https://orcid.org/0000-0002-4568-9732
Figures
Abstract
This research manuscript aims to study a novel implicit differential equation in the non-singular fractional derivatives sense, namely Atangana-Baleanu-Caputo () of arbitrary orders belonging to the interval (2, 3] with respect to another positive and increasing function. The major results of the existence and uniqueness are investigated by utilizing the Banach and topology degree theorems. The stability of the Ulam-Hyers (
) type is analyzed by employing the topics of nonlinear analysis. Finally, two examples are constructed and enhanced with some special cases as well as illustrative graphics for checking the influence of major outcomes.
Citation: Rezapour S, Thabet STM, Rafeeq AS, Kedim I, Vivas-Cortez M, Aghazadeh N (2024) Topology degree results on a G-ABC implicit fractional differential equation under three-point boundary conditions. PLoS ONE 19(7): e0300590. https://doi.org/10.1371/journal.pone.0300590
Editor: Abuzar Ghaffari, University of Education, PAKISTAN
Received: December 27, 2023; Accepted: March 1, 2024; Published: July 1, 2024
Copyright: © 2024 Rezapour 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: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
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/1445 to I. K.).
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The derivatives of non-integer orders, or fractional derivatives, are a mathematical concept that extends differentiation beyond integer orders. These types of derivatives have a large domain of applications in numerous fields, including physics, engineering, and finance, for additional details see these manuscripts [1–8]. In the realm of fractional calculus has seen the advent of a fresh fractional operator by Atangana and Baleanu () [9], which is free from singularity kernels. This operator is established through the Mittag-Lefler function in the Caputo and Riemann-Liouville contexts. The non-singular fractional operators are well-behaved and allow for more accurate modeling of the underlying system. The
operator can be used to describe phenomena such as diffusion, wave propagation, and viscoelasticity, and has applications in many fields such as image processing, signal analysis, and control theory. Non-singular fractional operators are important tools for researchers and practitioners seeking to understand and manipulate complex systems in a variety of contexts and they have stimulated a great deal of interest among researchers in their applicability to diverse problems, we indicate the readers to these works [10–19] and the references therein. Subsequently, authors of this work [20] popularized the
definition to contain differentiation and integration with respect to non-negative, non-decreasing function, leading to the development of the
operator. In 2023, Abdeljawad et al. [21] expanded this operator to higher-order fractional derivatives and integrals. Furthermore, this type of fractional derivative is a generalization of the traditional derivative, where the derivative is taken with respect to a function rather than a variable. This type of derivative is commonly used in fractional calculus with various operators to describe complex systems with non-integer order dynamics. In fact, by taking this fractional derivative, the researchers can better model the behavior of these systems and gain a deeper understanding of their underlying dynamics, see for example [22–24] and references cited therein.
An implicit differential equation involving a fractional derivative of an unknown function that appears implicitly in the equation has several benefits, including the ability to model complex systems with memory effects, non-local interactions, and anomalous diffusion. These equations also accurately describe physical phenomena such as transport in porous media or viscoelastic materials [25–28]. In particular, Thabet and Kedim [29] studied a Hilfer fractional snap dynamic system on an infinite interval. Authors [30] discussed stability analysis of fractional pantograph implicit differential equations with initial boundary and impulsive conditions. Also, fractional derivative used to investigate the stability of implicit differential problem by authors [31].
Recently in 2022, Shah et al. [32] used degree theory to establish qualitative results for the following differential equation:
(1.1)
Very recently, authors [33] extended the above equation (1.1) to the Caputo fractional order derivative for discussing the qualitative results and some types of
stability as in the following form:
(1.2)
Inspired by the research mentioned above articles, in this current work, we study the qualitative properties of the solution for implicit fractional differential equation (IFDE) of the following form:
(1.3)
where
denotes the
fractional derivatives of arbitrary order μ ∈ (2, 3] and the function
is continuous. Additionally,
be a non-decreasing and non-negative function with
, such that
and
Furthermore,
and
.
In this situation, we would like to indicate that our contributions are interesting and the Eq (1.3) is new in the framework of fractional order derivatives which include
derivative as a special case when
. Moreover, an approach analysis in this work is different about methods used in these works [32, 33], and the Eq (1.3) covers many problems available in the literature studies, for instance,
(i) the Eq (1.3) reduces to problem (1.1) if μ → 3, and the implicit term omitted;
(ii) the Eq (1.3) returns to problem (1.2) if we replace the operator by
with omitting the implicit term.
The remaining parts of this paper are arranged as follows: Sec.2 is devoted to recalling the basic background materials related to fractional calculus and nonlinear analysis. Sec.3 discusses the existence and uniqueness theorems by using . Sec.4 is investigated
stability. Finally, Sec.5 is dedicated to testing the effectiveness of main outcomes.
2 Preliminaries
In this situation, we present essential background material. Consider the space of continuous functions denoted by which is Banach space gifted with the norm ‖y‖ = supυ∈[ι,ρ]|y(υ)|.
Definition 2.1 [34] Let and μ > 0, then the equation
is μth order of
-Riemann–Liouville fractional integral, where Γ is Gamma function.
Definition 2.2 [21]. The fractional derivatives of
with order μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, is defined as
where
and
. If
, then
. Furthermore,
is the Mittag-Leffler function
and Φ(μ) denotes the normalization function endowed by Φ(0) = Φ(1) = 1.
Definition 2.3 [21]. The following relation:
is
fractional integral of a function y with order μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, where
is
-Riemann–Liouville fractional integral.
Lemma 2.4 [21]. For μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, and and
. Then
Lemma 2.5 [21]. For μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, ϵ > 0, and with
Then,
- (i)
;
- (ii)
.
Now, we about to introduce a definition of the Kuratowski’s measure of noncompactness χ(⋅) as follows:
where
, and
is a bounded subset of the Banach space Ω. It is clear that
[35].
Definition 2.6 [35] Let be bounded and continuous with
. Then,
will be χ-Lipschitz if ∃ ϵ ≥ 0, so that
As well as,
is named as strict χ-contraction when ϵ < 1 holds.
Definition 2.7 [35] A function is χ-condensing if
So,
gives χ(B) = 0. Also,
is Lipschitz for ϵ > 0 such that
If ϵ < 1, in this case
is called a strict contraction.
Lemma 2.8 [35] is χ-Lipschitz with constant ϵ = 0 iff
is compact.
Lemma 2.9 [35] A function is χ-Lipschitz with constant ϵ iff
is Lipschitz with Lipschitz constant ϵ.
Theorem 2.10 [36] Let be an χ-condensing and
If
is a bounded subset contained in Ω, i.e., a constant k > 0 exists with
then
for all ζ ∈ [0, 1]. Therefore,
has a fixed point and the set
belongs to Bk(0).
3 Existence and uniqueness analysis
We introduce an equivalent integral fractional equation of the IFDE (1.3). Regarding this, we first derive the following lemma:
Lemma 3.1 Let
and
. Then, the
fractional differential problem:
(3.1)
is equivalent to
Proof At the beginning, we apply on both sides of Eq (3.1) and using Lemma 2.4, we get
So, due to the condition y(ι) = 0, we deduce that e0 = 0. Thus, by substituting the value of e0 and by taking the first derivative with respect to a function
, we find
and due to the boundary condition
, we have e1 = 0, which yields that
Next, by applying the condition one has
which implies that
Hence, we deduce that
Therefore, the proof is finished.
As a consequence of the above lemma, we present the following essential result:
Lemma 3.2 Let
and
. Then, the
IFDE (1.3) has a solution equivalent to
(3.2)
Now, to achieve the required existence and uniqueness theorems, according to Lemma 3.2, the solution of the IFDE (1.3) is a fixed point of the operator ℵ: Ω → Ω which is defined as:
(3.3)
For working analysis, we state the following conditions:
- (AS1). There are the constants δ1, δ2, δ3 > 0, such that
for any
and υ ∈ J.
- (AS2). There are the constants ℓ1 > 0 and ℓ2 ∈ (0, 1), for any
and υ ∈ J, satisfy
Theorem 3.3 Under assumption (AS1) with δ3 ≠ 1. The mapping ℵ: Ω → Ω is continuous and satisfy the growth condition ‖ℵy‖ ≤ Π1 + Π2‖y‖.
Proof We define a bounded ball as
. Regarding to show the continuity of ℵ, let us taking the convergence sequence
to y in the ball ℧ς as n → ∞. Thus, by continuity of ϖ and by applying Lebesgue dominated convergence theorem, one has
Hence, ℵ is continuous.
Next, regarding to the growth condition, by applying (AS1), we find
(3.4)
Since, then
which implies that
(3.5)
Therefore, in view of the Eqs (3.4) and (3.5), and taking supremum, one has
Hence, ‖ℵy‖ ≤ Π1 + Π2‖y‖ and this finishes the proof.
Theorem 3.4 Under assumption (AS1) with δ3 ≠ 1, the mapping ℵ: Ω → Ω is compact and consequently is χ-Lipschitz with the Lipschitz’s constant zero.
Proof The boundedness of ℵ implied from Theorem 3.3. It remains to prove that ℵ is an equi-continuous mapping. Therefore, by the assumption (AS1), for any and υ1, υ2 ∈ J with υ1 < υ2, we get
Obviously, |(ℵy)(υ2) − (ℵy)(υ1)| → 0 whenever υ2 → υ1 and thus
is equi-continuous. Hence, due to Arzelá-Ascoli theorem,
is compact and in view of Lemma 2.8, the mapping ℵ is χ-Lipschitz with the Lipschitz’s constant ϵ = 0.
Theorem 3.5 Under assumption (AS2), the IFDE (1.3) possesses an one solution on condition of
(3.6)
Proof Let us take the mapping ℵ as given in (3.3). For any and υ ∈ J, we find
Thus,
. Therefore, by condition (3.6), ℵ is contraction mapping and based on the Banach contraction theorem, ℵ possesses a unique fixed point which is a solution of the
IFDE (1.3).
Theorem 3.6 Under the assumptions (AS1) and (AS2), the IFDE (1.3) admits a solution such that Π2 < 1. Furthermore, the set containing solutions of the
IFDE (1.3) is bounded.
Proof According Theorem 3.5, ℵ is Lipschitz mapping and by Lemma 2.9, ℵ is χ-Lipschitz which implies that ℵ is χ-condensing.
Now, due to Theorem 2.10, it remains to show that the set is bounded, where
For end this, let
, therefore for each υ ∈ J for some ζ ∈ [0, 1], and by Theorem 3.3, we can derive that
Thus,
which implies that
is a bounded set contained in Ω. In view of Theorem 2.10, implies that ℵ has at least one fixed point, which are act solutions of the
IFDE (1.3), and consequently
contains solutions of the Eq (1.3) is a bounded subset of Ω.
4 Stability analysis
Here, we will discuss the stability of type. So, we need to state the definitions of
stability:
Definition 4.1 [37] Let there is a real constant Ξϖ > 0, such that for all ς > 0. Then the IFDE (1.3), is called
stable when
is satisfying the relation
(4.1)
hence there is one function y ∈ Ω satisfying the Eq (1.3), provided
(4.2)
Moreover, the solution y ∈ Ω of the Eq (1.3) is called generalized
(
) stable, if there is a function
satisfied
(4.3)
Remark 4.2 The function
satisfying the inequality (4.1), iff there is a function σ ∈ Ω, where
1) |σ(υ)| ≤ ς, υ ∈ [ι, ρ], ς > 0;
2)
.
Theorem 4.3 Let the arguments of Theorem 3.5 are satisfied. Then, the solution of
IFDE (1.3) is
and consequently
stable.
Proof Suppose that satisfying the Ineq. (4.1), then by applying (4.2), we get
According to Eq (3.2), one has
(4.4)
which gives
(4.5)
Next, for , by utilizing Eqs (4.4) and (4.5) and (AS2), we have
which further implies
(4.6)
where
Thus, yields that
Hence, the
fractional implicit differential problem (1.3) is
stable. In addition, there is a non-decreasing function Θ: (0, ∞) → (0, ∞),
where Θ(0) = 0, so by (4.6), we find
Therefore, the
IFDE (1.3) is
stable.
5 Applications
This section concerns the applications of the essential results using two comprehensive examples with illustrative graphics and tables.
Example 5.1 Consider the
IFDE as follows:
(5.1)
Here,
and
Thus, we get
hence, ℓ1 = 0.0065, ℓ2 = 0.0131385. So, we have
Then, according to Theorem 3.5, the
IFDE (5.1) has one solution. Furthermore, based on Theorem 4.3 the such solution is
stable with
and consequently is
stable.
Additionally, Fig 1, represents the graphics of Π3, which are less than 1, and Table 1, shows the computation values of Π3, and Ξϖ, whenever the function , on υ ∈ [1, e], for the problem (5.1). Also, Fig 2, represents the graphics of Π3, which are less than 1, and Table 2, shows the computation values of Π3 whenever the function
, and various μ ∈ (2, 3] on υ ∈ [1, e] for problem (5.1).
Example 5.2 Consider the
IFDE as follows:
(5.2)
where,
and
Thus, we get
hence, ℓ1 = 0.0123456, ℓ2 = 0.0327538. So, we have
Then, in view of Theorem 3.5, the
IFDE (5.2) has one solution. Furthermore, based on Theorem 4.3 the such solution is
stable with
and consequently is
stable.
Moreover, Fig 3, represents the graphics of Π3, which are less than 1, and Table 3, shows the computation values of Π3, and Ξϖ, whenever the function , on υ ∈ [1, 2], for the problem (5.2). In addition, Fig 4, represents the graphics of Π3, which are less than 1, and Table 4, shows the computation values of Π3 whenever the function
, and various μ ∈ (2, 3] on υ ∈ [1, 2] for problem (5.2). According to Fig 4 and Table 4, we observe that Π3 ≥ 1 for some values μ at function
, thus for this reason and only at these values we can’t say that the problem (5.2) has one solution.
6 Conclusions
This manuscript dealt with a new class of -IFDE (1.3) with higher orders belonging to the interval (2, 3]. The fundamental conditions of the existence and uniqueness of the solution for Eq (1.3) were established by Banach and topology degree theories. Moreover, the
stability with its generalized was discussed. Finally, two application examples with illustrative graphics and tables were provided to check the effectiveness of the main results with compare the main parameters.
The results of this study can be employed in new problems as special cases of the main Eq (1.3) by taking various functions of . Furthermore, the
-IFDE (1.3) covers some problems are existing in the literature; for instance (i) the Eq (1.3) can be reduced to problem (1.1) if μ → 3 and the implicit term omitted; (ii) the Eq (1.3) can be returned to problem (1.2) if we replace the operator
by
with omitting the implicit term.
References
- 1.
Zhou Y. Basic theory of fractional differential equations. vol. 6. Singapore: World Scientific; 2014.
- 2. Thabet S T M and Dhakne M B. Nonlinear fractional integro-differential equations with two boundary conditions. Advanced studies in contemporary mathematics, 2016, 26(3): 513–526.
- 3. Liao F, Zhang L and Hu X. Conservative finite difference methods for fractional Schrödinger-Boussinesq equations and convergence analysis. Numerical Methods for Partial Differential Equations, 2019, 35(4): 1305–1325.
- 4.
Hilfer R. Applications of fractional calculus in physics. vol. 35, Singapore: World Scientific; 2000.
- 5. Liu J G, Yang X J, Geng L L and Yu X J. On fractional symmetry group scheme to the higher-dimensional space and time fractional dissipative Burgers equation. International Journal of Geometric Methods in Modern Physics, 2022, 19(11): 2250173.
- 6. Liu J G, Zhang Y F and Wang J J. Investigation of the time fractional generalized (2+1)-dimensional zakharov–kuznetsov equation with single-power law nonlinearity. Fractals, 2023, 31(5): 2350033.
- 7.
Yang X J. General fractional derivatives: Theory, methods and Applications, 1st ed., New York: CRC Press; 2019.
- 8. Thabet S T M, Al-Sádi S, Kedim I, Rafeeq A S and Rezapour S. Analysis study on multi-order ϱ–Hilfer fractional pantograph implicit differential equation on unbounded domains. AIMS Mathematics, 2023, 8(8): 18455–18473.
- 9. Atangana A and Baleanu D. New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm. Sci., 2016, 20(2): 763–769.
- 10. Abdeljawad T. A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel. J. Inequal. Appl., 2017, 2017,130: 1–11. pmid:28680233
- 11. Abdeljawad T and Baleanu D. Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. J. Nonlinear Sci. Appl., 2017, 9: 1098–1107.
- 12. Ayari M I and Thabet S T M. Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator. Arab Journal of Mathematical Sciences, 2023.
- 13. Khan M, Ahmad Z, Ali F, Khan N, Khan I and Nisar K S. Dynamics of two-step reversible enzymatic reaction under fractional derivative with Mittag-Leffler Kernel. PLoS ONE, 2023, 18(3): e0277806. pmid:36952579
- 14. Abdo M S, Abdeljawad T, Kucche K D, Alqudah M A, Ali S M and Jeelani M B. On nonlinear pantograph fractional differential equations with Atangana—Baleanu—Caputo derivative. Advances in Difference Equations, 2021, 2021,65: 1–17.
- 15. Abdo M S, Abdeljawad T, Shah K and Jarad F. Study of impulsive problems under Mittag–Leffler power law. Heliyon, 2020: 6(10): 6e05109. pmid:33072909
- 16. Jarad F, Abdeljawad T and Hammouch Z. On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative. Chaos Solitons Fractals, 2018, 117: 16–20.
- 17. Ali G, Shah K, Abdeljawad T, Khan H, Ur Rahman G and Khan A. On existence and stability results to a class of boundary value problems under Mittag-Leffler power law. Advances in Difference Equations, 2020, 2020,407: 1–13.
- 18. Abdo M S, Abdeljawad T, Ali S M and Shah K. On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions. Advances in Difference Equations, 2021, 2021,37: 1–21.
- 19. Ahmad Z, Ali F, Khan N and Khan I. Dynamics of fractal-fractional model of a new chaotic system of integrated circuit with Mittag-Leffler kernel. Chaos, Solitons and Fractals, 2021, 153: 111602.
- 20.
Fernandez A and Baleanu D. Differintegration with respect to functions in fractional models involving Mittag-Leffler functions. SSRN Electron. J., 2018.
- 21. Abdeljawad T, Thabet S T M, Kedim I, Ayari M I and Khan A. A higher-order extension of Atangana-Baleanu fractional operators with respect to another function and a Gronwall-type inequality. Boundary Value Problems, 2023, 2023,49: 1–16.
- 22. Boutiara A, Etemad S, Thabet S T M, Ntouyas S K, Rezapour S and Tariboon J. A mathematical theoretical study of a coupled fully hybrid (k, ϕ)-fractional order system of BVPs in generalized Banach spaces. Symmetry, 2023, 15, 1041.
- 23. Sousa J V C and Oliveira E C. On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul., 2018, 60: 72–91.
- 24. Sousa J V C and Oliveira E C. On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator. Journal of Fixed Point Theory and Applications, 2018, 20: 1–21.
- 25. Ahmed I, Kumam P, Shah K, Borisut P, Sitthithakerngkiet K and Demba M A. Stability results for implicit fractional pantograph differential equations via ϕ-Hilfer fractional derivative with a nonlocal Riemann-Liouville fractional integral condition. Mathematics, 2020, 8: 94.
- 26. Alnahdi A S, Jeelani M B, Abdo M S, Ali S M and Saleh S. On a nonlocal implicit problem under Atangana–Baleanu–Caputo fractional derivative. Boundary Value Problems, 2021, 2021,104: 1–18.
- 27. Ahmed A M S. Implicit Hilfer-Katugampula-type fractional pantograph differential equations with nonlocal Katugampola fractional integral condition. Palestine Journal of Mathematics, 2022, 11(3): 74–85.
- 28. Thabet S T M, Vivas-Cortez M and Kedim I. Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function. AIMS Mathematics, 2023, 8(10): 23635–23654.
- 29. Thabet S T M, Vivas-Cortez M, Kedim I, Samei M E and Ayari M I. Solvability of a ϱ-Hilfer fractional snap dynamic system on unbounded domains. Fractal Fract., 2023, 2023(7): 607.
- 30. Ali A, Mahariq I, Shah K, Abdeljawad T and Al-Sheikh B. Stability analysis of initial value problem of pantograph-type implicit fractional differential equations with impulsive conditions. Advances in Difference Equations, 2021, 2021,55: 1–17.
- 31. Asma , Shabbir S, Shah K and Abdeljawad T. Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative. Advances in Difference Equations, 2021, 2021,395. pmid:34456987
- 32. Shah K, Sher M, Ali A and Abdeljawad T. On degree theory for non-monotone type fractional order delay differential equations. AIMS Math., 2022, 7(5): 9479–9492.
- 33. Ertürk V S, Ali A, Shah K, Kumar P and Abdeljawad T. Existence and stability results for nonlocal boundary value problems of fractional order. Boundary Value Problems, 2022, 2022,25: 1–15.
- 34. Almeida R, Malinowska A B and Monteiro M T T. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Mathematical Methods in the Applied Sciences, 2018, 41(1): 336–352.
- 35.
Guo D, Lakshmikantham V and Liu X. Nonlinear integral equations in abstract spaces, mathematics and its applications. Netherlands: Kluwer Academic Publishers; 1996.
- 36. Isaia F. On a nonlinear integral equation without compactness. Acta. Math. Univ. Comen., 2006, 75: 233–240.
- 37. Wang C and Xu T. Hyers–Ulam stability of fractional linear differential equations involving Caputo fractional derivatives. Appl. Math., 2015, 60: 383–393.