1. Introduction
The applications of Sturm–Liouville equations in mathematical physics, science, and engineering are numerous and expanding all the time. The majority of second-order differential equations eventually rewrite as Sturm–Liouville type equations. The eigenvalues and eigenvectors are arranged according to the Sturm–Liouville theory, and they cover an orthogonal basis. The best way for us to gain insight into the spectrum of a dynamical system has been advantageously demonstrated [
1]. As a substitute strategy, differential equations of fractional order models are currently overly used since the data from exploratory and area of measurement studies cannot be strictly characterized by differential equations of integer order. In recent times, it has been discovered that the fractional Sturm–Liouville types offer more accurate system solutions than the traditional type [
2,
3].
The fractional Sturm–Liouville operator is widely used in quantum mechanics, as well as in applied mathematics, physics, science, and engineering, for example [
2]. As an extension of the common Sturm–Liouville operator, Klimek and Agrawal [
3] developed the fractional Sturm–Liouville operator for the first time. Since then, many versions have been introduced using the same style but with somewhat different formatting [
4,
5,
6]. They looked at the eigenvalues and eigenfunctions characteristics of the fractional Sturm–Liouville operators and addressed variational qualities. Many authors have expressed interest in it (see [
7,
8] and the references therein).
On the other hand, the infinite systems of differential equations take an extremely important role in describing physical phenomena such as the branching process, neural nets and dissociation of polymers [
9]. Solving of PDEs by numerical methods very often instructs an investigation of infinite systems of ODEs. Additionally, there is another example related to using semidiscretization to solve several challenging problems for parabolic PDEs [
10]. Due to this, other writers became interested in studying some of its principles and qualities, Refs. [
11,
12,
13,
14]. This research used a measure of noncompactness approach as their foundation. A mapping from a set of all nonempty and bounded subsets to a particular Banach space that is realized with the condition that it is equal to zero for relatively compact subsets is the measure of noncompactness. It has significant applications in several forms of nonlinear analysis, optimization, the differential, integral, and integro-differential equations, among other areas, and plays a very important role in fixed point theory [
15,
16].
Many mathematicians and physicians have focused their attention on thoroughly investigating the boundary value problems presented based on such equations due to the high visibility of fractional differential equations and their outstanding importance and verity applications, particularly the investigation of the existence and uniqueness of solutions in Banach spaces (see [
17,
18,
19]).
To demonstrate our problem in abstract form, we assume
,
and
are sequences of real functions, where
is the left Caputo fractional derivative of order
. The main objective of the current work is to discuss the existence of solutions to the following infinite system related to the fractional Sturm–Liouville operator
where
is an absolutely continuous function,
with
are supposed functions for all
under certain assumptions, which will be mentioned later, and
and
are the left and right Caputo fractional derivatives of orders
and
, respectively. Here,
c is a sequence space of all convergent sequence of real functions. The problem subjects to the conditions
where
,
and
are constants for all
.
Because the Sturm–Liouville operator and infinite system play such a significant role in the theory of differential equations, we dedicate this contribution to a discussion of the boundary value problems (
1) and (
2). Our research relies on using the measure of noncompactness technique in a sequence space connected to the space
c (the space of all convergent sequences) with the assistance of the Darbo and Meir–Keeler fixed point theorems.
Several contributors have attempted to apply the well-known Banach contraction principle in their publications. Darbo’s fixed point theorem is a well-known generalization of the Banach contraction principle. There are many generalizations of the Meir–Keeler condensing operator that use the measure of non-compactness to verify several new fixed point theorems and to analyze the solvability of a system of Volterra type functional integral equations. In addition, there are many expansions of Darbo’s fixed point theorem, as well as some conclusions on the existence of coupled fixed points for a specific class of operators in a Banach space which can be used to investigate the existence of a solution for a system of nonlinear functional integral and differential equations as applications [
20,
21,
22]. Due to the main role of these fixed point theorems, we choose them to be the basic tools with which to investigate our problem.
2. Basic Definitions and Lemmas
In this part, we present some fundamental concepts and identities for fractional integrals and derivatives that are covered in the book’s first and second chapters [
23,
24,
25]. Additionally, we provide helpful lemmas related to our topic.
Definition 1. Let be a finite interval and be the space of all absolutely continuous functions. Then, if and only if the exist a function (the space of primitives of Lebesgue summable functions) and a constant Q such that Remark 1. The previous definition implies that the absolutely continuous function has a summable derivative a.e. on the interval and . Therefore, we can denote by to the space of complex-valued functions which have continuous derivatives up to order on such that with . According to Lemma 1.1 in [23], the function for all if and only if there exists such thatwhere are constants. Definition 2. Let , and . Then, the left and right fractional derivatives in the Caputo sense, respectively, are given bywhere and are the left and right fractional integrals rendered by Lemma 1. Let and such that , and . Then, we havewhere and are constants for all . Lemma 2. Let such that and . Then, we haveand if , we haveFor , we have . Lemma 3. Let and . Then,
- (1)
if or where ;
- (2)
if where .
Proof. By helping Definition 1 and Remark 1, we can introduce the proofs as follows.
- (1)
Let
. Then,
It is obvious that , which implies that the function , and so by virtue of the last integral in Remark 1, and noting that the product of two Lebesgue integrable functions is also Lebesgue integrable, we get .
Now, let
, there exists
such that
and so
Plainly, the th term belongs for , then .
- (2)
Since
for
, then
and so by the first statement and the fact
, we get
These lead to the desired results. □
Lemma 4. Let , , , and be an absolutely continuous function. Then, the linear fractional Sturm–Liouville equation,subjected to the conditionshas the unique solutionwheresuch that . Proof. Let
,
and
satisfy the fractional differential Equation (
3) and its boundary conditions (
4). According to the last statement of Lemma 1 with operating
on both sides of (
3), we get
where
K is a constant. It is obvious that the fraction integral is well-defined due to
. Since the function
p has values in
, its reciprocal is defined and absolutely continuous. Hence, we have
According to the penultimate statement of Lemma 1 with operating
on both sides, we get
The boundary condition
leads to
which implies that the relation (
5) is realized.
Conversely, assume that (
5) is verified and the function
. It is easy to see that the solution (
5) satisfies the initial and boundary conditions
and
.
It is known that if
and its values in
, then
. Now, the function
can be rewritten as
which, in view of the first statement of Lemma 3, we find that
. Additionally, we have
, which implies
. In the same way, we get
which concludes that
and
is well-defined over the interval
. Now, we have
, which implies that
. Since
, then
, so it is not difficult to see that the (
5) satisfies the initial condition
. By using some results of Lemmas 1 and 2 operating by
on both sides of (
5), we get
which can be rewritten as
Again, Remark 1 tells us that
, which implies that
. Thus,
is well-defined over the interval
. Operating by
on both sides of the last equation using some results of Lemmas 1 and 2, we get the Equation (
3). This completes the proof. □
In the sequel, we need the following lemma,
Lemma 5. Let and ; then, we havewhere is the incomplete beta function defined by In particular, where is the beta function.
3. Measure of Noncompactness
Consider that is a Banach space, is the nonempty and bounded subset and is the subset of all relatively compact.
Definition 3 ([9]). A mapping is said to be a measure of noncompactness in if for all for all . Then, the following assertions hold:
- (1)
The set and ;
- (2)
if , then ;
- (3)
;
- (4)
for , we have ;
- (5)
if is a sequence of closed subsets of with and , then .
Definition 4 ([9]). The measure of noncompactness β is called sublinear if it verifies
- (i)
for all and , , (homogeneous measure);
- (ii)
for all , , (subadditive measure).
Definition 5 ([9]). The measure of noncompactness β is called regular if it is sublinear and verifies
- (iii)
for all , , (maximum property);
- (iv)
, (full measure).
Theorem 1 (Darbo’s Theorem [15]). Let Ω be a nonempty, bounded, closed and convex subset of a Banach space . Suppose that is a continuous map, such that there exists a constant with the property , then has a fixed point in Ω.
Definition 6 ([26]). Consider is a nonempty subset of a Banach space and β is a measure of noncompactness on . Then the operator is called a Meir–Keeler condensing operator if, for all , there exists such that for all bounded subset , we have Theorem 2 (Meir–Keeler Theorem [26]). Consider that is a nonempty, bounded, closed and convex subset of a Banach space and β is a measure of noncompactness on . If is a continuous and Meir–Keeler condensing operator, then the operator has at least one fixed point and the set of all fixed points of in is compact.
4. Main Results
Let
be a sequence space of all real sequences
, the space of all convergent sequences
c is defined as [
9]
equipped with the norm
It is known that
is a Banach space. According to Theorem 3.1 in [
27], for all
, the quantity
is a regular measure of noncompactness in the sequence space
.
Define the space
,
as
with
where
. Consider the space
equipped with the norm
The following lemma shows that the space is a Banach space.
Lemma 6. The space is a Banach space.
Proof. Let
be a Cauchy sequence in
, which means that
for all
and
for all
. It is known that
is a Banach space [
9] and so there is
such that
as
. According to the second statement of Lemma 3, we get
for all
. From the Definition 2 of the Caputo derivative and Lebesgue’s dominated convergence theorem, we can easily deduce that
exists for all
and
, which implies that the sequence
. We claim that
is a Cauchy sequence in the space
c, which means that there is
such that
as
. To prove that, let
such
. Then,
By virtue of
for all
, we get
for all
, which means that
are continuous for all
and attain their maximum in the interval
. Thus, there exist positive constants
such that
for all
and
. Therefore,
which implies that our claim is true.
Since
for all
, according to Remark 1, there are
such that
By Lebesgue’s dominated convergence theorem, we can deduce that
where
as
, which means that
for all
and so
. In the same way,
.
It suffices to prove that
. To do this, by using Lemma 1, we have
Since
as
uniformly on
, then we find that
as
uniformly on
. Hence, by using Lemma 1, we find that
as
, which leads to
where
k is a constant. Operating by
on both sides we obtain
. These conclude for any
that there exists an
such that
and
for
. Therefore,
This ends the proof. □
Let us introduce the following quantity,
for all nonempty bounded subsets
, where
Lemma 7. The quantity is a sublinear and full measure of noncompactness in the space .
Proof. Since
, then we have
and
, which mean, according to the regular measure of noncompactness (
6), that the quantities
and
are regular measures of noncompactness on the space
. This means that both
and
satisfy all identities mentioned in Definitions 3–5. It is clear that
. From the identities (1) in Definition 3 and
in Definition 5, we get
, which means that the identity (1) in Definition 3 and identity
in Definition 5 hold. It is not difficult to verify the identities (2)–(5) in Definition 3 and the identities
and
in Definition 4, which imply that
is a sublinear measure of noncompactness. □
Remark 2. Let with and , then we havewhich means that β has no maximum property in general. Indeed, we havewhich conclude that The discussion of the existence results for the infinite system (
1) and (
2) will be studied under the following suppositions:
- ()
The functions are absolutely continuous functions for all and ;
- ()
There exists a nonnegative sequence of functions
such
satisfies the inequality.
for all
, where
and
are real sequences of
and
for all
, respectively.
- ()
Let
be large enough, then we get
where
for all
,
and
.
- ()
There are positive constants
where
Remark 3. It is obvious that the function is increasing on due to the positivity of the function , which implies that for all . Additionally, for all , we haveIn particular, if , we have To simplify the calculations, we provide
where
We say that the sequence
is a solution of the initial value problems (
1) and (
2) if
satisfies Equation (
1) and boundary conditions (
2) for all
. From Lemma 4,
has a unique representation,
where
According to Lemma 4, and for all . In view of the assumption () and Lebesgue’s dominated convergence theorem, we get that exists for all which implies that .
Let
be operators defined, for all
and
, by
and its fractional derivatives of orders
for all
, using the third statement in Lemma 1, which can be evaluated as
As above, it is clear that
and
for all
. In view of the assumption (
) and Lebesgue’s dominated convergence theorem, we get that
exist for all
, which allows us to define the sequence operator
where
.
Lemma 8. Under the hypotheses ()–(), the operator is bounded and continuous on the closed ball.with fixed radius r satisfying the inequality provided that where , and are given in (8)–(10), respectively. Proof. It is easy using the hypothesis (
) to show that
For all
, in view of Remark 3, we get
According to the definition of the norm in the space
, we find that
Therefore, the operator is bounded and .
In order to prove the continuity on
, let
exist for all
and
such that
and
. Then,
which implies that
In the same way, we can deduce that
This ends the proof. □
Lemma 9. Under the hypotheses ()–(), the operator is equicontinuous on the interval .
Proof. Let
. Then, we see that
Since
is an absolutely continuous function, then
is also absolutely continuous on
and there exists a positive constant
P such that
. This implies that
which uniformly tends to zero as
. Since
and
, then
and so we have three cases:
- Case I.
If
, then we obtain
- Case II.
If
, then we obtain
- Case III.
If
, then we obtain
These can be used by applying the same technique and the results in Lemma 5 to show that
which uniformly tends to zero as
. Similarly, it can be proven that
which concludes that the operator
is equicontinuous on the interval
. □
Theorem 3. Under the hypotheses ()–(), the infinite systems (1) and (2) have at least one solution in the closed ball defined in (13) provided that where is given in (9). Proof. Based on the results obtained in the previous two Lemmas, it is sufficient to calculate the measure of noncompactness
given in (
7). In order to do this: Let
be large enough; by using the hypotheses (
) and (
), we get
We first compute the quantity
as follows:
Similarly,
which concludes that
We complete the proof using two different theorems as follows:
Darbo’s Theorem: In view of Darbo’s Theorem 1 and the assumption
, the infinite system of the fractional Sturm–Liouville operators (
1) and (
2) has at least one solution in
.
Meir–Keeler Theorem: Suppose that for all
, there exists
such that
and
. In view of the Meir–Keeler Theorem 2 and the assumption
, the infinite system of fractional Sturm–Liouville operators (
1) and (
2) has at least one solution in
.
The proof is done. □
5. Illustrative Example
Let us introduce the following example:
with the boundary conditions
Additionally, we take
It is obvious that the partial derivatives of
with respect to
t are continuous and so
are absolutely continuous functions for all
and
, which is fully compatible with the assumption (
). In order to verify the assumption (
), let
, noting that
Then, we have
where
and
is the Riemann zeta function.
Now, let
be large enough and
, then we can find that
where
and
which is fully coincident with the assumption (
).
Additionally, we have
and
By carrying out of simple calculations in Mathematica 11, we can estimate the following:
These lead to
∼
, which satisfies all assumptions of the theorem. Therefore, the infinite system of (
1) and (
2) has at least one solution in
.