Adv Syst Sci Appl 2022; 01:176–191
Published online at https://ijassa.ipu.ru.

On Order Covering Set-Valued Mappings and Their
Applications to the Investigation of Implicit Differential
Inclusions and Dynamic Models of Economic Processes

Evgenii O. Burlakov1,2, Elena A. Panasenko3,4, Irina D. Serova1, Evgeny S. Zhukovskiy2,3*

1University of Tyumen, Tyumen, Russia
2V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

3Derzhavin Tambov State University, Tambov, Russia
4Leonhard Euler International Mathematical Institute, St. Petersburg, Russia

Abstract: The present work is devoted to investigation of operator inclusions in partially
ordered spaces and application of the obtained results to differential inclusions. We consider
the inclusion Υ(x, x) 3 y with respect to the unknown x ∈ X, where Υ : X ×X ⇒ Y is a set-
valued mapping, X and Y are partially ordered spaces. It is assumed that the mapping Υ is order
covering with respect to the first argument and antitone with respect to the second argument.
We prove that for any x0 ∈ X, if the set G(x0) contains an element y0 such that y � y0, then
there exists a solution to the inclusion under consideration, which satisfies the inequality x � x0.
This statement is applied to investigation of a Cauchi problem for the differential inclusion
f(t, x, ẋ, ẋ) 3 0 with a bound for the derivative of the unknown function ẋ(t) ∈ B(t) (here
f : [a, b]× Rn × Rn × Rn ⇒ Rm, B : [a, b] ⇒ Rn). We obtain conditions of solvability in the
space of absolutely continuous functions, conditions of existence of a solution with the least
derivative, and derive the solutions estimates. The latter results are applied to the analysis of the
dynamic Walrasian-Evans-Samuelson model of economic processes, which can be reduced to a
system of implicit differential inclusions. We establish the existence of the equilibrium and obtain
estimates of the equilibrium prices.

Keywords: operator inclusion, covering mapping of partially ordered spaces, implicit
differential equation, supply-and-demand model

1. INTRODUCTION

The present work is devoted to the problem of solvability of operator inclusions in partially
ordered spaces. The research uses the notion of order covering introduced for the “standard
single-valued” mappings in [1, 2], and for set-valued mappings in [3, 4]. For a set-valued
mapping G acting from a partially ordered space (X,�) to a partially ordered space (Y,�),
the property of covering means that if for any u ∈ X and y ∈ Y , the set G(u) contains an
element v such that y � v, then there exists a solution x of the inclusion G(x) 3 y, which
satisfies the inequality x � u. In the cited papers, the existence of a coincidence point of a
covering mapping and an isotone mapping (for the cases of single- and set-valued mappings)
is demonstrated.

In the present work, the stability of the solvability property of inclusions with respect to
antitone perturbations is investigated. This problem is formalised in terms of the inclusion

∗Corresponding author: zukovskys@mail.ru



ON ORDER COVERING SET-VALUED MAPPINGS AND THEIR APPLICATIONS 177

Υ(x, x) 3 y with a set-valued mapping Υ : X ×X ⇒ Y that is order covering with respect
to the first argument and antitone with respect to the second argument. An analogous problem
for “standard single-valued” mappings was considered in the papers [5–8]. Based on the
statements on antitone perturbations proved in these works, comparison theorems (of the
same type with the well-known Chaplygin theorem [9] on differential inequality) for the
unresolved with respect to the derivative (i.e. implicit) differential equations were obtained
(see [5, 10]). The results on antitone perturbations of order covering mappings presented
in this article are applied to the investigation of implicit differential inclusions. Conditions
of existence and estimates of solutions of a Cauchi problem are obtained. These results
are applied to the analysis of the dynamic Walrasian-Evans-Samuelson supply-and-demand
model (see [11–13]) that is reduced to a system of implicit differential inclusions. The
existence of the equilibrium state is established and the estimates of the equilibrium prices
are obtained. These statements extend the results on the models of economic processes of
A.V. Arutyunov, N.G. Pavlova, S.E. Zhukovskiy, A.A. Shananin (see [14–17]) which employ
single-valued functions of supply and demand. The fundamental mathematical tools for these
works were represented by the theorems of the papers [18–21] on coincidence points of
single-valued covering mapping of metric spaces.

2. A COMPARISON THEOREM FOR OPERATOR INCLUSIONS IN PARTIALLY
ORDERED SPACES

Let partially ordered spaces (X,�), (Y,�) be given. For any u, v ∈ X andU ⊂ X , we denote

[u, v]X
.
= {x ∈ X : v � x � u}, OX(u)

.
= {x ∈ X : x � u}, OX(U)

.
=
⋃
∀u∈U

OX(u).

Consider a set-valued mappingG : X ⇒ Y, i.e. a mapping that puts to any element x ∈ X
into the correspondence a non-empty set G(x) ⊂ Y. If for any x ∈ X , the set G(x) contains
only one element, then the mapping G becomes a “standard single-valued” mapping (we
conventionally denote such a mapping as G : X → Y ). Thus, set-valued mappings naturally
generalise “standard” mappings.

Let us remind definitions of some properties of set-valued mappings used in the present
work.
Definition 2.1:
A mappingG : X ⇒ Y is called antitone (isotone) on the set U ⊂ X, if for any x, u ∈ U such
that x � u and for z ∈ G(u), there exists y ∈ G(x) satisfying the inequality y � z (y � z).
If a mapping is antitone (isotone) on the whole space X , it is called antitone (isotone).

The given definition of the antitone property for a mapping G : X → Y means that

∀x, u ∈ U x � u ⇒ G(x) � G(u),

matches the definition of the antitone property for single-valued mappings. The given isotone
property in the case of single-valued mappings also matches the “classical” isotone property.
Definition 2.2:
A mapping G : X ⇒ Y order covers the set V ⊂ Y if

∀u ∈ X OY

(
G(u)

)
∩ V ⊂ G

(
OX(u)

)
. (2.1)

The definition of order covering for set-valued mappings was introduced in [3, 4] for the
case V = Y.

Note that (2.1) is equivalent to the relation(
∀u ∈ X ∀v ∈ G(u) ∀y ∈ V y � v

)
⇒

(
∃x ∈ X y ∈ G(x) and x � u

)
.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



178 E.S. ZHUKOVSKIY, I.D. SEROVA, E.A. PANASENKO, E.O. BURLAKOV

Moreover, if V is a one-element set, i.e. V = {ŷ}, then the relation (2.1) is equivalent to the
implication

∀u ∈ X ŷ ∈ OY

(
G(u)

)
⇒ ŷ ∈ G

(
OX(u)

)
.

Thus, the property of order covering of the set {ŷ} means that for any u ∈ X such that the
set G(u) contains an element v � ŷ, the inclusion

ŷ ∈ G(x) (2.2)

has a solution x ∈ X satisfying the inequality x � u. In the same manner, a single-valued
mapping G : X → Y order covers the set {ŷ} if

∀u ∈ X G(u) � ŷ ⇒ ∃x ∈ OX(u) G(x) = ŷ.

We consider the problem of stability of the solvability property of the inclusion (2.2)
under antitone perturbations of the order covering mapping. Let a “perturbed” mapping
Υ : X ×X ⇒ Y that order covers the set {ŷ} with respect to the first argument and antitone
with respect to the second argument be given. Define the mapping F : X ⇒ Y by the relation

∀x ∈ X F (x) = Υ(x, x)

and consider the inclusion
ŷ ∈ F (x) (2.3)

(with respect to x ∈ X).
Let U ⊂ X. In order to formulate a theorem on existence of solutions to (2.3), we define

the set S(Υ, U, ŷ) of all chains S ⊂ U such that the following relations take place:

∀x ∈ S ∃y ∈ Υ(x, x) y � ŷ,

∀x, u ∈ S x ≺ u ⇒ ∃ξ ∈ [x, u] ŷ ∈ Υ(ξ, u).
(2.4)

Theorem 2.1:
Let there exist u0 ∈ X and y0 ∈ Y such that

y0 ∈ Υ(u0, u0), y0 � ŷ, (2.5)

and the follwing conditions are satisfied:

(a1) for any x ∈ OX(u0), the mapping Υ(·, x) : X ⇒ Y order covers the set {ŷ};
(a2) for any x ∈ OX(u0), the mapping Υ(x, ·) : X ⇒ Y is antitone on the set [x, u0]X ;
(a3) any infinite chain S ∈ S

(
Υ,OX(u0), ŷ

)
is bounded from below, and for some its lower

bound ω ∈ X , one can find z ∈ Υ(ω, ω) satisfying the inequality z � ŷ.

Then the inclusion (2.3) has a solution and the set of solutions possesses a minimal
element belonging to the set OX(u0).

Proof
We introduce the set

U0 = {x ∈ OX(u0) : ∃y ∈ F (x) y � ŷ}.

Note that U0 6= ∅ as u0 ∈ U0. We define binary relations � and � on U0 as follows:

∀v, u ∈ U0 v � u ⇔
(
v ≺ u and ∃ξ ∈ [v, u] ŷ ∈ Υ(ξ, u)

)
,

∀v, u ∈ U0 v � u ⇔
(
v = u or v � u

)
.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ON ORDER COVERING SET-VALUED MAPPINGS AND THEIR APPLICATIONS 179

Let us demonstrate that � and � are strict and non-strict order relations, respectively.
As the relation ≺ is antisymmetric, the relation � is antisymmetric as well. Prove that � is
transitive. For v, w, u ∈ U0 such that v � w � u, it holds true that

v ≺ w ≺ u and ∃ξ ∈ [w, u] ⊂ [v, u] ŷ ∈ Υ(ξ, u). (2.6)

Thus, we have v � u,, which means that the relation � is transitive. Obviously, the relation �
is also antisymmetric and transitive, and, moreover, reflexive. We point out that the relation
(2.6) also holds true for v, w, u ∈ U0 such that v ≺ w � u,, i.e. the relation v � u takes place
in this case as well.

Let us consider the partially ordered set (U0,�). According to the Hausdorff maximal
principle (see e.g. [22, Chapter 1]), this set possesses a maximal chain S that contains u0. We
first prove the theorem statement in the case when the chain S contains the least element ω,
i.e. ω � x for any x ∈ S. If ω is not a solution of inclusion (2.3), then for some z ∈ Υ(ω, ω),
it holds true that z � ŷ and, hence, by the virtue of the condition (a1), there exists an element
υ ∈ X such that ŷ ∈ Υ(υ, ω) and υ ≺ ω. According to (a2), there exists y ∈ Υ(υ, υ) such
that y � ŷ. So, υ ∈ U0 and υ � ω. We obtained that υ � x for any x ∈ S which contradicts
with the maximality of the chain S. Thus, ω is a solution to (2.3).

Consider now the case when the chain S does not possess the least element, i.e. for any
x ∈ S, there exists u ∈ S, u� x. Obviously, this chain is infinite. Note that S is a chain
with respect to the initial order � in X and belongs to S

(
Υ,OX(u0), ŷ

)
. The assumption

(a3) implies that this chain has a lower bound ω (with respect to the initial order �), for
which there exists z ∈ Υ(ω, ω) such that z � ŷ, i.e. ω ∈ U0 and ω /∈ S. For any x ∈ S, there
is u ∈ S, u� x. As ω ≺ u, we get ω � x. The latter contradicts with the maximality of the
chain S, so the situation where the maximal chain S does not possess the least element is not
possible.

Let us demonstrate that the obtained element ω ∈ OX(u0) is minimal in the set of
solutions of the inclusion (2.3). Assume the contrary. Then one can find z ≺ ω such that
ŷ ∈ Υ(z, z). Hence, z ∈ U0 and z � ω. However, this inequality is not possible by the virtue
of the maximality of the chain S in the space (U0,�).

The theorem 2.1 implies the theorem on perturbations of an order covering single-valued
mapping Υ : X ×X → Y. The corresponding properties of order covering and monotonicity
of a single-valued mapping are defined above, and the relations (2.4) can be written as:

∀x ∈ S Υ(x, x) � ŷ,

∀x, u ∈ S x ≺ u ⇒ ∃ξ ∈ [x, u] ŷ = Υ(ξ, u). (2.7)

The statement on perturbations of order covering single-valued mappings is analogous to the
results obtained in the papers [5, 6]. These works employed a less restrictive assumptions on
the chains S compared to (2.7), namely:

∀x, u ∈ S x ≺ u ⇒ ŷ � Υ(ξ, u),

so the condition corresponding to (a3) was more strict.

3. CONDITIONS OF ORDER COVERING FOR THE NEMYTSKII OPERATOR

In the application of Theorem 2.1 to concrete functional inclusions, the most of the difficulties
connected with the verification of the order covering property of the corresponding set-valued
operators, primarily the superposition operator that is also called the Nemytskii operator. Here
we demonstrate that the Nemytskii operator is order covering provided that the corresponding
set-valued function possesses this property with respect to the corresponding argument.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



180 E.S. ZHUKOVSKIY, I.D. SEROVA, E.A. PANASENKO, E.O. BURLAKOV

Typically, the verification of the order covering property for functions does not involve
significant difficulties.

Below we assume that the space Rn is equipped with the natural order, i.e for x =
(x1, ..., xn) ∈ Rn and u = (u1, ..., un) ∈ Rn, we have x ≤ u if xi ≤ ui for all i = 1, n. We
denote by C(Rn) and K(Rn) the sets of all non-empty closed and all non-empty compact
subsets of the space Rn, respectively. We denote by CC(Rn) and KC(Rn) the sets of all
non-empty connected closed and all non-empty connected compact subsets of the space Rn,
respectively. Let a set-valued mapping B : [a, b] ⇒ Rn having closed images B(t) ⊂ Rn

for any t ∈ [a, b] be given. We denote such a mapping as B : [a, b]→ C(Rn). Let this
mapping be measurable. We define W (B) to be the set of all Lebesgue measurable functions
x : [a, b]→ Rn such that x(t) ∈ B(t) for almost all t ∈ [a, b]. Thus, the set W (B) is the set
of all measurable selections of the set-valued mapping B. We endow the set W (B) with
an order by assuming that for any two its elements x, u, the inequality x ≤ u holds true if
x(t) ≤ u(t) for almost all t ∈ [a, b]. In the case B = Rn, this set of all measurable functions
x ∈ [a, b]→ Rn we denote by W n.

Consider a set-valued mapping g : [a, b]× Rn → C(Rm). Assume that it satisfies the
Caratheodori conditions (i.e. it is measurable with respect to the first argument and continuous
in the Hausdorff metric in the second argument). These assumptions allow to define the set-
valued Nemytskii operator

Ng : W n ⇒ Wm, ∀x ∈ W n Ng x =
{
y ∈ Wm : y(t) ∈ g(t, x(t)) for almost all t ∈ [a, b]

}
.

We denote ∆ =
{

(t, x) : t ∈ [a, b], x ∈ B(t)
}

and define g∆ : ∆→ C(Rm) to be the
restriction of the set-valued mapping g to the set ∆, and by Ng∆

: W (B) ⇒ Wm — the
restriction of the operator Ng to W (B).

The following statement establishes connection between the property of order covering
of the set-valued mapping g∆ with respect to the second argument and the order covering
property of the corresponding Nemytskii operator Ng∆

.
Let a measurable function ŷ : [a, b]→ Rm be given.

Theorem 3.1:
If at almost all t ∈ [a, b], a set-valued mapping g∆(t, ·) : B(t)→ C(Rm) order covers the set
{ŷ(t)} ⊂ Rm, then the operator Ng∆

: W (B) ⇒ Wm order covers the set {ŷ} ⊂ Wm.

Proof
Let the mapping g∆(t, ·) order cover the set {ŷ(t)} ⊂ Rm. assume that for some measurable
function u ∈ W (B), there exists y ∈ Ng∆

u such that y ≥ ŷ. Thus, y(t) ∈ g∆

(
t, u(t)

)
and

y(t) ≥ ŷ(t) for almost all t ∈ [a, b]. We define the set-valued mappings O,U : [a, b] ⇒ Rn

by the relations:

O(t)
.
= ORn(u(t)), U(t)

.
= O(t)

⋂
B(t) =

{
x ∈ B(t) : x ≤ u(t)

}
, t ∈ [a, b].

For almost all t ∈ [a, b], the closedness of the sets ORn(u(t)) and B(t) in Rnimplies the
closedness of the set U(t), i.e. U : [a, b]→ C(Rn). The measurability of O,B : [a, b]→
C(Rn) guarantees the measurability of the mapping U (see [23, § 1.5.1, § 1.5.8]).

As g∆(t, ·) order covers the set {ŷ(t)} ⊂ Rm, for almost all t ∈ [a, b], the inclusion
ŷ(t) ∈ g∆

(
t, U(t)

)
takes place for almost all t ∈ [a, b]. Therefore, according to Filippov’s

lemma on implicit function (see [23, § 1.5.15]), there exists x̂ ∈ W n such that x̂(t) ∈ U(t)
and ŷ(t) ∈ g∆

(
t, x̂(t)

)
for almost all t ∈ [a, b]. Thus, the relations ŷ ∈ Ng∆

x̂ and x̂ ≤ u are
fulfilled, i.e. the operator Ng∆

order covers the set {ŷ} ⊂ Wm.

Example 2.1:
Let ordinary single-valued” functions q0, q1 : [a, b]→ R be given. Consider the function

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ON ORDER COVERING SET-VALUED MAPPINGS AND THEIR APPLICATIONS 181

g : [a, b]× R→ R defined by the formula

g(t, x)
.
= q0(t) + 2q1(t)x− x2, t ∈ [a, b], x ∈ R. (3.8)

Obviously, g satisfies the Caratheodori conditions. Let r > 0, B(t)
.
= [q1(t)− r, q1(t) + r]

and ∆ =
{

(t, x) : t ∈ [a, b], |x− q1(t)| ≤ r
}
, respectively. The values of g∆(t, ·) : B(t)→

R constitute the segment
[
q0(t) + q2

1(t)− r2, q0(t) + q2
1(t)
]
. If for the measurable function

ŷ : [a, b]→ R, it holds true that

ŷ(t) ≥ q0(t) + q2
1(t)− r2, (3.9)

then g∆(t, ·) : B(t)→ R order covers the set {ŷ(t)} ⊂ R at almost all t ∈ [a, b] and, hence,
by the virtue of Theorem 3.1, the Nemytskii operator Ng∆

: W (B) ⇒ W order covers the set
{ŷ} ⊂ W. In the particular case whenB(t) ≡ R, the operatorNg : W ⇒ W is order covering
(i.e. order covers the whole space W ).

Now we pick δ > 0 and define two set-valued mappings g+, g− : [a, b]× R→ K(R) by
the formulas:

g+(t, x)
.
=
[
g(t, x), g(t, x)+δ

]
, g−(t, x)

.
=
[
g(t, x)−δ, g(t, x)

]
, t ∈ [a, b], x ∈ R. (3.10)

The images of the set-valued functions g+
∆(t, ·) : B(t)→ K(R) and g−∆(t, ·) : B(t)→ K(R)

are the sets
[
q0(t) + q2

1(t)− r2, q0(t) + q2
1(t) + δ

]
, and

[
q0(t) + q2

1(t)− r2 − δ, q0(t) +

q2
1(t)
]
, respectively.

Using Theorem 3.1 one can easily demonstrate that if the measurable function ŷ : [a, b]→
R satisfies the inequality (3.9) for almost all t ∈ [a, b], then the operator Ng+

∆
: W (B) ⇒

C(W ) order covers the set {ŷ(t)} ⊂ R. Theorem 3.1 also implies that the operator Ng−
∆

:

W (B) ⇒ C(W ) order covers the set {ŷ(t)} ⊂ R provided that the inequality

ŷ(t) ≥ q0(t) + q2
1(t)− r2 − δ,

is valid almost everywhere on [a, b],

4. IMPLICIT DIFFERENTIAL INCLUSION

In this section, based on Theorems 2.1 and 3.1 we investigate differential inclusions not
resolved with respect the derivative (which are also called implicit differential inclusions).

Let a measurable set-valued mapping B : [a, b]→ C(Rn) be given. Denote by L(B) the
space of all integrable selections of the set-valued mapping B and denote by AC(B) the
space of all absolutely continuous functions x : [a, b]→ Rn such that ẋ ∈ L(B). In the case
B(t) ≡ Rn, we denote these spaces these spaces by Ln and ACn.

We will require the fulfillment of the following analogue of the one-sided continuity
property known for “ordinary single-valued” functions applicable to a set-valued mapping
g : R→ K(Rm), which was defined in [24]. Such mapping is called right continuous at the
point x0 ∈ R if for any ε > 0, there exists δ > 0 such that for all x ∈ (x0, x0 + δ), it holds
true that hRm

(
G(x0), G(x)

)
< ε (hereinafter the symbol hRm denotes the Hausdorff distance

between the sets in the space Rm).
Let a set-valued function f : [a, b]× Rn × Rn × Rn → K(Rm) and a vector γ ∈ Rn

be given. We assume that for any x, v, u ∈ Rn, the function f(·, x, v, u) : [a, b]→ K(Rm)
is measurable; for almost all t ∈ [a, b] and all v, u ∈ Rn, the function f(t, ·, v, u) : Rn →
K(Rm) is right continuous in each of the arguments x1, . . . , xn; for almost all t ∈ [a, b] and
any x, u ∈ Rn, the function f(t, x, ·, u) : Rn → K(Rm) is right continuous in each of the

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182 E.S. ZHUKOVSKIY, I.D. SEROVA, E.A. PANASENKO, E.O. BURLAKOV

arguments v1, . . . , vn; for almost all t ∈ [a, b] and any x, v ∈ Rn, the function f(t, x, v, ·) :
Rn → K(Rm) is continuous.

We consider the differential inclusion

f(t, x, ẋ, ẋ) 3 0, t ∈ [a, b], (4.11)

with the following additional restriction on the derivative of the unknown function:

ẋ(t) ∈ B(t), t ∈ [a, b]. (4.12)

A solution of the system of inclusions (4.11), (4.12) is understood to be a function x ∈
AC(B) that satisfies the inclusion (4.11) for almost all t ∈ [a, b]. Let us formulate a statement
on solvability and estimates of solutions to the Cauchi problem for the system (4.11), (4.12)
with the initial condition

x(a) = γ. (4.13)

Denote Ω =
{

(t, x, v, u) : t ∈ [a, b], x ∈ Rn, v ∈ B(t), u ∈ B(t)
}

and define fΩ : Ω→
C(Rm) to be the restriction of the set-valued mapping f to the set Ω.

Theorem 4.1:
Let a function v0 ∈ AC(B) such that v0(a) ≥ γ and

f
(
t, v0(t), v̇0(t), v̇0(t)

)
∩ Rn

+ 6= ∅ for almost all t ∈ [a, b] (4.14)

be given. Let the set of measurable selections of the set-valued mapping B(·) ∩ ORn

(
v̇0(·)

)
:

[a, b]→ C(Rn) be integrally bounded from below (i.e. there exists a number C such that
for any measurable function u ∈ W n, satisfying the relations u(t) ∈ B(t) and u(t) ≤ v̇0(t)

for almost all t ∈ [a, b], it holds true that
∫ b

a
u(t)dt ≥ C) and the following conditions are

fulfilled:

(b1) for almost all t ∈ [a, b], any x ∈ Rn and v ∈ B(t), the mapping fΩ(t, x, v, ·) : B(t)→
K(Rm) order covers the set {0} ⊂ Rm;

(b2) for almost all t ∈ [a, b] and any u ∈ B(t), the mapping fΩ(t, ·, ·, u) : Rn ×B(t)→
K(Rm) is antitone.

Then there exists a solution x ∈ AC(B) to the problem (4.11), (4.12), (4.13) such that
ẋ(t) ≤ v̇0(t) for almost all t ∈ [a, b].

Proof
The problem (4.11), (4.12), (4.13) with respect to the unknown function x ∈ AC(B) can be
rewritten in the form of the following inclusion

fΩ

(
t, γ +

∫ t

a

u(s)ds, u(s), u(s)
)
3 0, t ∈ [a, b], (4.15)

with respect to the unknown function u = ẋ ∈ L(B). Let us demonstrate that the inclusion
(4.15) can be represented in the form of the operator inclusion (2.3), that can be investigated
using Theorem 2.1.

According to [24, Theorem 2.1], the assumptions made for the set-valued function f
provide its superpositional measurability, hence, for any x ∈ ACn, the set-valued function
f
(
·, x(·), ẋ(·), ẋ(·)

)
: [a, b]→ K(Rm) is measurable. Superpositional measurability of f

allows to define the mapping Υ : L(B)× L(B) ⇒ Wm,

∀u, v ∈ L(B) Υ(u, v)
.
= fΩ

(
t, γ +

∫ t

a

v(s)ds, v(t), u(t)
)
, t ∈ [a, b], (4.16)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



ON ORDER COVERING SET-VALUED MAPPINGS AND THEIR APPLICATIONS 183

and the corresponding mapping F : L(B) ⇒ Wm, F (u)
.
= Υ(u, u). Let us verify the

conditions of Theorem 2.1 (where we put ŷ = 0 ∈ Wm) for these mappings.
First of all, there exists a measurable function y ∈ Wm satisfying the relations

y(t) ∈ f
(
t, v0(a) +

∫ t

a

v̇0(s)ds, v̇0(t), v̇0(t)
)

= fΩ

(
t, v0(a) +

∫ t

a

v̇0(s)ds, v̇0(t), v̇0(t)
)
,

y(t) ≥ 0 for almost all t ∈ [a, b].

These relations, according to (b2), imply the existence of a measurable function y0 ∈ Wm

such that

y0(t) ∈ fΩ

(
t, γ +

∫ t

a

v̇0(s)ds, v̇0(t), v̇0(t)
)
, y0(t) ≥ y(t) ≥ 0 for almost all t ∈ [a, b].

Thus, for the mapping Υ defined by (4.16), we have Υ(v̇0, v̇0) 3 y0, y0 ≥ 0, i.e. the relations
(2.5) where u0 = v̇0 ∈ L(B) are fulfilled.

Secondly, for any v ∈ L(B), the mapping Υ(·, v) : L(B) ⇒ Wm is the Nemytskii
operator generated by the restriction g∆ to the set ∆ =

{
(t, u) : t ∈ [a, b], u ∈ B(t)

}
of the

function g : [a, b]× Rn → K(Rm),

g(t, u)
.
= fΩ

(
t, γ +

∫ t

a

v(s)ds, v(t), u
)

for almost all t ∈ [a, b] and any u ∈ Rn.

The assumption (b1) implies that for almost all t ∈ [a, b], the function g∆(t, ·) : B(t)→
K(Rm) order covers the set {0} ∈ Rm. According to Theorem 3.1, for any v ∈ L(B), the
mapping Υ(·, v) order covers the set {0} ∈ Wm. Thus, the condition (a1) of Theorem 2.1 is
satisfied.

Next, due to the assumption (b2), for any u ∈ L(B), the mapping Υ(u, ·) : L(B) ⇒ Wm

is antitone, so the condition (a2) of Theorem 2.1 is fulfilled as well.
In order to verify the assumption (a1) of Theorem 2.1, we consider an arbitrary chain S ∈

S
(
Υ,OL(B)(v̇0), 0

)
and demonstrate that it possesses an infimum u

.
= inf S and, moreover,

there exists a decreasing sequence {un} ⊂ S having the same lower bound inf{un} =
inf S = u. For any u ∈ S, we define the number lu .

=
∫ a

0
u(s)ds. The set lS .

= {lu, u ∈ S}
is a chain in R, which is bounded from below due to the integral boundedness of the set of
measurable selections of the set-valued mapping B(·) ∩ ORn

(
v̇0(·)

)
: [a, b]→ C(Rn) from

below. Thus, there exists λ .
= inf lS, and, hence, exists a decreasing sequence {un} ⊂ S such

that limn→∞ lun = λ. In the space L, any decreasing norm bounded sequence has an infimum
(see e.g. [25, p. 257]), so there exists u = inf{un}, and, moreover, for almost all t ∈ [a, b], it
holds true that u(t) = inf{un(t)} = limn→∞ un(t). Therefore, for almost all t ∈ [a, b], due to
closedness of the set B(t) ⊂ Rn, the inclusion u(t) ∈ B(t) takes place, so u ∈ L(B). Let us
now demonstrate that inf S = u. For any u ∈ S, it holds true that lu > λ, so for some number
n, we get lu > lun, hence u > un > u. Thus, u is a lower bound of S, and any function that
is greater than u, is not a lower bound of the sequence {un} and moreover, not a lower bound
of the chain S. So, u = inf S and u(t) = inf{un(t)} = limn→∞ un(t) for almost all t ∈ [a, b].

It remains now to prove the existence of a non-negative measurable selection of the set-
valued mapping Υ(u, u) : [a, b]→ K(Rm) to verify the condition (a3). By the definition of
the chain S, for any natural n, there exists a measurable function yn ∈ Wm such that yn ∈
Υ(un, un) and yn ≥ 0.As the mapping Υ(un, ·) is antitone, there exists a measurable function
ζn ∈ Wm such that ζn ∈ Υ(un, u) and ζn ≥ 0. The mapping Υ(·, u) : L(B) ⇒ Wm is the
restriction to L(B) of the Nemytskii operator generated by the function g : [a, b]× Rn →
K(Rm),

g(t, u)
.
= f

(
t, γ +

∫ t

a

u(s)ds, u(t), u
)

for almost all t ∈ [a, b] and any u ∈ Rn,

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184 E.S. ZHUKOVSKIY, I.D. SEROVA, E.A. PANASENKO, E.O. BURLAKOV

that satisfies the Caratheodori conditions. Therefore

hRm

(
g(t, un(t)), g(t, u(t))

)
→ 0. (4.17)

For almost all t ∈ [a, b], it holds true that g(t, un(t)) ∩ Rm
+ 6= ∅ (as ζn(t) ∈ g(t, un(t))).

Assume that the set-valued mapping g(·, u(·)) does not possess non-negative selections. Then
for some set T ⊂ [a, b] of a positive measure, it holds true that g(t, u(t)) ∩ Rm

+ = ∅. In this
case, for t ∈ T there is ε > 0 such that the ε-neighborhood of the set g(t, u(t)) does not
have intersections with the cone Rm

+ . The latter implies that hRm

(
g(t, un(t)), g(t, u(t))

)
> ε

for any n, which contradicts with the convergence (4.17). Thus, the condition (a3) is also
fulfilled.

Theorem 2.1 implies the existence of a solution u ∈ L(B) of the inclusion (4.15) and,
hence, the existence of a solution x ∈ AC(B) of the problem (4.11), (4.12), (4.13) (which is
defined by the formula x = γ +

∫ t

a
u(s)ds).

Let us now illustrate applications of Theorem 4.1 to investigation of concrete differential
equations and inclusions.
Example 3.1:
We denote by χ the Heaviside function, i.e. χ(x) = 1 for x ≥ 0 and χ(x) = 0 for x < 0. Let
a measurable function q0 : [a, b]→ R, a measurable integrable function q1 : [a, b]→ R, and
a positive number r be given. We consider the differential equation

ẋ2 − 2q1(t)ẋ+ χ(x)− q0(t) = 0, t ∈ [a, b], (4.18)

together with the additional condition

ẋ ∈ B(t)
.
= [q1(t)− r, q1(t) + r], t ∈ [a, b]. (4.19)

Let us demonstrate that if the inequalities

− 1 ≤ q0(t) + q2
1(t) ≤ r2, t ∈ [a, b], (4.20)

are fulfilled, the Cauchi problem for the system (4.18), (4.19) with the initial condition (4.13)
for any γ has a solution x ∈ AC(B) such that ẋ(t) ≤ q1(t), t ∈ [a, b].

We treat the equation (4.18) as the inclusion (4.11) with the single-valued mapping
f : [a, b]× R× R→ R defined by the relation

f(t, x, u) = g(t, u)− χ(x), t ∈ [a, b], x, u ∈ R,

where g is given in the Example 2.1 by the formula (3.8). Obviously, for all x, u, the function
f(·, x, u) is measurable, for almost all t and any u, the function f(t, ·, u) decreases (i.e.
the condition (b2) of Theorem 4.1 is fulfilled) and right continuous, and for almost all t
and any x, the function f(t, x, ·) is continuous. As it was demonstrated in Example 2.1,
the inequality q0(t) + q2

1(t) ≤ r2 implies the condition (b1) of Theorem 4.1. If one takes
v0(t) = γ +

∫ b

a
p(s)ds, t ∈ [a, b], the relations

f
(
t, v0(t), v̇0(t)

)
= −v̇2

0(t) + 2q1(t)v̇0(t)− χ(vo(t)) + q0(t) ≥ q2
1(t)− 1 + q0(t) ≥ 0

take place for all t ∈ [a, b]. Thus, the condition (4.14) is also satisfied, and, according to
Theorem 4.1, the problem (4.18), (4.19), (4.13) has a solution x ∈ AC(B) satisfying the
estimate ẋ(t) ≤ q1(t), t ∈ [a, b].

Consider now the following differential inclusion

g+(t, ẋ)− χ(x) 3 0, t ∈ [a, b], (4.21)

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ON ORDER COVERING SET-VALUED MAPPINGS AND THEIR APPLICATIONS 185

where the set-valued mapping g+ : [a, b]× R→ K(R) is given by the first formula in
(3.10) (see Example 2.1). In the investigation of this inclusion, one can also make use of
Theorem 4.1. In order to do this, one should use the condition that guarantees that the
mapping g+(t, ·) order covers the set {0} ⊂ R, which was derived in Example 2.1. One thus
concludes that if the inequalities

− 1− δ ≤ q0(t) + q2
1(t) ≤ r2, t ∈ [a, b], (4.22)

are satisfied, the problem (4.21), (4.19), (4.13) possesses a solution for any γ.
Finally, we consider the differential inclusion

g−(t, ẋ)− χ(x) 3 0, t ∈ [a, b], (4.23)

with the set-valued function g− : [a, b]× R→ K(R) given by the second formula in (3.10).
For this inclusion, we analogously find that the validity of the inequalities

− 1 ≤ q0(t) + q2
1(t) ≤ r2 + δ, t ∈ [a, b], (4.24)

guarantees the solvability of the problem (4.23), (4.19), (4.13) for any γ.
Consider a particular case of the inclusion (4.11), which allows to obtain the existence of

a lower solution to the corresponding Cauchi problem.
Let m = n and the components of f be given by the functions fi : [a, b]× Rn × Rn ×

R→ KC(R), i = 1, n (having compact and connected values, i.e. the real line segments).
Let us assume that for any x, v ∈ Rn, z ∈ R, the function fi(·, x, v, z) : [a, b]→ KC(R) is
measurable; for almost all t ∈ [a, b], any v ∈ Rn and z ∈ R, the function fi(t, ·, v, z) : Rn →
KC(R) is right continuous in each of the arguments x1, . . . , xn; for almost all t ∈ [a, b], any
x ∈ Rn and z ∈ R, the function fi(t, x, ·, z) : Rn → KC(R) is right continuous in each of the
arguments v1, . . . , vn; for almost all t ∈ [a, b] and any x, v ∈ Rn, the function fi(t, x, v, ·) :
R→ KC(R) is continuous. We investigate the following system:

fi(t, x, ẋ, ẋi) 3 0, t ∈ [a, b], i = 1, n. (4.25)

Theorem 4.2:
Let functions u0, v0 ∈ ACn such that u0(a) ≤ γ ≤ v0(a),

−fi
(
t, u0(t), u̇0(t), u̇0i(t)

)
∩ R+ 6= ∅ for almost all t ∈ [a, b], i = 1, n,

fi
(
t, v0(t), v̇0(t), v̇0i(t)

)
∩ R+ 6= ∅ for almost all t ∈ [a, b], i = 1, n,

be given. Define a set-valued mapping B : [a, b]→ CC(Rn) as follows:

B(t)
.
=
[
v̇0(t), u̇0(t)

]
Rn , t ∈ [a, b]. (4.26)

Assume that for any i = 1, n, for almost all t ∈ [a, b] and all u ∈ B(t), the mapping
fi(t, ·, ·, u) : Rn × Rn → KC(R) is antitone on the set Rn ×B(t). Then there exists a
solution x ∈ AC(B) to the problem (4.25), (4.12), (4.13). Moreover, the set of solutions to
the problem (4.25), (4.12), (4.13) contains a solution with the least derivative.
Remark 4.1:
By the virtue of the definition of the mapping B by (4.26), the fact that a solution x
belongs to AC(B) means that this absolutely continuous function satisfies the inequalities
u̇0(t) ≤ ẋ(t) ≤ v̇0(t) for almost all t ∈ [a, b].

Proof
We will denote the restrictions of mappings by the same symbols as the initial mappings

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186 E.S. ZHUKOVSKIY, I.D. SEROVA, E.A. PANASENKO, E.O. BURLAKOV

provided that leads to no ambiguity. We denote the components of the set-valued mappingB :
[a, b]→ CC(Rn) as Bi : [a, b]→ CC(R), i.e. Bi(t)

.
=
[
v̇0i(t), u̇0i(t)

]
, t ∈ [a, b]. We define

the set Ti ⊂ [a, b] of such t ∈ [a, b] that for all x ∈ Rn, v ∈ B(t), the mapping fi(t, x, v, ·) :
Bi(t)→ KC(R) is continuous, and−∞ < v̇0(t), u̇0(t) < +∞. The Lebesgue measure of this
set equals to b− a, and for all t ∈ Ti, the set Bi(t) is a segment of the real line.

Let us verify the condition (b1) of Theorem 4.1. Let for any i = 1, n, for some t ∈ Ti,
there exist x ∈ Rn, v ∈ B(t) such that the mapping fi(t, x, v, ·) : Bi(t)→ KC(R) does not
order cover the set {0} ⊂ R. Then there exists z0 ∈ R such that the set fi(t, x, v, z0) contains
some positive number, and for any z ∈ [v̇0i(t), z0], zero does not belong to the set fi(t, x, v, z).
Let us demonstrate that there exists δ > 0 such that for any z ∈ [v̇0i(t), z0], it holds true that

fi(t, x, v, z) ∩ (−δ, δ) = ∅. (4.27)

Otherwise, there exists a sequence {zk}∞k=1 ⊂ [v̇0i(t), z0] whose elements satisfy the
inequality |zk| < 2−k. This sequence is compact, so it contains a subsequence converging
to some z ∈ [v̇0i(t), z0]. Due to continuity of the mapping fi(t, x, v, ·) at z, the inclusion
0 ∈ fi(t, x, v, z) holds true, which contradicts with the assumptions made.

The relation (4.27), due to the connectedness of the values fi(t, x, v, z), implies that the
segment U .

= [v̇0i(t), z0] is a union of the following two sets:

U+
.
=
{
z ∈ U : fi(t, x, v, z) ⊂ [δ,+∞)

}
, U−

.
=
{
z ∈ U : fi(t, x, v, z) ⊂ (−∞, δ]

}
.

Both these sets should be closed as the mapping fi(t, x, v, ·) is continuous. However, this
is not possible by the virtue of the connectedness of the segment U. Thus, it is proved that
for any t ∈ Ti, for all x ∈ Rn and v ∈ B(t), the mapping fi(t, x, v, ·) : Bi(t)→ KC(R) order
covers the set {0} ⊂ R. Considering the fact that the Lebesgue measure of the set Ti equals
to b− a, the condition (b1) is satisfied.

The validity of the rest of the conditions of Theorem 4.1 under the assumptions of the
statement being proved is obvious. Thus, there exists a solution x ∈ AC(B) of the problem
(4.25), (4.12), (4.13).

We now prove the existence of a solution with the least derivative. Let us remind the
reader (see the proof of Theorem 4.1) that the derivatives of the solutions of the problem
(4.25), (4.12), (4.13) are the solutions of the following system of inclusions

fi
(
t, γ +

∫ t

a

u(s)ds, u(t), ui(t)
)
3 0, t ∈ [a, b], i = 1, n. (4.28)

Thus, it remains to prove that the set of solutions of the system (4.28) possesses the least
element.

Denote by R the set (in the space L(B)) of solutions of the system (4.28). According to
Theorem 2.1, the set R possesses a minimal element û ∈ L(B). Assume that this element
is not the least in R. Then there exists a solution z ∈ R of (4.28) such that z � û. For each
i = 1, n, we define the sets

Ei+ =
{
t ∈ [0, 1] : ûi(t) ≤ zi(t)

}
, Ei− =

{
t ∈ [0, 1] : ûi(t) > zi(t)

}
.

Define a measurable function ẑ with the components equal to

ẑi(t) = min
{
zi(t), ûi(t)

}
=

{
ûi(t) for t ∈ Ei+ ,
zi(t) for t ∈ Ei− ,

i = 1, n.

Thus, the inequality ẑ < û is fulfilled. Hence, for almost all t ∈ Ei+, the inclusion

0 ∈ fi
(
t, γ +

∫ t

a

û(s)ds, û(t), ûi(t)
)
, i = 1, n,

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ON ORDER COVERING SET-VALUED MAPPINGS AND THEIR APPLICATIONS 187

due to antitonicity of the mapping fi(t, ·, ·, u) on Rn ×B(t), implies that for any i = 1, n, the
set

fi
(
t, γ +

∫ t

a

ẑ(s)ds, ẑ(t), ẑi(t)
)

= fi
(
t, γ +

∫ t

a

ẑ(s)ds, ẑ(t), ûi(t)
)

contains a non-negative number. Analogously, for any i = 1, n, we get that for almost all
t ∈ Ei−, some non-negative number belongs to the set

fi
(
t, γ +

∫ t

a

ẑ(s)ds, ẑ(t), ẑi(t)
)
.

By Theorem 2.1, there exists a solution ξ ∈ R of the system (4.28) such that ξ ≤ ẑ < û.
The latter, however, contradicts with the minimality of û inR.

Finally, we note that the equation (4.18) considered in Example 3.1 under the restriction
(4.20) satisfies not only the assumptions of Theorem 4.1, but also the assumptions of
Theorem 4.2 if one takes

v0(t) = γ +

∫ b

a

q1(s)ds, u0(t) = γ +

∫ b

a

(
q1(s)− r

)
ds, t ∈ [a, b].

Therefore, the set of solutions of the problem (4.18), (4.19), (4.13) such that

q1(t)− r ≤ ẋ(t) ≤ q1(t), t ∈ [a, b], (4.29)

contains a solution with the least derivative.
Analogously, for the system of inclusions (4.21), (4.19) with the initial condition (4.13),

which was considered in Example 3.1, if the relations (4.22) take place, the set of solutions
that satisfy the inequalities (4.29) contains a solution with the least derivative. The same
statement is valid for the Cauchi problem (4.23), (4.19), (4.13). Namely, if the condition
(4.24) takes place, the set of solutions satisfying the inequalities (4.29) possesses a solution
with the least derivative.

5. EXISTENCE AND ESTIMATES OF EQUILIBRIUM PRICES IN DYNAMICAL
CONTINUOUS SUPPLY-AND-DEMAND MODELS

In this section, Theorems 4.1 and 4.2 are applied to investigation of equilibrium in dynamical
supply-and-demand models. The model of economic processes we consider has its roots in the
classical works of G.C. Evans [11], P.A. Samuelson [12], and R. Allen [13]. A considerable
advance in the investigation of models describing equilibrium processes in market models
were the results of A.V. Arutyunov, N.G. Pavlova, S.E. Zhukovskiy, A.A. Shananin (see
[14–17]) obtained using the results on covering mappings of metric spaces. In particular,
in [17] the authors managed to investigate the models with the mappings of supply and
demand that are dependent not only on the market prices, but also on the the prices change
rates. In this section, the models under consideration contain set-valued mappings of supply
and demand. Such generalisation is natural as the values of these mappings are defined in the
economic problems as the set of solutions of constraint minimisation problems (see e.g. [23,
§4.2.2]). In the present research, in contrast to the cited works that base on the results on
mappings of metric spaces, methods of analysis of mappings of partially ordered spaces are
employed.

Let us formulate the problem.
Let us have n types of goods, whose prices at any time t ∈ [a, b] we denote by pi(t),

i = 1, n. Let these prices be known at the initial moment of time

pi(a) = γi, i = 1, n. (5.30)

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188 E.S. ZHUKOVSKIY, I.D. SEROVA, E.A. PANASENKO, E.O. BURLAKOV

We define the vector p(t) =
(
p1(t), . . . , pn(t)

)
∈ Rn

+ of prices. Assume that there are some
restrictions on the change rates of the prices, i.e. there is a measurable set-valued mapping
B : [a, b]→ K(Rn) and the inclusion

ṗ(t) =
(
ṗ1(t), . . . , ṗn(t)

)
∈ B(t) for almost all t ∈ [a, b] (5.31)

takes place.
Next, let the set-valued mappings D : [a, b]× Rn

+ × Rn × Rn → K(Rn
+) and S : [a, b]×

Rn
+ × Rn → K(Rn

+) be given. We assume that for any x ∈ Rn
+ and any v, u ∈ Rn, the

functions D(·, x, v, u), S(·, x, v) : [a, b]→ K(Rn
+) are measurable; for almost all t ∈ [a, b]

and any v, u ∈ Rn, the functions D(t, ·, v, u), S(t, ·, v) : Rn
+ → K(Rn

+) are right continuous
in each of the arguments x1, . . . , xn; for almost all t ∈ [a, b] and any x ∈ Rn

+, u ∈ Rn, the
functionsD(t, x, ·, u), S(t, x, ·) : Rn → K(Rn

+) are right continuous in each of the arguments
v1, . . . , vn; for almost all t ∈ [a, b] and any x ∈ Rn

+, v ∈ Rn, the function D(t, x, v, ·) : Rn →
K(Rn

+) is continuous. We assume that the aggregate consumer demand at time t is defined
by the formula D

(
t, p(t), ṗ(t), ṗ(t)

)
and aggregate manufacturer’s supply — by the formula

S
(
t, p(t), ṗ(t)

)
.

For the defined above set-valued mappings of supply and demand, we consider the
problem of existence of an equilibrium — a solution p ∈ AC(B) (i.e. absolutely continuous
function, whose derivative satisfies the restriction (5.31)) of the inclusion

D
(
t, p(t), ṗ(t), ṗ(t)

)
− S

(
t, p(t), ṗ(t)

)
3 0, (5.32)

with the initial condition (5.30). Here the difference D − S of sets in the space Rn is the set
of all vectors x− y, where x ∈ D, y ∈ S.

We will denote the restrictions of mappings by the same symbols as the initial mappings
indicating their domains and ranges. Applying Theorem 4.1 to the mapping f : [a, b]× Rn

+ ×
Rn × Rn → K(Rn

+) defined by the relation

f(t, x, v, u)
.
= D(t, x, v, u)− S(t, x, v) for almost all t ∈ [a, b] and any x ∈ Rn

+, v, u ∈ Rn
+,

we get the following statement.
Corollary 5.1:
Let a non-negative function p ∈ AC(B) such that p(a) ≥ γ and(

D
(
t, p(t), ṗ(t), ṗ(t)

)
− S

(
t, p(t), ṗ(t)

))
∩ Rn

+ 6= ∅ for almost all t ∈ [a, b],

be given. Let the set of measurable selections of the set-valued mapping B(·) ∩ ORn

(
p(·)
)

:
[a, b]→ C(Rn) be integrally bounded from below (i.e. there exists a number C such that
for any measurable function u ∈ W n, satisfying the relations u(t) ∈ B(t) and u(t) ≤ ṗ(t)

for almost all t ∈ [a, b], it holds true that
∫ b

a
u(t)dt ≥ C) and the following conditions take

place:

(c1) for almost all t ∈ [a, b], any x ∈ Rn
+ and v ∈ B(t), the mapping D(t, x, v, ·) : B(t)→

K(Rn
+) order covers the set {0} ⊂ Rn;

(c2) for almost all t ∈ [a, b] and any u ∈ B(t), the mapping D(t, ·, ·, u) : Rn
+ ×B(t)→

K(Rn
+) is antitone, and the mapping S(t, ·, ·) : Rn

+ → K(Rn
+) is isotone.

Then there exists a solution p ∈ AC(B) of the problem (5.32), (5.31), (5.30) such that

ṗ(t) ≤ ṗ(t) for almost all t ∈ [a, b]. (5.33)

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ON ORDER COVERING SET-VALUED MAPPINGS AND THEIR APPLICATIONS 189

Note that the assumptions (c2) have natural economic sense: a decrease of the prices and
the growth rates of prices result in an increase of the demand and a decrease of supply.

The main distinctions of Corollary 5.1 from the results of [14–17] are: set-valued
mappings of supply and demand, omission of the additional condition of essentially
boundedness of the derivative ṗ of the prices function, and the estimate of the equilibrium
price derivative of the form (5.33). Corollary 5.1 has the type of the Chaplygin comparison
theorem for the differential inclusion modelling the economic processes under consideration.

Finally, let us formulate a statement allowing to obtain two-sided estimates of the
equilibrium prices and guarantee the existence of the equilibrium price having the least rate
of change.

Let the componentsDi, Si of the mappingsD,S be the functionsDi : [a, b]× Rn
+ × Rn ×

R→ KC(R), Si : [a, b]× Rn
+ × Rn → KC(R), i = 1, n (having compact connected values).

We assume that for any x ∈ Rn
+, v ∈ Rn, z ∈ R, the functionsDi(·, x, v, z) : [a, b]→ KC(R),

Si(·, x, v) : [a, b]→ KC(R) are measurable; for almost all t ∈ [a, b], any v ∈ Rn and z ∈ R,
the functions Di(t, ·, v, z) : Rn

+ → KC(R), Si(t, ·, v) : Rn
+ → KC(R) are right continuous in

each of the arguments x1, . . . , xn; for almost all t ∈ [a, b], any x ∈ Rn
+ and z ∈ R, the

functions Di(t, x, ·, z) : Rn → KC(R), Si(t, x, ·) : Rn → KC(R) are right continuous in each
of the arguments v1, . . . , vn; for almost all t ∈ [a, b] and any x ∈ Rn

+, v ∈ Rn, the function
Di(t, x, v, ·) : R→ KC(R) is continuous. Consider a particular case of the inclusion (5.32)
— the system

Di(t, x, ẋ, ẋi)− Si(t, x, ẋ) 3 0, t ∈ [a, b], i = 1, n. (5.34)
Theorem 4.2 implies the following statement.

Corollary 5.2:
Let functions u0, v0 ∈ ACn such that u0(a) ≤ γ ≤ v0(a),(
Si

(
t, u0(t), u̇0(t)

)
−Di

(
t, u0(t), u̇0(t), u̇0i(t)

))
∩ R+ 6=∅ for almost all t ∈ [a, b], i = 1, n,(

Di

(
t, v0(t), v̇0(t), v̇0i(t)

)
− Si

(
t, v0(t), v̇0(t)

))
∩ R+ 6= ∅ for almost all t ∈ [a, b], i = 1, n,

be given. Define a set-valued mapping B : [a, b]→ CC(Rn) by the relation (4.26). Assume
that for any i = 1, n, for almost all t ∈ [a, b] and all u ∈ B(t), the mapping Di(t, ·, ·, u) :
Rn

+ × Rn → KC(R) is antitone on the set Rn ×B(t) and the mapping Si(t, ·, ·) : Rn
+ ×

Rn → KC(R) is isotone on the set Rn ×B(t). Then there exists a solution p ∈ AC(B)
of the problem (5.34), (5.31), (5.30). Moreover, the set of solutions of the problem
(5.34), (5.31), (5.30) contains a solution with the least derivative.

ACKNOWLEDGEMENTS

The reported study was funded by RFBR, project number 20-04-60524. The
results in Section 4 were obtained with the support of Ministry of Science and Higher
Education of the Russian Federation (agreement no. 075-15-2019-1619). The results
in Sections 2 and 3 were obtained with the support of the Russian Science Foundation
(grant no. 20-11-20131) in V.A. Trapeznikov Institute of Control Sciences of RAS.

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Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)


	Introduction
	A comparison theorem for operator inclusions in partially ordered spaces
	Conditions of order covering for the Nemytskii operator
	Implicit Differential Inclusion
	Existence and estimates of equilibrium prices in dynamical continuous supply-and-demand models