Adv Syst Sci Appl 2023; 03:48–65
Published online at https://ijassa.ipu.ru.

On a Random Topological Characteristic
for Inclusions with Nonlinear Fredholm Operators:

Applications to Some Classes of Feedback Control Systems

Valeri Obukhovskii*, Sergey Kornev, Ekaterina Getmanova

Voronezh State Pedagogical University, Voronezh, Russia

Abstract: We define and study an oriented random coincidence index for a pair consisting of a
nonlinear zero index Fredholm operator f and a nonconvex - valued random multivalued map G
which is fundamentally restrictible with respect to f. It is shown how this characteristic can be
used for the justification of the existence of random coincidence points. We present an application
of developed results to the existence of a random solution for a control system whose dynamics
is governed by an implicit integro-differential equation and the feedback is realized by a random
differential inclusion.

Keywords: feedback control system, random differential inclusion, random coincidence index,
random coincidence point, nonlinear Fredholm operator, random multivalued map

1. INTRODUCTION

Topological methods provide strong and useful tools to deal with problems in the control
theory and optimization (among many others, let us mention monographs [3], [7], [8], [9],
[15], [16], [19], [22] and references therein). An important place in these applications is
occupied by the effective use of various topological characteristics. In the present paper we
define a random coincidence index for a pair consisting of a nonlinear Fredholm operator
and a random nonconvex-valued multimap and use it for the study of a control system whose
dynamics is presented by an implicit integro-differential equation and the feedback is realized
by a random differential inclusion.

It is worth noting that the problem of a coincidence of nonlinear Fredholm operators and
their perturbations of various types is of a great mathematical interest and finds contentable
applications in many fields of contemporary mathematics (see, for example, [5], [27], [29],
[32], [33] and references therein). It should be mentioned that the authors of the works
[20], [25], [26], [30], [31] introduced topological coincidence index for various classes
of multivalued perturbations of nonlinear Fredholm operators and used it to obtain some
applications.

In the recent years the topological degree theory was successfully extended to the case of
random maps and new random fixed point results and their applications were obtained (see,
for example, [1], [11], [24], [28]). In particular, in papers [24], [28] the random coincidence
degree theory was developed for some classes of multivalued perturbations of linear Fredholm
operators.

The present paper is organized in the following way. In the next section we collect
necessary preliminaries. The third section is devoted to the construction of an oriented
random coincidence index, the description of its properties and its usage to random

∗Corresponding author: valerio-ob2000@mail.ru



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 49

coincidence points. In the forth section we present an application to the problem of existence
of a solution for a feedback control system.

2. PRELIMINARIES

2.1. Nonlinear Fredholm operators
By the symbols E,E ′ we will denote real Banach spaces. Everywhere, by Y we will denote
an open bounded set U ⊂ E (case (i)) or U∗ ⊂ E × [0, 1] (case (ii)). We recall some notions
(see, e.g. [5], [33]).

Definition 2.1:
A C1-map f : Y → E ′ is called a Fredholm operator of index k ≥ 0 (f ∈ ΦkC

1 (Y )) if for
every y ∈ Y the Frechet derivative f ′ (y) is a linear Fredholm operator of index k, that is,
dimKer f

′
(y) <∞, dimCo ker f

′
(y) <∞ and

dimKer f
′
(y)− dimCo ker f

′
(y) = k .

Definition 2.2:
A map f : Y → E ′ is proper if f−1 (K) is compact for every compact set K ⊂ E ′.

We recall now the notion of oriented Fredholm structure on Y .
An atlas {(Yi,Ψi)} on Y is said to be Fredholm if, for each intersecting charts (Yi,Ψi)

and (Yj,Ψj) and every y ∈ Yi ∩ Yj it is(
Ψj ◦Ψ−1

i

)′
(Ψi (y)) ∈ CG

(
Ẽ
)
,

where Ẽ is the corresponding model space, and CG
(
Ẽ
)

denotes the collection of all linear

invertible operators in Ẽ of the form I +K, where I is the identity map and K is a compact
linear operator.

The set CG
(
Ẽ
)

is divided into two connected components. The component containing

the identity map will be denoted by CG+
(
Ẽ
)

.
Two Fredholm atlases are said to be equivalent if their union is still a Fredholm atlas. The

class of equivalent atlases is called a Fredholm structure.
A Fredholm structure on U is associated to a Φ0C

1-map f : U → E ′ if it admits an atlas
{(Yi,Ψi)} with model space E ′ for which(

f ◦Ψ−1
i

)′
(Ψi (y)) ∈ LC (E ′)

at each point y ∈ U , where LC (E ′) denotes the collection of all linear operators in E ′ of the
form: identity plus a compact map. Let us note that each Φ0C

1-map f : U → E ′ generates a
Fredholm structure on U associated to f .

A Fredholm atlas {(Yi,Ψi)} on Y is said to be oriented if for each intersecting charts
(Yi,Ψi) and (Yj,Ψj) and every y ∈ Yi ∩ Yj it is true that(

Ψj ◦Ψ−1
i

)′
(Ψi (y)) ∈ CG+ (E) .

Two oriented Fredholm atlases are called orientally equivalent if their union is an oriented
Fredholm atlas on Y . The equivalence class with respect to this relation is said to be the
oriented Fredholm structure on Y .

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



50 V. OBUKHOVSKII, S. KORNEV, E. GETMANOVA

2.2. Multivalued maps
We describe now some notions of the theory of multivalued maps that will be used in the
sequel (details can be found, e.g. in [2], [4], [11], [13], [15], [22]).

Let X, Z be metric spaces, K(Z) [C(Z)] denote the collection of all nonempty compact
[resp., closed] subsets of Z.

A multimap F : X → C (Z) is said to be upper semicontinuous (u.s.c.) [lower
semicontinuous (l.s.c.)] if for every open [closed] set V ⊂ Z, the set F−1

+ (V ) =
{x ∈ X : F (x) ⊂ V } is open [resp. closed] in X.

If a multimap F : X → C (Z) is both u.s.c and l.s.c. it is called continuous.
A multimap F : X → C (Z) is said to be closed if its graph

ΓF = {(x, z) ∈ X × Z : z ∈ F(x)}
is a closed subset of X × Z.

To present the class of multimaps which will be considered, we recall some notions.
Definition 2.3 (see, e.g. [21], [4], [11], [12]):
A nonempty compact subset A of a metric space Z is said to be aspheric (or UV ∞, or ∞-
proximally connected) if for every ε > 0 there exists δ, 0 < δ < ε, such that for each n =
0, 1, 2, . . . every continuous map g : Sn → Oδ (A) can be extended to a continuous map g̃ :
Bn+1 → Oε (A), where Sn = {x ∈ Rn+1 : ∥x∥ = 1} and Bn+1 = {x ∈ Rn+1 : ∥x∥ ≤ 1}.
Definition 2.4 (see [11]):
A u.s.c. multimap F : X → K (Z) is said to be a J-multimap (F ∈ J (X,Z)) if every value
F (x), x ∈ X is an aspheric set.

We recall (see, for example, [6]) that a metric space M is said to be an absolute retract
(an AR-space) (respectively, an absolute neighborhood retract (an ANR-space)), if for each
homeomorphism h, which maps it to a closed subset of the metric space Z the set h(M) is a
retract ofZ (respectively, of some its open neighborhood inZ). Notice that the class ofANR-
spaces is sufficiently wide: in particular, a compact subset of a finite-dimensional space is an
ANR-space if and only if it is locally contractible. In turn, this means that compact polyhedra
and compact finite-dimensional manifolds are ANR-spaces. The union of a finite number of
closed convex subsets of a normed space is also an ANR-space.
Definition 2.5 ([14]):
A nonempty compact space A is said to be an Rδ-set, if it can be presented as the intersection
of a decreasing sequence of compact AR-spaces.
Proposition 2.1 (see [11]):
Let Z be an ANR-space. In each of the following cases a u.s.c. multimap F : X → K (Z) is
a J-multimap:
for each x ∈ X the value F (x) is

a) a convex set;

b) a contractible set;

c) an Rδ-set;

d) an AR-space.
In particular, every continuous map σ : X → Z is a J-multimap.
Definition 2.6:
By CJ (X,X ′) we will denote the collection of all multimaps G : X → K (X ′) of the form
G = φ ◦ F, where F ∈ J (X,Z) for some metric space Z, φ : Z → X ′ is a continuous map.
The composition φ ◦ F will be called the representation (or decomposition, cfr. [11]) of G.
We will denote G = (φ ◦ F) ∈ CJ (X,X ′).

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 51

It is worth noting that a multimap can admit different representations (see [11]).

2.3. An Oriented Coincidence Index for Compact Triplets
We will start with the following notion.
Definition 2.7:
A map f : Y → E ′, a multimap G = (φ ◦ F) ∈ CJ

(
Y ,E ′) and the space Y form a compact

triplet
(
f,G, Y

)
C

if the following conditions are satisfied:

h1) f is a continuous proper map, f|Y ∈ ΦkC
1 (Y ) with k = 0 in case (i), k = 1 in case (ii),

and the Fredholm structure on Y generated by f is oriented;

h2) G is compact, i.e., G
(
Y
)

is a relatively compact subset of E ′;

h3) Coin (f,G) ∩ ∂Y = ∅, where

Coin (f,G) = {y ∈ Y : f(y) ∈ G(y)}
is the coincidence point set of f and G.

Let us mention that from hypotheses (h1), (h2) it follows that the coincidence point set
Q = Coin (f,G) is compact.

As earlier, let U be an open bounded subset of a Banach space E. For a compact triplet(
f,G, U

)
C

the oriented coincidence index

Ind
(
f,G, U

)
C
,

the integer-valued topological characteristic with the following basic properties can be
defined (see [26]).
Proposition 2.2 (The coincidence point property):
If Ind

(
f,G, U

)
C
̸= 0, then ∅ ≠ Coin (f,G) ⊂ U .

To formulate the topological invariance property of the coincidence index, we will give
the following definition.
Definition 2.8:
Two compact triplets(

f0, G0 = (φ0 ◦ F0) , U0

)
C

and
(
f1, G1 = (φ1 ◦ F1) , U1

)
C

are said to be homotopic(
f0, G0 = (φ0 ◦ F0) , U0

)
C
∼

(
f1, G1 = (φ1 ◦ F1) , U1

)
C

if there exists a compact triplet
(
f∗, G∗, U∗

)
C

, where U∗ ⊂ E × [0, 1] is an open set, such
that:

a) Ui = U∗ ∩ (E × {i}), i = 0, 1;

b) f∗|Ui
= fi, i = 0, 1;

c) G∗ has the form
G∗ (x, λ) = φ∗ (F∗ (x, λ) , λ)

where F∗ ∈ J
(
U∗, Z

)
, φ∗ : Z × [0, 1] → E ′ is a continuous map, and

F∗ |U i
= Fi , φ∗ |Z×{i}= φi , i = 0, 1 .

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



52 V. OBUKHOVSKII, S. KORNEV, E. GETMANOVA

Proposition 2.3 (The homotopy invariance property):
If (

f0, G0, U0

)
C
∼

(
f1, G1, U1

)
C
,

then ∣∣Ind (
f0, G0, U0

)
C

∣∣ = ∣∣Ind (
f1, G1, U1

)
C

∣∣ .
Remark 2.1:
If the Fredholm operator f is constant under the homotopy, i.e., U∗ has the form U∗ =
U × [0, 1] where U ⊂ E is an open set and f∗ (x, λ) = f (x) for all λ ∈ [0, 1], where f ∈
Φ0C

1 (U), then
Ind

(
f,G0, U

)
C
= Ind

(
f,G1, U

)
C
.

Proposition 2.4 (Additive dependence on the domain property):
Let U0 and U1 be disjoint open subsets of an open bounded set U ⊂ E and

(
f,G, U

)
C

be a
compact triplet such that

Coin (f,G) ∩
(
U \ (U0 ∪ U1)

)
= ∅ .

Then
Ind

(
f,G, U

)
C
= Ind

(
f,G, U0

)
C
+ Ind

(
f,G, U1

)
C

2.4. Random Multivalued Maps and Random Coincidence Points
Let (Ω,Σ) be a measurable space and X, Z separable metric spaces. Let B(X) be the σ-
algebra of all Borel subsets of X and Σ⊗ B(X) denote the minimal σ-algebra containing the
sets A×B, where A ∈ Σ, B ∈ B(X).

Definition 2.9:
A multimap Φ: Ω×X → C(Z) is called random if it is measurable with respect to the σ-
algebra Σ⊗ B(X), i.e.,

Φ−1
+ (V ) ∈ Σ⊗ B(X)

for each open V ⊂ Z.

As example of a random multimap we can consider a Carathéodory type multimap
Φ: Ω×X → C(Z), i.e., it is supposed that Φ is such that: 1) Φ(·, x) : Ω → C(Z) is
measurable w.r.t. Σ for each x ∈ X; 2)Φ(ω, ·) : X → C(Z) is continuous for each ω ∈ Ω
(see, e.g., [13], Proposition 7.9).
Definition 2.10:
For a multimap Φ: Ω×X → C(Z) and a map ψ : X → Z, a measurable map ξ : Ω → X is
called a random coincidence point of ψ and Φ if it satisfies the following inclusion

ψ(ξ(ω)) ∈ Φ(ω, ξ(ω))

for each ω ∈ Ω.

The following assertion on a random coincidence point holds true.
Proposition 2.5:
Let (Ω,Σ) be a complete measurable space (see, e.g., [13], Definition 1.29); a space X is
complete; Φ: Ω×X → C(Z) a random multimap; a map ψ : X → Z be measurable w.r.t.
B(X). If for each ω ∈ Ω the coincidence point set

Coinω(ψ,Φ) = {x ∈ X : ψ(x) ∈ Φ(ω, x)}

is nonempty, then ψ and Φ have a random coincidence point.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 53

Proof
The measurable map ψ : X → Z can be naturally extended to the random map ψ̃ : Ω×X →
Z if we set ψ̃(ω, x) = ψ(x) for all ω ∈ Ω. In fact, for each open V ⊂ Z we will have then
ψ̃−1(V ) = Ω× ψ−1(V ) ∈ Σ× B(X).

Notice that the function µ : Ω×X → R,

µ(ω, x) = dist(ψ(x),Φ(ω, x)) = dist(ψ̃(ω, x),Φ(ω, x))

is random (see [10]).
Define the multimap F : Ω ⊸ X as

F(ω) = Coinω(ψ,Φ).

Then for the graph ΓF of the multimap F we have ΓF = µ−1(0) ∈ Σ× B(X) and applying to
F the Aumann selection theorem for a multifunction with a measurable graph (see, e.g., [13],
Theorem 2.2.14) we conclude that F admits a measurable selection ξ : Ω → X which is the
desirable random coincidence point.

3. A RANDOM COINCIDENCE INDEX FOR FUNDAMENTALLY
RESTRICTIBLE QUADRIPLES

3.1. A Completely Fundamentally Restrictible Triplet
At first we recall some notions (see, e.g. [15]). Let again Y = U ⊂ E, or U∗ ⊂ E × [0, 1] be
open bounded sets, f : Y → E

′ a map; G : Y → K (E ′) a multimap.
Definition 3.1:
A convex, closed subset T ⊂ E

′
is said to be fundamental for a triplet

(
f,G, Y

)
if:

(i) G (f−1 (T )) ⊆ T ;

(ii) for any point y ∈ Y , the inclusion f (y) ∈ co (G (y) ∪ T ) implies f (y) ∈ T.

The entire space E ′ and coG
(
Y
)

are natural examples of fundamental sets for
(
f,G, Y

)
.

It is easy to verify the following properties of a fundamental set.
Proposition 3.1:
The following holds:

a) The set Coin (f,G) is included in f−1 (T ) for each fundamental set T of
(
f,G, Y

)
;

b) Let T be a fundamental set of
(
f,G, Y

)
, and P ⊂ T , then the set T̃ =

co (G (f−1 (T )) ∪ P ) is also fundamental;

c) Let {Tα} be a system of fundamental sets of
(
f,G, Y

)
. The set T = ∩αTα is also

fundamental.

Definition 3.2:
We will say that a triple

(
f,G, Y

)
is completely fundamentally restrictible if it possesses a

nonempty, compact fundamental set T.

The fact that a triple
(
f,G, Y

)
is completely fundamentally restrictible will be denoted

as
(
f,G, Y

)
F
. It is clear that each compact triple

(
f,G, Y

)
C

is completely fundamentally
restrictible since we can take coG

(
Y
)

as a compact fundamental set. To obtain more valuable

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



54 V. OBUKHOVSKII, S. KORNEV, E. GETMANOVA

example of a completely fundamentally restrictible triplet, we will need the following notions.

Denote by P (E ′) the collection of all nonempty subsets of a Banach spaceE ′
. Let (A,≥)

be a partially ordered set.

Definition 3.3:
A map β : P (E ′) → A is called a measure of noncompactness (MNC) in E

′
if

β (coD) = β (D) for every D ∈ P (E ′) .

A MNC β is called:

(i) monotone, if D0, D1 ∈ P (E ′) , D0 ⊆ D1 implies β (D0) ≤ β (D1) ;

(ii) nonsingular, if β ({a} ∪D) = β (D) for every a ∈ E
′
, D ∈ P (E ′);

(iii) real, if A = R+ = [0,+∞] with the natural ordering, and β (D) < +∞ for every
bounded set D ∈ P (E ′) .

Among the known examples of MNC satisfying all the above properties we can consider
the Hausdorff MNC

χ (D) = inf {ε > 0 : D has a finite ε-net} .

and the Kuratowski MNC

α (D) = inf {d > 0 : D has a finite partition with sets of diameter less than d} .

Definition 3.4:
Maps f , G and the space Y form a β-condensing triplet,

(
f,G, Y

)
β

if they satisfy conditions
(h1) and (h3) in Definition 2.7 and

h2β) a multimap G = φ ◦ F ∈ CJ
(
Y ,E

′)
is β-condensing w.r.t. f , i.e.,

β (G (D)) ≱ β (f (D))

for every D ⊆ Y such that G (D) is not relatively compact.

Proposition 3.2:
Each β-condensing triplet

(
f,G, U

)
β
, where β is a monotone, nonsingular MNC is

completely fundamentally restrictible.

Proof
Consider the collection {Tα} of all fundamental sets of

(
f,G, U

)
β

containing an arbitrary
point a ∈ E

′ . This collection is nonempty since it contains E ′ . Then, taking T = ∩αTα ̸= ∅
we obviously have

T = co
(
G
(
f−1 (T )

)
∪ {a}

)
and hence

β
(
f
(
f−1 (T )

))
≤ β (T ) = β

(
G
(
f−1 (T )

))
,

so G (f−1 (T )) is relatively compact and T is compact.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 55

3.2. A Random Coincidence Index
Let (Ω,Σ) be a complete measurable space.

Definition 3.5:
A map f : Y → E ′, a multimapG : Ω× Y → K(E ′) and the spaces Ω and Y form a random
fundamentally restrictible quadriple

(
f,G,Ω, Y

)
F

if the following conditions are satisfied:

g1) f is a continuous proper map, f|Y ∈ ΦkC
1 (Y ) with k = 0 in case (i), k = 1 in case (ii),

and the Fredholm structure on Y generated by f is oriented;

g2) G is a random multimap;

g3) for each ω ∈ Ω the multimap Gω = G(ω, ·) = (φω ◦ Fω) ∈ CJ
(
Y ,E ′) ;

g4) for each ω ∈ Ω the triplet (f,Gω, Y ) is completely fundamentally restrictible and
Coin (f,Gω) ∩ ∂Y = ∅.

Now our purpose is to define an oriented random coincidence index for a random
fundamentally restrictible quadriple

(
f,G,Ω, U

)
F
. To this end, define for each ω ∈ Ω a

coincidence index of a completely fundamentally restrictible triplet
(
f,Gω, U

)
F
.

Let T ω be a nonempty compact fundamental set of a triplet
(
f,Gω, U

)
F

and ρ : E ′ → T ω

be any retraction. Consider the multimap G̃ω = ρ ◦ φω ◦ Fω ∈ CJ
(
U,E

′). From Proposition
3.1(a) it follows that

Coin
(
f, G̃ω

)
= Coin (f,Gω) . (3.1)

Hence, f, G̃ω, and U form a compact triplet
(
f, G̃ω, U

)
C

. We will say that
(
f, G̃ω, U

)
C

is a compact approximation of the triplet
(
f,Gω, U

)
F

.

Definition 3.6:
The oriented coincidence index of a completely fundamentally restrictrictible triplet(
f,Gω, U

)
F

is defined by the equality

Ind
(
f,Gω, U

)
F
:= Ind

(
f, G̃ω, U

)
C
,

where
(
f, G̃ω, U

)
C

is a compact approximation of
(
f,Gω, U

)
F

.

To prove the consistency of the above definition, consider two nonempty, compact
fundamental sets T0 and T1 of the triplet

(
f,Gω = φω ◦ Fω, U

)
F

with retractions ρ0 : E
′ →

T0 and ρ1 : E
′ → T1 respectively.

If T0 ∩ T1=∅, then by Proposition 3.1 (a), (c), Coin
(
f, G̃ω0

)
= Coin

(
f, G̃ω1

)
=

Coin
(
f, G̃ω

)
= ∅, where G̃ωi = ρi ◦ φω ◦ Fω, i = 0, 1. Hence, by Proposition 2.2,

Ind
(
f, G̃ω0, U

)
C
= Ind

(
f, G̃ω1, U

)
C
= 0.

Otherwise, we can assume, w.l.o.g., that T0 ⊆ T1. In this case, consider the map φω : Zω ×
[0, 1] → E

′ , given by φω (z, λ) = ρ1 ◦ (λφω (z) + (1− λ) ρ0 ◦ φ− ω (z)) and the multimap
Gω ∈ CJ

(
U × [0, 1] , E

′), Gω (x, λ) = φω (Fω (x) , λ).

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



56 V. OBUKHOVSKII, S. KORNEV, E. GETMANOVA

The compact triplet
(
f,Gω, U × [0, 1]

)
C

realizes the homotopy(
f, G̃ω0, U

)
C
∼

(
f, G̃ω1, U

)
C
.

Indeed, the only fact that we need to verify is that

Coin
(
f,Gω

)
∩ (∂U × [0, 1]) = ∅,

where f (x, λ) ≡ f (x) is the natural extension.
To the contrary, suppose that there exists (x, λ) ∈ ∂U × [0, 1] such that

f (x) = ρ1 ◦ (λφω (z) + (1− λ) ρ0 ◦ φω (z))

for some z ∈ Fω (x). But in this case, x ∈ f−1 (T1) and hence φω (z) ∈ T1. Since also
ρ0 ◦ φω (z) ∈ T1 we have

λφω (z) + (1− λ) ρ0 ◦ φω (z) ∈ T1

and so
f (x) = λφω (z) + (1− λ) ρ0 ◦ φω (z) ∈ co (Gω (x) ∪ T0)

and we obtain that f (x) ∈ T0 and x ∈ f−1 (T0), implying φω (z) ∈ T0 and ρ0 ◦ φω (z) =
φω (z). We conclude that f (x) = φω (z) ∈ Gω (x) giving the contradiction.

Definition 3.7:
For a given random fundamentally restrictible quadriple(
f,G,Ω, U

)
F

the oriented random coincidence index is defined as the following collection
of integers:

Ind
(
f,G,Ω, U

)
F
= {Ind

(
f,Gω, U

)
F
: ω ∈ Ω}.

By definition we set Ind
(
f,G,Ω, U

)
F
̸= 0 under the condition that Ind

(
f,Gω, U

)
F
̸= 0

for all ω ∈ Ω.

From Proposition 2.5 we obtain the following random coincidence point property.

Theorem 3.1:
If Ind

(
f,G,Ω, U

)
F
̸= 0 then the quadriple

(
f,G,Ω, U

)
F

has a random coincidence point,
i.e., there exists a measurable map ξ : Ω → U such that

f(ξ(ω)) ∈ G(ω.ξ(ω)), ∀ω ∈ Ω.

Definition 3.8:
Two random fundamentally restrictible quadriples(
f0, G0 = (φ0 ◦ F0) ,Ω, U0

)
F

and
(
f1, G1 = (φ1 ◦ F1) ,Ω, U1

)
F

are said to be homotopic(
f0, G0 = (φ0 ◦ F0) ,Ω, U0

)
F
∼

(
f1, G1 = (φ1 ◦ F1) ,Ω, U1

)
F

if there exists a random completely fundamentally restrictible quadriple
(
f∗, G∗,Ω, U∗

)
F

,
where U∗ ⊂ E × [0, 1] is an open set, such that:

a) Ui = U∗ ∩ (E × {i}), i = 0, 1 ;

b) f∗ |Ui
= fi, i = 0, 1 ;

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 57

c) for each ω ∈ Ω, the multimap G∗ω = G∗(ω, ·, ·) has the form

G∗ω (x, λ) = φ∗ω (F∗ω (x, λ) , λ)

where F∗ω ∈ J
(
U∗, Zω

)
, φ∗ω : Zω × [0, 1] → E ′ is a continuous map, and

F∗ω |U i
= Fiω , φ∗ω |Zω×{i}= φiω , i = 0, 1 .

We get the following homotopy invariance property.

Theorem 3.2:
If (

f0, G0,Ω, U0

)
F
∼

(
f1, G1,Ω, U1

)
F
,

then ∣∣∣Ind (f0, G0,Ω, U0

)
F

∣∣∣ = ∣∣∣Ind (f1, G1,Ω, U1

)
F

∣∣∣
in the sense that ∣∣∣Ind((f0, G0ω, U0

)
F

∣∣∣ = ∣∣∣Ind (f1, G1ω, U1

)
F

∣∣∣
for all ω ∈ Ω.

Proof
For a given ω ∈ Ω, let T ω

∗ be a nonempty compact fundamental set of the triplet(
f∗, G∗ω = (φ∗ω ◦ F∗ω) , U∗

)
F
. It is easy to see that T ω

∗ is fundamental also for the triplets(
fi, Giω, U i

)
F

, i = 0, 1.

Let ρ∗ω : E
′ → T ω

∗ be any retraction, and
(
f∗, G̃∗ω = ρ∗ω ◦ φ∗ω ◦ F∗ω, U∗

)
C

the correspond-

ing compact approximation of
(
f∗, G∗ω, U∗

)
F

. Then
(
f∗, G̃∗ω, U∗

)
C

realizes a compact

homotopy connecting the triplets
(
fi, ρ∗ω ◦ φiω ◦ Fiω, U i

)
C

, i = 0, 1 which are compact
approximations of

(
fi, Giω, U i

)
F

, i = 0, 1 respectively.
By Proposition 2.3 we have∣∣Ind (

f0, ρ∗ω ◦ φ0ω ◦ F0ω, U0

)
C

∣∣ = ∣∣Ind (
f1, ρ∗ω ◦ φ1ω ◦ F1ω, U1

)
C

∣∣
giving the desired.

Remark 3.1:
In connection with Remark 2.1 if the operator f is constant under the homotopy, i.e., U∗ has
the form U∗ = U × [0, 1], where U ⊂ E is an open set and f∗(x, λ) = f(x) for all λ ∈ [0, 1]
with f ∈ Φ0C

1(U) then

Ind
(
f,G0,Ω, U

)
F
= Ind

(
f,G1,Ω, U

)
F

Let us formulate the additive dependence on domain property for random fundamentally
restrictible quadriples which follows from Proposition 2.4.

Theorem 3.3:
Let U0 and U1 be disjoint open subsets of an open bounded set U ⊂ E and

(
f,G,Ω, U

)
F

be
a fundamentally restrictible quadriple such that

Coin (f,Gω) ∩
(
U \ (U0 ∪ U1)

)
= ∅

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



58 V. OBUKHOVSKII, S. KORNEV, E. GETMANOVA

for each ω ∈ Ω. Then

Ind
(
f,G,Ω, U

)
F
= Ind

(
f,G,Ω, U0

)
F
+ Ind

(
f,G,Ω, U1

)
F

in the sense that

Ind
(
f,Gω, U

)
F
= Ind

(
f,Gω, U0

)
F
+ Ind

(
f,Gω, U1

)
F

for each ω ∈ Ω.

Analogs of the above notions and assertions for the condensing case can be formulated in
the natural way. Let β be a monotone, nonsingular MNC in E ′.
Definition 3.9:
A map f : Y → E ′, a multimapG : Ω× Y → K(E ′) and the spaces Ω and Y form a random
β-condensing quadriple

(
f,G,Ω, Y

)
β

if they satisfy conditions (g1)− (g3) of Definition 3.5
and the following condition

g4β) for each ω ∈ Ω the triplet (f,Gω, Y ) is β-condensing and Coin (f,Gω) ∩ ∂Y = ∅.

For a given ω ∈ Ω, the oriented coincidence index of a β-condensing triplet (f,Gω, U)β
can be defined via its compact approximation as in Definition 3.6 and the oriented random
coincidence index of a random β-condensing quadriple (f,G,Ω, U)β also is defined in
accordance with Definition 3.7:

Ind
(
f,G,Ω, U

)
β
= {Ind

(
f,Gω, U

)
β
: ω ∈ Ω}.

The notion of homotopy for random β-condensing quadriples follows Definition 3.8 with
the substitution of ”completely fundamentally restrictible” with ”β-condensing” and the
corresponding homotopy invariance property also holds true.
Remark 3.2:
Let us mention that in case of constant f and U :

U∗ = U × [0, 1]
f∗ (x, λ) ≡ f (x) , ∀λ ∈ [0, 1] ,

the condition of β-condensivity for a triplet
(
f,G∗ω,U × [0, 1]

)
β

may be weakened: for the
existence of a nonempty, compact fundamental set T ω it is sufficient to demand that

β (G∗ω (D × [0, 1])) ≱ β (f (D))

for every D ⊆ U such that G∗ω (D × [0, 1]) is not relatively compact.
In fact, it is enough to notice that in this case f−1

∗ (T ) = f−1 (T )× [0, 1] and to follow the
line of reasoning of Proposition 3.2.
Taking into consideration the corresponding property of compact triplets, we can precise the
above property of homotopy invariance.
If
(
f,G∗ω,U × [0, 1]

)
β

is a β-condensing triplet, where G∗ω has the form (c) of Definition
2.8, then

Ind
(
f,G0ω, U

)
β
= Ind

(
f,G1ω, U

)
β

where Giω = G∗ω (·, {i}), i = 0, 1.

As an example of the application of a random coincidence point property, let us consider
the following assertion.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 59

Theorem 3.4:
Let f ∈ Φ0C

1 (E,E ′) be odd; G : Ω× E → K(E ′) a random multimap such that for
each ω ∈ Ω, Gω ∈ CJ (E,E ′) and β-condensing w.r.t. f on bounded subsets of E, i.e.
β (Gω (D)) ≱ β (f (D)) for every bounded set D ⊂ E such that G (D) is not relatively
compact.
If the set of solutions of two-parameter family of operator inclusions

f (x) ∈ λGω (x) (3.2)

is a priori bounded, then there exists a random coincidence point of f and G.

Proof
From the condition it follows that there exists a ball B ⊂ E centered at the origin whose
boundary ∂B does not contain solutions of (3.2).
Let φω ◦ Fω be a representation of Gω. If G∗ω : B × [0, 1] → K (E ′) has the form

G∗ω (x, λ) = φ∗ω (Fω (x) , λ) , (x, λ) ∈ B × [0, 1] ,
φ∗ω (z, λ) = λφω (z)

then f,G∗ω and B × [0, 1] form a β-condensing triplet
(
f,G∗ω,B × [0, 1]

)
β
.

In fact, suppose that β (G∗ω (D)) ≥ β (f (D)) for some D ⊂ B. Since G∗ω (D × [0, 1]) =
co (Gω (D) ∪ {0}) we have that β (Gω (D)) ≥ β (f (D)) implying that Gω (D) and hence
G∗ω (D × [0, 1]) is relatively compact.
So the triplet

(
f,G∗ω,B × [0, 1]

)
β

induces a homotopy connecting the triplets
(
f,Gω,B

)
β

and
(
f, 0,B

)
β
. Since the triplet

(
f, 0,B

)
β

is finite dimensional, from the odd condition on f
and the odd field property of the Brouwer degree, it follows that

(
f, 0,B

)
β

is an odd number.
Then, from the equality Ind

(
f,Gω,B

)
β
= Ind

(
f, 0,B

)
β

it follows that Ind
(
f,Gω,B

)
β
̸=

0 for all ω ∈ Ω and hence Ind
(
f,G,Ω,B

)
β
̸= 0 and we can apply the coincidence point

property.

4. AN APPLICATION: EXISTENCE OF A SOLUTION FOR A RANDOM
FEEDBACK CONTROL SYSTEM

Let (Ω,Σ, µ) be a σ-finite measure space. Consider a random feedback control system
governed by the following relations:

A (t, x (t) , x′ (t)) = B

(
t, x (t) , x′ (t) ,

∫ t

0

yω (s) ds

)
, t ∈ [0, a]; (4.3)

yω (τ) ∈ C (ω, τ, x (τ)) , a.e. τ ∈ [0, a], ∀ω ∈ Ω; (4.4)

x (0) = x0, (4.5)

where ω ∈ Ω, A : [0, a]× Rn × Rn → Rn, B : [0, a]× Rn × Rn × Rm → Rn are continu-
ous maps; C : Ω× [0, a]× Rn ⊸ Rm is a multimap, and x0 ∈ Rn.

By a random solution of problem (4.3)-(4.5) we mean a pair (xω, yω) consisting of a
trajectory function xω(·) such that ω ∈ Ω → xω ∈ C1 ([0, a] ;Rn) is a measurable map and
the function xω satisfies for each ω ∈ Ω relations (4.3)-(4.5), whereas the control function
yω(·) is such that ω ∈ Ω → yω ∈ L1 ([0, a] ;Rm) is a measurable map and the function yω
satisfies for each ω ∈ Ω relations (4.3)-(4.4).

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



60 V. OBUKHOVSKII, S. KORNEV, E. GETMANOVA

Our aim is to show that, under appropriate conditions, the problem of finding of a random
solution to systrm (4.3)-(4.5) can be reduced to the study of a condensing quadriple of the
above mentioned form (see Section 3).

Consider the following condition:
(A) For each (t, u, v) ∈ [0, a]× Rn × Rn there exist continuous partial derivatives

A′
u (t, u, v), A

′
v (t, u, v) and moreover, detA′

v (t, u, v) ̸= 0.

The following assertion holds true (see [26], Proposition 5.1).
Proposition 4.1:
Under condition (A) a map f : C1 ([0, a] ;Rn) → C ([0, a] ;Rn)× Rn defined as

f (x) (t) = (A (t, x (t) , x′ (t)) , x (0))

is a Fredholm map of index zero, whose restriction to each closed bounded set D ⊂
C1 ([0, a] ;Rn) is proper.

Now we will describe the assumptions on the map B and the multimap C.

Denoting by the symbol Kv (Rm) the collection of all nonempty compact convex subsets
of Rm, we suppose that the multimap C : Ω× [0, a]× Rn → Kv (Rm) satisfies the following
conditions:
(C1) C is a random multimap, i.e., it is measurable w.r.t. the σ-algebra Σ⊗ L⊗ B(Rn),

where L is the σ-algebra of Lebesgue subsets of [0, a];

(C2) for all (ω, t) ∈ Ω× [0, a] the multimap C (ω, t, ·) : Rn → Kv (Rm) is upper
semicontinuous;

(C3) for each r > 0 there exists a function γr : Ω× [0, a] → R+ such that: (i) γr(ω, ·) ∈
L1(0, a) for each ω ∈ Ω; (ii) the function γr(·, t) : Ω → R+ is measurable for a.e.
t ∈ [0, a] and the following estimate holds for all ω ∈ Ω and u ∈ Rn, ∥u∥ ≤ r :

∥C (ω, t, u)∥ := sup {∥c∥ : c ∈ C (ω, t, u)} ≤ γr (ω, t) a.e. t ∈ [0, a].

Notice that from condition (C1) it follows that for each (ω, u) ∈ Ω× Rn the
multifunction

C(ω, ·, u) : [0, a] → Kv(Rm)

is Lebesgue measurable.
It is known (see, e.g. [8], [11], [13], [15], [22]) that under given conditions for each ω ∈ Ω

the superposition multioperator P(ω, ·) : C([0, a];Rn) ⊸ L1([0, a];Rm) :

P(ω, x) = {ψ ∈ L1([0, a];Rm) : ψ(t) ∈ C(ω, t, x(t)) a.e. t ∈ [0, a]

is well defined. Moreover, if we will consider the composition Π: Ω× C([0, a];Rn) ⊸
C([0, a];Rm)

Π(ω, x) = j ◦ P(ω, x),

where j : L1([0, a];Rm) → C([0, a];Rm) is the integral operator j(ψ)(t) =
∫ t

0
ψ(s)ds, then

Π(ω, ·) : C([0, a];Rn) ⊸ C([0, a];Rm) is a closed multioperator for each ω ∈ Ω. Further,
since the embedding C1([0, a];Rn) ↪→ C([0, a];Rn) is completely continuous, the restriction
of Π(ω, ·) toC1([0, a];Rn) is locally compact and hence upper semicontinuous (see, e.g. [22],
Theorem 1.2.32). It is easy to see also that the multimap Π has compact convex values and
hence this restriction is a J-multimap.

At last, from [23], Theorem 3 it follows that Π is a random multimap.

Now, we will assume that the maps A and B satisfy the following Lipschitz type
condition:

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 61

(AB) there exists a constant q, 0 ≤ q < 1 such that

|B (t, u, v, w)−B (t, u, v, w)| ≤ q |A (t, u, v)− A (t, u, v)|

for all t ∈ [0, a] , u, v, v ∈ Rn, w ∈ Rm.

Consider the continuous map ξ : C1 ([0, a] ;Rn)× C ([0, a] ;Rm) → C ([0, a] ;Rn)
defined as

ξ (x, z) (t) = B (t, x (t) , x′ (t) , z (t))

and the multimap F̃ : Ω× C1 ([0, a] ;Rn) → K (C1 ([0, a] ;Rn)× C ([0, a] ;Rm)),
F̃ (ω, x) = {x} × Π(ω, x).

Consider the composition G̃ = ξ ◦ F̃ : Ω× C1 ([0, a] ;Rn) → K (C ([0, a] ;Rn)). It is
easy to see that the multimap F̃ and hence G̃ are random. Further, from [22], Theorem
1.3.17 it follows that for each ω ∈ Ω the multimap F̃ω = F̃(ω, ·) is u.s.c. and hence it is a J-
multimap, and therefore the composition G̃ω = ξ ◦ F̃ω : C1 ([0, a] ;Rn) → K (C ([0, a] ;Rn))

is a CJ-multimap. It is clear that the set G̃ω (x) consists of all functions of the form
B (t, x (t) , x′ (t) , πω (t)) where πω ∈ Π(ω, x) .

Define now the multimap G : Ω× C1 ([0, a] ;Rn) → K (C ([0, a] ;Rn)× Rn) by

G (ω, x) = G̃ (ω, x)× {x0} .

It is easy to see that G is a random multimap and Gω = G(ω, ·) is a CJ-multimap for each
ω ∈ Ω.

Now, in accordance with Proposition 2.5 the existence of a trajectory for system (4.3)-
(4.5) is equivalent to the existence of a coincidence point xω ∈ C1 ([0, a] ;Rn) for the pair
(f,Gω) for each ω ∈ Ω.

If U ⊂ C1 ([0, a] ;Rn) is an open bounded set, then to show, for a given ω ∈ Ω that(
f,Gω, U

)
form a condensing triplet w.r.t. the Kuratowski MNC, it is sufficient to prove the

following statement.
Proposition 4.2:
The triplet

(
f̃ , G̃ω, U

)
, where f̃ : C1 ([0, a] ;Rn) → C ([0, a] ;Rn) , is defined as

f̃ (x) (t) = A (t, x (t) , x′ (t))

is α-condensing w.r.t. the Kuratowski MNC α in the space C ([0, a] ;Rn) for each ω ∈ Ω.

Proof
Take any subset D ⊂ U , and let α

(
f̃ (D)

)
= d. From the definition of Kuratowski

MNC it follows that taking an arbitrary ε > 0 we may find a partition of the setf̃ (D)

into subsets f̃ (Di) , i = 1, · · · , s such that diam
(
f̃ (Di)

)
≤ d+ ε. Since the embedding

C1 ([0, a] ;Rn) ↪→ C ([0, a] ;Rn) is completely continuous, the image DC of D under this
embedding is relatively compact. It is known (see, e.g. [13], [15], [22]) that a u.s.c. compact-
valued multimap sends compact sets to compact sets, then we can conclude that the set
Πω (D) is relatively compact. It means that taking a fixed δ > 0 and any Di we may divide the
sets DiC and Πω (D) into a finite number of subsets DijC , j = 1, · · · , pi, and balls Bik (zik),
k = 1, · · · , ri, centered at zik ∈ C ([0, a] ;Rm) respectively, such that for each t ∈ [0, a] ;

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



62 V. OBUKHOVSKII, S. KORNEV, E. GETMANOVA

u1 (·) , u2 (·) ∈ DijC , v ∈ Rn; w1 (·) , w2 (·) ∈ Bik (zik) we have that

|A (t, u1 (t) , v)− A (t, u2 (t) , v)| < δ (4.6)

|B (t, u1 (t) , v, w1 (t))−B (t, u2 (t) , v, w2 (t))| < δ. (4.7)

Now, the set G̃ω (D) is covered by a finite numbers of sets Γijk, i = 1, · · · , s; j = 1, · · · , pi;
k = 1, · · · , ri of the form

Γijk = {B (·, x (·) , x′ (·) , y (·)) : x ∈ DijC , y ∈ Bik (zik)} .

Let us estimate the diameters of these sets. Taking arbitrary x1, x2 ∈ DijC and y1, y2 ∈
Bik (zik) and applying (4.6), (4.7) and condition (AB), for any t ∈ [0, a] we have

|B (t, x1 (t) , x
′
1 (t) , y1 (t))−B (t, x2 (t) , x

′
2 (t) , y2 (t))|

< |B (t, x1 (t) , x
′
1 (t) , zik (t))−B (t, x2 (t) , x

′
2 (t) , zik (t))|+ 2δ

≤ |B (t, x1 (t) , x
′
1 (t) , zik (t))−B (t, x1 (t) , x

′
2 (t) , zik (t))|+

+ |B (t, x1 (t) , x
′
2 (t) , zik (t))−B (t, x2 (t) , x

′
2 (t) , zik (t))|+ 2δ

≤ q |A (t, x1 (t) , x
′
1 (t))− A (t, x1 (t) , x

′
2 (t))|+ 3δ

≤ q |A (t, x1 (t) , x
′
1 (t))− A (t, x2 (t) , x

′
2 (t))|+

+ q |A (t, x2 (t) , x
′
2 (t))− A (t, x1 (t) , x

′
2 (t))|+ 3δ

< q (d+ ε) + qδ + 3δ.

Now, if q = 0 it means, by the arbitrariness of the choice of δ > 0 that α
(
G̃ (D)

)
= 0 and

then the triplet
(
f̃ , G̃, U

)
and therefore

(
f,G, U

)
is compact. Otherwise, let us take ε > 0

and δ > 0 so small that
qε+ (q + 3) δ < (1− q) d.

Then, q (d+ ε) + qδ + 3δ = µd where 0 < µ < 1 and, hence diamΓijk ≤ µd, implying that

α
(
G̃ (D)

)
≤ µα

(
f̃ (D)

)
.

Now, if we suppose that the trajectory function xω(·) for system (4.3)-(4.5) is found
already, the existence of the corresponding control yω(·) follows from the next reasonings.

For a given trajectory xω, let a measurable function h : Ω → C([0, a];Rn) be given as

h(ω)(t) = A(t, xω(t), x
′
ω(t)).

Further, consider the random map g : Ω× L1([0, a];Rm),

g(ω, y) = B(t, xω(t), x
′
ω(t),

∫ t

0

y(s)ds)

and a multimap V : Ω ⊸ L1([0, a];Rm), V (ω) = P(ω, xω). Applying again Theorem 3 from
[23] we conclude that the multimap V is measurable.

Since xω is a trajectory, we get that

h(ω) ∈ g(ω, V (ω)), ∀ω ∈ Ω.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



ON A RANDOM TOPOLOGICAL CHARACTERISTIC 63

From a version of the Filippov implicit function lemma (see Proposition 2.2.25 of [13]) we
conclude that there exists a measurable function v : Ω → L1([0, a];Rm) such that

h(ω) = g(ω, v(ω)), ∀ω ∈ Ω.

It is clear that we can set yω(·) = v(ω) as the desired control function.

The above arguments yield that the random coincidence index theory, developed in the
previous sections, can be applied to the study of the solvability of problem (4.3)-(4.5). For
example, the direct application of Theorem 3.4 yields the following assertion.
Proposition 4.3:
Under above conditions, suppose that the map A is odd: A (t,−u,−v) = A (t, u, v) for all
t ∈ [0, a]; u, v ∈ Rn and the set of functions x ∈ C1 ([0, a] ;Rn) satisfying the two-parameter
family of relations

A (t, x (t) , x′ (t)) = λB

(
t, x (t) , x′ (t) ,

∫ t

0

yω (s) ds

)
, t ∈ [0, a], λ ∈ [0, 1]; (4.8)

yω (τ) ∈ C (ω, τ, x (τ)) , a.e. s ∈ [0, a], ∀ω ∈ Ω; (4.9)
x (0) = x0, (4.10)

is a priori bounded. Then, there exists a random solution of problem (4.3)–(4.5).

ACKNOWLEDGEMENTS

The results of V. Obukhovskii and E. Getmanova were supported by the Russian Science
Foundation (project number 22-71-10008). The research of S. Kornev was supported by the
Ministry of Education of the Russian Federation within the framework of the state task in the
field of science (topic No FZGF-0640-2020-0009).

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Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)


	Introduction
	Preliminaries
	Nonlinear Fredholm operators
	Multivalued maps
	An Oriented Coincidence Index for Compact Triplets
	Random Multivalued Maps and Random Coincidence Points

	A Random Coincidence Index for Fundamentally  Restrictible Quadriples
	A Completely Fundamentally Restrictible Triplet
	A Random Coincidence Index

	An application: existence of a solution for a random feedback control system