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
Γuniquely 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 dL-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
xX
to
yX
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
)
(
t2
)) =
|t1t2|
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 . For
AX
, 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
be 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