Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 46, pp. 1–5.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.46

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF

INTERIOR POINTS FOR POSITIVE CONES

MOHAMMED SAID EL KHANNOUSSI, ABDERRAHIM ZERTITI

Abstract. Using partial ordering methods we give a sufficient condition for

a positive cone to have nonempty interior.

1. Introduction

Let (E, ∥ · ∥E) be a real Banach space and P be a nonempty closed convex set
in E. P is called a cone if it satisfies the following two conditions:

(i) x ∈ P and λ ≥ 0 imply λx ∈ P ,
(ii) x ∈ P and −x ∈ P implies x = θ, where θ denotes the zero element in E.

A cone P is said to be generating (or reproducing) if E = P − P , i.e., every
element x ∈ E can be represented in the form x = u− v where u, v ∈ P .

A cone P is called solid if there exists an element u0 which belongs to the interior
of the cone P , that is, there exists positive constant r such that

B(u0, r) = {x ∈ E : ∥u0 − x∥ ≤ r} ⊂ P.

A cone P defines a linear ordering in E by

x ≤ y if and only if y − x ∈ P.

A cone P is said to be normal if there exists a constant N > 0 such that

θ ≤ x ≤ y =⇒ ∥x∥ ≤ N∥y∥, x, y ∈ P.

We denote by u0 some fixed non-zero element of P . Our main result reads as
follows.

Theorem 1.1. If u0 be a non-zero element of P such that for any x ∈ E there
exists positive constant αx > 0 such that x ≤ αxu0, then u0 belongs to the interior
of the cone P . That is, there exists positive constant r such that

B(u0, r) = {x ∈ E : ∥u0 − x∥ ≤ r} ⊂ P.

In Section 3 we introduce the u0-norm and the space Eu0 , where u0 is a given

nonzero element of P . It is well-known that if P is a solid cone and u0 ∈ P̊ ,
then E = Eu0 . In this paper we shall study the converse statement and give an
improvement and generalization of [2, Theorem 1.5.1].

2020 Mathematics Subject Classification. 54F05, 47L07, 46B40.
Key words and phrases. Positive cone; solid cone; interior points; u0-norm;

Baire Hausdorf’s Theorem.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted November 3, 2023. Published August 21, 2024.

1



2 M. S. EL KHANNOUSSI, A. ZERTITI EJDE-2024/46

2. Proof of Theorem 1.1

To prove Theorem 1.1 we establish the following two lemmas. The first one is
based on [2, Lemma 1.4.2].

Lemma 2.1. Let u0 be a non-zero element of P such that for any x ∈ E there
exists positive constant αx > 0 satisfying x ≤ αxu0. Then a constant τ > 0 can
be found such that for any x ∈ E there exists positive constant β(x) > 0 such that
x ≤ β(x)u0 and ∥β(x)u0∥ ≤ τ∥x∥.

Proof. It is clear that E = ∪∞
n=1En, where

En = {x ∈ E : there is β(x) > 0 such that x ≤ β(x)u0 and ∥β(x)u0∥ ≤ n∥x∥},
for n = 1, 2, 3, . . . . By the Baire-Hausdorff’s Theorem (that is, a nonempty com-
plete metric space is a second Baire set), there exist positive integer n1, x0 ∈ E
and R > r > 0 satisfying

B0 = {x ∈ E : r < ∥x− x0∥ < R} ⊂ En1
.

Let β0 > 0 and n2 be a positive integer such that −x0 ≤ β0u0, and ∥β0u0∥ ≤
n2∥x0∥. Let B = {x ∈ E : r < ∥x∥ < R}, and choose an integer n3 satisfying

n3 > n1 +
1

r
(n1 + n2)∥x0∥.

In what follows, we prove that B ⊂ En3
. Indeed, for any x ∈ B, we have y =

x0 + x ∈ B0, then there exists a sequence {xi} ⊂ En1
such that xi → y as i → ∞.

Clearly, we can assume that xi ∈ B0 for i = 1, 2, 3, . . . . Take constants βi > 0 such
that xi ≤ βiu0 and ∥βiu0∥ ≤ n1∥xi∥. Then we obtain xi − x0 ≤ (βi + β0)u0 and

∥(βi + β0)u0∥ ≤ n1∥xi∥+ n2∥x0∥
≤ (n1 + n2)∥x0∥+ n1∥xi − x0∥

≤
[
(n1 + n2)

∥x0∥
r

+ n1

]
∥xi − x0∥

≤ n3∥xi − x0∥.
from which it follows that xi − x0 ∈ En3

for n = 1, 2, 3, . . . . From the fact that
xi − x0 → y − x0 as i → ∞ we obtain x ∈ En3

. Therefore B ⊂ En3
.

Clearly, from x ∈ En3
, we can easily prove that tx ∈ En3

, for all t ≥ 0. Conse-
quently, E = En3 .

Finally, we show that E = E3n3 . Taking x ∈ E such that x ̸= θ, then there
exists x1 ∈ En3

satisfying

∥x− x1∥ <
1

2
∥x∥.

Since x1 ∈ En3
, there exists β1 > 0 such that

x1 ≤ β1u0, ∥β1u0∥ ≤ n3∥x1∥.
Similarly, there exist x2 ∈ En3

and β2 > 0 such that

∥x− x1 − x2∥ <
1

22
∥x∥, x2 ≤ β2u0, ∥β2u0∥ ≤ n3∥x2∥.

Inductively, we find sequences {xk} ⊂ En3
and {βk} > 0, k = 1, 2, . . . , satisfying

∥x− x1 − x2 − · · · − xk∥ <
1

2k
∥x∥, xk ≤ βku0, and ∥βku0∥ ≤ n3∥xk∥ ,



EJDE-2024/46 EXISTENCE OF INTERIOR POINTS FOR POSITIVE CONES 3

for k = 1, 2, 3, . . . .
Clearly, x =

∑∞
k=1 xk and

∥xk∥ ≤ ∥x−
k−1∑
i=1

xi∥+ ∥x−
k∑

i=1

xi∥ <
3∥x∥
2k

k = 1, 2, . . . .

From which it follows that
∞∑
k=1

∥βku0∥ ≤ n3

∞∑
k=1

∥xk∥ ≤ 3n3∥x∥ < ∞.

Consequently the series
∑∞

k=1 βk converges to some constant β > 0. Clearly

x =

∞∑
k=1

xk ≤
∞∑
k=1

βku0 = βu0,

and

∥βu0∥ ≤
∞∑
k=1

∥βku0∥ ≤ 3n3∥x∥.

Therefore, x ∈ E3n3
, which implies that E = E3n3

. □

As a consequence of the previous lemma we have.

Lemma 2.2. Let u0 be a non-zero element of P such that for each x ∈ E there
exists positive constant αx > 0 satisfying x ≤ αxu0. Then there is a constant β > 0,
not depending on x, such that for every x ∈ E satisfying ∥x∥ ≤ 1 we have x ≤ βu0.

Proof. By using Lemma 2.1, for every x ∈ E satisfying ∥x∥ ≤ 1 there exists positive
constant β(x) > 0 such that x ≤ β(x)u0 and ∥β(x)u0∥ ≤ τ∥x∥ ≤ τ . Then for all
constant β > τ

∥u0∥ we have x ≤ βu0. □

Proof of Theorem 1.1. By Lemma 2.2 there is a constant β > 0 such that for every
x ∈ E satisfying ∥x∥ ≤ 1 we have x ≤ βu0. By taking r = 1

β we have for every

x ∈ E satisfying ∥x∥ ≤ 1, u0 − rx ≥ 0. Taking an element x ∈ E (x ̸= u0) such
that ∥u0 − x∥ ≤ r, we obtain

x = u0 − (u0 − x)

= u0 − r
∥u0 − x∥

r

u0 − x

∥u0 − x∥
≥ 0.

Consequently x ∈ P , which completes the proof. □

3. Space Eu0

In what follows, we suppose that P is a cone in E and let u0 be a non-zero
element of P . We define the space Eu0 and u0-norm as follows (see [7]),

Eu0
= {x ∈ E : there exists λ > 0 such that − λu0 ≤ x ≤ λu0},

∥x∥u0
= inf{λ > 0 : −λu0 ≤ x ≤ λu0}, x ∈ Eu0

.

It is easy to see that Eu0
is a normed linear space with the norm ∥·∥u0

. Then ∥x∥u0

is called a u0-norm of x ∈ Eu0
(see [7] for more details). The following theorem

can be found in [2, Theorem 1.5.1]

Theorem 3.1. If P is a normal cone, then:



4 M. S. EL KHANNOUSSI, A. ZERTITI EJDE-2024/46

(i) The space Eu0
is a Banach space.

(ii) Pu0 = P ∩ Eu0 is a normal solid cone in space Eu0 and

P̊u0
= {x ∈ Eu0

: there exists τ > 0 such that x ≥ τu0}
= {x ∈ E : there exists λ > τ > 0 such thatτu0 ≤ x ≤ λu0}.

Remark 3.2. If x ∈ E and there exists positive constant αx > 0 such that x ≤
αxu0, then from the inequality −x ≤ α−xu0, for some α−x > 0 one has −α−xu0 ≤
x ≤ αxu0. Then x ∈ Eu0

and thus E = Eu0
.

Theorem 3.3. A necessary and sufficient condition for a cone P to be solid is that
E = Eu0

.

Proof. Suppose that E = Eu0
then for any x ∈ E there exists λ > 0 such that

x ≤ λu0 hence by Theorem 3.1, u0 ∈ P̊ and thus P is a solid cone.
Conversely, suppose that u0 ∈ P̊ , then there exists positive constant r > 0 such

that B(u0, r) = {x ∈ E : ∥u0 − x∥ ≤ r} ⊂ P . For each x ∈ E, (x ̸= 0), we

have u0 ± r
∥x∥x ∈ P and then −∥x∥

r u0 ≤ x ≤ ∥x∥
r u0. Therefore, x ∈ Eu0 and

E = Eu0 . □

In what follows, we assume that P is a normal cone.

Theorem 3.4. If P is a solid cone, then u0 ∈ P̊ if and only if the u0−norm ∥ · ∥u0

is equivalent to the original norm ∥ · ∥.

Proof. Suppose that u0 ∈ P̊ , then there exists positive constant r > 0 such that
B(u0, r) = {x ∈ E : ∥u0 − x∥ ≤ r} ⊂ P . For each x ∈ E, (x ̸= 0), we have

−∥x∥
r u0 ≤ x ≤ ∥x∥

r u0. Then

∥x∥u0 ≤ 1

r
∥x∥, x ∈ E.

On the other hand, for each x ∈ Eu0 , we have −αu0 ≤ x ≤ αu0, where α = ∥x∥u0 ,
and then 0 ≤ x+ αu0 ≤ 2αu0. Thus, by the normality of P , we obtain

∥x+ αu0∥ ≤ 2αN∥u0∥,

where N is the normal constant of P , which implies that

∥x∥ ≤ ∥x+ αu0∥+ ∥ − αu0∥ ≤ M∥x∥u0
,

where M = (2N + 1)∥u0∥. Consequently, the u0-norm ∥ · ∥u0
is equivalent to the

original norm ∥ · ∥.
Conversely, suppose that for any x ∈ E there exist two positive constants c and

C satisfying

c∥x∥u0
≤ ∥x∥ ≤ C∥x∥u0

then it is easy to show that E = Eu0 , and then by Theorem 3.3, u0 ∈ P̊ . □

Remark 3.5. Theorem 3.3 does not assume P to be normal.

Remark 3.6. It is well-known that if P is a solid cone and u0 ∈ P̊ , then E = Eu0

and the u0-norm ∥ · ∥u0 is equivalent to the original norm ∥ · ∥. But here we have
studied the converse statement and then our work improves and generalizes [2,
Theorem 1.5.1].



EJDE-2024/46 EXISTENCE OF INTERIOR POINTS FOR POSITIVE CONES 5

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[3] D. Guo, Y. Cho, Z. Jiang, Partial Ordering Methods in Nonlinear Problems, Nova Science

Publishers, New York, 2004.

[4] M. S. El Khannoussi, A. Zertiti; Bounds for the spectral radius of positive operators, Elec-
tronic Journal of Differential Equations, 2022 (2022), no. 29 1-7.

[5] M.S. El Khannoussi, A. Zertiti; Topological methods in the study of positive solutions for
operator equations in ordered Banach spaces. Electronic Journal of Differential Equations,

2016 (2016) no. 171, 1-13.

[6] M. A. Krasnosel’skii, P. P. Zabreiko; Geometrical Methods of Nonlinear Analysis, Springer-
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[7] M. A. Krasnosel’skii; Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.

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Mohammed Said El Khannoussi

Université Abdelmalek Essaadi, Faculté des sciences, Département de Mathématiques,

BP 2121, Tétouan, Morocco
Email address: said 774@hotmail.com

Abderrahim Zertiti
Université Abdelmalek Essaadi, Faculté des sciences, Département de Mathématiques,

BP 2121, Tétouan, Morocco
Email address: abdzertiti@hotmail.fr


	1. Introduction
	2. Proof of Theorem 1.1
	3. Space Eu0
	References