Adv Syst Sci Appl 2025; 02:58–74
Published online at https://ijassa.ipu.ru.

A Cauchy Problem for Functional Inclusions
with Causal Operators in Banach Spaces

Garik Petrosyan*, Maria Soroka

Voronezh State University, Voronezh, Russia

Abstract: In this paper, we consider the Cauchy problem for functional inclusions containing
the sum of n causal single-valued operators and a multivalued causal operator in Banach spaces.
The peculiarity of single-valued operators is that, starting from the second term, each operator
is represented by an integral with a variable upper limit of the previous term. Such functional
inclusions generalize the Cauchy problem for semilinear differential equations and inclusions of
arbitrary order n, as well as a Cauchy-type problem in the case of inclusions and equations of
fractional order not exceeding n. To solve the problem, we will apply the theory of topological
degree for multivalued condensing mappings. To prove the existence of a solution, we will
construct a resolving multivalued operator in the space of continuous functions corresponding
to the problem. Based on the properties of the resolving operator, we will prove a theorem on the
existence of solutions to this problem.

Keywords: causal multioperator, functional inclusion, Cauchy type problem, differential
inclusion, fractional derivative, measure of non-compactness, fixed point,
topological degree, condensing multioperator

1. INTRODUCTION

The study of boundary value problems for differential equations and inclusions is a complex
and extremely important section of modern mathematical science, which has numerous
applications and currently attracts the attention of many scientists. Many physical, economic,
biological and engineering problems, primarily related to the flow of processes in dynamical
systems, lead to the need for new research in the field of differential equations and inclusions
in abstract spaces. In turn, the development of the theory of differential inclusions is due to
the fact that they are a convenient and natural tool for describing control systems of various
classes, systems with discontinuous characteristics studied in various sections of optimal
control theory, mathematical physics, radio physics, acoustics, etc. One of the best tools for
studying this kind of problem, provide multivalued and nonlinear analysis methods that stand
out as very powerful, efficient and useful.

Recently, the attention of many researchers (see [1], [2], [3], [4] and references therein)
has been attracted to generalizations of differential equations and inclusions, namely to the
class of functional equations and inclusions with causal operators. The term of a causal or
Volterra operator in the sense of A.N. Tikhonov (see [5]), was used in mathematical physics to
solve problems of differential equations, integro-differential equations, functional-differential
equations with a finite or infinite delay, integral equations of Volterra type, functional
equations of a neutral type, etc. (see, for example, [6]). The papers [7], [8], [9], [10], [11]
among others are devoted to the study of equations and inclusions with causal operators of

∗Corresponding author: garikpetrosyan@yandex.ru



A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 59

various types, theorems on the existence of solutions, description of qualitative properties of
solutions and various applications.

In this paper we study a Cauchy type problem in Banach spaces for various classes
of functional inclusions with causal operators. Basing on the topological degree theory for
condensing multimaps we prove a global theorem on the existence of trajectories for systems
governed by functional inclusions. As an application, we obtain generalizations of existence
theorems for solutions of a Cauchy type problem for semilinear differential inclusions an
arbitrary order n and semilinear differential inclusions of a fractional order n− 1 < q < n.

2. PRELIMINARIES

2.1. Measures of Noncompactness
We denote a Banach space by E and introduce the following notation:

• P (E) = {A ⊆ E : A ̸= ∅} is the collection of all non-empty subsets of E ;
• Pb(E) = {A ∈ P (E) : A is bounded} ;
• Pv(E) = {A ∈ P (E) : A is convex} ;
• C(E) = {A ∈ P (E) : A is closed} ;
• Cv(E) = Pv(E) ∩ C(E);
• K(E) = {A ∈ P (E) : A is compact} ;
• Kv(E) = Pv(E) ∩K(E).

Definition 2.1:
(See [12]). Let (A,≥) be a partially ordered set. A function β : Pb(E) → A is called the
measure of noncompactness (MNC) in E if for each Ω ∈ Pb(E) we have:

β(coΩ) = β(Ω),

where coΩ denotes the closure of the convex hull of Ω.
A measure of noncompactness β is called:

1) monotone, if for each Ω0,Ω1 ∈ Pb(E), from Ω0 ⊆ Ω1 follows β(Ω0) ≤ β(Ω1);
2) nonsingular, if for each a ∈ E and each Ω ∈ Pb(E) we have β({a} ∪ Ω) = β(Ω).

If A is a cone in a Banach space, the MNC β is called:

3) regular, if β(Ω) = 0 is equivalent to the relative compactness of Ω ∈ Pb(E);
4) real, if A is the set of all real numbers R with the natural ordering.

As the example of a real MNC obeying all above properties, we can consider the Hausdorff
MNC χ(Ω):

χ(Ω) = inf{ε > 0, for which Ω has a finite ε-net in E }.
As other examples, consider the measures of noncompactness defined in the space of

continuous functions C([a, b]; E) with values in the Banach space E :

(1) the modulus of fiber noncompactness:

φ(Ω) = sup
t∈[a,b]

χE(Ω(t)),

where χE is the Hausdorff MNC in E and Ω(t) = {y(t) : y ∈ Ω};
(2) the fading modulus of fiber noncompactness:

γ(Ω) = sup
t∈[a,b]

e−LtχE(Ω(t)),

where L > 0 is a given number;

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60 G. PETROSYAN, M. SOROKA

(3) the modulus of equicontinuity:

modC (Ω) = lim
δ→0

sup
y∈Ω

max
|t1−t2|≤δ

∥y (t1)− y (t2)∥ .

These measures of noncompactness satisfy all the above properties, except for the
regularity.

2.2. A Family of Cosine Operator Functions
Definition 2.2:
(See [13]). A family of bounded operators {C(t)}t∈R in a Banach spaceE is called a strongly
continuous family of cosine operator functions if:

(1) C(0) = I;
(2) C(s+ t) + C(s− t) = 2C(s)C(t) for all t, s ∈ R;
(3) t→ C(t)x is continuous for all x ∈ E.

The family of strongly continuous sine operator functions associated with the family of
cosine operator functions {C(t)}t∈R is the family of operators {S(t)}t∈R such that

S(t)x =

∫ t

0

C(s)xds, x ∈ E, t ∈ R.

The operator A is a generator a family of cosine operator functions {C(t)}t∈R if:

Ax =
d2

dt2
C(t)x|t=0,

for all x ∈ D(A) for which the last expression is well defined.

2.3. Multivalued Maps
Let X be a metric space and Y be a normed space. Let us recall some notations (see, for
example, [14], [15]).
Definition 2.3:
A multivalued map (multimap) F : X → P (Y ) is said to be upper semicontinuous (u.s.c.) at
a point x ∈ X, if for every open set V ⊂ Y such that F(x) ⊂ V, there exists a neighborhood
U(x) of x such that F(U(x)) ⊂ V.

Definition 2.4:
A multivalued map F : X → P (Y ) is called closed if its graph GF = {(x, y) : x ∈ X, y ∈
F(x)} is a closed subset of X × Y.

Definition 2.5:
A multivalued map F : X → P (Y ) is called quasicompact if its restriction to each compact
subset A ⊂ X is compact.
Definition 2.6:
For a given p ≥ 1, a multifunction G : [0, T ] → K(Y ) is called:

• Lp–integrable if it admits an Lp–Bochner integrable selection, i.e., there exists a
function g ∈ Lp ([0, T ];Y ) such that g(t) ∈ G(t) for a.e. t ∈ [0, T ];

• Lp–integrably bounded if there exists a function ξ ∈ Lp([0, T ]) such that

∥G(t)∥Y ≤ ξ(t)

for a.e. t ∈ [0, T ].

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A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 61

Definition 2.7:
A multimap F : X ⊆ E → K(E) is called condensing with respect to a MNC β (or β–
condensing) if for each bounded set Ω ⊆ X which is not relatively compact, we have:

β(F(Ω)) ̸≥ β(Ω).

Let D ⊂ E be a non-empty closed convex subset, V be a non-empty bounded open subset
of D, β is a monotone nonsingular MNC in E and F : V → Kv (D) be a u.s.c. β-condensing
map such that x /∈ F (x) for all x ∈ ∂V , where V and ∂V denote the closure and the boundary
of the set V in the relative topology of D.

In such a setting, the (relative) topological degree

degD
(
i−F , V

)
of the corresponding vector field i−F , satisfying the standard properties is well defined (see,
for example, [14], [15]). In particular, the condition

degD
(
i−F , V

)
̸= 0

implies that the fixed points set FixF = {x : x ∈ F(x)} is a nonempty subset of V.
Application of topological degree theory leads to the following fixed point principles,

which will be used in the that follows.
Theorem 2.1:
(See [14], Corollary 3.3.1). Let M be a convex closed bounded subset of E and F : M →
Kv(M) a β–condensing multimap, where β is a monotone nonsingular MNC in E . Then the
fixed point set FixF is non-empty.
Theorem 2.2:
(See [14], Theorem 3.3.4). Let V ⊂ D be a bounded open neighborhood of a point a ∈ V
and F : V → Kv(D) a u.s.c. β-condensing multimap, where β is a monotone nonsingular
MNC in E , satisfying the boundary condition

x− a /∈ λ(F(x)− a)

for all x ∈ ∂V and 0 < λ ≤ 1. Then FixF ̸= ∅ is a non-empty compact set.

2.4. Causal Multioperators
Let E be a separable Banach space. By Lp ([0, T ];E) , 1 ≤ p ≤ ∞, we denote the Banach
space of all Bochner summable functions f : [0, T ] → E with the usual norm.

For each subset N ⊂ Lp ([0, T ];E) and τ ∈ (0, T ) we define restriction N on [0, τ ] as

N |[0,τ ]= {f |[0,τ ]: f ∈ N}.

Definition 2.8:
A multivalued map Q : C ([0, T ];E) ⊸ Lp ([0, T ];E) is said to be a causal multioperator,
if for each τ ∈ (0, T ] and for every u, v ∈ C ([0, T ];E) the condition u |[0,τ ]= v |[0,τ ] implies
that Q(u) |[0,τ ]= Q(v) |[0,τ ] .

Let us give examples of causal multioperators.
Example 2.1:
We assume that the multimap F : [0, T ]× E → Kv (E) satisfies the following conditions:

(F1) for each ϕ ∈ E the multifunction F (·, ϕ) : [0, T ] → Kv (E) admits a measurable
selection;

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62 G. PETROSYAN, M. SOROKA

(F2) for a.e. t ∈ [0, T ] the multifunction F (t, ·) : E → Kv (E) is u.s.c.;
(F3) there exists a function α ∈ Lp

+[0, T ], 1 ≤ p ≤ ∞, such that

∥F (t, ϕ)∥E := sup{∥z∥E : z ∈ F (t, ϕ)} ≤ α(t)(1 + ∥ϕ∥E)
for a.e. t ∈ [0, T ] and ϕ ∈ E.

From above conditions (F1)− (F3) it follows that the multimap PF : C([0;T ];E) →
P (Lp([0, T ];E)), given in the following way

PF (x) = {f ∈ Lp([0, T ];E) : f(t) ∈ F (t, x(t)) for a.e. t ∈ [0, T ]}
is well defined (see, for example, [14], [15]). It is clear that the multioperator PF is causal.
Example 2.2:
Let F : [0, T ]× E → Kv(E) be a multimap satisfying conditions (F1)− (F3) from
Example 2.1. Suppose that {K(t, s) : 0 ≤ s ≤ t ≤ T} is a continuous (with respect to
the corresponding norm) family of bounded linear operators in E and m ∈ L1([0, T ];E)
is a given function. Consider the Volterra integral multioperator V : C ([0, T ];E) ⊸
L1 ([0, T ];E) defined as

V(u)(t) = m(t) +

∫ t

0

K(t, s)F (s, u)ds,

i.e.,

V(u) =
{
y ∈ L1 ([0, T ];E) : y(t) = m(t) +

∫ t

0

K(t, s)f(s)ds : f ∈ PF (u)

}
.

It is also clear that the multioperator V is causal.

3. CAUCHY TYPE PROBLEM FOR FUNCTIONAL INCLUSIONS
WITH THE CAUSAL OPERATORS

We will assume that the causal operator Q : C ([0, T ];E) → C (Lp ([0, T ];E)) satisfies the
following conditions:
(Q1) the operator Q is weakly closed in the following sense: conditions {un}∞n=1 ⊂

C ([0, T ];E) , {fn}∞n=1 ⊂ Lp ([0, T ];E) , 1 ≤ p ≤ ∞, fn ∈ Q(un), n ≥ 1, un → u0,

fn
L1

⇀ f0 implies f0 ∈ Q(u0);
(Q2) there exists a function α ∈ L∞

+ ([0, T ]) such that

∥Q (u) (t) ∥E ≤ α (t) (1 + ∥u(t)∥E) , for a.e. t ∈ [0, T ],

for all u ∈ C([0, T ];E);
(Q3) there exists a function ω : [0, T ]× R+ → R+ such that

(ω1) for all x ∈ R+ : ω(·, x) ∈ Lp
+([0, T ]), 1 ≤ p ≤ ∞;

(ω2) for a.e. t ∈ [0, T ] a function ω(t, ·) : R+ → R+ is continuous, nondecreasing and
quasihomogeneous in the sense that ω(t, λx) ≤ λω(t, x) for all x ∈ R+ and λ ≥ 0;

(ω3) for each bounded set ∆ ⊂ C ([0, T ];E) we have

χ (Q (∆) (t)) ≤ ω

(
t, sup

s∈[0,t]
φ (∆(s))

)
for a.e. t ∈ [0, T ],

where the set ∆(s) = {y(s) : y ∈ ∆} ⊂ E and φ is the modulus of fiber
noncompactness in C ([0, T ];E) .

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A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 63

Note that the condition (ω2) means that ω(t, 0) = 0 for a.e. t ∈ [0, T ] and as an example
of such a function we can consider ω(t, x) = k(t) · x, where k(·) ∈ Lp

+([0, T ]).

Consider a linear operator S : Lp([0, T ];E) → C([0, T ];E), which is causal in the sense
that for every τ ∈ (0, T ] and f, g ∈ Lp([0, T ];E) condition f(t) = g(t) for a.e. t ∈ [0, τ ]
implies (Sf) (t) = (Sg) (t) for all t ∈ [0, τ ]. Following [14], we impose the next conditions
on operator S :

(S1) for 1 ≤ p <∞ there exist D ≥ 0 such that

∥Sf(t)− Sg(t)∥pE ≤ D

∫ t

0

∥f(s)− g(s)∥pEds

for all f, g ∈ Lp([0, T ];E) and 0 ≤ t ≤ T ;
if p = ∞ there exist D1 ≥ 0 such that

∥Sf(t)− Sg(t)∥E ≤ D1

∫ t

0

∥f(s)− g(s)∥Eds

for all f, g ∈ L∞([0, T ];E) and 0 ≤ t ≤ T.
(S2) for an arbitrary compact set K ⊂ E and a sequence {fn}∞n=1 ⊂ Lp ([0, T ];E) , 1 ≤ p ≤

∞, such that {fn(t)}∞n=1 ⊂ K for all t ∈ [0, T ] the weak convergence fn
L1

⇀ f0 implies
Sfn → Sf0 in C([0, T ];E).

Also we suppose that S satisfies the relation:

(S3) (Sf) (0) = 0 for each function f ∈ Lp([0, T ];E).

Notice, that condition (S1) implies that the operator S satisfies the Lipschitz condition:

(S1′) ∥Sf − Sg∥C ≤ D∥f − g∥L1 .

Consider following important examples of special cases of the operator S.
(i) The operator M1 : L

1([0, T ];E) → C([0, T ];E) defined as

M1f(t) =

∫ t

0

eA(t−s)f(s)ds,

where a closed linear operator A : D (A) ⊂ E → E is the infinitesimal generator of a
uniformly bounded C0-semigroup {eAt}t≥0.

Remark 3.1:
The Laplace transform for a real-valued function f

L[f ](λ) =
∫ ∞

0

e−λsf(s)ds, λ ∈ R, λ > 0,

is a special cases of the operator S. Notice, that the inverse Laplace transform is a
causal operator also and satisfy conditions (S1)− (S3).

(ii) The operator M2 : L
1([0, T ];E) → C([0, T ];E) defined as

M2f(t) =

∫ t

0

(t− s)sh(At) f(s)ds,

where a closed linear operator A : D (A) ⊂ E → E is a generator of a uniformly
bounded family of strongly continuous cosine operator functions {ch(At)}t≥0.

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64 G. PETROSYAN, M. SOROKA

(iii) The operator L1 : L
p([0, T ];E) → C([0, T ];E), p > 1/q, defined as

L1f(t) =

∫ t

0

(t− s)q−1T1(t− s)f(s)ds, 0 < q < 1,

where

T1(t) = q

∫ ∞

0

θξq(θ)U(t
qθ)dθ, ξq(θ) =

1

q
θ−1− 1

qΨq(θ
−1/q),

Ψq(θ) =
1

π

∞∑
n=1

(−1)n−1θ−qn−1Γ(nq + 1)

n!
sin(nπq), θ ∈ R+,

andA : D(A) → E is a closed linear operator in E generating a uniformly bounded C0-
semigroup {U(t)}t≥0 .

(iv) The operator L2 : L
p([0, T ];E) → C([0, T ];E), p ≥ 1, defined as

L2f(t) =

∫ t

0

(t− s)q−1T2(t− s)f(s)ds, 0 < q ≤ 1,

where
T2(t) = q

∫ ∞

0

θξq(θ)S(t
qθ)dθ, ξq(θ) =

1

q
θ−1− 1

qΨq(θ
−1/q),

Ψq(θ) =
1

π

∞∑
n=1

(−1)n−1θ−qn−1Γ(nq + 1)

n!
sin(nπq), θ ∈ R+.

and A : D(A) ⊂ E → E is a closed linear operator in E generating a uniformly
bounded family of strongly continuous cosine operator functions {C(t)}t≥0 .

Let a closed linear operator A : D (A) ⊂ E → E be a generator of a uniformly bounded
family operator functions {G0(t)}t≥0. We suppose the following conditions:

(G1) For each t ∈ [0, T ], G0(·) are linear bounded operators, more precisely, for each x ∈ E

∥G0(t)x∥E ≤M ∥x∥E
where M = sup{∥G0(t)∥; t ∈ [0,+∞)}.

(G2) The operator function G0(·) are strongly continuous, i.e., functions t ∈ [0, T ] → G0(t)x
are continuous for each x ∈ E.

Consider the operator functions Gi, i = 1, 2, ..., n− 1 defined as

Gi(t) =

∫ t

0

Gi−1 (s)ds.

Lemma 3.1:
Under conditions (G1), (G2) the operator functions Gi, (i = 1, · · · , n− 1) possess the
following properties:

1) for each t ∈ [0, T ], Gi are linear bounded operators, more precisely, for each x ∈ E we
have

∥Gi(t)x∥E ≤M ∥x∥E
ti

i!
;

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A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 65

2) the operator functions tiGi(·) are strongly continuous, i.e., functions t ∈ [0, T ] →
tiGi(t)x are continuous for each x ∈ E.

Proof
1) For every fixed t ≥ 0 and for each x ∈ E, we prove the statement using the principle of
mathematical induction. Show it is true for i = 1. By using condition (G1) we have

∥G1(t)x∥E ≤
∫ t

0

∥G0(s)x∥E ds ≤M∥x∥E t.

Let us assume that

∥Gk(t)x∥E ≤M ∥x∥E
tk

k!

is true. Now, we prove that it is true for i = k + 1.

∥Gk+1(t)x∥E ≤
∫ t

0

∥Gk(s)x∥E ds ≤M∥x∥E
1

k!

∫ t

0

sk ds =M ∥x∥E
tk+1

(k + 1)!
.

2) For any x ∈ E, i = 1, · · · , n− 1, and any 0 ≤ t1 ≤ t2 we have

∥Gi(t1)x− Gi(t2)x∥E ≤
∫ t2

t1

∥Gi−1(s)x∥Eds ≤∫ t2

t1

M ∥x∥E
si−1

(i− 1)!
ds ≤M ∥x∥E

(t2 − t1)
i

i!
.

Then if t2 → t1 we have ∥Gi(t1)x− Gi(t2)x∥E → 0 for every x ∈ E.

By using Lemma 3.1 it is easy to see that the operator functions Gi, i = 0, 1, · · · , n, are
satisfy conditions (S1)− (S3).

Consider a system governed by a functional inclusion with causal operators Q and S, of
the following form:

x(t) ∈ G0(t)x0 + G1(t)x1 + · · ·+ Gn−1(t)xn−1 + S ◦ Q(x)(t), t ∈ [0, T ] (3.1)

x(0) = x0, x′(0) = x1, · · · , x(n−1)(0) = xn−1. (3.2)

Definition 3.1:
A function x ∈ C([0, T ];E) is called a mild solution to problem (3.1)-(3.2), if it satisfies
inclusion (3.1) and condition (3.2).

Consider the multioperator Γ : C([0, T ];E) ⊸ C([0, T ];E) defined as

Γ(x) = {x ∈ C([0, T ];E) : x(t) = G0(t)x0 + G1(t)x1 + · · ·+ Gn−1(t)xn−1 + S ◦ Q(x)}.

It is clear that if the function x is a fixed point of the multioperator Γ, then x is a solution
to the problem (3.1) - (3.2), therefore, our goal is to prove the existence of fixed point of the
multioperator Γ.

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66 G. PETROSYAN, M. SOROKA

Definition 3.2:
A sequence of functions {ξn} ⊂ Lp([0, T ];E) is called Lp–semicompact if it is Lp–integrably
bounded, i.e.,

∥ξn(t)∥E ≤ v(t) for a.e. t ∈ [0, T ] and for all n = 1, 2, ...,

where v ∈ Lp([0, T ]), and the set {ξn(t)} is relatively compact in E for a.e. t ∈ [0, T ].

Lemma 3.2:
(See. [14], Proposition 4.2.1.). Every Lp–semicompact sequence is weakly compact in
L1([0, T ];E).

We need the following properties of the multioperator S ◦ Q. Since for every 1 < p ≤
∞ : Lp([0, T ];E) ⊂ L1([0, T ];E), we can formulate a modification of Theorem 5.1.1 from
[14] in the following form.

Lemma 3.3:
Let S : Lp([0, T ];E) → C([0, T ];E) be an operator satisfying conditions (S1) and (S2).
Then for every Lp-semicompact sequence {fn}∞n=1 ⊂ Lp([0, T ];E), the sequence {Sfn}∞n=1

is relatively compact in C([0, T ];E) and, moreover, the weak convergence fn
L1

⇀ f0 implies
that Sfn → Sf0 in C([0, T ];E).

Theorem 3.1:
(See [4]). Let the multioperator Q satisfy conditions (Q1)–(Q3) and the operator S satisfy
(S1), (S2). Then the composition S ◦ Q : C([0, T ];E) ⊸ C([0, T ];E) is a u.s.c. multimap
with compact values.

Let us proceed to finding conditions under which the multioperator S ◦ Q will be
condensing with respect to a corresponding MNC. For this we need the following statements.

Lemma 3.4:
(See [4]). Let a sequence of functions {fn}∞n=1 ⊂ Lp([0, T ];E) be Lp-integrally bounded and
there exist a function υ ∈ Lp

+([0, T ]) such that

χ ({fn (t)}∞n=1) ≤ υ(t) for a.e. t ∈ [0, T ].

If an operator S satisfies conditions (S1) and (S2), then for 1 ≤ p <∞ we have

χ ({Sfn (t)}∞n=1) ≤
(
4pD

∫ t

0

υp(s)ds

)1/p

,

and for p = ∞

χ ({Sfn (t)}∞n=1) ≤ 2D1

∫ t

0

υ(s)ds,

where D,D1 are the constants from condition (S1).
Consider the measure of noncompactness ν in the space C([0, T ];E) with values in the

cone R2
+. On a bounded subset of Ω ⊂ C([0, T ];E) we define the values of ν as follows:

ν(Ω) = (γ (Ω) ,modC (Ω)) ,

where modC is the modulus of equicontinuity, γ is the fading modulus of fiber
noncompactness

γ(Ω) = sup
t∈[0,T ]

e−Ltχ(Ω(t)).

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A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 67

The constant L > 0 is chosen so that

max{q1, q2} < 1,

where

q1 = sup
t∈[0,T ]

(
4D1/p

(∫ t

0

e−pL(t−s)ωp (s, 1) ds

)1/p
)
,

q2 = sup
t∈[0,T ]

(
2D1

∫ t

0

e−L(t−s)ω (s, 1) ds

)
,

where the constants D,D1 are from condition (S1), ω is a function from condition (Q3).
It is easy to see that the MNC ν is monotone, nonsingular, and algebraically semi-additive.

It follows from the Arzela-Ascoli theorem that it is also regular.
Theorem 3.2:
Let a causal multioperator Q : C([0, T ];E) ⊸ Lp ([0, T ];E) satisfy conditions (Q2) and
(Q3) and for a causal operator S : Lp ([0, T ];E) → C ([0, T ];E) the conditions (S1)–(S3)
be satisfied. Then the multioperator Γ is ν-condensing.

Proof
By Lemma 3.1 it is suffices to prove the assertion of the theorem for the multioperator S ◦ Q.
Let Ω ⊂ C([0, T ];E) be a bounded set such that

ν (S ◦ Q (Ω)) ≥ ν (Ω) . (3.3)

Let us show that the set Ω is relatively compact.
Inequality (3.3) means that

γ({S ◦ Q (Ω)}) ≥ γ(Ω). (3.4)

Applying the condition (Q3) and by using the properties of the function ω, we obtain for
a.e. t ∈ [0, T ]

χ ({f(t) : f ∈ Q (Ω)}) ≤ ω

(
t, sup

s∈[0,t]
φ ({y(s) : y ∈ Ω})

)
= ω (t, φ ({y : y ∈ Ω})) =

ω
(
t, eLte−Ltφ ({y : y ∈ Ω})

)
≤ ω

(
t, eLtγ (Ω)

)
≤ ω

(
t, eLt

)
· γ (Ω) .

At first, we consider the case 1 ≤ p <∞. By Lemma 3.4 we have for each t ∈ [0, T ] :

χ ({Sf(t) : f ∈ Q (Ω)}) ≤
(
4pD

∫ t

0

ωp
(
s, eLs

)
ds · γp (Ω)

)1/p

≤

4D1/p

(∫ t

0

epLsωp (s, 1) ds

)1/p

· γ (Ω) .

Inequality (3.4) and the last inequality imply the following estimate

γ(Ω) ≤ sup
t∈[0,T ]

(
4D1/p

(∫ t

0

e−pL(t−s)ωp (s, 1) ds

)1/p
)
γ (Ω) = q1 · γ (Ω) ,

therefore
γ (Ω) = 0,

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



68 G. PETROSYAN, M. SOROKA

thus
φ (Ω) = 0

for all t ∈ [0, T ].
Let us turn to the case p = ∞. By Lemma 3.4 we have for each t ∈ [0, T ] :

χ ({S(f)(t) : f ∈ Q (Ω)}) ≤ 2D1

∫ t

0

ω
(
s, eLs

)
ds · γ (Ω) ≤ 2D1

∫ t

0

eLsω (s, 1) ds · γ (Ω) ;

Inequality (3.4) and the last inequality imply the following

γ(Ω) ≤ sup
t∈[0,T ]

(
2D1

∫ t

0

e−L(t−s)ω (s, 1) ds

)
γ (Ω) = q2 · γ (Ω) ,

therefore
γ (Ω) = 0,

thus
φ (Ω) = 0

for each t ∈ [0, T ].
Now we will show that the set Ω is equicontinuous. We take sequences {yn}∞n=1 ⊂ Ω, n ≥

1, and {fn}∞n=1, fn ∈ Q(yn). From conditions (Q2) and (Q3) it follows that the sequence
{fn}∞n=1 is Lp-semicompact, and therefore by Lemma 3.3 the sequence {Sfn}∞n=1 is relatively
compact. Hence

modC({Sfn}∞n=1) = 0.

Thus
ν (S ◦ Q (Ω)) = (0, 0),

but then it follows from the inequality (3.3) that

ν(Ω) = (0, 0),

and the last expression yields that the set Ω is relatively compact.

To prove the main theorem of the paper, we need the following statements, known as the
Gronwall - Bellman Lemma and the generalized Gronwall - Bellman Lemma (see [16]).
Lemma 3.5:
Let v(t) and f(t) be nonnegative continuous functions on the segment [a, b], moreover

v(t) ≤ c+

∫ t

a

f(s)v(s)ds, t ∈ [a, b],

where c is a positive constant. Then for each t ∈ [a, b] the inequality

v(t) ≤ ce
∫ t
a f(s)ds,

holds.
Lemma 3.6:
Let h(t), u(t) and v(t) be nonnegative functions integrable on [a, b] satisfying the inequality:

v(t) ≤ u(t) +

∫ t

a

h(s)v(s)ds, t ∈ [a, b].

Then the following inequality holds:

v(t) ≤ u(t) +

∫ t

a

e
∫ t
a h(θ)dθh(s)u(s)ds, t ∈ [a, b].

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 69

Theorem 3.3:
Let a causal multioperator Q : C([0, T ];E) → Cv(Lp([0, T ];E)), 1 ≤ p ≤ ∞, satisfy
conditions (Q1)–(Q3) and a linear causal operator S : Lp([0, T ];E) → C([0, T ];E) satisfy
conditions (S1)–(S3). Then the set Σ of all solutions to problem (3.1)-(3.2) is a non-empty
compact set.

Proof

Let us show that the set of all solutions x ∈ C([0, T ];E) of a one-parameter inclusion

x ∈ λΓ(x), λ ∈ [0, 1], (3.5)

is a priori bounded. We divide the proof into three cases: p = 1, 1 < p <∞, p = ∞.
Let p = 1, if a function x ∈ C([0, T ];E) satisfies condition (3.5), then by Lemma 3.1 for

each t ∈ [0, T ], we have the following estimates:

∥x(t)∥E ≤ ∥G0(t)x0∥E + ∥G1(t)x1∥E + · · ·+ ∥Gn−1(t)xn−1∥E +D

∫ t

0

∥f(s)∥Eds ≤

M∥x0∥E +Mt∥x1∥E + · · ·+M
tn−1

(n− 1)!
∥xn−1∥E +D

∫ t

0

∥f(s)∥Eds ≤

M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E +D

∫ t

0

∥f(s)∥Eds,

where f ∈ Q(x) and, therefore, by condition (Q2) :

∥f(s)∥E ≤ α(s)(1 + ∥x(s)∥E),

we have

∥x(t)∥E ≤M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E+

D

∫ t

0

α(s)(1 + ∥x(s)∥E)ds ≤M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E+

D∥α∥L∞T +D∥α∥L∞

∫ t

0

∥x(s)∥Eds.

The last expression is a non-decreasing function of t, so we get

∥x(t)∥E ≤M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E+

D∥α∥L∞T +

∫ t

0

D∥α∥L∞∥x(s)∥Eds.

This means that the function v(t) = ∥x(t)∥E satisfies the estimate

v(t) ≤M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E+

D∥α∥L∞T +

∫ t

0

D∥α∥L∞v(s)ds.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



70 G. PETROSYAN, M. SOROKA

Applying Lemma 3.5, we obtain the required a priori boundedness:

v(t) = ∥x(t)∥E ≤ UeD∥α∥L∞ = γ1,

where

U =M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E + ∥xn∥E +D∥α∥L∞T.

Then ∥x∥C = supt∈[0,T ] ∥x(t)∥E ≤ γ1.
For the case 1 < p <∞, we have

∥x(t)∥E ≤M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E +

(
D

∫ t

0

∥f(s)∥pEds
) 1

p ≤

M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E +

(
D

∫ t

0

αp(s)(1 + ∥x(s)∥E)pds
) 1

p ≤

M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E+(

D

∫ t

0

αp(s)ds+D

∫ t

0

αp(s)∥x(s)∥pEds
) 1

p ≤

M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E+(

D

∫ t

0

αp(s)ds
)1/p

+
(
D

∫ t

0

αp(s)∥x(s)∥pEds
) 1

p
.

Let us introduce the following notation:

c0 =M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E +D1/p∥α∥Lp ,

h(s) = D1/pα(s).

Then we get:

∥x(t)∥E ≤ c0 +

(∫ t

0

hp(s)∥x(s)∥pEds
)1/p

.

Let v(t) = ∥x(t)∥pE, then from the last inequality we obtain the estimate:

v(t) ≤ 2pcp0 + 2p
∫ t

0

hp(s)v(s)ds.

Now applying Lemma 3.6 to the last inequality, we get

v(t) = ∥x(t)∥pE ≤ 2pcp0

(
1 +

∫ t

0

e2
p
∫ t
0 hp(θ)dθhp(s)ds

)
.

Then we have the final estimate for 1 < p <∞ :

∥x(t)∥E ≤ 2c0
p

√
1 +

∫ t

0

e2
p
∫ t
0 hp(θ)dθhp(s)ds = γ2.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 71

Then ∥x∥C = supt∈[0,T ] ∥x(t)∥E ≤ γ2.
For the case p = ∞, in the same way as in the case p = 1, the following estimate holds:

∥x∥C ≤ U1e
D1∥α∥L∞ = γ3,

where

U1 =M∥x0∥E +MT∥x1∥E + · · ·+M
T n−1

(n− 1)!
∥xn−1∥E +D1∥α∥L∞T.

Now, if we take R ≥ max{γ1, γ2, γ3}, we can guarantee that the set V ⊂ C([0, T ];E),
given as

V = {x ∈ C([0, T ];E) : ∥x∥C < R},
contains all solutions of inclusion (3.5). Thus, the multioperator Γ satisfies on ∂V the
condition of Theorem 2.2 with a = 0, hence the set of its fixed points is nonempty and
compact.

4. EXAMPLES

We will need the following notion (see [17, 18]).
Definition 4.1:
The Gerasimov-Caputo fractional derivative of an order q ≥ 0 of a function g ∈
Cn([0, T ];E) is the function CDq

0g of the following form:

CDq
0g(t) =

1

Γ(n− q)

∫ t

0

(t− s)n−q−1g(n)(s) ds, n = [q] + 1,

where Γ is the Euler’s gamma-function

Γ(q) =

∫ ∞

0

xq−1e−xdx.

Notice, that the last definition includes the case, when q = n ∈ N.
Consider the following system governed by a differential inclusion in a separable Banach

space E :
CDq

0y (t) ∈ Ay (t) + F (t, y(t)) , t ∈ [0, T ] , n− 1 < q ≤ n, (4.6)

y (0) = y0, y′(0) = y1, ..., y(n−1)(0) = yn−1. (4.7)
Let a multimap F : [0, T ]× E → Kv(E) is such that:

(F1) for each ψ ∈ C([0, T ];E) the multifunction F (·, ψ) : [0, T ] → Kv (E) admits a
measurable selection;

(F2) for a.e. t ∈ [0, T ] the multimap F (t, ·) : E → Kv (E) is u.s.c.;
(F3) there exists a function α ∈ L∞

+ [0, T ] such that

∥F (t, ψ)∥E := sup{∥z∥E : z ∈ F (t, ψ)} ≤ α(t)(1 + ∥ψ∥E)

for a.e. t ∈ [0, T ] and for all ψ ∈ E;
(F4) there exists a function ωF : [0, T ]× R+ → R+ satisfying the conditions (ω1)-(ω3) such

that for each bounded set Ω ⊂ C([0, T ];E) we have

χ (F (t,Ω)) ≤ ωF (t, φ (Ω)) for a.e. t ∈ [0, T ].

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



72 G. PETROSYAN, M. SOROKA

The fact that the superposition multioperator PF : C([0, T ];E) ⊸ Lp([0, T ];E), p >
1/q, satisfies condition (Q1) can be verified as in paper [19]. Conditions (Q2) and (Q3)
for PF follow from (F3) and (F4), respectively.

If 0 < q ≤ 1 we will suppose that

(AI) A : D (A) ⊂ E → E is a linear closed operator generating a uniformly bounded C0-
semigroup {U(t)}t≥0.

Then

G0(t) =

∫ ∞

0

ξq(θ)U(t
qθ)dθ,

and a function y ∈ C([0, T ];E) is a mild solution to problem (4.6)–(4.7), if it can be
presented in the form:

y(t) = G0(t)y0 +

∫ t

0

(t− s)q−1T1(t− s)f(s)ds,

where f ∈ PF (y).
We can consider the relation (4.6) as a special case of functional inclusion (3.1) with

Q = PF , and the Cauchy type operator S = L1.
As a direct consequence of Theorem 3.3, we obtain the following result.

Theorem 4.1:
Suppose that conditions (AI), (F1)–(F4) hold and 0 < q ≤ 1. Then the set of solutions to
problem (4.6)-(4.7) is a non-empty compact subset of the space C([0, T ];E).

Now, let 1 < q ≤ 2. We will suppose that

(AII) A : D (A) ⊂ E → E be a closed linear operator in E generating a uniformly bounded
family of strongly continuous cosine operator functions {C(t)}t≥0 .

Then

G0(t) =

∫ ∞

0

ξq(θ)C(t
q/2θ)dθ, G1(t) =

∫ t

0

G0(s)ds,

and a function y ∈ C([0, T ];E) is a mild solution to problem (4.6)–(4.7), if it can be
presented in the form:

y(t) = G0(t)y0 + G1(t)y1 +

∫ t

0

(t− s)q/2−1T2(t− s)f(s)ds,

where f ∈ PF (y).
We can consider the relation (4.6) as a special case of functional inclusion (3.1) with

Q = PF , and the Cauchy type operator S = L2.
As a direct consequence of Theorem 3.3, we obtain the following result.

Theorem 4.2:
Suppose that conditions (AII), (F1)–(F4) hold and 1 < q ≤ 2. Then the set of solutions to
problem (4.6)-(4.7) is a non-empty compact subset of the space C([0, T ];E).

Finally, in the general case n− 1 < q < n, we will suppose that

(AIII) A : D (A) ⊂ E → E generating a uniformly bounded family operator functions
{G0(t)}t≥0 such that

(G01) G0(0) = I;

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



A CAUCHY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH CAUSAL OPERATORS... 73

(G02) the operator function G0(·) is strongly continuous, i.e., the function t→ G0(t)x, t ≥ 0
is continuous for each x ∈ E.

Then, the operator functions Gi, i = 1, 2, ..., n− 1, will be define as

Gi(t) =

∫ t

0

Gi−1 (s)ds.

A function y ∈ C([0, T ];E), is a mild solution to problem (4.6)–(4.7), if it can be
presented in the form:

y(t) = G0(t)y0 + G1(t)y1 + · · ·+ Gn−1(t)yn−1 + L−1
[
(λq − A)−1L [f ] (λ)

]
(t),

where f ∈ PF (y), L is a Laplace transform and λq ∈ ρ(A).
We can consider the relation (4.6) as a special case of functional inclusion (3.1) with

Q = PF , and the Cauchy type operator S = L−1 [(λq − A)−1L [f ] (λ)] .
As a direct consequence of Theorem 3.3, we obtain the following result.

Theorem 4.3:
Suppose that conditions (AIII), (F1)–(F4) hold. Then the set of solutions to problem (4.6)-
(4.7) is a non-empty compact subset of the space C([0, T ];E).

ACKNOWLEDGEMENTS

The work was supported by the Russian Science Foundation (project no. 23-71-10026).

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Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)


	Introduction
	Preliminaries
	Measures of Noncompactness
	A Family of Cosine Operator Functions
	Multivalued Maps
	Causal Multioperators

	Cauchy type problem for functional inclusions  with the causal operators
	Examples