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