1. Introduction
Fractional differential equations (FDEs) play a crucial role in addressing non-standard dynamic behaviors intertwined with elements of short and long memory and hereditary influences, as highlighted in literature reviews of earlier research [
1,
2,
3,
4,
5]. Recently, Vanterler and Capelas de Oliveira proposed a novel fractional derivative referred to as the
-Hilfer fractional derivative, which brings together various fractional definitions; a concise explanation can be found in the following referenced studies: [
6,
7,
8]. Given that it comprises a large class of fractional derivatives, the reasonable inference is that it has a lot of advantages over classical derivatives and is better suited for practical applications. Recent improvements in FDEs using the
-Hilfer fractional derivative include those seen in [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18].
One significant and unique topic in fractional calculus is the exploration of pantograph differential equations (PEs), which involve proportional delays. This concept arose from research on the electric current in the pantograph system of electric trains, conducted by Tayler and Ockendonare [
19]. Over time, it has found applications in numerous fields of both pure and applied mathematics, such as quantum physics, electrodynamics, number theory, control systems, and probability theory. The pantograph, often utilized in electric trains and electric cells [
20,
21,
22], is commonly seen as a device for measurement and graphing. As a result, there have been various efforts to develop pantograph-type fractional differential equations with a variety of specific boundary conditions [
23,
24,
25,
26,
27,
28,
29,
30,
31,
32]. Importantly, we can focus on problems with linear functional conditions [
33,
34,
35,
36,
37,
38], which greatly generalize and extend many specific boundary findings.
As far as we know, the pantograph equation boundary value problems that we talked about earlier with
-Hilfer fractional derivatives rarely mention resonance scenarios. Keeping the previous findings in perspective, we will introduce in this paper a set of
-Hilfer fractional functional boundary value problems related to pantograph equations:
where
denotes the left
-Hilfer fractional derivative of order
and type
,
and
. A boundary value problem is said to be at resonance if its corresponding homogeneous boundary value problem has a non-trivial solution; so,
are continuous linear functionals with the resonance condition
:
.
This research contains several key findings: To begin with, we examine functional boundary conditions in a highly general manner. Next, there is an effective integration of the
-Hilfer fractional derivative
. Additionally, the pantograph equation we explore has substantial practical applications, which we detail. Lastly, we address the resonance issue for this category of equations, which adds to the initial boundary value challenge for
-Hilfer fractional derivatives with pantograph equations.
The structure of this document is laid out as follows:
Section 2 includes some introductions and essential concepts regarding linear operators, the coincidence degree continuation theorem, general fractional calculus, and supporting Banach spaces and lemmas. In
Section 3, we present three existence findings concerning the various “smoothness” criteria of the unknown function related to resonance, relying on Mawhin’s coincidence theory, and we include a numerical example to illustrate our main findings.
Definition 1. We say that
(please refer to Section 2 for the definition of X) is a solution to the functional boundary value problem (FBVP) (Equation (1)) if u satisfies the equation and boundary conditions in (Equation (1)). 2. Basic Definitions and Preliminaries
We first restate the following definitions, theorem, and auxiliary lemmas related to fractional calculus theory; for details, see [
6,
7,
8].
Definition 2 ([
6,
7,
8]).
The left-sided ψ-Riemann–Liouville fractional integral for an integrable function
with respect to another function
, for all
, is defined as follows: Definition 3 ([
6,
7,
8]).
The left-sided ψ-Riemann–Liouville fractional derivative of order
for a function
with respect to another function
, for all
, is defined as follows: Definition 4 ([
6]).
Let
be an interval such that
and
for all
. The (left-sided) ψ-Hilfer fractional derivative of function
of order α and type
is determined as follows: where
Lemma 1 ([
6,
7,
8]).
Assume
and
; then,
Lemma 2 - (1)
If
and
, thenwhere
- (2)
Assume
and
; then,and
Example 1.
Definition 5 ([
39,
40]).
Let X, Z be real Banach spaces and be a linear operator. X is said to be the Fredholm operator of index zero provided that the following are true:- (i)
is a closed subset of Z;
- (ii)
.
Let and
be continuous projectors such that
,
,
, and
. It follows that
is reversible. We denote the inverse of the mapping as
(the generalized inverse operator of L). If
is an open bounded subset of X such that
, the mapping
will be called L-compact on
, if
is bounded and
is compact.
Theorem 1 (see [
39,
40] for the Mawhin continuation theorem).
Let
be a Fredholm operator of index zero and
be L-compact on
. We assume that the following conditions are satisfied:- (i)
for every
;
- (ii)
for every
;
- (iii)
, where
is a continuous projection such that
.
Then, the equation
has at least one solution in
.
As is usual, we shall use the classical Banach space
with norm
and the Banach space
with norm
. Then, we can define and easily verify that the space
with norm
is the Banach space.
We define the linear operator as
, the nonlinear operator
as
and
where
.
Then, the
(Equation (1)) is equivalent to the operator equation
To achieve the outcomes we want, we need to use the following hypotheses.
Hypothesis 1 (
H1).
The linear functionals
satisfy
where
.
Hypothesis 2 (
H2).
The functionals
are continuous and linearly independent, with the respective norms
; that is,
.
Hypothesis 3 (
H3).
There exists a function
such that the equation The next statement shows that there is indeed a function
for which the assumption in
is true.
Lemma 3. Assume that
holds. Then, there exists
such that Proof. For convenience, we set
, and
Then, there must exist
such that
.
If not, ; that is,
. From
. Thus,
for every ‘polynomial’
. It is easy to verify that
is dense in X, thus we obtain
. This is a contradiction (as
and
are linearly independent on X).
There must be some
such that
. For these i, we simply take
Thus, there exists an
satisfying
. □
3. Main Results
Lemma 4. If assumptions
,
, and
hold, then the mapping
is a Fredholm mapping of index zero.
Proof. If
, and
we can deduce that
,
and
which means that
Let
; then, there exists
such that
and it follows that
which, with
leads to
Then, we can deduce from
that
Therefore,
That is,
If
then
It is easy to find that
and
which implies that
Hence, we obtain Equation (2).
We define
as follows:
where
is introduced in condition
.
From the property of
in
, we know that
So,
is a continuous linear projector such that
and
It is clear that
and
; that is, L is a Fredholm mapping of index zero. □
For
, set
. It is not hard to check that
, since
and
It is quite simple to verify that
; moreover, we take
If, in addition,
, then
We must obtain
by setting
i.e.,
. So,
.
Lemma 5. The mapping
is defined byand is the generalized inverse operator of L. Proof. In view of Definition 4 and Lemma 2, one can show that
for all
. Meanwhile, it is suffices to show that
and
In accordance with the equations above,
. If
, from Lemma 2(1) and
it can be deduced that
□
The next lemma provides the norm estimates needed for the following results.
Lemma 6. For
, we now show thatand Proof. For
, we can deduce that
which means that
Based on the assumption of
, we can derive the following:
and
Hence, we can conclude that
which yields our desired result. □
Lemma 7. N is L-compact on
if
where Ω is an open and bounded subset of X.
Proof. Since
is a bounded set and functions
and
are continuous for
, there exist constants
such that
and
hold and
. Invoking
and Hypothesis
, it is easy to see that
hence,
is bounded.
Now, we will prove that
is compact. The proof process is divided into two steps:
For each
one can show that
To begin with, one knows that
and
As
, it is easy to find that
Next, from the definition of
, we can derive the following:
Taking
, from the continuity of
and
for
, we can deduce that Equation (8)
and Equation (9)
.
By taking the prior estimation (Steps 1–2) into account and applying the Arzelà–Ascoli theorem, it holds that the operator
is compact and
is bounded, i.e., N is L-compact. □
Lemma 8. Assume
and the following hypotheses hold:
Hypothesis 4 (
H4).
There exists a constant
such that if
, for all
, then Hypothesis 5 (
H5).
There exist some positive functions
, with
, and
, such that for all
provided that
Then, the set
, for some
is bounded. Proof. Take
; then,
thus we obtain
This, together with
, means that there exists
such that
.
Write
where
and
Thus,
In light of (Equation (5)) in Lemma 7, one obtains
Now,
so
and
where
C is the same as in (Lemma 6). Recall that
, where
is introduced for the sake of brevity. Hence, it follows from Equation (
11) that
.
Furthermore, it is straightforward to derive that the following inequalities:
and
which leads to
where
and
which together with Equation (10) and
, yield
As a result,
is bounded. This proof is thus complete. □
Since
L is a Fredholem mapping of index zero, there exists an isomorphism
. For this, the operator
J is defined by
Obviously,
If
then
, where
h the same as in
, and
Thus,
is an isomorphism.
Lemma 9. Assume
and
and the following hypothesis holds:
Hypothesis 6 (
H6).
There exists a constant
such that either for each
orwhere
. Then, sets
and
are bounded, where
is a homeomorphism with
: Proof. If
, it is easy to find that
and
which represents
,
Now, let us estimate the norm of
in X.
and
By using the above estimates, we obtain
; that is,
is bounded.
For
and
If
then
If
then
i.e.,
which follows from the proof of the boundedness of
as
If
, we claim that
and then, taking
we obtain
which, with multiplying by
, leads to
which is impossible! Thus, we have
. So, the boundedness of
follows
. □
Theorem 2. Assume (H) and
–
hold. Then, the functional boundary value problem (Equation (1))has at least one solution in X.
Proof. Let be open and bounded such that
. In light of the boundedness of
and
in Lemmas 8 and 9, we can derive
for
and
Thus,
and
of Theorem 1 hold.
Let
; noticing the boundedness of
in Lemma 9 and
, we know that
.
For
, we find that
and
To achieve this, via Lemma 9, we know that
Thus, by calculating the invariance of degree under a homotopy, we find that
therefore, the condition
of Theorem 1 holds, and the existence result for
(Equation (1)) is provided in
. □
Example 2. We introduce an example in order to demonstrate the application of the previous result. Consider the nonlocal boundary value problems of the following pantograph equation: It is easy to see that
and
The problem is seen at resonance and when
and
. At this point, we can take
to ensure that
, i.e.,
holds.
Also,
which represents
. Then, we can directly calculate
Hence,
holds.
If
then
.
If
, then
.
Hence,
provided that
satisfies
since
and
So, Hypothesis
holds.
Last, for
, where
, one can choose
such that
, which shows that
is confirmed, since
and
It follows from Theorem 2 that there must be at least one solution in X.
Theorem 3. Assume that
with
or
,
) (of Lemma 9), and the following conditions hold:
Hypothesis 4′ (
H4′).
There exists a constant
such that if
, for all
, then Hypothesis 5′ (
H5′).
There exist some positive functions
, with
, and
, such that for all
provided that
Then, the functional boundary value problem (Equation (1)) has at least one solution in X. Proof. As in the proof of Lemma 8,
means
From (H4′), we know that there exists a constant
such that
. □
Remark 1. Similarly to Lemma 8, in this case
does not come for free.
Similarly,
where
and
Thus,
As in the proof of Lemma 8,
Now,
. In view of (Equation (5)), it is easy to deduce that
and
As in the proof of Lemma 8,
. Hence, it follows from (15) that
by
in
. Then,
and
This implies that
where
and
which reveals that
From (H5′), we know that
is bounded and the rest of the proof is identical to that of Theorem 2.
Theorem 4. Assume that (H), (
–
) with
or
,
(of Lemma 9), and the following conditions hold:
Hypothesis 4″ (
H4″).
There exists a constant
such that if
, for all
, then Hypothesis 5″ (
H5″).
There exist some positive functions
, with
, and
, such that for all
provided that
Then, the functional boundary value problem (Equation (1)) has at least one solution in X. Proof. As in the proof of Lemma 8,
means
From (H4″), we know that there exists a constant
such that
. □
Remark 2. Note that we do not readily obtain
, which follows directly from (H4) and (H4′).
Similarly,
where
and
Thus,
As in the proof of Lemma 8,
Now,
and we can invoke Equation (3) to show that
As in the proof of Lemma 8,
. From (16), one obtains
by solving
in
. Then,
and
which enables us to obtain
where
and
Furthermore, it holds that
From (H5″), we know that
is bounded. The remaining part of the proof follows the same steps as those in Theorem 2.