Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 36, pp. 1–9.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.36

EXISTENCE OF PSEUDOSOLUTIONS FOR DYNAMIC

FRACTIONAL DIFFERENTIAL EQUATIONS

ANETA SIKORSKA-NOWAK

Abstract. In this article, we consider the existence of pseudosolutions for

boundary value problem for fractional differential equations of the form

C
T ∆αx(t) = f(t, x(t)), for t ∈ Ia = [0, a] ∩ T,

x(0) = x0, x0 ∈ E,

where C
T ∆αx(t), α ∈ (0, 1] denotes the Caputo fractional derivative, T denotes

a time scale, and the function f is weakly-weakly sequentially continuous with

values in a Banach space E and satisfies some boundary conditions and con-
ditions expressed in terms of measures of weak non-compactness.

1. Introduction

The research on fractional calculus and fractional differential equations in Ba-
nach spaces, with a special focus on weak topology initiated in 2005 by Salem
and his team (see [23] for Riemann-Liouville type fractional calculus and [22] for
Hadamard type), marked the beginning of a new era in this field of mathematics.
The publication of articles [22, 23] triggered significant interest, as evidenced by
numerous citations such as [1, 4, 6, 10, 11, 16, 17, 19, 21, 24, 25, 26], leading to
the development of a series of studies on initial and boundary value problems for
various types of fractional differential equations. The introduction of fractional
derivatives on time scales, allowing for the simultaneous modeling of discrete and
continuous phenomena, provided additional flexibility in modeling phenomena that
do not change in a linear manner or at a constant rate. These mathematical tools
have found applications in many fields, from physics and engineering to control
theory and quantum mechanics, opening up new possibilities in financial market
modeling and population dynamics. In 1967, the Italian mathematician Caputo
introduced the differential operator known as the Caputo operator, allowing for a
deeper understanding of and solutions to problems related to viscoelasticity using
fractional derivatives. The relationship between the Caputo Fractional Derivative
and other fractional derivatives, such as Riemann-Liouville or Atangana-Baleanu,
highlighted the significance of generalized Mittag-Leffler functions in mathemat-
ical modeling. By utilizing specific mathematical models, it became possible to

2020 Mathematics Subject Classification. 34K40, 34K42, 34A08, 34G20.
Key words and phrases. Fractional differential equations; fixed point; time scales;

Caputo fractional derivative; delta HKP integral.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted May 20, 2024. Published June 20, 2024.

1



2 A. SIKORSKA-NOWAK EJDE-2024/36

enhance outcomes and solve problems previously considered difficult or impossible
to address, opening new horizons in the theoretical and applied aspects of frac-
tional calculus. In this paper, we consider the existence of a pseudosolution for the
boundary value problem for fractional differential equations of the form

C
T∆

αx(t) = f(t, x(t)), for t ∈ Ia = [0, a] ∩ T,

x(0) = x0, x0 ∈ E,
(1.1)

where C
T∆

αx(t), α ∈ (0, 1] is the Caputo fractional derivative, T denotes a time
scale. We assume that the function f is weakly-weakly sequentially continuous with
values in a Banach space and satisfies some regularity conditions expressed in terms
of the De Blasi measure of weak noncompactness. We introduce a weakly sequen-
tially continuous operator associated with an integral equation that is equivalent
to the initial problem. There exist many important examples of mappings that are
weakly sequentially continuous but not weakly continuous. The relations between
weakly sequentially continuous and weakly continuous mappings are studied by Ball
[5]. Adopting the fixed point theorem for weakly sequentially continuous mappings
given by Kubiaczyk [14], and the properties of measures of weak noncompactness,
we are able to study the existence results for the problem.

2. Preliminaries

Let (E, ∥ · ∥) be a Banach space and let E∗ be the dual space. Denote by
(C(Ia, E), ω) the space of all continuous functions from Ia to E endowed with
the topology σ(C(Ia, E), C(Ia, E)∗), and by Crd(Ia, E) denote the space of all rd-
continuous functions from the time scale interval Ia to E. By µ∆ we denote the
Lebesgue measure on time scale T . For a precise definition and basic properties of
this measure we refer the reader to [8].

We now gather some well-known definitions and results from the literature, which
we will use throughout this article.

I. To enable the reader to understand the so-called dynamic equations and to
follow this paper easily, we present some preliminary definitions and notations of
time scales which are very common in the literature (see [2, 3, 7, 12, 13] and
references therein). A time scale T is a nonempty closed subset of real numbers
R, with the subspace topology inherited from the standard topology of R. By an
interval we mean the time scale interval

Ia = [0, a] ∩ T = {t ∈ T : 0 ≤ t ≤ a} = [0, a]T .

Definition 2.1. The forward jump operator σ : T → T and the backward jump
operator ρ : T → T as σ(t) = inf{s ∈ T : s > t} and ρ(t) = sup{s ∈ T : s < t},
respectively. We put inf ∅ = inf T (i.e. ρ(m) = m if T has a minimum m).

The jump operators σ and ρ allow the classification of points in time scale in the
following way: t is called right dense, right scattered, left dense, left scattered, dense
and isolated if σ(t) = t, σ(t) > t, ρ(t) = t, ρ(t) < t, ρ(t) = t = σ(t) respectively.

Definition 2.2. We say that is right-dense continuous (rd-continuous) if k is con-
tinuous at every right-dense point t ∈ T and lims→t− k(s) exists and is finite at
every left-dense point t ∈ T .



EJDE-2024/36 EXISTENCE OF PSEUDOSOLUTIONS 3

Definition 2.3. Fix t ∈ T . Let f : Ia → E. Then we define ∆-derivative of f by

f∆(t) = lim
s→t

f(σ(t))− f(s)

σ(t)− s
.

The function f is called ∆-differentiable on T , if for each t ∈ T there exists f∆(t).

Note that

(1) f∆ = f ′ is the usual derivative if T = R,
(2) f∆ = ∆f , is the usual forward difference operator if T = Z,
(3) f∆ = Dqf is the q-derivative if T = qN0 = {qt : t ∈ N0, q > 1}.

Hence, the time scale allows us to consider the unification of differential, difference
and q-difference equations as particular cases. However, our results also hold for
more exotic time scales, which appear in fields such as mathematical biology or
economics (see [7], for instance).

II. As in classical case, we need to introduce vector - valued Henstock-Kurzweil
∆-integrals. Definitions and basic properties of non absolute integrals were pre-
sented in [9]. We will use the notation η(t) := σ(t) − t(t) where η is called the
graininess function and v(t) := t− ρ(t), where v is called the left - graininess func-
tion. We say that δ = (δL, δR) is a ∆-gauge for time scale interval [a, b] provided
δL(t) > 0 on (a, b], δR(t) > 0 on [a, b), δL(t) ≥ 0, δR(t) ≥ 0 and δR(t) ≥ η(t) for all
t ∈ [a, b). We say that a partition D for a time scale interval [a, b] given by

D = {a = t0 ≤ ξ1 ≤ t1 ≤ · · · ≤ tn−1 ≤ ξn ≤ tn = b}
with ti > ti−1, for 1 ≤ i ≤ n and ti, ξi ∈ T is δ-fine if ξi − δL(ξi) ≤ ti−1 < ti ≤
ξi + δR(ξi), for 1 ≤ i ≤ n.

Definition 2.4. A function f : [a, b] → E is the Henstock-Kurzweil ∆-integrable
on [a, b] (HK ∆-integrable in short) if there exists a function F : [a, b] → E, defined
on the subintervals of [a, b], satisfying the following property: given ϵ > 0 there
exists a positive function δ on [a, b] such that D = {[u, v], ξ} is δ-fine division of a
[a, b], we have

∥
∑
D

f(ξ)(v − u)− (F (v)− F (u))∥ < ϵ

Definition 2.5. A function f : Ia → E is Henstock-Kurzweil-Pettis ∆-integrable
(HKP ∆-integrable for short) if

(1) for all x∗ ∈ E∗, x∗f is Henstock-Kurzweil ∆-integrable on Ia,

(2) forall t ∈ Ia and all x∗ ∈ E∗, x∗g(t) = (Delta-HK)
∫ t

0
x∗f(s)∆s.

The function g will be called a primitive of f and by g(t) = (Delta-HK))
∫ t

0
f(s)∆s

we will denote the Henstock-Kurzweil-Pettis ∆-integral of fon the interval Ia.

In [9] the author give examples of Henstock-Kurzweil-Pettis ∆-integrable func-
tions which are not integrable in the sense of Pettis and Henstock-Kurzweil on time
scales.

Theorem 2.6. Suppose that f, fn : [a, b] → E, n = 1, 2, . . . are HKP∆-integrable
functions. Let Fn be a primitive of fn. If one assumes that:

(1) for all x∗ ∈ E∗, x∗fn(x) → x∗f(x) µ∆ almost everywhere on Ia,
(2) for all x∗ ∈ E∗ the family G = {x∗Fn : n = 1, 2, . . . } is uniformly ACG∗

on Ia (i.e., weakly uniformly ACG∗ on Ia),
(3) for each x∗ ∈ E∗ the set G is equicontinuous on Ia



4 A. SIKORSKA-NOWAK EJDE-2024/36

then f is ∆-HKP integrable on Ia and
∫ t

0
fn(s)∆s tends weakly in E to

∫ t

0
f(s)∆s

for each t ∈ Ia.

Theorem 2.7 ((Mean Value Theorem). For each ∆-subinterval [c, d] ⊂ [a, b], if

the integral (∆-HKP)
∫ d

c
y(s)∆s exists, then we have

(∆-HKP)

∫ d

c

y(s)∆s ∈ µ∆([c, d]) · convy([c, d]),

where convy([c, d]) denotes the close convex hull of the set y([c, d]).

For completeness we introduce the definitions of the Caputo derivative of frac-
tional order.

Definition 2.8. Suppose that T is a time scale. The Caputo fractional derivative
of g is defined by

C
T∆

αg(t) =
1

Γ(n− α)

∫ t

0

(t− s)n−α−1g∆
n

(s)∆s, t ∈ Ia,

where n = [α] + 1 and [α] denote the integer part of α and integral is taken in the
sense of Delta-HKP , Γ is the Gamma function.

Definition 2.9. Suppose that T is a time scale, g : I → E is ∆-HKP integrable
function. The fractional ∆-HKP integral of the order α ∈ R+ of g is defined by

Iαg(t) =

∫ t

a

(t− s)α−1

Γ(α)
g(s)∆s,

where integral is taken in the sense of ∆-HKP and Γ is the Gamma function.

III. Our fundamental tools is the deBlasi measure of weak noncompactness β(A).
The deBlasi measure of weak noncompactness β(A) is defined by

β(A) = inf{t > 0 : there exists C ∈ Kω such that A ⊂ C + tB0}

where Kω is the set of weakly compact subsets of E and B0 is the norm unit ball
in E. The properties of the measure of noncompactness β(A) are as follows:

(i) if A ⊂ B then β(A) ≤ β(B);
(ii) β(A) = 0 if and only if A is relatively weakly compact;
(iii) β(A ∪B) = max{β(A), β(B)};
(iv) β(Āω) = β(A), where Āω denotes the weak closure of A
(v) β(λA) = |λ|β(A), (λ ∈ R);
(vi) β(A+B) ≤ β(A) + β(B);
(vii) β(conv(A)) = β(A), where conv(A) denotes the convex extension of A.

Theorem 2.10 ([15]). Let H ⊂ C(Ia, E) be a family of strongly equicontinuous
functions. Let H(t) = {h(t) ∈ E, h ∈ H}, for t ∈ Ia and H(Ia) =

⋃
t∈Ia

H(t).
Then

βC(H) = sup
t∈Ia

β(H(t)) = β(H(Ia))

where βC(H) denotes the measure of weak noncompactness in C(Ia, E), and the
function t 7→ β(H(t)) is continuous.



EJDE-2024/36 EXISTENCE OF PSEUDOSOLUTIONS 5

Definition 2.11. A function f : Ia → E is said to be weakly continuous if it is
continuous from Ia to E endowed with its weak topology. A function g : E → E1

where E and E1 are Banach spaces, is said to be weakly sequentially continuous
if for each weakly convergent sequence (xn) in E, the sequence (g(xn)) is weakly
convergent in E1. When the sequence xn tends weakly to x0 in E, we will write

xn
ω→ x0.

Definition 2.12 ([12]). A family F of functions F is said to be uniformly absolutely
continuous in the restricted sense on A or in short uniformly AC∗(A), if for every ϵ >
0 there is η > 0, such that for every F in F and for every finite or infinite sequence
of nonoverlapping intervals {[ai, bi]} with ai, bi ∈ A,and satisfying

∑
i |bi − ai| < η,

we have
∑

i ω(F, [ai, bi]) < ϵ where ω denotes the oscillation of F over [ai, bi].

A family F of functions F is said to be uniformly generalized absolutely contin-
uous in the restricted sense on [a, b] or uniformly ACG∗([a, b]) if [a, b] is the union
of a sequence of closed sets Ai such that on each Ai the function F is uniformly
AC∗(Ai).

In the proof of the main theorem we will apply the following fixed point theorem.

Theorem 2.13 ([14]). Let X be a metrizable locally convex topological vector space.
Let D be a closed convex subset of X, and let F be a weakly-weakly sequentially
continuous map from D into itself. If for some x ∈ D the implication that

V̄ = conv({x} ∪ F (V )) =⇒ V is relatively weakly compact, (2.1)

holds for every subset V of D, then F has a fixed point.

3. Main problem

Now we will consider the integral problem

x(t) = x0 +
1

Γ(α)

∫ t

0

(t− s)α−1f(s, x(s))∆s, for t ∈ Ia, (3.1)

where f : Ia × E → E, T denotes a time scale (nonempty closed subset of real
numbers R), 0 ∈ T , Ia denotes a time scale interval, (E, ∥ ·∥) is a Banach space and
integral is taken in the sense of ∆−HKP . Fix x∗ ∈ E∗ and consider the problem

C
T∆

α(x∗x)(t) = x∗(f(t, x(t))) (3.2)

Definition 3.1. Let F : I → E and let A ⊂ I. The function f : A → E is a
fractional pseudo ∆-derivative of F on A if for each x∗ ∈ E∗ the real-valued function
x∗F is C

T∆
α-differentiable µ∆ almost everywhere on A and C

T∆
α(x∗F ) = x∗f µ∆

almost everywhere on A.

Regarding the above definition it is clear that the left-hand side of (3.2) can
be rewritten to the form x∗(CT∆

αx(t)), where C
T∆

α denotes the fractional pseudo
∆-derivative.

To obtain the existence result for our problem it is necessary to define a notion
of a solution.

Definition 3.2. A function x : Ia → E is said to be a pseudosolution of problem
(1.1) if it satisfies the following conditions:

(1) x(·) is ACG∗ function,
(2) x(0) = x0,



6 A. SIKORSKA-NOWAK EJDE-2024/36

(3) for each x∗ ∈ E∗ there exists a set A(x∗) with µ∆ measure zero, such that
for each t /∈ A(x∗),

C
T∆

α(x∗x)(t) = x∗(f(t, x(t))) .

Definition 3.3. A continuous function x : Ia → E is said to be a solution to
problem (3.1) if it satisfies (3.1) for every t ∈ Ia.

Let

B = {x ∈ E : ∥x∥ ≤ ∥x0∥+ p, p > 0},

B̃ = {x ∈ (C(Ia, E), ω) : x(0) = x0, ∥x∥ ≤ ∥x0∥+ p, p > 0},

F (x)(t) = x0 +
1

Γ(α)

∫ t

0

(t− s)α−1f(s, x(s))∆s, for t ∈ Ia,

K = {F (x) : x ∈ B} .

Theorem 3.4. Assume that for each ACG∗ function x : Ia → E, f(·, x(·)) is frac-
tionale ∆-HKP integrable, f(t, ·) is weakly-weakly sequentially continuous. Suppose,
that there exists a constans c > 0 such that

β(f(I ×X)) ≤ cβ(X), 0 <
c

Γ(α)

∫ t

0

(t− s)α−1∆s < 1, t ∈ I, (3.3)

for each bounded subset X ⊂ B and for each subinterval I of Ia. Suppose that
the set K is equicontinuous, equibounded and weakly uniformly ACG∗ on Ia. Then
there exists at least one pseudo solution of problem (1.1) on Id, for some number
d ∈ T , 0 < d ≤ a.

Proof. We will prove, in fact, the existence of a solution for problem (3.1) because
each solution of problem (3.1) is a solution of problem (1.1). Let x be a continuous
solution of (3.1).

Fix an arbitrary p ≥ 0. Recall, that the set K of continuous function F (x) ∈ K
defined on a time scale interval Ia is equicontinuous on Ia if for each ϵ > 0 there
exists δ > 0 such that ∥F (x)(t)− F (x)(τ)∥ < ϵ for all x ∈ B̃ whenever |t− τ | < δ,
t, τ ∈ Ia, for each F (x) ∈ K. Thus, for each ϵ > 0 there exists δ > 0 such that

∥
∫ t

τ
(t−s)α−1f(s, x(s))∆s∥ < ϵ, for all x ∈ B̃, whenever |t−τ | < δ and t, τ ∈ Ia. As

a result, there exists a number d, 0 < d ≤ a, such that ∥
∫ t

0
(t−s)α−1f(s, x(s))∆s∥ ≤

p, t ∈ Id, x ∈ B̃.
We will show that the operator F is well defined and maps B̃ into B̃. To see

this, note for any x∗ ∈ E∗, such that ∥x∗∥ ≤ 1, for each x ∈ B̃ and t ∈ Id we have

|x∗F (x)(t)| = |x∗x0|+
∣∣x∗

( 1

Γ(α)

∫ t

0

(t− s)α−1f(s, x(s))∆s
)∣∣

≤ ∥x∗∥∥x0∥+ ∥x∗∥
∥∥ 1

Γ(α)

∫ t

0

(t− s)α−1f(s, x(s))∆s
∥∥

≤ ∥x0∥+ | 1

Γ(α)
|p ≤ ∥x0∥+ p

So
sup{|x∗F (x)(t)| : x∗ ∈ E∗, ∥x∗∥ ≤ 1} ≤ ∥x0∥+ p.

and as a result ∥F (x)(t)∥ ≤ ∥x0∥ + p. Thus F (x)(t) ∈ B̃. We will show, that the
operator F is weakly-weakly sequentially continuous. By [18, Lemma 9] a sequence



EJDE-2024/36 EXISTENCE OF PSEUDOSOLUTIONS 7

xn(·) is weakly convergent in C(Id, E) to x(·) if and only if xn(t) tends weakly to

x(t) for each t ∈ Id, so if xn
ω→ x in C(Id, E) then f(s, xn(s))

ω→ f(s, x(s)) in
E for t ∈ Id and by Theorem 2.6 we have F (xn)(t) → F (x)(t) weakly in E for
each t ∈ Id, so F (xn) → F (x) in C(Id, E) with its weak topology. Suppose that

V ⊂ B̃ satisfies the condition V = conv({x} ∪ F (V )). We will prove that V is

relatively weakly compact and so (2.1) is satisfied. Since V ⊂ B̃, F (V ) ⊂ K. Then
V ⊂ V = conv({x}∪F (V )) is equicontinuous. By Theorem 2.10 t 7→ v(t) = β(V (t))
is continuous on Id. For fixed t ∈ Id we divide the interval [0, t] into m parts in the
following way:

t0 = 0, t1 = sups∈Ia{s : s ≥ t0, s− t0 < δ},
t2 = sup

s∈Ia

{s : s ≥ t1, s− t1 < δ}, . . . ,

tm = sup
s∈Ia

{s : s ≥ tm−1, s− tm−1 < δ}.

Since T is closed, we have ti ∈ Ia. If some ti+1 = ti then ti+2 = {inf t ∈ T : t ≥
ti+1}.

F (x)(t) = x0 +
1

Γ(α)

∫ t

0

(t− s)α−1f(s, x(s))∆s

= x0 +
1

Γ(α)

m−1∑
i=0

∫
Ji

(t− s)α−1f(s, x(s))∆s

∈ x0 +
1

Γ(α)

m−1∑
i=0

µ∆(Ji) sup
si∈Ji

(t− si)
α−1conv(f(Ji, V (Ji)))

where Ji = [ti, ti+1], i = 0, 1, . . . ,m− 1. Using (3.3) and properties of the measure
of weak noncompactness we obtain

β(F (V (t))) ≤ 1

Γ(α)

m−1∑
i=0

µ∆(Ji)(t− qi)
α−1β(f(Ji, V (Ji)))

≤ 1

Γ(α)

m−1∑
i=0

µ∆(Ji)(t− qi)
α−1 · c · β(V (Id))

≤ c · β(V (Id))

Γ(α)

∫ t

0

(t− s)α−1∆s .

Since V ⊂ V = conv({x} ∪ F (V )), β(V (t)) ≤ c·β(V (Id))
Γ(α)

∫ t

0
(t − s)α−1∆s. Using

Theorem 2.6 we obtain

β(V (Id)) ≤
c · β(V (Id))

Γ(α)

∫ t

0

(t− s)α−1∆s

Since c
Γ(α)

∫ t

0
(t − s)α−1∆s < 1, we obtain v(t) = β(V (t)) = 0, for t ∈ Id. Using

Ascoli‘s theorem, V is relatively weakly compact. By Theorem 2.13 the operator
F has a fixed point. Then there exists a pseudosolution to problem (1.1). □

References

[1] Abbas, S.; Benchohra, M.; Henderson, J.; Weak Solutions for Implicit Fractional Differential
Equations of Hadamard Type. Adv. Dyn. Syst. Appl., vol. 11 (2016), pp. 1–13.



8 A. SIKORSKA-NOWAK EJDE-2024/36

[2] Agarwal, R. P.; Bohner, M.; Basic Calculus on Time Scales and Some of Its Applications.

Result Math., vol. 35 (1999), pp. 3-22.

[3] Agarwal, R. P.; Bohner, M.; Peterson, A.; Inequalities on Time Scales: A Survey. Math.
Inequal. Appl., vol. 4, no. 4 (2021), pp. 535-557.

[4] Agarwal, R. P.; Lupulescu, V.; O’Regan, D.; Rahman, G.; Weak Solutions for Fractional

Differential Equations in Nonreflexive Banach Spaces via Riemann–Pettis Integrals. Math.
Nachr., vol. 289 (2016), pp. 395–409. DOI 10.1002/mana.201400010

[5] Ball, J. M.; Weak Continuity Properties of Mappings and Semigroups. Proc. Royal Soc.

Edinburgh Sect. A, vol. 72 (1979), pp. 275–280.
[6] Belmor, S.; Ravichandran, C.; Jarad, F.; Nonlinear Generalized Fractional Differential Equa-

tions with Generalized Fractional Integral Conditions. J. Taibah Uni. Sci., vol. 14 (2020), pp.

114–123. DOI 10.1080/16583655.2019.1709265
[7] Bohner, M.; Peterson, A.; Advances in Dynamic Equations on Time Scales. Birkhäuser,

Boston, 2003.
[8] Cabada, A.; Vivero, D. R.; Expression of the Lebesgue ∆-Integral on Time Scales as a Usual

Lebesgue Integral; Applications of the Calculus of ∆-Antiderivatives. Math. Comp. Modell.,

vol. 43 (2006), pp. 194-207.
[9] Cichoń, M.; On Integrals of Vector-Valued Functions on Time Scales. Commun. Math. Anal.,

vol. 11, no. 1 (2011), pp. 94-110.

[10] Cichoń, M.; Salem, H. A. H.; On the Solutions of Caputo–Hadamard Pettis-Type Fractional
Differential Equations. RACSAM Rev. R. Acad. Cienc. Exactas F́ıs. Nat. Ser. A Mat., vol.

113 (2019), pp. 3031–3053. DOI 10.1007/s13398-015-0228-4

[11] Gou, H.; Li, Y.; Weak Solutions for Fractional Differential Equations via Hen-
stock–Kurzweil–Pettis Integrals. IJNSNS, 2019. DOI 10.1515/ijnsns-2018-0174

[12] Henstock, R.; The General Theory of Integration. Oxford Math. Monographs, Clarendon

Press, Oxford, 1991.
[13] Hilger, S.; Ein Makettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. thesis,

Universität Würzburg, Germany, 1988.
[14] Kubiaczyk, I.; Kneser Type Theorems for Ordinary Differential Equations in Banach Spaces.

Journal of Differential Equations, vol. 45, no. 2 (1982), pp. 139-146.

[15] Kubiaczyk, I.; Sikorska-Nowak, A.; Existence of Solutions of the Dynamic Cauchy Problem
of Infinite Time Scale Intervals. Discuss. Math. Differ. Incl., vol. 29 (2009), pp. 113–126.

[16] Kumar, V.; Malik, M.; Existence and Stability Results of Nonlinear Fractional Differential

Equations with Nonlinear Integral Boundary Condition on Time Scales. AAM, vol. 15, no.
3 (2020), Special Issue 6, art. 10.

[17] Lazreg, J. E.; Benkhettou, N.; Benchora, M.; Karapinar, E.; Neutral Functional Sequential

Differential Equations with Caputo Fractional Derivative on Time Scales. Fixed Point Theory
Algorithms Sci. Eng., , Article 6 (2022). DOI 10.1186/s13663-022-00716-9

[18] Mitchell, A. R.; Smith, C.; An Existence Theorem for Weak Solutions of Differential Equa-

tions in Banach Spaces. In: Nonlinear Equations in Abstract Spaces, Proceedings of the
International Symposium, University of Texas, Arlington, Texas, 1977, V. Lakshmikantham

(Ed.), pp. 387-403, Academic Press, New York, NY, USA, 1978.
[19] Nisar, K. S.; Jothimani, K.; Ravichandran, C.; Baleanu, D.; Kumar, D.; New Approach on

Controllability of Hilfer Fractional Derivatives with Nondense Domain. AIMS Math., vol. 7

(2022), pp. 10079–10095.DOI 10.3934/math.2022561
[20] Podlubny, I.; Fractional Differential Equations: An Introduction to Fractional Derivatives,

Fractional Differential Equations, to Methods of Their Solution and Some of Their Applica-
tions. San Diego: Academic Press, 1999.

[21] Ravichandran, C.; Sowbakiya, V.; Nisar, K. S.; Study on Existence and Data Dependence

Results for Fractional Order Differential Equations. Chaos, Solitons, Fractals, vol. 160 (2022),

112232. DOI 10.1016/j.chaos.2022.112232
[22] Salem, H. A. H.; Hadamard-Type Fractional Calculus in Banach Spaces. RACSAM Rev.

R. Acad. Cienc. Exactas F́ıs. Nat. Ser. A Mat., vol. 113 (2019), pp. 987–1006, DOI
10.1007/s13398-018-0531-y

[23] Salem, H. A. H.; El-Sayed, A. M. A.; Moustafa, O. L.; A Note on the Fractional Calculus in

Banach Spaces. Studia Scientiarum Mathematicarum Hungarica, vol. 42 (2005), pp. 115–130.

[24] Salem, H. A. H.; On the Fractional Order M-Point Boundary Value Problem in Reflexive
Banach Spaces and Weak Topologies. J. Comput. Appl. Math., vol. 224 (2009), pp. 565–572.



EJDE-2024/36 EXISTENCE OF PSEUDOSOLUTIONS 9

[25] Salem, H. A. H.; On the Fractional Calculus in Abstract Spaces and Their Applications to

the Dirichlet-Type Problem of Fractional Order. Comput. Math. Appl., vol. 59 (2010), pp.

1278–1293.
[26] Salem, H. A. H.; Cichoń, M.; On Solutions of Fractional Order Boundary Value Problems

with Integral Boundary Conditions in Banach Spaces. J. Funct. Spaces Appl., vol. 2013

(2013), Article ID 428094. DOI 10.1155/2013/428094

Aneta Sikorska-Nowak
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwer-

sytetu Poznańskiego 4, 61-614 Poznań, Poland

Email address: anetas@amu.edu.pl


	1. Introduction
	2. Preliminaries
	3. Main problem
	References