Adv Syst Sci Appl 2024; 01:40–57
Published online at https://ijassa.ipu.ru.

A Cauchy Type Problem for Causal
Functional Inclusions in Banach Spaces

Garik Petrosyan*, Maria Soroka

Voronezh State University, Voronezh, Russia

Abstract: In this paper, we study a Cauchy type problem in Banach spaces for various classes
of functional inclusions with causal multioperators. Based 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 a Cauchy type problem for semilinear second order differential inclusions of this
type and semilinear differential inclusions of the fractional order 1 < q < 2.

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

1. INTRODUCTION

Recently, the attention of many researchers (see [2], [19], [20], [23] 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 [29]), was used in mathematical
physics to solve problems of differential equations, integro-differential equations, functional-
differential equations with finite or infinite delay, integral equations of Volterra type,
functional equations of neutral type, etc. (see, for example, [6]). The papers [5], [7], [8], [13]
among others are devoted to the study of equations and inclusions with causal multioperators
of various types, theorems on the existence of solutions, description of qualitative properties
of solutions and various applications.

In recent decades, the interest to the theory of fractional-order differential equations has
significantly increased, thanks to applications in various branches of applied mathematics,
physics, engineering, biology, economics, etc. (see, for example, monographs [17], [27]
papers [1], [4], [9], [21], etc.). The boundary value problems of various types for fractional
differential equations and inclusions were considered in the works [10], [15], [16], [24], [25].

In this paper, we study a Cauchy type problem in Banach spaces for various classes of
functional inclusions with causal multioperators. Based 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 second order semilinear differential
inclusions and semilinear differential inclusions of the fractional order 1 < q < 2. In this
paper we use standard notation, the symbol “⊸” denotes a multimap.

∗Corresponding author: garikpetrosyan@yandex.ru



A CAUCHY TYPE PROBLEM... 41

2. PRELIMINARIES

2.1. Measures of Noncompactness
We denote by E a Banach space 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 [3]). 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 it 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 E , 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 satisfying 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;
(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.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



42 G. PETROSYAN, M. SOROKA

2.2. Multivalued Maps
Let X be a metric space and Y be a normed space. Let us recall some notations (see, for
example, [14], [22]).

Definition 2.2:
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 2.3:
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 2.4:
A multivalued map F : X → P (Y ) is called quasicompact if its restriction to each compact
subset A ⊂ X is compact.

Definition 2.5:
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 ].

Definition 2.6:
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 defined (see, for
example, [14], [22]). 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 what follows.

Theorem 2.1:
( [14], Corollary 3.3.1). Let M be a convex closed bounded subset of E and F : M →
Kv(M) be a β–condensing multimap, where β is a monotone nonsingular MNC in E . Then
the fixed point set FixF is non-empty.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 43

Theorem 2.2:
( [14], Theorem 3.3.4). Let V ⊂ D be a bounded open neighborhood of a point a ∈ V and
F : V → Kv(D) be 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.

2.3. Family of Cosine Operator Functions
Definition 2.7:
(See [12], [18], [28]) A family of bounded operators {C(t)}t∈R in a Banach space E 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 generates 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 defined.

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 some 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;

(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.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



44 G. PETROSYAN, M. SOROKA

From the 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], [22]). 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 MULTIOPERATORS

We will assume that the causal operator Q : C ([0, T ];E) → C (Lp ([0, T ];E)) satisfies the
following conditions:
(Q1) 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 imply
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, 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) .

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 :

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 45

(S1) for 1 ≤ p <∞ there exists 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 = ∞ then there exists 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 the following important examples.

(i) Let a closed linear operator A : D (A) ⊂ E → E generates a family of strongly
continuous cosine operator functions {cos(At)}t≥0. The operator L : L1([0, T ];E) →
C([0, T ];E) defined as

Lf(t) =
∫ t

0

sin(At) f(s)ds

is a special case of the operator S.
Note that taking A = 0 we obtain, as a special case, the usual integral operator LI :
L1([0, T ];E) → C([0, T ];E),

LIf(t) =

∫ t

0

f(s)ds.

It is easy to verify the following statement.
Lemma 3.1:
The operator L satisfies conditions (S1)− (S3).

(ii) Consider the operator functions

G(t) =
∫ ∞

0

ξq(θ)C(t
qθ)dθ, K(t) =

∫ t

0

G(s)ds, T (t) = q

∫ ∞

0

θξq(θ)S(t
qθ)dθ,

where

ξq(θ) =
θ−1− 1

q

q
Ψq(θ

−1/q), Ψq(θ) =
∞∑
n=1

(−1)n−1θ−qn−1Γ(nq + 1)

πn!
sin(nπq), θ ∈ R+.

Lemma 3.2:
(see [30]). The operator functions G,K and T possess the following properties:

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



46 G. PETROSYAN, M. SOROKA

1) For each t ∈ [0, T ], G(t),K(t) and T (t) are linear bounded operators, more
precisely, for each x ∈ E we have

∥G(t)x∥E ≤M ∥x∥E , ∥K(t)x∥E ≤M ∥x∥E t, ∥T (t)x∥E ≤ M

Γ(2q)
∥x∥E t

q,

where M = sup{∥C(t)∥; t ∈ [0, T ]}.
2) the operator functions G(·),K(·) and tq−1T (·) are strongly continuous, i.e.,

functions t ∈ [0, T ] → G(t)x, t ∈ [0, T ] → K(t)x and t ∈ [0, T ] → tq−1T (t)x are
continuous for each x ∈ E.

By using Lemma 3.2 it is easy to see that the operator functions G,K and T are also
satisfy the conditions (S1)− (S3).

(iii) Let A : D(A) ⊂ E → E be a closed linear operator E generating a family of strongly
continuous cosine operator functions {C(t)}t≥0 . The operator G : Lp([0, T ];E) →
C([0, T ];E), p ≥ 1, defined as

Gf(t) =

∫ t

0

(t− s)q−1T (t− s)f(s)ds, 1 < q ≤ 2,

is a special case of the operator S.
Lemma 3.3:
The operator G satisfies conditions (S1)− (S3).

Proof
(S1) Let 1 ≤ p <∞. By using the Holder inequality, we get:

∥G(f)(t)−G(g)(t)∥E ≤
∫ t

0

(t− s)q−1 ∥T (t− s)(f(s)− g(s))∥E ds ≤

≤ M

Γ(2q)

[∫ t

0

(t− s)
(2q−1)p

p−1 ds

] p−1
p
[∫ t

0

∥f(s)− g(s)∥pE ds
] 1

p

.

Then

∥G(f)(t)−G(g)(t)∥pE ≤ D

∫ t

0

∥f(s)− g(s)∥pE ds,

where

D =

[
p− 1

2qp− 1

]p−1
MpT 2qp−1

Γp(2q)
.

For p = ∞, we have:

∥G(f)(t)−G(g)(t)∥E ≤
∫ t

0

(t− s)q−1 ∥T (t− s)(f(s)− g(s))∥E ds ≤

≤ MT q−1

Γ(2q)

∫ t

0

∥f(s)− g(s)∥E ds.

Then

∥G(f)(t)−G(g)(t)∥E ≤ D1

∫ t

0

∥f(s)− g(s)∥E ds,

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 47

where

D1 =
MT q−1

Γ(2q)
.

(S2) LetK ⊂ E be a compact set and a sequence {fn}∞n=1 ⊂ Lp ([0, T ];E) , 1 ≤ p ≤ ∞,

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

⇀ f0. Applying Lemma 3.2, we obtain:

χ

({
G (fn) (t)

})
≤
∫ t

0

(t− s)q−1χE({T (t− s) fn})ds = 0.

This means that the sequence {G (fn) (t)}∞n=1 ⊂ E is relatively compact for each t ∈
[0, T ].

From the other side, if we take ϵ > 0 and t1, t2 ∈ [0, T ] such that 0 < t1 < t2 ≤ T , then
for each fn, we have: ∥∥∥∥∥G (fn) (t2)−G (fn) (t1)

∥∥∥∥∥
E

=

∥∥∥∥∥
∫ t2

0

(t2 − s)q−1 T (t2 − s) fn(s)ds−
∫ t1

0

(t1 − s)q−1 T (t1 − s) fn(s)ds

∥∥∥∥∥
E

≤

∥∥∥∥∥
∫ t2

t1

(t2 − s)q−1 T (t2 − s) fn(s)ds

∥∥∥∥∥
E

+

+

∥∥∥∥∥
∫ t1

0

(
(t2 − s)q−1 T (t2 − s)− (t1 − s)q−1 T (t1 − s)

)
fn(s)ds

∥∥∥∥∥
E

= Z1 + Z2,

where

Z1 =

∥∥∥∥∥
∫ t2

t1

(t2 − s)q−1 T (t2 − s) fn(s)ds

∥∥∥∥∥
E

,

Z2 =

∥∥∥∥∥
∫ t1

0

(
(t2 − s)q−1 T (t2 − s)− (t1 − s)q−1 T (t1 − s)

)
fn(s)ds

∥∥∥∥∥
E

.

Since {fn(t)}∞n=1 ⊂ K then there exists a constant N > 0, such that ∥fn(t)∥E ≤ N.
By using Lemma 3.2 and condition (F2), for each ϵ > 0 we can choose δ1 > 0 such that

|t2 − t1| < δ1, implies the following estimate:

Z1 ≤
MN(t2 − t1)

2q

Γ (2q + 1)
<
ϵ

2
.

Taking into account that the family of operators tq−1T (t) is strongly continuous, for each
ϵ > 0 we can choose γ = γ(ϵ) > 0 and δ2 > 0 such that the inequality |t2 − t1| < δ2 implies∥∥(t2 − s)q−1 T (t2 − s)x− (t1 − s)q−1 T (t1 − s)x

∥∥
E
< γ, x ∈ E,

we get the following estimate:
Z2 ≤ γT <

ϵ

2
.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



48 G. PETROSYAN, M. SOROKA

Therefore, for each ϵ > 0 we may choose δ = min {δ1, δ2} such that∥∥∥∥∥G (fn) (t2)−G (fn) (t1)

∥∥∥∥∥
E

≤ Z1 + Z2 <
ϵ

2
+
ϵ

2
= ϵ.

So, the sequence {G (fn)} is equicontinuous. By the Arzela-Ascoli theorem, we conclude
that the sequence {G (fn)} ⊂ C ([0, T ] ;E) is relatively compact.

From (S1) it follows that the operator G is linear and bounded, therefore it is continuous
with respect to the topology of weak sequential convergence, hence from the weak
convergence fn ⇀ f0 in L1([0, T ];E) implies G(fn)⇀ G(f0). Since the sequence {G(fn)}
is relatively compact, we conclude that G(fn) → G(f0) in C([0, T ];E).

The satisfaction of the condition (S3) is obvious.

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

x(t) ∈ G(t)x0 +K(t)x1 + S ◦ Q(x), t ∈ [0, T ] (3.1)

x(0) = x0, x′(0) = x1, (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 the
conditions:

(1) the function x ∈ C([0, T ];E) and it satisfies inclusion (3.1);
(2) x(0) = x0, x

′(0) = x1.

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

Γ(x) = {x ∈ C([0, T ];E) : x(t) = G(t)x0 +K(t)x1 + 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 a fixed point of
the multioperator Γ.
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 3.4:
(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 3.5:
Let S : Lp([0, T ];E) → C([0, T ];E) be an operator satisfying the 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).

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 49

Theorem 3.1:
(See [2]). Let a multioperator Q satisfy the conditions (Q1)–(Q3) and a 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 find conditions for the multioperator S ◦ Q to be condensing with respect to a
corresponding MNC. For this we need the following statements.

Lemma 3.6:
(See [2]) Let a sequence of functions {fn}∞n=1 ⊂ Lp([0, T ];E) be Lp-integrally bounded and
there exists 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 the 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 the fiber
noncompactness

γ(Ω) = sup
t∈[0,T ]

e−Ltχ(Ω(t)).

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 the condition (S1), ω is a function from the 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 the 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.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



50 G. PETROSYAN, M. SOROKA

Proof
By Lemma 3.2, it 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]
χ ({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) .

First, we consider the case 1 ≤ p <∞. By Lemma 3.6 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

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,

and
φ (Ω) = 0

for all t ∈ [0, T ].
Let us turn to the case p = ∞. By Lemma 3.6 we have for each t ∈ [0, T ] :

χ ({Sf(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,

and
φ (Ω) = 0

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 51

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 the conditions (Q2) and (Q3) it follows that the sequence
{fn}∞n=1 is Lp-semicompact, and therefore by Lemma 3.5 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.
Lemma 3.7:
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 3.8:
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].

Theorem 3.3:
Let a causal multioperator Q : C([0, T ];E) → Cv(Lp([0, T ];E)), 1 ≤ p ≤ ∞, satisfy the
conditions (Q1)–(Q3) and a linear causal operator S : Lp([0, T ];E) → C([0, T ];E) satisfy
the 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 3.2

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

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



52 G. PETROSYAN, M. SOROKA

∥x(t)∥E ≤ ∥G(t)x0∥E + ∥K(t)x1∥E +D

∫ t

0

∥f(s)∥Eds ≤

M∥x0∥E +Mt∥x1∥E +D

∫ t

0

∥f(s)∥Eds ≤M∥x0∥E +MT∥x1∥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).

Then

∥x(t)∥E ≤M∥x0∥E +MT∥x1∥E +D

∫ t

0

α(s)(1 + ∥x(s)∥E)ds ≤

M∥x0∥E +MT∥x1∥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 +D∥α∥L∞T +

∫ t

0

D∥α∥L∞∥x(t)∥Eds.

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

v(t) ≤M∥x0∥E +MT∥x1∥E +D∥α∥L∞T +

∫ t

0

D∥α∥L∞v(s)ds.

Applying Lemma 3.7, we obtain the required a priori boundedness:

v(t) = ∥x(t)∥E ≤ UeD∥α∥L∞ = γ1,

where
U =M∥x0∥E +MT∥x1∥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 +
(
D

∫ t

0

∥f(s)∥pEds
)1/p

≤M∥x0∥E +MT∥x1∥E +
(
D

∫ t

0

αp(s)(1 + ∥x(s)∥E)pds
)1/p

≤M∥x0∥E +MT∥x1∥E +
(
D

∫ t

0

αp(s)ds+D

∫ t

0

αp(s)∥x(s)∥pEds
)1/p

≤

≤M∥x0∥E +MT∥x1∥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 +D1/p∥α∥Lp ,

h(s) = D1/pα(s).

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 53

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 3.8 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.

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 +D1∥α∥L∞T.

Now, if we take R ≥ max{γ1, γ2, γ3}, then 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 non-empty and
compact.

4. CAUCHY TYPE PROBLEMS FOR SEMILINEAR
DIFFERENTIAL INCLUSIONS

4.1. A Cauchy Type Problem for a Second Order Semilinear Differential Inclusions
Consider the following system governed by a differential inclusion in a separable Banach
space E :

y′′ (t) ∈ Ay (t) + F (t, y(t)) , t ∈ [0, T ] , (4.6)

y(0) = y0, y′(0) = y1, (4.7)
where F : [0, T ]× E → Kv(E) is a multivalued map, y0, y1 ∈ E are given functions.
Suppose that

(A) A : D (A) ⊂ E → E generates a family of strongly continuous cosine operator
functions {cos(At)}t≥0.

A multimap F : [0, T ]× E → Kv(E) is such that:

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

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



54 G. PETROSYAN, M. SOROKA

(F2) for a.e. t ∈ [0, T ] the multimap F (t, ·) : E → Kv (E) is upper semicontinuous (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 ].

Notice that when q = 2 :

G(t) = cos(At), K(t) = sin(At), T (t) =
sin(At)

t
.

In accordance with [11], a function y ∈ C([0, T ];E) is a mild solution to problem (4.6)-(4.7),
if it can be represented in the form:

y(t) = cos(At)y0 + sin(At)y1 +

∫ t

0

sin(A(t− s))f(s)ds,

where f ∈ PF (y), PF is a superposition multioperator (see Example 2.1).
The fact that the superposition multioperator PF : C([0, T ];E) ⊸ L1([0, T ];E) satisfies

the condition (Q1) can be verified by Lemma 5.1.1 from [14]. Conditions (Q2) and (Q3)
for PF follow from (F3) and (F4), respectively. Taking into account Lemma 3.1, we can
consider relation (4.6) as a special case of functional inclusion (3.1) with Q = PF , and S = L
is the Cauchy operator.

As a direct consequence of Theorem 3.3, we obtain the following result.
Theorem 4.1:
Suppose that the conditions (A), (F1)–(F4), hold true. Then the set of solutions to problem
(4.6)-(4.7) is a non-empty compact subset of the space C([0, T ];E).

4.2. A Cauchy Type Problem for Fractional Semilinear Differential Inclusions
Let us recall the notion of the Caputo fractional derivative.
Definition 4.1:
The Caputo fractional derivative of the order q ∈ (1, 2) of a function g ∈ C2([0, T ];E) is the
function CDq

0g of the following form:

CDq
0g(t) =

1

Γ(2− q)

∫ t

0

(t− s)1−qg′′(s) ds,

where Γ is Euler’s gamma-function

Γ(q) =

∫ ∞

0

xq−1e−xdx.

Consider the following system governed by a functional differential inclusion in a
separable Banach space E :

CDq
0y (t) ∈ Ay (t) + F (t, y(t)) , t ∈ [0, T ] , (4.8)

y (0) = y0, y′(0) = y1. (4.9)
Suppose that

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 55

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

Assume also that a multimap F : [0, T ]× E → Kv(E) satisfies the conditions (F1)–
(F4) from section 4.1.

A function y ∈ C([0, T ];E) is a mild solution to problem (4.8)–(4.9), if it can be
presented in the form:

y(t) = G(t)y0 +K(t)y1 +

∫ t

0

(t− s)q−1T (t− s)f(s)ds,

where f ∈ PF (y).
The fact that the superposition multioperator PF : C([0, T ];E) ⊸ Lp([0, T ];E), p > 1/q

satisfies the condition (Q1) can be verified as in the paper [26]. Conditions (Q2) and (Q3)
for PF follow from (F3) and (F4), respectively. Taking into account Lemma 3.3, we can
consider the relation (4.8) as a special case of functional inclusion (3.1) with Q = PF , and
S = G is the Cauchy type operator.

As a direct consequence of Theorem 3.3, we obtain the following result.
Theorem 4.2:
Suppose that the conditions (A), (F1)–(F4) hold true. Then the set of solutions to problem
(4.8)-(4.9) 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).

REFERENCES

1. Afanasova, M., Liou, Y. Ch., Obukhoskii, V., Petrosyan, G. (2019). On controllability
for a system governed by a fractional-order semilinear functional differential inclusion
in a Banach space, Journal of Nonlinear and Convex Analysis, 20(9), 1919–1935.

2. Afanasova, M., Obukhovskii, V., Petrosyan G. (2021). A controllability problem for
causal functional inclusions with an infinite delay and impulse conditions, Advances in
Systems Science and Applications, 21(3), 40–62.

3. Ahmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.
(1992). Measures of Noncompactness and Condensing Operators. Boston–Basel–
Berlin, Birkhauser.

4. Appell, J., Lopez, B., Sadarangani, K. (2018). Existence and uniqueness of solutions for
a nonlinear fractional initial value problem involving Caputo derivatives, J. Nonlinear
Var. Anal., 2, 25–33.

5. Bulgakov, A.I., Maximov, V.P. (1981). Functional and functional-differential inclusions
with Volterra operators, Differential equations, 17(8), 1362–1374.

6. Corduneanu, C. (2002). Functional Equations with Causal Operators. Stability and
Control: Theory, Methods and Applications. London, UK: Taylor and Francis.

7. Drici, Z., McRae, F.A., Vasundhara Devi, J. (2005). Differential equations with causal
operators in a Banach space, Nonlinear Anal., 62(2), 301–313.

8. Drici, Z., McRae, F.A., Vasundhara Devi, J. (2006). Monotone iterative technique for
periodic boundary value problems with causal operators, Nonlinear Anal., 64(6), 1271–
1277.

9. Gomoyunov, M.I. (2018). Fractional derivatives of convex Lyapunov functions and
control problems in fractional order systems. Fract. Calc. Appl. Anal., 21, 1238–1261.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



56 G. PETROSYAN, M. SOROKA

10. Gomoyunov, M.I. (2019). Approximation of fractional order conflict-controlled
systems, Prog. Fract. Differ. Appl., 5, 143–155.

11. Hartman, P. (1964). Ordinary Differential Equations. New York, NY: John Wiley and
Sons.

12. He, J.W., Liang, Y., Ahmad, B., Zhou, Y. (2019). Nonlocal Fractional Evolution
Inclusions of Order α ∈ (1, 2), Mathematics, 7(2), 1–17.

13. Jankowski, T. (2008). Boundary value problems with causal operators, Nonlinear Anal.,
68(12), 3625–3632.

14. Kamenskii, M., Obukhovskii, V., Zecca, P. (2001). Condensing Multivalued Maps
and Semilinear Differential Inclusions in Banach Spaces. Berlin, Germany: Walter de
Gruyter.

15. Kamenskii, M., Obukhoskii, V., Petrosyan, G., Yao, J.-C. (2019). On a Periodic
Boundary Value Problem for a Fractional Order Semilinear Functional Differential
Inclusions in a Banach Space, Mathematics, 7(12), 5–19.

16. Kamenskii, M., Obukhoskii, V., Petrosyan, G., Yao, J.-C. (2021). On the Existence of
a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space,
Mathematics, 9(2), 136–154.

17. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. (2006). Theory and Applications
of Fractional Differential Equations. Amsterdam, Netherlands: North-Holland
Mathematics Studies, Elsevier Science B.V.

18. Kostin, V.A., Kostin, D.V., Kostin, A.V. (2019). Operator cosine functions and boundary
value problems, Reports of the Academy of Sciences, 486(5), 531–536.

19. Kulmanakova, M.M., Ulyanova, E.L. (2019). O razreshimosti kauzalnih funkcionalnih
vkluchenii s beskonechnim zapazdivaniem [On the solvability of causal functional
inclusions with infinite delay]. Tambov University Reports. Series: Natural and
Technical Sciences, 24(127), 293–315, [in Russian].

20. Lupulescu, V. (2008). Causal functional differential equations in Banach spaces,
Nonlinear Anal., 69(12), 4787–4795.

21. Mainardi, F. (1994). On the Initial Value Problem for the Fractional Diffusion-Wave
Equation. In Rionero, S. and Ruggeri, T., Eds., Waves and Stability in Continuous Media,
World Scientific, Singapore, 246–251.

22. Obukhovskii, V., Gel’man, B. (2020). Multivalued Maps and Differential Inclusions.
Elements of Theory and Applications. Hackensack, NJ: World Scientific Publishing Co.
Pte. Ltd.

23. Obukhovskii, V., Zecca, P. (2011). On certain classes of functional inclusions with
causal operators in Banach spaces, Nonlinear Anal., 74(8), 2765–2777.

24. Petrosyan, G. (2021). Antiperiodic boundary value problem for a semilinear differential
equation of fractional order, The Bulletin of Irkutsk State University. series:
Mathematics, 34, 51–66.

25. Petrosyan, G. (2020). On antiperiodic boundary value problem for a semilinear
differential inclusion of fractional order with a deviating argument in a Banach space,
Ufa Mathematical Journal, 12(3), 69–80.

26. Petrosyan, G.G. (2015). Teorema o slaboi zamknutosti superpozicionnogo multiopera-
tora [A theorem on the weak closure of superposition multioperator], Tambov University
Reports. Series: Natural and Technical Sciences, 20(5), 1355–1358, [in Russian].

27. Podlubny, I. (1999). Fractional Differential Equations. San Diego, CA: Academic Press.
28. Travis, C.C., Webb, G.F. (1978). Cosine families and abstract nonlinear second order

differential equations, Acta Math.Hungar, 32, 75–96.
29. Tychonoff, A.N. (1938). Sur les equations fonctionnelles de Volterra et leurs

applications a certains problemes de la physique mathematique, Bulletin de I’Universite
d’etat de Moscou. Ser. internationale. Section A. Mathematiques et mecanique, 1(8),
1–25, [in French].

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



A CAUCHY TYPE PROBLEM... 57

30. Zhou, Y., He, J.W. (2019). New results on controllability of fractional evolution systems
with order α ∈ (1, 2), Evol. Eq. Control Theory, 10(3), 491–509.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)


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

	Cauchy type problem for functional inclusions  with the causal multioperators
	Cauchy type problems for semilinear  differential inclusions
	 A Cauchy Type Problem for a Second Order Semilinear Differential Inclusions
	 A Cauchy Type Problem for Fractional Semilinear Differential Inclusions