https://orcid.org/0000-0002-4568-9732
Figures
Abstract
In this paper, we investigate the generalized Langevin-Sturm-Liouville differential problems involving Caputo-Atangana-Baleanu fractional derivatives of higher orders with respect to another positive, increasing function denoted by ρ. The fixed point theorems in the framework of Kransnoselskii and Banach are utilized to discuss the existence and uniqueness of the results. In addition, the stability criteria of Ulam-Hyers, generalize Ulam-Hyers, Ulam-Hyers-Rassias, and generalize Ulam-Hyers-Rassias are investigated by non-linear analysis besides fractional calculus. Finally, illustrative examples are reinforced by tables and graphics to describe the main achievements.
Citation: Thabet STM, Boutiara A, Samei ME, Kedim I, Vivas-Cortez M (2024) Efficient results on fractional Langevin-Sturm-Liouville problem via generalized Caputo-Atangana-Baleanu derivatives. PLoS ONE 19(10): e0311141. https://doi.org/10.1371/journal.pone.0311141
Editor: Joshua Kiddy K. Asamoah, Kwame Nkrumah University of Science and Technology, GHANA
Received: March 13, 2024; Accepted: September 11, 2024; Published: October 2, 2024
Copyright: © 2024 Thabet et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: Pontificia Universidad Cat´olica del Ecuador, Proyecto T´ıtulo: “Algunos resultados Cualitativos sobre Ecuaciones diferenciales fraccionales y desigualdades integrales” Cod UIO2022, [to M. V-C.]. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1446), [to I. K.]. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The powerful effect of fractional derivatives and integrals in checking the existence of solutions for differential equations (s) is very impressive, see books [1–4] and references therein. In addition, the combination of nonlinear analysis and fixed point theorems (FPTs) with nonlinear fractional differential equations (
s) along with their stability analysis has been well presented by the authors in the published articles [5–8].
In 2016, Atangana and Baleanu introduced the following new fractional derivative with non-local and non-singular kernel
(1)
presented some useful properties of the derivative and applied to solve the fractional heat transfer model [9]. In 2018, Baleanu and Fernandez [10] introduced
fractional derivatives with respect to another function ρ for order belongs to the interval (0, 1). Recently in 2023, the authors [11] extended it to higher fractional order with respect to another function. Thabet et al. studied a new category of implicit ρ-Caputo-Atangana-Baleanu (
) fractional pantograph systems as form
(2)
under the conditions
,
, with
, 1 < ϑ ≤ 2 and continuous functions
[12]. In 2024, Rezapour et al. [13] used the topology degree approach to study implicit
with three-point boundary conditions via the same
derivatives.
The Langevin-Sturm-Liouville problems have attracted attention of many researchers, and used to describe dynamical systems in engineering and physics fields. Additionally, these equations are studied under various fractional derivatives by several approaches. For example, see these articles [14–17].
Inspired by the above research works, the current article focuses on studying the qualitative properties of the solutions for the fractional generalized Langevin-Sturm-Liouville problem
of the form:
(3)
where
,
and
denote the
fractional derivatives of arbitrary orders ϑ1 ∈ (m, m + 1] and ϑ2 ∈ (0, 1] respectively, the functions
,
,
, with |μ(υ)| ≥ λ > 0 and
be a non-decreasing and non-negative function with
, such that ρ′(υ) ≠ 0 on J.
The innovation and contributions of this work are that the (3) is new and for the first time studied under the
fractional derivatives of higher order. Moreover, the
(3) can be reduced to the
fractional derivatives sense if ρ(υ) = υ. Furthermore, the
(3) can be reduced to the Langevin problem if μ(υ) = 1 and
as follows:
(4)
also returns to the Sturm-Liouville problem if γ(υ) = 0 as follows:
(5)
The structure of this article is as: In Sec 2, we present the basics and background materials. In Sec 3, we prove the qualitative results by helping the classical FPTs. In Sec 4, we discuss the Ulam-Hyers (UH) stability and the Ulam-Hyers-Rassias (UHR) stability. Finally, in Sec 5, we provide numerical examples with graphics and tables to test major results.
2 Preliminaries
We are in a position to present several major definitions and notations. Assume that Banach space denoted by χ ≔ C(J) endowed with norm ‖ω‖ = supυ∈J |ω(υ)|.
Definition 2.1 ([18]). For ϑ > 0, the ρ-Riemann–Liouville fractional integral of integrable function is defined as
(6) where
and Γ is Gamma function.
Lemma 2.2 ([18]). Let and ϑ, μ > 0. Then
-
;
-
;
-
.
Definition 2.3 ([11]). Let ϑ ∈ (m, m + 1], ν = ϑ − m, m = 0, 1, 2, …. Then, the fractional derivative and the
fractional integral of a function
are defined as
(7) and
(8) respectively, where
(9)
is the Mittag-Leffler function given by
(10)Λ(ϑ) satisfied Λ(0) = Λ(1) = 1 and is called normalization function, and
is defined as
(11)
If , then
Lemma 2.4 ([11]). Let ω ∈ χ, and ρ ∈ Cm(J). For ν = ϑ − m, ϑ ∈ (m, m + 1], m = 0, 1, 2, …, then (12)
Lemma 2.5 ([11]). Let ρ ∈ Cm(J), ρ′(υ) ≠ 0. For ν = ϑ − m, ϑ ∈ (m, m + 1], α ≥ m + 1, m = 0, 1, 2, …. Then,
-
;
-
.
Theorem 2.6 (Krasnoselskii’s FPT [19]). Let K be a Banach space, and H ⊆ K be a nonempty, convex, and closed set. Suppose that ψ1 and ψ2 are two mappings where (i) ψ1ζ1 + ψ2ζ2 ∈ H, for all ζ1, ζ2 ∈ H; (ii) ψ1 is continuous and compact; (iii) ψ2 is contraction. Then, there is υ ∈ H, where υ = ψ1υ + ψ2υ.
3 Existence and uniqueness criteria
We are in a position to derive the equivalent solution of the
(3). Thus, we start to present the next lemma.
Lemma 3.1 Let
and ω ∈ χ. Then, the equivalent solution form of the
:
(13)
is given by
(14)
Proof. By taking
on both sides of Eq (13) and using Lemma 2.4, we get
(15)
By taking i-th derivatives with respect to a function ρ where i = 0, 1, …, m, we have
(16)
Now, by the condition
, 0 ≤ i < m, we deduce that
. Then, we obtain
(17)
which implies that
(18)
Next, by taking
on both sides of Eq (18), and via Lemma 2.4, we find
(19)
where
Thus, by the boundary condition
, we get
(20)
Therefore, by substituting the value of c0 into Eq (19), one have
(21)
Hence, the proof is finished.
As a consequence of Lemma 3.1, we conclude the following interesting lemma.
Lemma 3.2. Let ϑ1 ∈ (m, m + 1], ϑ2 ∈ (0, 1] and ω ∈ χ. Then, the equivalent solution form of the
(3), is
(22)
In this position, we need to give the following hypotheses:
- (G1)
;
- (G2) There is a positive function α with bounds ‖α‖, such that
(23)
- (G3) There exists a constant φ ≥ 0, with
.
Now, we investigate the uniqueness results by helping Banach contraction theorem.
Theorem 3.3. Let the hypotheses (G1)–(G2) fulfilled. If
(24)
then there is an exactly one solution of the
(3) on J, where μ* = infυ∈J |μ(υ)|, γ* = supυ∈J |γ(υ)|, and
(25)
Proof. The idea of the proof is transform the
(3) into a fixed point of the operator
given by
(26)
Notice that
is well defined. Let Br = {ω ∈ χ : ‖ω‖ ≤ r} be a closed, convex and bounded subset of χ, where the fixed constant r satisfies
(27)
where
, and
(28)
Next, we prove that
and by using the triangle inequality
(29)
we have
In the following, we go to prove that
is a contraction mapping. Let
, for υ ∈ J, and using (G2), it follows that
which implies
. In view of Δ < 1, we have
is a contraction. Therefore, by Banach FPT the
possesses an exactly one fixed point and acts a solution of the
(3).
In follows, we discuss the existence result by utilizing Krasnoselskii’s FPT.
Theorem 3.4. let the hypotheses (G1)–(G3) fulfilled. Then, (3) possesses at least one solution, on condition of
(30)
Proof. According to the hypothesis (G3), we set
(31)
where BP = {ω ∈ χ : ‖ω‖ ≤ P}. By dividing the mapping
given by Eq (26) as
where
,
are given by
(32)
for υ ∈ J. The producers of our proof will be divide into several steps:
Step 1:
. Let ω ∈ BP and υ ∈ J. Then
Hence,
, which shows that
.
Step 2:
is a contration mapping on BP. The mapping
is a contraction on condition of
, according to the contractility of
as in Theorem 3.3.
Step 3:
is completely continuous on BP. By continuity of
, it implies that
is continuous. For ω ∈ BP
(33)
(34)
We get
, which yields that
is uniformly bounded on BP.
Step 4: The mapping is compactness. For ω ∈ BP and υ ∈ J, we can evaluate the mapping derivative as below:
(35)
where we used the fact
and use definition
from Eq (9) for k = 0, 1, …, m − 1. Hence, for each υ1, υ2 ∈ J with
and for ω ∈ BP, we get
(36)
which tends to zero independent of ω as υ2 → υ1. So,
is equicontinuous. In view of the Arzelà–Ascoli theorem,
is compact mapping on
. Therefore, the conditions of the Krasnoselskii’s FPT 2.6 satisfy, thus there is at least one solution of
(3) on J.
4 Stability criteria
In this section, the stability analysis in the sense of the UH and UHR are established with their generalized form for solutions of the
(3). For more details about the following definitions, we refer the readers to these works [20, 21].
Definition 4.1. The
(3) is
- UH stable if there is
where, for all ϵ > 0 and for every ω*(υ) ∈ χ satisfying
(37) there is ω(υ) ∈ χ satisfying the
(3) with
, for each υ ∈ J;
- Generalized UH stable if there is
with σg(0) = 0 where, for all ϵ > 0 and for every ω*(υ) ∈ χ satisfying (37), then there is ω(υ) ∈ χ satisfying the system (3) with
.
- UHR stable if there are
, and
, where, for all ϵ > 0 and for every ω*(υ) ∈ χ satisfying
(38) there is ω(υ) ∈ χ satisfying the
(3) with
, for each υ ∈ J;
- Generalized UHR stable if there are
, and
, with σg(0) = 0, and for every ω*(υ) ∈ χ satisfying (38), then there is ω(υ) ∈ χ satisfying the system (3) with
.
Remark 4.2. We note the following:
- If ω*(υ) ∈ χ is a solution for (37) if and only if there is σ ∈ χ depending on ω* such that, for each υ ∈ J, |σ(υ)| < ϵ and
(39)
- If ω*(υ) ∈ χ is a solution for (38) if and only if there is
depending on ω* such that, for each υ ∈ J,
and
(40)
- There are real number Zϕ > 0, and increasing function ϕ(υ) ∈ χ such that
Theorem 4.3. Let the hypotheses (G1)–(G2) and Δ < 1 are hold. Then, the solution of (3) is UH and generalized UH stable.
Proof. Let ϵ > 0, and be a function which verifying the inequality (37) and consider ω ∈ χ the unique solution of the problem
(41)
with boundary conditions in (3). Now, Lemma 3.2 implies that
(42)
Since
verifying (37). Thus by Remark 4.2, we have
(43)
with the boundary conditions:
(44)
Again, Lemma 3.2 implies that
(45)
Now, for each υ ∈ J, we find
(46)
Hence, by using the Remark 4.2 and (G2) we can get
, where Δ is defined in (24) and Θ given by
(47)
In consequence, it follows that
. Therefor, the UH stability condition is satisfied whenever
. More generally, for
,
, the condition of generalized UH stability is also satisfied. Thus the proof is completed.
Theorem 4.4. Let the hypotheses (G1)–(G2) and Δ < 1 are hold. Then, the solution of (3) is UHR and generalized UHR stable.
Proof. Let ϵ > 0, and be a function which verifying the inequality (38) and consider ω ∈ χ the unique solution of the problem
(48)
with boundary conditions in (3). Now, Lemma 3.2 implies that
(49)
Since
verifying (38). Thus by Remark 4.2, we have
(50)
with the boundary conditions:
(51)
Then, by Lemma 3.2, we get
(52)
Now, for each υ ∈ J, we find
(53)
Thus, by using the Remark 4.2 and (G2) we can get
, where Δ is defined in (24).
In consequence, it follows that . Hence, the UHR stability condition is satisfied whenever
. More generally, for ϵ = 1, the condition of generalized UHR stability is also satisfied. Thus the proof is completed.
5 Numerical examples with discussion
Here, we present different version of the function ρ, and various modes ϑ1, ϑ2.
Example 5.1. Based on the problem (3), we consider the following
on domain J = [1, 2.25], with
,
and m = 3 as form
(54)
when
(55)
under conditions
and
(56)
Clearly,
,
, m = 3,
,
,
,
,
and
. We define
,
,
(57)
and consider the normalization function
. There is no doubt about the continuity of function
, this means condition (G1) is satisfied. In addition, for
, we have
where
with ‖α‖ ≃ 0.5154. Indeed, the correctness of assumption (G2) can be accepted. Also, by considering the
, hypothesis (G3) holds:
(58)
and
(59)
Now, by using Eq (25), we obtain
(60)
(61)
and by employing Eq (24) we get
(62)
Table 1 present the numeric values of
,
and Δ for υ ∈ [1, 2.25]. The values of
and Δ are also shown in Figs 1 and 2 for three cases of ϑ1, respectively. The variable
is not dependent on ϑ1. Therefore, conditions of Theorem 3.3 are hold for all cases of order ϑ1. Therefore,
(54) admits unique solution on domain J = [1, 2.25].
In the example 5.2, we check the validity of Theorem 3.3 for four cases of function ρ(υ).
Example 5.2. We consider the same
(54) defined in Example 5.1 as form
(63)
with
is fixed, but
(64)
on domain J = [1, 2.25], under the same conditions (56). There we showed that assumptions (G1)–(G3) are valid. Just check the values of
,
and Δ calculations again. In this case, by employing Eq (25), we obtain
(65)
(66)
and by using Eq (24) we get
(67)
We plot the numerical results in Figs 3–5 for four case of function ρ(υ), respectively. Also, one can see the data in Tables 2 and 3 for different cases of the function ρ(υ) on υ ∈ [1, 2.25]. Hence, conditions of Theorem 3.3 are hold, and so, the
(63) admits unique solution on domain J = [1, 2.25] for all of four cases of ρ(υ). To find the appropriate value for r, we use
and relation (28):
(68)
Table 4 shows the obtained numerical results. Because the parameter
is not dependent on the function ρ(υ), it is displayed in one column in Table 4. Thus
(69)
The curve of the minimum suitable value for r is drawn in Fig 6.
In this example, all parts of Theorem 3.3 were examined along with its proof. The curves drawn in Fig 6 show that for the linear function ρ(υ) = υ, we should choose a value of r greater than 9.2.
In the next example, we consider the extracted results for changes in order ϑ2, and we show that in addition to Theorem 3.3, Theorem 3.4 also confirms the existence of the solution.
Example 5.3. We consider the following
(54) with a few changes which is defined in Example 5.1 as form
(70)
with order
and
are fixed but
(71)
on domain J = [1, 2.25], under conditions
and
There we showed that assumptions (G1)–(G3) are valid. We just check the values of
,
and Δ calculations again. Then, we calculate the values of
,
, Inequality (30) and P. In this case, Eq (25) implies that
(72)
(73)
and by using Eq (24) we get
(74)
Tables 5 and 6 show the numerical results of ,
and Δ for υ ∈ [1, 2.25] and for different cases of ϑ2. These values are also shown in Figs 7–9 for four cases of ϑ2, respectively. So, conditions of Theorem 3.3 are verified, which this ensure that for all of four cases of ϑ2, the
(70) admits unique solution on domain J = [1, 2.25]. Now, to find the suitable value for P, we use
and relation (28):
(75)
(76)
Tables 7 and 8, show the obtained numerical results for ,
and
. It can be seen in Figs 10–12 that by reducing the order of the derivative ϑ2 to less than one, the value of
should also increase, but it is less than 1. Indeed, Inequality (30) holds. Thus,
(77)
The curve of the minimum suitable value for P is drawn in Fig 13. In this example, all parts of Theorem 3.4 were examined along with its proof. The curves drawn in Fig 13 show that when ϑ2 is equal to 1, P can have its lowest value.
6 Conclusion
In this paper, we investigated a new class of higher order
with the
fractional derivatives. To confirm the sufficient conditions of the existence and uniqueness criterion, we used the valid Krasnoselskii and Banach FPT. Furthermore, we discussed the UH, the generalized UH, the UHR, and the generalized UHR stabilities for the proposed problem (3). In the end, we presented four comprehensive examples, supported by graphics and tables for different cases of fractional orders ϑ1, ϑ2, and respected function ρ, to checking our major findings. Of course, our results are valid to the
in the sense of many types of derivatives. For instance
fractional derivatives for ρ(υ) = υ, Hadamard version for ρ(υ) = ln(υ), Katugampola form for ρ(υ) = υp, p > 0, and ordinary derivatives for ρ(υ) = υ and
, ϑ2 = 1.
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