ
(
u
)
⊕τξ
(
u
)
.
Since
ζ|∂S
=
ξ|∂S
, it follows that
u
=
ξ
(
u
), which is a contradiction to the
fact that
ξ∈K∂S
(
S, T
). Hence
u
= (1
−τ
)
ζ
(
u
)
⊕τξ
(
u
)
,
for each (
u, τ
)
∈∂S ×
[0
,
1]
.
Hence by using
Theorem 2, we have ζ≃ξ. Thus we have ξu =uand ζ≃ξ.
Conversely assume that there exists
ξ∈K∂S
(
S, T
)with
ξ
(
u
)
=
u
for all
u∈S
and
ζ≃ξ
in
K∂S
(
S, T
).
Let
H
:
S×
[0
,
1]
→T
with
Ht∈K∂S
(
S, T
)for all
τ∈
[0
,
1] be a continuous compact map with
H0
=
ζ
and H1=ξ. Consider
M=u∈S:u=H(u, τ)for some τ∈[0,1].
https://doi.org/10.17993/3ctic.2022.112.99-106
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
102
1. ρ(Γ(x, y, τ),Γ(u,v,s)) = |τ−s|ρ(u, v)for all τ,s ∈[0,1]
2. Γ(u, v, 0) = uand Γ(u, v, 1) = v
Also, it is enough to consider
Γ
(
u,v,τ
)=
γu,v
(
τρ
(
u, v
)), i.e., it is a point on
γ
at a distance
τρ
(
u, v
)
from uand we denote it by (1 −τ)u⊕τv.
Definition 3. [1] Let
X
and
Y
be two metric spaces. A map
ζ
:
X→Y
is called compact if
ζ
(
X
)is
contained in a compact subset of Y.
Definition 4. [1] Let
T
be a closed convex subset of a Banach space
X
and
S
be a closed subset of
T
.
Denote set of all continuous, compact maps
ζ
:
S→T
by
K
(
S, T
)and set of all maps
ζ∈K
(
S, T
)
with x=ζ(x)by K∂S(S, T )for x∈∂S.
A map
ζ∈K∂S
(
S, T
)is essential in
K∂S
(
S, T
)if for every map
ξ∈K∂S
(
S, T
)with
ζ|∂S
=
ξ|∂S
there
exists
x∈intS
with
x
=
ξ
(
x
). Otherwise
ζ
is inessential in
K∂S
(
S, T
), that is, there exists a fixed point
free ξ∈K∂S(S, T )with ζ|∂S =ξ|∂S.
Definition 5. [1] Two maps
ζ,ξ ∈K∂S
(
S, T
)are homotopic in
K∂S
(
S, T
), written
ζ≃ξ
in
K∂S
(
S, T
),
if there exists a continuous, compact mapping
H
:
S×
[0
,
1]
→T
such that
Ht
(
x
) :=
H
(
·,τ
):
S→T
belongs to K∂S(S, T )for each τ∈[0,1] with H0=ζand H1=ξ.
Definition 6. [3] AΓ−uniquely geodesic space is said to have property (Q) if
lim
ε→0sup{ρ((1 −τ)x⊕τy, (1 −τ)x⊕τz):τ∈[0,1], x, y, z ∈X, ρ(y, z)≤ε}=0.
Definition 7. [3] Let
S
be a nonempty subset of a Γ
−
uniquely geodesic space. We say that
S
is
Γ−convex if γx,y ⊆Sfor all x, y ∈S
Remark 2. [3] Let (
X, ∥.∥
)be a normed linear space, Γ
L
the family of linear segments and let
ρ
denote
the metric induced by the norm ∥.∥. Then (X, ρ)is a ΓL-uniquely geodesic space with property (Q).
Definition 8. [3] A metric space (
X, ρ
)is a hyperbolic space (in the sense of Reich-Shafir [19]) if
X
is Γ−uniquely geodesic and the following inequality holds
ρ(1
2x⊕1
2y, 1
2x⊕1
2z)≤1
2ρ(y, z).
Many authors have developed various extensions, and modifications to Schauder’s fixed point theorem
[10,15]. In [3], Schauder-type fixed point theorem in the setting of geodesic spaces with the property
(Q) is proved as follows:
Theorem 1. [3] Let (
X, ρ
)be a Γ
−
uniquely geodesic space with property (Q), and all balls are
Γ-convex. Let
K
be a nonempty, closed, Γ
−
convex subset of (
X, ρ
). Then, any continuous mapping
T:K→Kwith compact range T(K)has at least one fixed point in K.
Throughout this paper, we denote a closed Γ
−
convex subset of the geodesic space
X
by
T
and interior
of
S
by
int S
, where
S
is a closed subset of
T
. We denote the set of all continuous, compact maps
ζ:S→Tby K(S, T )and set of all maps ζ∈K(S, T )with x=ζ(x)by K∂S(S, T )for x∈∂S.
3 RESULTS
Theorem 2. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q) and
ζ,ξ ∈K∂S
(
S, T
).
Suppose that for all (u, τ)∈∂S ×[0,1],
u= (1 −τ)ζ(u)⊕τξ(u)
i.e., geodesic segment joining ζ(u)and ξ(u)does not contain u. Then ζ≃ξin K∂S(S, T ).
https://doi.org/10.17993/3ctic.2022.112.99-106
Proof. Let H(u, τ)=Ht(u)=Γ(ζ(u),ξ(u),τ) = (1 −τ)u⊕τξu. Clearly, His continuous.
We show that H:S×[0,1] →Tis a compact map.
Let
{xi}
be a sequence in
S
. Since
ζ,ξ
:
S→T
are compact maps,
ζ
(
xi
)
→u
and
ξ
(
xi
)
→v
as
i→∞
for some subsequence
S
of natural numbers and
u, v ∈T
. Let
τ∈
[0
,
1] be such that
τ
is the limit of
some sequence τi∈[0,1]. Now using Remark 1,
ρH(xi,τ
i),(1 −τ)u⊕τv=ρ
Γ
(ζ(xi),ξ(xi),τ
i),(1 −τ)u⊕τv
≤ρ(1 −τi)ζ(xi)⊕τiξ(xi),(1 −τ)ζ(xi)⊕τξ(xi)
+ρ(1 −τ)ζ(xi)⊕τξ(xi),(1 −τ)u⊕τv
≤|τi−τ|ρ(ζ(xi),ξ(xi))
+ρ((1 −τ)ζ(xi)⊕τξ(xi),(1 −τ)ζ(xi)⊕τv)
+ρ((1 −τ)ζ(xi)⊕τv, (1 −τ)u⊕τv).(1)
Since τi→τ, we get
|τi−τ|ρ(ζ(xi),ξ(xi)) →0.(2)
Since ξ(xi)→v, we have ρ(ξ(xi),v)≤ϵfor every ϵ>0. Thus using property (Q), we get,
ρ(1 −τ)ζ(xi)⊕τξ(xi),(1 −τ)ζ(xi)⊕τv→0.(3)
Similarly,
ρ(1 −τ)ζ(xi)⊕τv, (1 −τ)u⊕τv→0.(4)
Using (2), (3) and (4) in (1), we have
{H(xi,τ
i)}→(1 −τ)u⊕τv, for ti∈[0,1].(5)
Since Tis Γ−convex, (1 −τ)u⊕τv ∈T. Hence His compact.
But it is given that
u
= (1
−τ
)
ζ
(
u
)
⊕τξ
(
u
)for (
u, τ
)
∈∂S ×
[0
,
1]. Hence
Hτ
(
u
)
∈K∂S
(
S, T
),
H(u, 0) = ζ(u)and H(u, 1) = ξ(u). Therefore ζ≃ξin K∂S(S, T ).
Following result in [1] is a corollary to our theorem.
Corollary 1. [1, Theorem 6.1] Let
X
be a Banach space,
T
a closed, convex subset of
X
,
S
a closed
subset of Tand ζ,ξ ∈K∂S(S, T ). Suppose that for all (u, λ)∈∂S ×[0,1],
u= (1 −λ)ζ(u)+λξ(u).
Then ζ≃ξin K∂S(S, T ).
Proof. Using Remark 2,
X
is a Γ
L
-uniquely geodesic space. Hence by Theorem 2, the result follows.
Next result is a characterization of inessential maps in K∂S(S, T )in Γ−uniquely geodesic spaces.
Theorem 3. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q) and let
ζ∈K∂S
(
S, T
).
Then
ζ
is inessential in
K∂S
(
S, T
)if and only if there exists
ξ∈K∂S
(
S, T
)with
ξ
(
u
)
=
u
for all
u∈S
and ζ≃ξin K∂S(S, T ).
Proof. Assume that
ζ
is inessential in
K∂S
(
S, T
). Hence there exists a map
ξ∈K∂S
(
S, T
)such that
ξ
(
u
)
=
u
for all
u∈S
and
ζ|∂S
=
ξ|∂S
by definition. Suppose there exists (
u, τ
)
∈∂S ×
[0
,
1] such that
u
= (1
−τ
)
ζ
(
u
)
⊕τξ
(
u
)
.
Since
ζ|∂S
=
ξ|∂S
, it follows that
u
=
ξ
(
u
), which is a contradiction to the
fact that
ξ∈K∂S
(
S, T
). Hence
u
= (1
−τ
)
ζ
(
u
)
⊕τξ
(
u
)
,
for each (
u, τ
)
∈∂S ×
[0
,
1]
.
Hence by using
Theorem 2, we have ζ≃ξ. Thus we have ξu =uand ζ≃ξ.
Conversely assume that there exists
ξ∈K∂S
(
S, T
)with
ξ
(
u
)
=
u
for all
u∈S
and
ζ≃ξ
in
K∂S
(
S, T
).
Let
H
:
S×
[0
,
1]
→T
with
Ht∈K∂S
(
S, T
)for all
τ∈
[0
,
1] be a continuous compact map with
H0
=
ζ
and H1=ξ. Consider
M=u∈S:u=H(u, τ)for some τ∈[0,1].
https://doi.org/10.17993/3ctic.2022.112.99-106
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
103
There arise two cases.
Case-1: M=∅.
If
M
=
∅
then
Ht
(
u
)
=
u
for all
u∈S
and
τ∈
[0
,
1]
.
In particular,
ζ
(
u
)=
H0
(
u
)
=
u
for all
u∈S
.
Hence the inessentiality of ζin K∂S(S, T )follows.
Case-2: M=∅.
Let
{xi}∈M
such that
{xi}→u
. Then
xi
=
H
(
xi,τ
i
). Using the continuity of
H
, we get
u
=
H
(
u, τ
).
Thus Mis a closed subset of S.
Now, suppose
M∂S
=
∅
. If
u∈M∂S
, then
u∈M
and
u∈∂S
, which implies
u
=
Ht
(
u
)for
u∈∂S
, a contradiction since
Ht∈K∂S
(
S, T
). Hence by the Urysohn’s lemma, there exist
ξ
:
S→
[0
,
1]
continuous, with ξ(M)=1and ξ(∂S)=0.
Define
f
:
S→T
by
f
(
u
)=
H
(
u, ξ
(
u
)). Clearly,
f
is a continuous compact map. If
u∈∂S
,
f(u)=H(u, 0) = ζ(u).Thus f|∂S =ζ|∂S.
If
u
=
f
(
u
)for some
u∈S
then,
u
=
f
(
u
)=
H
(
u, ξ
(
u
)). Thus
u∈M
and hence
ξ
(
u
)=1. Hence
u
=
f
(
u
)=
H
(
u, ξ
(
u
)) =
H
(
u,
1) =
ξ
(
u
). Thus
f∈K∂S
(
S, T
)with
u
=
f
(
u
)with
f|∂S
=
ζ|∂S
.
Therefore
ξ
has a fixed point, which is a contradiction to our assumption. Thus
u
=
f
(
u
). Hence
ζ
is
inessential in K∂S(S, T ), by Definition 4. Hence the proof.
As a consequence of Theorem 3, we obtain the following corollary.
Corollary 2. Let
X
be a Hyperbolic space (in the sense of [19]),
T
a closed, convex subset of
X
,
S
a
closed subset of Tand ζ∈K∂S(S, T ). Then the following are equivalent:
(i) ζis inessential in K∂S(S, T ).
(ii) There exists ξ∈K∂S(S, T )with ξ(u)=ufor all u∈Sand ζ≃ξin K∂S(S, T ).
Proof. Every Hyperbolic space is a Γ-uniquely geodesic space with property (Q). Hence the result
follows from Theorem 3.
Theorem 4. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q). Suppose that
ζ,ξ ∈K∂S
(
S, T
)with
ζ≃ξ
in
K∂S
(
S, T
). Then
ζ
is essential in
K∂S
(
S, T
)if and only if
ξ
is essential
in K∂S(S, T ).
Proof. Suppose
ζ
is inessential in
K∂S
(
S, T
). Then from Theorem 3, there exists
T∈K∂S
(
S, T
)
with
ζ≃T
in
K∂S
(
S, T
)such that
T
(
u
)
=
u
for all
u∈S
. Hence,
ξ≃T
in
K∂S
(
S, T
). Thus
ζ≃ξ
and
ξ≃T
implies
ξ≃T
in
K∂S
(
S, T
). Therefore by Theorem 3,
ξ
is inessential in
K∂S
(
S, T
). Hence
the proof.
Corollary 3. [20] Let
X
be a Hyperbolic space (in the sense of [19]),
S
be a closed, convex subset
of
X
,
S
be a closed subset of
T
and
ζ,ξ ∈K∂S
(
S, T
)with
ζ≃ξ
in
K∂S
(
S, T
). Then
ζ
is essential in
K∂S(S, T )if and only if ξis essential in K∂S(S, T ).
The above corollary is a special case of Theorem 4.
Theorem 5. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space with all balls are Γ
−
convex and satisfies
property (Q). Let w∈int S. Then the map ζ(S)=wis essential in K∂S(S, T ).
Proof. Consider the continuous compact map
ξ
:
S→T
which agrees with
ζ
on
∂S
. It is enough to
show that ξ(u)=ufor some u∈int S. Let ζ:T→Tbe given by
ζ(u)=ξ(u),u∈S;
w, u ∈T\S.
Then
ζ
is continuous and compact (since
ξ
and
w
are continuous and compact and on
∂S
,
ζ
(
u
)=
ξ
(
u
)=
w
). Thus by Theorem 1,
ζ
(
u
)=
u
for some
u∈S
. If
u∈T\S
, we get
ζ
(
u
)=
w
. Hence
https://doi.org/10.17993/3ctic.2022.112.99-106
ζ
(
w
)=
w
, which is a contradiction, since
w∈intS
. Clearly
u∈int S
. Hence
x
is a fixed point of
ξ
.
Therefore ζis essential in K∂S(S, S).
The following result in [1] is a corollary to our theorem.
Corollary 4. [1, Theorem 6.5] Let
X
be a Banach space,
T
a closed, convex subset of
X
,
S
a closed
subset of Tand u∈int S. Then the constant map ζ(S)=wis essential in K∂S(S, T ).
Proof. We know that all balls in Banach spaces are convex. Hence the result follows from theorem 5.
As a consequence of the above theorems, authors proved the Leray Schauder principle in Γ
−
uniquely
geodesic spaces in [20], which generalizes the Leray Schauder principle in Hyperbolic spaces proved
in [3].
Theorem 6. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q), and all balls are
Γ−convex. Suppose that ζ:S→Tis a continuous compact map. Then either
(i) ζhas a fixed point in S, or
(ii) There exists (x0,τ)∈∂S ×[0,1] such that x0= (1 −τ)u⊕τζ(x0).
Proof. Suppose that (
ii
)does not hold and
ζ
(
u
)
=
u
for all
u∈∂S
. Define
ξ
:
S→T
by
ξ
(
u
)=
w
for all u∈S. Consider the map H:S×[0,1] →Tdefined by
H(u, τ) := (1 −τ)w⊕τζ(u),
which is continuous and compact. Also, for all
u∈∂S
,
Ht
(
u
) = (1
−τ
)
w⊕τζ
(
u
)
=
u
for a fixed
τ
(since we assumed that condition (
ii
)does not hold). Hence by Theorem 2,
ζ≃w
in
K∂S
(
S, T
)
.
But
we know that
w
is essential in
K∂S
(
S, T
)by theorem 5. Hence
ζ
is essential in
K∂S
(
S, T
)by Theorem
4. Now using Definition 4, ζ(u)=ufor some u∈int S.
The following Theorem is a consequence of Theorem 6.
Corollary 5. [3, Theorem 23] Let
X
be a Hyperbolic space(in the sense of [19]),
x0∈X
and
r>
0.
Suppose that T:B[x0,r]→Xbe a continuous mapping with T(B[x0,r]) compact. Then either
(i) Thas at least one fixed point in B[x0,r], or
(ii) There exists (u, λ)∈∂B[x0,r]×[0,1] with u= (1 −λ)x0⊕λT (u).
Proof. Hyperbolic spaces are geodesic spaces with property (Q), and balls are Γ-convex. Hence the
result follows from the above theorem.
4 CONCLUSIONS
In this review article, we considered continuation results in Γ-uniquely geodesic spaces and the Leray-
Schauder principle in Γ
−
uniquely geodesic spaces proved in [20]. Geodesic spaces having the property (Q)
include Busemann spaces, linear spaces,
CAT
(
κ
)spaces with diameters smaller than
Dk/
2, hyperbolic
spaces (in the sense of [19], etc., and balls are Γ-convex in these spaces. All the results in this paper
are thus applicable to these spaces as well. One can attempt to develop similar results for multivalued
mappings in geodesic spaces and establish similar results for d-essential maps, d-L essential maps, and
other general classes of maps [2,18].
ACKNOWLEDGMENT
The first author is highly grateful to University Grant Commission, India, for providing financial
support in the form of Junior/Senior Research fellowship.
https://doi.org/10.17993/3ctic.2022.112.99-106
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
104
There arise two cases.
Case-1: M=∅.
If
M
=
∅
then
Ht
(
u
)
=
u
for all
u∈S
and
τ∈
[0
,
1]
.
In particular,
ζ
(
u
)=
H0
(
u
)
=
u
for all
u∈S
.
Hence the inessentiality of ζin K∂S(S, T )follows.
Case-2: M=∅.
Let
{xi}∈M
such that
{xi}→u
. Then
xi
=
H
(
xi,τ
i
). Using the continuity of
H
, we get
u
=
H
(
u, τ
).
Thus Mis a closed subset of S.
Now, suppose
M∂S
=
∅
. If
u∈M∂S
, then
u∈M
and
u∈∂S
, which implies
u
=
Ht
(
u
)for
u∈∂S
, a contradiction since
Ht∈K∂S
(
S, T
). Hence by the Urysohn’s lemma, there exist
ξ
:
S→
[0
,
1]
continuous, with ξ(M)=1and ξ(∂S)=0.
Define
f
:
S→T
by
f
(
u
)=
H
(
u, ξ
(
u
)). Clearly,
f
is a continuous compact map. If
u∈∂S
,
f(u)=H(u, 0) = ζ(u).Thus f|∂S =ζ|∂S.
If
u
=
f
(
u
)for some
u∈S
then,
u
=
f
(
u
)=
H
(
u, ξ
(
u
)). Thus
u∈M
and hence
ξ
(
u
)=1. Hence
u
=
f
(
u
)=
H
(
u, ξ
(
u
)) =
H
(
u,
1) =
ξ
(
u
). Thus
f∈K∂S
(
S, T
)with
u
=
f
(
u
)with
f|∂S
=
ζ|∂S
.
Therefore
ξ
has a fixed point, which is a contradiction to our assumption. Thus
u
=
f
(
u
). Hence
ζ
is
inessential in K∂S(S, T ), by Definition 4. Hence the proof.
As a consequence of Theorem 3, we obtain the following corollary.
Corollary 2. Let
X
be a Hyperbolic space (in the sense of [19]),
T
a closed, convex subset of
X
,
S
a
closed subset of Tand ζ∈K∂S(S, T ). Then the following are equivalent:
(i) ζis inessential in K∂S(S, T ).
(ii) There exists ξ∈K∂S(S, T )with ξ(u)=ufor all u∈Sand ζ≃ξin K∂S(S, T ).
Proof. Every Hyperbolic space is a Γ-uniquely geodesic space with property (Q). Hence the result
follows from Theorem 3.
Theorem 4. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q). Suppose that
ζ,ξ ∈K∂S
(
S, T
)with
ζ≃ξ
in
K∂S
(
S, T
). Then
ζ
is essential in
K∂S
(
S, T
)if and only if
ξ
is essential
in K∂S(S, T ).
Proof. Suppose
ζ
is inessential in
K∂S
(
S, T
). Then from Theorem 3, there exists
T∈K∂S
(
S, T
)
with
ζ≃T
in
K∂S
(
S, T
)such that
T
(
u
)
=
u
for all
u∈S
. Hence,
ξ≃T
in
K∂S
(
S, T
). Thus
ζ≃ξ
and
ξ≃T
implies
ξ≃T
in
K∂S
(
S, T
). Therefore by Theorem 3,
ξ
is inessential in
K∂S
(
S, T
). Hence
the proof.
Corollary 3. [20] Let
X
be a Hyperbolic space (in the sense of [19]),
S
be a closed, convex subset
of
X
,
S
be a closed subset of
T
and
ζ,ξ ∈K∂S
(
S, T
)with
ζ≃ξ
in
K∂S
(
S, T
). Then
ζ
is essential in
K∂S(S, T )if and only if ξis essential in K∂S(S, T ).
The above corollary is a special case of Theorem 4.
Theorem 5. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space with all balls are Γ
−
convex and satisfies
property (Q). Let w∈int S. Then the map ζ(S)=wis essential in K∂S(S, T ).
Proof. Consider the continuous compact map
ξ
:
S→T
which agrees with
ζ
on
∂S
. It is enough to
show that ξ(u)=ufor some u∈int S. Let ζ:T→Tbe given by
ζ(u)=ξ(u),u∈S;
w, u ∈T\S.
Then
ζ
is continuous and compact (since
ξ
and
w
are continuous and compact and on
∂S
,
ζ
(
u
)=
ξ
(
u
)=
w
). Thus by Theorem 1,
ζ
(
u
)=
u
for some
u∈S
. If
u∈T\S
, we get
ζ
(
u
)=
w
. Hence
https://doi.org/10.17993/3ctic.2022.112.99-106
ζ
(
w
)=
w
, which is a contradiction, since
w∈intS
. Clearly
u∈int S
. Hence
x
is a fixed point of
ξ
.
Therefore ζis essential in K∂S(S, S).
The following result in [1] is a corollary to our theorem.
Corollary 4. [1, Theorem 6.5] Let
X
be a Banach space,
T
a closed, convex subset of
X
,
S
a closed
subset of Tand u∈int S. Then the constant map ζ(S)=wis essential in K∂S(S, T ).
Proof. We know that all balls in Banach spaces are convex. Hence the result follows from theorem 5.
As a consequence of the above theorems, authors proved the Leray Schauder principle in Γ
−
uniquely
geodesic spaces in [20], which generalizes the Leray Schauder principle in Hyperbolic spaces proved
in [3].
Theorem 6. [20] Let (
X, ρ
)be a Γ-uniquely geodesic space satisfying property (Q), and all balls are
Γ−convex. Suppose that ζ:S→Tis a continuous compact map. Then either
(i) ζhas a fixed point in S, or
(ii) There exists (x0,τ)∈∂S ×[0,1] such that x0= (1 −τ)u⊕τζ(x0).
Proof. Suppose that (
ii
)does not hold and
ζ
(
u
)
=
u
for all
u∈∂S
. Define
ξ
:
S→T
by
ξ
(
u
)=
w
for all u∈S. Consider the map H:S×[0,1] →Tdefined by
H(u, τ) := (1 −τ)w⊕τζ(u),
which is continuous and compact. Also, for all
u∈∂S
,
Ht
(
u
) = (1
−τ
)
w⊕τζ
(
u
)
=
u
for a fixed
τ
(since we assumed that condition (
ii
)does not hold). Hence by Theorem 2,
ζ≃w
in
K∂S
(
S, T
)
.
But
we know that
w
is essential in
K∂S
(
S, T
)by theorem 5. Hence
ζ
is essential in
K∂S
(
S, T
)by Theorem
4. Now using Definition 4, ζ(u)=ufor some u∈int S.
The following Theorem is a consequence of Theorem 6.
Corollary 5. [3, Theorem 23] Let
X
be a Hyperbolic space(in the sense of [19]),
x0∈X
and
r>
0.
Suppose that T:B[x0,r]→Xbe a continuous mapping with T(B[x0,r]) compact. Then either
(i) Thas at least one fixed point in B[x0,r], or
(ii) There exists (u, λ)∈∂B[x0,r]×[0,1] with u= (1 −λ)x0⊕λT (u).
Proof. Hyperbolic spaces are geodesic spaces with property (Q), and balls are Γ-convex. Hence the
result follows from the above theorem.
4 CONCLUSIONS
In this review article, we considered continuation results in Γ-uniquely geodesic spaces and the Leray-
Schauder principle in Γ
−
uniquely geodesic spaces proved in [20]. Geodesic spaces having the property (Q)
include Busemann spaces, linear spaces,
CAT
(
κ
)spaces with diameters smaller than
Dk/
2, hyperbolic
spaces (in the sense of [19], etc., and balls are Γ-convex in these spaces. All the results in this paper
are thus applicable to these spaces as well. One can attempt to develop similar results for multivalued
mappings in geodesic spaces and establish similar results for d-essential maps, d-L essential maps, and
other general classes of maps [2,18].
ACKNOWLEDGMENT
The first author is highly grateful to University Grant Commission, India, for providing financial
support in the form of Junior/Senior Research fellowship.
https://doi.org/10.17993/3ctic.2022.112.99-106
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
105
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https://doi.org/10.17993/3ctic.2022.112.99-106
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