@
Appl. Gen. Topol. 23, no. 1 (2022), 107-119

doi:10.4995/agt.2022.15669

© AGT, UPV, 2022

Fixed point index computations for multivalued

mapping and application to the problem of

positive eigenvalues in ordered space

Vo Viet Tri

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam

(trivv@tdmu.edu.vn)

Communicated by E. Karapinar

Abstract

In this paper, we present some results on fixed point index calcula-
tions for multivalued mappings and apply them to prove the existence
of solutions to multivalued equations in ordered space, under flexible
conditions for the positive eigenvalue.

2020 MSC: 47H07; 47H08; 47H10; 35P30.

Keywords: multivalued operator; multivalued mapping; fixed point index;
eigenvalue; eigenvector.

1. Introduction

The theory of the fixed point index for the compact single-valued mapping
has achieved many brilliant achievements in studying the existence of solutions
of equations, through the existence results on fixed points of operators (see
e.g. [1, 9, 14–17]). This concept has been extended to multivalued mappings
very early in [10] and the references therein. Up to now, this topic has always
been interested by many mathematicians (see e.g. [2–8,12,18–23,28,30,31]). An
impressive achievement when extending this theory to multivalued mappings
can be found in [12], in which the authors established the concept of a fixed
point index with respect to the cone for multivalued operators acting on convex
Fréchet spaces with convex and closed values. This concept has remarkable

Received 22 May 2021 – Accepted 10 July 2021

http://dx.doi.org/10.4995/agt.2022.15669


V. V. Tri

properties including the fixed point index concept for compact single-valued
mappings.

When studying multivalued equations, we may face several difficulties due
to some strict properties appearing on the multivalued operator including pos-
sessing the value of a closed, convex set (which the single-valued mapping
obviously owns). A natural problem generating is to find an alternative one,
which is still feasible for the study on the existence of solutions. For exam-
ple, we can find a selection function f satisfying the condition f(x) ∈ T(x),
where T(x) is the multivalued operator. In this article, the strategy of finding
alternative functions can be described as follows. We choose functions which
can act as upper/lower bounds thanks to several relations between the two sets
and possess natural properties including continuous linear. In addition, we also
consider the condition which allows us to use a map with better properties in
the case the neighborhood of the origin is sufficiently small or large enough.

Let us recall a well-known result on the relationship between the concept of
spectral radius and the eigenvalues of a linear mapping, which is known as the
Krein-Rutman Theorem [24].

Theorem 1.1. Let E be a Banach space with the ordered by cone K and
ϕ : E → E be a positive completely continuous with spectral radius r(ϕ) > 0.
Then, r(ϕ) is eigenvalue of ϕ with respect to eigenvector x0. Further, if ϕ is
strongly positive and intK 6= ∅, then
1. x0 ∈ intK,
2. r(ϕ) is geometrically simple,
3. if λ 6= r(ϕ) is the eigenvalue of ϕ, |λ| ≤ r(ϕ).

The above results have been extended to some non-strong positive mapping
classes such as u0-positive [31], non-decomposable maps, etc, in the works of
Krasnoselskii and his students [25]. Recently, in the papers of Nussbaum [27],
K.Chang [11], Mahadevan [26], Krein’s theorem has been extended to the
increasing, positively 1-homogeneous mapping class. Following these works,
in [14], we have extended these concepts to positively 1-homogeneous positive-
homogeneous multivalued mappings. In [29], we evaluate the range of eigen-
values for multivalued operators, find a sufficient condition for existence of
eigenvalues for the dual operator of the multivalued mapping [30]. In this pa-
per, we continue to demonstrate a result that looks like the spectral radius of
a linear mapping.

We have structured our paper as follows. In the next section, we briefly
recall some useful preliminaries. Section 3 is divided in to two subsections with
two separate results. In Subsection 3.1, some results on the fixed point index
of the multivalued operator are established. In Subsection 3.2, some existence
results for the positive eigen-pair are stated.

2. Preliminaries

Let X be a Banach space and K be a cone in X, i.e, K is a closed convex
subset of X such that K + K ⊂ K, λK ⊂ K for λ ≥ 0 and K ∩−K = {θ} (θ

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 108



Fixed point index computations for multivalued mapping

is the zero element of X). A partial order in X is defined by a ≤ b (or, b ≥ a)
if and only if a− b ∈−K. For nonempty subsets A,B of X, we write A �1 B
(or, B �1 A) iff for every a ∈ A, there is b ∈ B satisfying a ≤ b (or, a ≥ b), and
write A �2 B (or, B �2 A) iff for every b ∈ B, there is a ∈ A satisfying a ≤ b
(or, a ≥ b). A mapping T : X → 2X\{∅} is said to be positive if T(K) ⊂ K.

Throughout this paper, we use the following notations if there is no appear-
ance of special cases. Let (X,K,‖.‖) be an ordered Banach space with cone
K, X∗ be the dual topology space of X, Ω ⊂ X be a convex neighbourhood of
the origin θ, cc(K) be the all nonempty closed convex subset of K,

�
K = K\{θ},

∂KΩ = K ∩∂Ω, where ∂Ω is boundary of Ω in X,
〈x〉+ = {αx : α > 0}, where x ∈ X,

B(x,r) = {y ∈ X : ‖x−y‖ < r}, where x ∈ X,r > 0;
K∗ = {f ∈ X∗ : f(x) ≥ 0 ∀x ∈ K} ,
S∗+ = K

∗ ∩{p ∈ X∗ : ‖p‖ = 1}

R+ = {x ∈ R : x ≥ 0};
�
R+ = R+\{0}.

A multivalued mapping T : K ∩ Ω → 2K\{∅} is said to be compact if T(E) is
relatively compact for any bounded subset E of K∩Ω, where T(E) = ∪x∈ET(x)
and Ω is the closure of Ω in X. T is called an upper semi-continuous (in short,
u.s.c.) if {x ∈ K∩Ω : T(x) ⊂ W} is open in K∩Ω for every open subset W of K.
Further, if x /∈ T(x) for all x ∈ ∂KΩ, the fixed point index of T in Ω with respect
to K is defined and we denote this integer index by iK(T, Ω) (see e.g. [12]). T is
said to be convex if its graph is convex subset in (X×X). Clearly, T is convex
iff (1 −λ)T(x) + λT(y) ⊂ T((1 −λ)x + λy) for all λ ∈ [0, 1] and x,y ∈ X.

In what follows, we present some useful properties, which are of importance
in constructing the main results in the next section.

Proposition 2.1 ( [12]). Let Ω be a bounded open and T : K ∩Ω :→ cc(K) be
an u.s.c compact satisfying x /∈ ∂KΩ. Then
1. If iK(T, Ω) 6= 0, then T has a fixed point,
2. If x0 ∈ Ω, then iK(x̂0, Ω) = 1, where x̂0 is a constant mapping with x̂0(x) =

x0 ,∀x ∈ K ∩ Ω.
3. If Ω1, Ω2 ⊂ Ω are onpen with Ω1 ∩ Ω2 = ∅ and x /∈ T(x) for all x ∈

K ∩ (Ω\(Ω1 ∪ Ω2)), then
iK(T, Ω1) + iK(T, Ω2) = iK(T, Ω).

4. If H : [0, 1] × (K ∩ Ω) :→ cc(K) is an u.s.c compact satisfying x /∈ H(α,x)
for all (α,x) ∈ [0, 1] ×∂KΩ, then

iK(H(1, .), Ω) = iK(H(0, .), Ω)

Proposition 2.2 ( [12, 14]). Let Ω be a bounded open subset of X, and T :
K∩Ω → cc(K) be an u.s.c compact such that x /∈ T(x) for all x ∈ ∂KΩ. Then

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 109



V. V. Tri

1. iK(T, Ω) = 0 if there is u ∈
�
K such that

x /∈ T(x) + λu for all (λ,x) ∈ (0,∞) ×∂KΩ.
2. iK(T, Ω) = 1 if

λx /∈ T(x) for all (λ,x) ∈ (1,∞) ×∂KΩ.

Let L : X → X be positive continuous linear operator, and u0 ∈
�
K. L is

said to be u0-positive if for every x ∈
�
K, there are α > 0, β > 0 and n,m ∈ N

satisfying αu0 ≤ Lnx and Lmx ≤ βu0.

Proposition 2.3 ( [31]). Let L1,L2 : X → X be positive continuous linear
operators, and one of them is u0-positive. Assume that L1u ≤ L2u for all
u ∈ K and there exists (λ,x) ∈

�
R+ ×

�
K, (µ,y) ∈

�
R+ ×

�
K such that

λx ≤ L1x and L2y ≤ µy.
Then, the following properties hold

1. λ ≤ µ,
2. 〈x〉 = 〈y〉 if λ = µ.

Proposition 2.4.

1. x ∈ K iff 〈f,x〉≥ 0 ,∀f ∈ K∗.

2. For x ∈
�
K, there exists f ∈ K∗ such that 〈f,x〉 > 0.

Proposition 2.5 ( [13]). Let X,Y be Banach spaces, T : Ω ⊂ X → 2Y\{∅} be
u.s.c. Assume that {(xn,yn)} is a consequence in graph(T) satisfying
limn→∞(xn,yn) = (x,y). Then, we have (x,y) ∈ graph(T) if T(x) is closed
subset of Y , where graph(T) = {(a,b) : a ∈ Ω,b ∈ T(a)}.

3. Abstract results

3.1. The fixed point index of the multivalued operator. In this subsec-
tion, we present several results on the fixed point index for multivalued map-
pings by using some useful tools including some continuous linear operators
and approximate mappings at the origin and the infinity.

Theorem 3.1. Let Ω be a bounded open subset of X, A : K ∩ Ω → cc(K) be
an u.s.c and compact operator.

1. iK(A, Ω) = 1 if there exists a continuous linear operator L with the spectral
radius r(L) ≤ 1 such that

(3.1) A(u) �1 Lu and u /∈ A(u) ∀u ∈ ∂KΩ.
2. Assume that X = K −K and there exists a continuous linear mapping and

u0-positive L with the spectral radius r(L) ≥ 1 such that
(3.2) Lu �2 A(u) and u /∈ A(u) ∀u ∈ K ∩∂Ω,

then iK(A, Ω) = 0.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 110



Fixed point index computations for multivalued mapping

Proof. 1. To use Proposition 2.2, we aim at showing that

(3.3) λu /∈ T(u) for all (λ,u) ∈ (1,∞) ×∂KΩ.
Assume that it is not true, then we can find (λ,u) ∈ (1,∞) ×∂KΩ satisfying
λu ∈ T(u). From (3.1), we have λu ≤ Lu, which implies that (I −λ−1L)−1 is
a positive continuous linear operaor. This gives u ≤ θ, which leads to u = θ.
A contradiction can be seen here obviously.

2. Let us choose x0 ∈
�
K, we will prove that

(3.4) u /∈ T(u) + λx0 ,∀(λ,u) ∈ (0,∞) ×∂KΩ.
Indeed, assume that (3.4) is not true, then u ∈ T(u) + λx0, for some (λ,u) ∈
(0,∞) × ∂KΩ. Then, from (3.2), one obtain u ≥ Lu. By the Krein-Rutman

theorem, we have r(L) is the eigen vallue of L, i.e, there exists y ∈
�
K such that

Ly = r(L)y. Using Proposition 2.3, we have r(L) = 1 and u ∈ 〈y〉+. By setting
u = αy (α > 0), one can see Lu = u which implies that

u ≥ Lu + λx0 = u + λx0.
This is impossible. By Proposition 2.2, we obtain iK(T, Ω) = 0. The proof is
complete. �

Theorem 3.2. Let Ω be a bounded open subset of X, T : K → cc(K) is an
u.s.c compact convex satisfying x /∈ T(x) for all x ∈ K. Then

1. iK(T, Ω) = 0 if there exists (λ0,x0) ∈ (1,∞) ×
�
K such that λ0x0 ∈ T(x0).

2. iK(T, Ω) = 1 if λx ∈ T(x) for all (λ,x) ∈ (1,∞) ×
�
K.

Proof. The second assertion can be seen as a consequence of Proposition 2.2.
To prove the first assertion, we will show that

(3.5) x ∈ T(x) + λx0 ∀(λ,x) ∈ (0,∞) ×∂KΩ.
Indeed, assume the contrary, namely, x /∈ T(x)+λx0, for some (λ,x) ∈ (0,∞)×
∂KΩ. Then, there exists y ∈ T(x),x = y + λx0. For arbitrary positive numbers
α,β we have

αλ0x + βx0 = αλ0y +

(
β

λ0
+ αλ

)
λ0x0.

Therefore, the following identity holds

(3.6) αλ0x + βx0 ∈ αλ0T(x) +
(
β

λ0
+ αλ

)
T(x0).

Let us choose α as follows

α =

(
λ0 +

λλ0
λ0 − 1

)−1
and β =

αλλ0
λ0 − 1

.

Then, it is clear that β satisfies

β =
β

λ0
+ αλ and αλ0 +

β

λ0
+ αλ = 1.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 111



V. V. Tri

Now, we set v = αλ0x + βx0, v ∈
�
K. Since T is convex, we have

αλ0T(x) +

(
β

λ0
+ αλ

)
T(x0) ⊂ T

(
αλ0x +

(
β

λ0
+ αλ

))
This together with (3.6) yields v ∈ T(v). This is a contradiction, hence
iK(T, Ω) = 0. �

Let F,ϕ : K → 2K\{∅}. For every x ∈ K, we denote
D (F(x),ϕ(x)) = sup{‖y −y′‖ : y ∈ F(x),y′ ∈ ϕ(x)} .

We consider the following conditions for the pair (F,ϕ).

(C0) : lim
x∈

�
K,‖x‖→0

D(F(x),ϕ(x))
‖x‖ = 0.

(C∞) : lim
x∈

�
K,‖x‖→∞

D(F(x),ϕ(x))
‖x‖ = 0.

In what follows, we aim at giving several relations between the aforemen-
tioned conditions and the results on the fixed point index for multivalued map-
pings. First of all, we are interested in giving an answer for the natural question

“When does the two aforementioned conditions for the pair (F,ϕ) are
guaranteed? ”

by presenting some simple illustrations for such pair.

Example 3.3. Let X = R, K = R+,B = [0, 1].
1. Let F,ϕ : K → 2K with

F(x) = x + x2B and ϕ(x) = x.

Then, D (F(x),ϕ(x)) = x2; hence, the pair (F,ϕ) satisfies the condition (C0).
2. We define F,ϕ : K → 2K by

F(x) =

{
{0}, x = 0,
x + B, x ∈ (0,∞)

and ϕ(x) = x. Then, for x 6= 0 we have
D (F(x),ϕ(x)) = sup{|y −x| : y = x + α,α ∈ B}

= 1.

Therefore the pair (F,ϕ) satisfies the condition (C∞).
3. Let F : R → R be a Fréchet differentiable function with F(0) = 0, ϕ be the
Fréchet differentiable of F with recpect to K at 0 (at ∞, resp.). Then, the pair
(F,ϕ) satisfies the condition (C0) ((C∞), resp.)

Theorem 3.4. Let F,ϕ : K → cc(K) be u.s.c and compact with x /∈ ϕ(x), for
all x ∈ K. Assume that ϕ(λx) = λϕ(x), for all λ > 0,x ∈ K (in order words,
A is positively 1-homogeneous). Then, it holds

iK(F,B(θ,r)) = iK(ϕ,B(θ,r)),

in the case the following conditions hold

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 112



Fixed point index computations for multivalued mapping

1. (F,ϕ) satisfies the condition (C0) if r is sufficiently small.
2. (F,ϕ) satisfies the condition (C∞) if r is sufficiently large.

Proof. For the sake of convenience, we denote

H(α,x) = αF(x) + (1 −α)ϕ(x), x ∈
�
K,α ∈ [0, 1].

The operator H(α,.) is u.s.c compact with closed convex values. For y ∈ F(x)
and y′ ∈ ϕ(x) we have

‖x−αy − (1 −α)y′‖ = ‖x−y′ −α(y −y′)‖
≥‖x−y′‖−α‖y −y′‖
≥‖x−y′‖−‖y −y′‖.(3.7)

Set b = inf{‖x−y‖ : x ∈ K,‖x‖ = 1,y ∈ ϕ(x)}. If b = 0, we can find sequences
{xn}⊂ K, {yn}⊂ K such that

‖xn‖ = 1, yn ∈ ϕ(xn) and ‖xn −yn‖→ 0.

Thanks to the compactness of ϕ(x), we can assume that limn→∞yn = y0 ∈ K,
hence ‖y0‖ = 1. Since ϕ is u.s.c, we have y0 ∈ ϕ(y0) which is contradictory with

the assumption. Thus, b > 0. Now, fix x ∈
�
K, write x = λx′ with λ = ‖x‖,

then x′ ∈
�
K and ‖x′‖ = 1. It follows from the positively 1-homogeneous

properties of ϕ that

infw∈ϕ(x) ‖x−w‖
‖x‖

=
inf 1

λ
w∈ϕ(x′) ‖x′ −

1
λ
w‖

‖x‖
≥ b.

This implies that

(3.8) inf
w∈ϕ(x)

‖x−w‖≥ b‖x‖.

From (3.8) and (3.7), we have

(3.9)
‖x−αy − (1 −α)y′‖

‖x‖
≥ b−

D (F(x),ϕ(x))

‖x‖
.

If (F,ϕ) satisfies (C0), there exists r > 0 such that b−
D(F(x),ϕ(x))

‖x‖ > 0 for all

x ∈
�
K with ‖x‖≤ r. From (3.9), it follows that

x ∈ H(α,x) for all x ∈ ∂KB(θ, 0).

Hence, we deduce that iK(F,B(0,r)) = iK(ϕ,B(0,r)). If (F,ϕ) satisfies (C∞),
we make the same argument as above. The proof is complete. �

3.2. Existence of a positive eigen-pair. In this section we present results
on the existence of the eigenvalue for multivalued operator.

Theorem 3.5. Let A : K → cc(K) be u.s.c compact. Assume that X =
K−K, Ω1, Ω2 are bounded open subsets of x, θ ∈ Ω1 ( Ω2 satisfy the following
conditions

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 113



V. V. Tri

1. There exist completely continuous linear maps P,Q : K → K with spectral
radius r(P),r(Q), respectively, and P is u0-positive such that either

(3.10) Px �2 A(x) ∀x ∈ ∂KΩ1, A(x) �1 Qx ∀x ∈ ∂KΩ2
or

(3.11) Px �2 A(x) ∀x ∈ ∂KΩ2, A(x) �1 Qx ∀x ∈ ∂KΩ1
2. 0 < r(Q) < r(P).

Then, for λ ∈ (r(Q),r(P)), the inclusion λx ∈ A(x) has a positive solution.

Proof. We assume that (3.10) is satisfied and x /∈ µA(x), for all x ∈ ∂Ω1 ∪ Ω2.
Denote by µ = λ−1. Then, we have

µA(u) �1 µQu ,∀u ∈ ∂KΩ2.

Since r(µQ) ≤ 1 by Theorem 3.1, we obtain iK(µA, Ω2) = 1. Similarly,
iK(µA, Ω1) = 0. It follows that iK(µA, Ω2\Ω1) = 1 by Proposition 2.1. Hence,
µA has a fixed point in Ω2\Ω1. By a similar argument as in the previous one
with the condition (3.11). �

Let ϕ : K → 2K\{∅}, we denote

r∗(ϕ) = sup

{
λ > 0 : ∃x ∈

�
K,λx ∈ ϕ(x)

}
, define sup ∅ = 0;

r∗(ϕ) = inf

{
λ > 0 : ∃x ∈

�
K,λx ∈ ϕ(x)

}
, define inf ∅ = ∞;

Theorem 3.6. Let A : K → cc(K) be u.s.c compact. Assume that there exist
positively 1-homogeneous convex operators P,Q : K → cc(K0) satisfying the
following conditions

1. (A,P) satisfies (C0) and (A,Q) satisfies (C∞),
2. 0 < r∗(P) < r∗(Q) < ∞ (or 0 < r∗(Q) < r∗(P) < ∞, resp.)
Then, if λ ∈ (r∗(P),r∗(Q)) (or λ ∈ (r∗(Q),r∗(P)) resp.), the equation λx ∈

A(x) has a solution in
�
K.

Proof. Denote by µ = λ−1, F = µA, ϕ1 = µP, ϕ2 = µQ. We first prove that
there are r1 > 0,r2 > 0 (r1 < r2) such that

(3.12) iK(F, Ω1) = iK(ϕ1, Ω1) and iK(F, Ω2) = iK(ϕ2, Ω2),

where Ω1 = B(θ,r1), Ω2 = B(θ,r2). Indeed, applying Theorem 3.4 for the
pair (F,ϕ1) we can find r1 > 0 (small enough) such that iK(F,B(θ,r1)) =
iK(ϕ1,B(θ,r1)). Similarly, there exists r2 > 0 (large enough) satisfying
iK(F,B(θ,r2)) = iK(ϕ2,B(θ,r2)). Now, we assume that 0 < r

∗(P) < r∗(Q).
By Theorem 3.2, iK(F, Ω2) = 0 and iK(F, Ω1) = 1, this leads to the assertion
that needs to be proved. In the case 0 < r∗(Q) < r∗(P) < ∞ the proof is
analogous to the one above. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 114



Fixed point index computations for multivalued mapping

Let A in a nonempty subset of K, for ervery p ∈ K∗ we define

σ(A,p) = {〈p,x〉 : x ∈ A} ,

where 〈p,x〉 is value of p at x. For u ∈
�
K we denote S = u+K. In the following

lemma, we present the eigenvalue for the bounded multivalued operator.

Lemma 3.7. Assume F : S → 2K\{∅} satisfying the conditions following
(i) σ(F(x),p) for all (p,x) ∈ S∗+ ×S,

(ii) F(S) is relatively compact in X,
(iii) there is (α,v) ∈ (0,∞) ×S such that αv �1 F(v).
Then

(3.13) 0 < sup
p∈S∗

+

(
inf
p∈S

〈p,x〉
σ(F(x),p)

)
< ∞.

Proof. Since the conditions (i) and (ii) are satisfied 0 < M := sup{σ(F(S),p) :
p ∈ S∗+} < ∞. By Proposition 2.4, there exists p0 ∈ S∗+ such that µ0 :=
〈p0,u〉 > 0. For any x ∈ S with x = u + y,y ∈ K, we have

〈p0,x〉 = 〈p0,u〉 + 〈p0,y〉≥ µ0.

Hence,
〈p0,x〉

σ(F(x),p0)
≥ µ0

M
∀x ∈ S, which gives

inf
x∈S

〈p0,x〉
σ(F(x),p0)

≥
µ0
M

> 0.

This implies that

(3.14) sup
p∈S∗

+

(
inf
p∈S

〈p,x〉
σ(F(x),p)

)
≥ inf
x∈S

〈p0,x〉
σ(F(x),p0)

≥
µ0
M

> 0.

From the condition (iii), we can find z ∈ F(v) such that αz ≤ z. Therefore

〈p,αv〉≤ 〈p,αz〉≤ σ(F(v),p),

so
〈p,v〉

σ(F(v),p)
≤ 1

α
for all p ∈ S∗+. It follows that

inf
y∈S

〈p,y〉
σ(F(y),p)

≤
1

α
for all p ∈ S∗+.

This implies that

(3.15) sup
p∈S∗

+

(
inf
p∈S

〈p,x〉
σ(F(x),p)

)
≤

1

α
.

The proof is complete. �

Theorem 3.8. Let F : S → cc(K) be an u.s.c convex multivalued operator
satisfying the conditions in Lemma 3.7. Then

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V. V. Tri

1. If λ0 is defined by

1

λ0
= sup
p∈S∗

+

(
inf
p∈S

〈p,x〉
σ(F(x),p)

)
,

there exists x0 ∈ S such that λ0x0 ∈ F(x0) and
1

λ0
= sup
p∈S∗

+

〈p,x0〉
σ(F(x0),p)

.

2. Further, if (λ,x) ∈ (0,∞) ×S with λx ∈ F(x), we have λ ≤ λ0.

Proof. We first see that λ0 is fine defined by Lemma 3.7.
We will prove the first assertion by steps following
Step 1. (Showing that (F − λ0I)(S) is convex subset of K). Assmue that
z,z′ ∈ (F −λ0I)(S) and α,β ∈ [0, 1] with α + β = 1. There is (x,x′) ∈ S ×S
such that z ∈ F(x) −λ0x and z′ ∈ F(x′) −λ0x′. We have

αz + βx ∈ αF(x) + βF(x′) −λ0(αx + βx′).
By the convexity of F, αF(x) + βF(x′) ⊂ F(αx + βx′). Since S is convex,
αx + βx′ ∈ S, hence αz + βz′ ∈ (F −λ0I)(S).
Step 2. (Showing that (F −λ0I)(S) is close subset of K). Assume {zn}n=1,2,...
is a sequence in (F − λ0I)(S) with limn→∞zn = z. We can find a sequence
{xn}⊂ S and {yn} satisfying zn ∈ F(xn) −λ0xn, yn ∈ F(xn) and
(3.16) zn = yn −λ0xn.
Since F(S) is relatively compact, we can assume limn→∞yn = y. Therefore,
there exists x ∈ S, limn→∞xn = x. On the other hand, F is u.s.c and F(x)
is closed set, by Proposition 2.5 it follows that y ∈ F(x). Letting n → ∞ in
(3.16) we obtain z = y −λ0x, thus z ∈ (F −λ0I)(S).
Step 3. (Proving θ ∈ (F − λ0I)(S)). Assume the contrary, that θ /∈ (F −
λ0I)(S). By applying separation Theorem for two sets {θ} and (F −λ0I)(S)
we can fine a number � > 0 and p1 ∈ X∗ with ‖p1‖ = 1 (for if not, we replace
p1 by

1
‖p1

p1) such that 〈p1,z〉 < −� ∀z ∈ (F −λ0I)(S), i.e,

〈p1,y〉−λ0〈p1,x〉 < −� ∀(x,y) ∈ S ×F(x).
This implies that

(3.17) σ(F(x),p1) −λ0〈p1,x〉≤−� for all x ∈ S.
We now will show that p1 ∈ S∗+. Indeed, if there exists y ∈ K such that
〈p1,y〉 < 0, using (3.17) for x = u + ny, (n = 1, 2, ...) we have
(3.18) σ(F(x),p1) −λ0〈p1,u〉−nλ0〈p1,y〉≤−�.
Set c = sup{σ(F(x),p1) : x ∈ S}. Since F(S) is relatively compact, c ∈
(−∞,∞). Letting n →∞ in (3.17) we obtain a contradiction, hence p1 ∈ S∗+.
From (3.17) it follows that

1

λ0
≤

〈p1,x〉
σ(F(x),p1)

−
�

λ0c
for all x ∈ S.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 116



Fixed point index computations for multivalued mapping

Thus
1

λ0
≤ inf
x∈S

〈p1,x〉
σ(F(x),p1)

−
�

λ0c
.

On the other hand,

inf
x∈S

〈p1,x〉
σ(F(x),p1)

≤ sup
p∈S∗

+

(
inf
x∈S

〈p,x〉
σ(F(x),p)

)
=

1

λ0
.

This implies 1
λ0
≤ 1

λ0
− �

λ0c
. We have a contradiction. Hence θ ∈ (F −λ0I)(S).

Therefore, there exists x0 ∈ S such that λ0x0 ∈ F(x0).
Step 4. (Showing that 1

λ0
= sup

p∈S∗
+

(
〈p,x0〉

σ(F(x0),p)

)
). For every p ∈ S∗+, we have

σ(F(x0),p) ≥〈p,λ0x0〉 = λ0〈p,x0〉. Thus,
1

λ0
≥

〈p,x0〉
σ(F(x0),p)

for all p ∈ S∗+.

On the other hand, we have

1

λ0
≥

〈p,x0〉
σ(F(x0),p)

≥ inf
x∈S

〈p,x〉
σ(F(x),p)

for all p ∈ S∗+.

This implies that

1

λ0
≥ sup
p∈S∗

+

〈p,x0〉
σ(F(x0),p)

≥ sup
p∈S∗

+

(
inf
x∈S

〈p,x〉
σ(F(x),p)

)
=

1

λ0
.

We deduce 1
λ0

= sup
p∈S∗

+

(
〈p,x0〉

σ(F(x0),p)

)
.

Now, we prove the second assertion. Assume that λx ∈ F(x) for some (λ,x) ∈
(0,∞) ×S. Then, we have

σ(F(x),p) ≥〈p,λx〉 = λ〈p,x〉.
Thus,

1

λ
≥

〈p,x〉
σ(F(x),p)

≥ inf
y∈S

〈p,y〉
σ(F(y),p)

.

It follows that
1

λ
≥ sup
p∈S∗

+

(
inf
x∈S

〈p,x〉
σ(F(x),p)

)
=

1

λ0
.

Hence λ ≤ λ0. The proof is complete. �

Remark 3.9.

1. In the proofs of our results, we have not used the cone condition with non-
empty interior (which is called the solid cone).Therefore, the case int(K) =
∅ is just a special case of the results in this work. In Theorem 3.1 and
Theorem 3.5 , we have used the condition that K is a reproducing cone.
A solid cone is a reproducing cone. However, the opposite is not true.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 117



V. V. Tri

For example, Let Ω be a bouned subset of RN, X = Lp(Ω). The set of
nonnegative functions K in X is a reproducing cone. However, it has empty
interior.

2. Same as above, the normal cone condition has not been used.

4. Conclusion

This paper is a continuation of the series works [14, 29, 30] of extending the
well-known result of Krein-Rutman Theorem. Initially, we investigate the fixed
point index for multivalued mappings by using some useful tools including some
continuous linear operators and approximate mappings at the origin and the
infinity. Lastly, distinct results on the existence of solutions to the multivalued
equations are constructed flexibly.

Acknowledgements. This paper was supported by Thu Dau Mot university
under grant number DT.21.1-014.

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