All articles published by MDPI are made immediately available worldwide under an open access license. No special
permission is required to reuse all or part of the article published by MDPI, including figures and tables. For
articles published under an open access Creative Common CC BY license, any part of the article may be reused without
permission provided that the original article is clearly cited. For more information, please refer to
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature
Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for
future research directions and describes possible research applications.
Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive
positive feedback from the reviewers.
Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world.
Editors select a small number of articles recently published in the journal that they believe will be particularly
interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the
most exciting work published in the various research areas of the journal.
In this study, we deal with an impulsive boundary value problem (BVP) for differential equations of variable fractional order involving the Caputo–Hadamard fractional derivative. The fundamental problems of existence and uniqueness of solutions are studied, and new existence and uniqueness results are established in the form of two fixed point theorems. In addition, Ulam–Hyers stability sufficient conditions are proved illustrating the suitability of the derived fundamental results. The obtained results are supported also by an example. Finally, the conclusion notes are highlighted.
The idea of fractional-order integration and differentiation goes back to sixteenth century [1,2]. Since then, the attention to fractional calculus greatly increased in relation to modeling and control of numerous real processes using functions with fractional derivatives. Such a generalization of the notion allows the use of the important features of fractional derivatives, such as more degrees of freedom and infinite memory. In fact, fractional-order systems are characterized by infinite memory, as opposed to integer-order systems. Additionally, fractional integrals can be used to describe the fractal media .
In recent years, fractional differential equations have been actively used as models of numerous real-world phenomena studied in science, biology, engineering, and economics. In fact, fractional-order differential equations are widely applied in material and quantum mechanics, signal processing and systems identification, anomalous diffusion, wave propagation, etc. [4,5]. The efficient use of these equations in mathematical modeling requires the development of their fundamental and qualitative theories [6,7,8]. The progress in this development is related to the investigation of various type of fractional derivatives, such as Riemann–Liouville, Caputo, Hadamard, Grunwald–Letnikov, Marchaud, and Riesz just to name a few . The books [6,8] provide an excellent summary on the subject.
Recently, the topic of fractional equations has been expanded and different new classes of equations have been introduced. One of the most extensively studied classes of fractional differential equations is the class of fractional equations with variable fractional order [9,10]. Different researchers studied properties of fractional equations with variable fractional order of Riemann–Liouville type [11,12,13], Caputo type [14,15,16], and Hadamard type [17,18,19]. The enormous interest in these equations is due mainly to the extended possibilities of their applications [20,21,22]. A very good overview of some main applications of variable-order fractional operators has been given in .
The applicability of the hybrid Caputo–Hadamard type fractional derivatives is the main reason for the research interest in their theory [24,25,26]. Very recently, a few authors have studied the properties of the extension of such derivatives to a variable order [27,28].
From the other side, the apparatus of impulsive differential equations has been widely used in the description of processes with abrupt changes during their evolution [29,30,31]. The theory of such equations is also very well applied in impulsive control problems .
Additionally, the theory of impulsive differential equations has been extended to the fractional-order case. Numerous impulsive fractional-order systems with constant fractional derivatives have been proposed and their dynamical properties have been studied [33,34,35,36,37,38].
In particular, Benchohra et al. studied in  the following problem
where are given functions, , illustrates the Caputo fractional derivative of a constant order , , given as
for a function and denotes the Gamma function.
Correspondingly, results on impulsive variable-order fractional differential equations are reported very seldom . In addition, the existing results on impulsive fractional differential equations involving constant Caputo–Hadamard type derivatives are very few [41,42]. There are no results reported for impulsive fractional Caputo–Hadamard fractional differential equations with variable order fractional derivatives. The aim of the presented research is to introduce some fundamental results for such equations. We expect that our contribution will motivate more researchers to develop the theory.
Inspired by [11,14,17,19,37,39,40,41], we deal with the following impulsive boundary value problem (BVP)
where , are continuous functions, and illustrate the Caputo–Hadamard derivative and the Hadamard integral operators of variable order , respectively, , and c are real constants with , , , and represent the right-hand side and left-hand side limits of at , respectively.
The main contributions of our research are:
1. We introduce a BVP for a class of impulsive Caputo–Hadamard fractional differential equations with fractional derivatives of variable order;
2. Existence and uniqueness criteria for the introduced BVPs are established;
3. As an application, results on Ulam–Hyers stability of the solutions are proposed;
4. An example is developed to demonstrate our results.
The organization of the rest of this paper is as follows. Some definitions and auxiliary results are given in Section 2. In Section 3, the main existence and uniqueness results for solutions of the BVP (1)–(3) of impulsive fractional Caputo–Hadamard fractional differential equations with fractional derivatives of variable order are proposed. The criteria proposed are presented in the form of two fixed point theorems. Section 4 is devoted to our main Ulam–Hyers stability results. One example is presented in Section 5, to show the efficiency and validity of the proposed results. Finally, some conclusion notes are given in Section 6.
2. Auxiliary Results
In this section, we list some definitions and propositions that are used in the following sections.
For we denote by the set
and for , , we denote by the set
Consider the sets of functions
, there exist and with ;
, there exist and with .
Note that the set is a Banach space with a norm defined as
For , we consider the mapping . The Hadamard fractional integral (HFI) of variable order for [9,27,28] is
and the Caputo-type Hadamard fractional derivative (CHFD) of variable order for  is
It is clear that, if is a constant function, , then HFI and CHFD are reduced to the classical Hadamard integral and Caputo-type Hadamard derivative , respectively [9,27].
Next, we will present some important properties of and .
(a) The interval I is called a generalized interval if it is either an interval or or ∅.
(b) A partition of I is a finite set such that each x in I lies in exactly one of the generalized intervals E in .
(c) A function is called piecewise constant with respect to the partition of I if for any , g is constant on E.
().(Arzela–Ascoli theorem) Let Λ be a subset of . Λ is relatively compact if:
Λ is uniformly bounded.
Λ is équicontinuous.
The following fixed point theorem will be used in the proof of our main results.
().(Schauder fixed point theorem) Let Λ be a convex subset of a Banach space E and be a continuous and compact map. Then, possesses a fixed point in Λ.
Finally, we will extend the definition in  as follows:
The BVP (1)–(3) is Ulam–Hyers (UH) stable if there exists such that for any and for every solution satisfying
there exists a solution of the BVP (1)–(3), such that
3. Main Existence and Uniqueness Results
Let us introduce the following assumption:
Let be a partition of the interval (with ) and let be a piecewise constant function with respect to and , i.e.,
where are constants, and
In addition, we will give the definition of the solution to the BVP (1)–(3).
The function is a solution of the BVP (1)–(3) if x fulfills the equation on and the conditions
First, we will analyze the Equation (1) of the BVP (1)–(3). For any , it becomes a Caputo–Hadamard fractional differential equation of a variable order for , with CHFD given by (5). Then, for the sum, we have
Thus, according to (9), the Equation (1) can be written for any in the form
In the case, when on , the Equation (1) is reduced to
We need the following auxiliary proposition.
Let be continuous. The solution of the following impulsive BVP
is given by
Let x be a solution of the BVP (11)–(13). If , by Proposition 1, we get
If , then Proposition 1 implies
If , by Proposition 1, we get
Then, the solution for can be written as
Applying the boundary conditions , we have
Conversely, we can easily show that x solves the BVP (11)–(12). □
Now, we present our first result, assuming that the following assumptions are satisfied:
For , the function is continuous and there exist constants , such that, for any and .
The operator S defined in (16) is well defined from the continuity of function and from the properties of fractional integrals.
Let the set
Clearly is non-empty, closed, convex, and bounded.
Step 1: Claim: .
For , we get
Step 2: Claim: S is continuous.
Let the sequence converges to x in . We will prove that
For , we have
Since is continuous, then
Then, S is continuous.
Step 3: Claim: S is compact.
By Step 1, we have for each , which means that is bounded. Now we will show that is equicontinuous.
For and , we have
As the right-hand side of the above inequality tends to zero. Hence . It implies that is equicontinuous.
Thus, by Theorem 2 the BVP (1)–(3) possesses a solution in . Since the claim of Theorem 3 is proved. □
Now, we will invoke the Banach contraction principle to verify the uniqueness of solutions for the BVP (1)–(3).
In addition to (A1) and (A2), assume that:
For , there exists such that, for any and,
the BVP (1)–(3) possesses a solution uniquely determined on .
For and , we have
Ergo, by (17), the operator S forms a contraction. Thus, S involves a fixed point uniquely which is the unique solution of the BVP (1)–(3). □
Theorems 3 and 4 offer existence and uniqueness results for impulsive systems of fractional differential equations of the hybrid Caputo–Hadamard type with variable order. These results extend the results for differential equations with variable Caputo–Hadamard type fractional derivatives [27,28] to the impulsive case. Additionally, our results extend and complement some recently published results on impulsive fractional differential equations involving Caputo–Hadamard type constant order derivatives [41,42] considering variable order fractional differential equations.
4. Ulam–Hyers Stability
To apply the obtained existence and uniqueness results, in this section we will consider the Ulam–Hyers stability of solutions of the BVP (1)–(3).
Consider the hypotheses of Theorem 4. Then, the BVP (1)–(3) is (UH) stable.
Assume satisfies the inequality (8). Then the integral inequality
According to Proposition 5, for the unique solution x of the BVP (1)–(3) is given by
In , the authors studied the Ulam–Hyers stability of a class of Hadamard fractional differential equations with integral boundary value condition and impulses. Our Ulam–Hyers stability result extend and generalized these results to the case of variable order of fractional derivatives considering the hybrid type of Caputo–Hadamard derivatives. In fact, fractional derivatives of variable order are more general and expand the possibilities for applications of fractional-order models. Additionally, the stability result presented in this section show the applicability of the existence and uniqueness criteria established in Theorems 3 and 4 in the investigation of the qualitative properties of the introduced BVP (1)–(3).
5. An Example
As an example, consider the following impulsive BVP,
For each and , we have
Thus, assumption (A2) is satisfied with and .
For all , we have
Then, the assumption (A4) holds with .
We will also check that assumption (17) is fulfilled with , , , , and . Indeed,
Hence, assumption (17) is satisfied. By Theorem 4, the BVP (18)–(20) has a unique solution on .
In addition, according to Theorem 5, the BVP (18)–(20) is (UH) stable.
The elaborated example again demonstrates the efficiency of our existence and uniqueness results. Additionally, it shows that the obtained fundamental criteria for differential equations with Caputo-type Hadamard derivatives of variable order and impulses can be easily applied in the study of their the Ulam–Hyers stability properties. Since the proposed criteria are represented as algebraic inequalities, they can be easily applied in the investigation of other qualitative properties of such equations.
In this paper, we introduce a BVP for impulsive differential equations with Caputo–Hadamard fractional derivatives of variable order. We study the existence and uniqueness of solutions of the proposed fractional BVP. The obtained new results extend and complement some existing results on Caputo–Hadamard differential equations with constant-order fractional derivatives. The proposed existence and uniqueness criteria are also applied to establish Ulam–Hyers stability results. One example is presented to show the validity and applicability of the obtained results. The fundamental results presented in this paper open up many possibilities for future research. The obtained results can be applied in the qualitative study of the introduced fractional-order systems, such as stability, periodicity, almost periodicity, oscillations, asymptotic behavior, etc. In addition, it is possible to extend the proposed results to the uncertain case and study robust stability of such systems with uncertain terms. An important future topic is to apply the derived Ulam–Hyers stability results to fractional neural networks with Caputo–Hadamard fractional derivatives of variable order. An interesting future direction of research is to extend and implement the developed results to the class of fractional-order octonion-valued bidirectional associative memory neural networks introduced in  considering impulsive perturbations and variable order Caputo–Hadamard fractional derivatives. In addition, an analysis on the global Mittag–Leffler stability and synchronization problems for the established class of impulsive fractional differential equations and related neural network systems can be provided.
Conceptualization, A.B. and M.S.S.; methodology, A.B., M.S.S., G.S. and I.S.; formal analysis, A.B., M.S.S., G.S. and I.S.; investigation, A.B., M.S.S., G.S. and I.S.; writing—original draft preparation, I.S. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
Das, A.; Hazarika, B.; Parvaneh, V.; Mursaleen, M. Solvability of generalized fractional order integral equations via measures of noncompactness. Math. Sci.2021, 15, 241–251. [Google Scholar] [CrossRef]
Katugampola, U.N. A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl.2014, 6, 1–15. [Google Scholar]
Tarasov, V.E. Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, 1st ed.; Springer: Beijing, China, 2015; ISBN 978-3-642-14003-7. [Google Scholar]
Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Fractional Calculus: Models and Numerical Methods, 1st ed.; World Scientific: Singapore, 2012; ISBN 978-981-4355-20-9. [Google Scholar]
Magin, R. Fractional Calculus in Bioengineering, 1st ed.; Begell House: Redding, CA, USA, 2006; ISBN 978-1567002157. [Google Scholar]
Kilbas, A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations, 1st ed.; Elsevier: New York, NY, USA, 2006; ISBN 9780444518323. [Google Scholar]
Petráš, I. Fractional-Order Nonlinear Systems, 1st ed.; Springer: Heidelberg, Germany; Dordrecht, The Netherlands; London, UK; New York, NY, USA, 2011; ISBN 978-3-642-18101-6. [Google Scholar]
Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications, 1st ed.; Gordon and Breach: Yverdon, Switzerland, 1993; ISBN 9782881248641. [Google Scholar]
Almeida, R.; Tavares, D.; Torres, D.F.M. The Variable-Order Fractional Calculus of Variations, 1st ed.; Springer: Cham, Switzerland, 2019; ISBN 978-3-319-94005-2. [Google Scholar]
Samko, S.G. Fractional integration and differentiation of variable order. Anal. Math.1995, 21, 213–236. [Google Scholar] [CrossRef]
Benkerrouche, A.; Souid, M.S.; Stamov, G.; Stamova, I. On the solutions of a quadratic integral equation of the Urysohn type of fractional variable order. Entropy2022, 24, 886. [Google Scholar] [CrossRef]
Zhang, S.; Hu, L. The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order. AIMS Math.2020, 5, 2923–2943. [Google Scholar] [CrossRef]
Zhang, S.; Sun, S.; Hu, L. Approximate solutions to initial value problem for differential equation of variable order. J. Fract. Calc. Appl.2018, 9, 93–112. [Google Scholar]
Benkerrouche, A.; Souid, M.S.; Sitthithakerngkiet, K.; Hakem, A. Implicit nonlinear fractional differential equations of variable order. Bound. Value Probl.2021, 2021, 64. [Google Scholar] [CrossRef]
Odzijewicz, T.; Malinowska, A.B.; Torres, D.F.M. Fractional variational calculus of variable order. In Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, 1st ed.; Almeida, A., Castro, L., Speck, F.O., Eds.; Birkhäuser: Basel, Switzerland, 2013; Volume 229, pp. 291–301. [Google Scholar]
Sarwar, S. On the existence and stability of variable order Caputo type fractional differential equations. Fractal Fract.2022, 6, 51. [Google Scholar] [CrossRef]
Benkerrouche, A.; Souid, M.S.; Karapinar, E.; Hakem, A. On the boundary value problems of Hadamard fractional differential equations of variable order. Math. Methods Appl. Sci.2022. [Google Scholar] [CrossRef]
Benkerrouche, A.; Souid, M.S.; Etemad, S.; Hakem, A.; Agarwal, P.; Rezapour, S.; Ntouyas, S.K.; Tariboon, J. Qualitative study on solutions of a Hadamard variable order boundary problem via the Ulam-Hyers-Rassias stability. Fractal Fract.2021, 5, 108. [Google Scholar] [CrossRef]
Refice, A.; Souid, M.S.; Stamova, I. On the boundary value problems of Hadamard fractional differential equations of variable order via Kuratowski MNC technique. Mathematics2021, 9, 1134. [Google Scholar] [CrossRef]
Samko, S. Fractional integration and differentiation of variable order: An overview. Nonlinear Dyn.2013, 71, 653–662. [Google Scholar] [CrossRef]
Kalidass, M.; Zeng, S.; Yavuz, M. Stability of fractional-order quasi-linear impulsive integro-differential systems with multiple delays. Axioms2022, 11, 308. [Google Scholar] [CrossRef]
Karthikeyan, P.; Arul, R. Uniqueness and stability results for non-local impulsive implicit Hadamard fractional differential equations. J. Appl. Nonlinear Dyn.2020, 9, 23–29. [Google Scholar] [CrossRef]
Kumar, V.; Kostić, M.; Malik, M.; Debouche, A. Controllability of switched Hilfer neutral fractional dynamic systems with impulses. IMA J. Math. Control Inf.2022, 39, 807–836. [Google Scholar] [CrossRef]
Kumar, V.; Malik, M.; Debouche, A. Stability and controllability analysis of fractional damped differential system with non-instantaneous impulses. Appl. Math. Comput.2021, 391, 125633. [Google Scholar] [CrossRef]
Stamova, I.M.; Stamov, G.T. Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, 1st ed.; Taylor & Francis Group: Boca Raton, FL, USA, 2017; ISBN 9781498764834. [Google Scholar]
Wang, J.; Feckan, M.; Zhou, Y. A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal.2016, 19, 806–831. [Google Scholar] [CrossRef]
Benchohra, M.; Seba, D. Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ.2009, 8, 1–14. [Google Scholar] [CrossRef]
Duc, T.M.; Hoa, N.V. Stabilization of impulsive fractional-order dynamic systems involving the Caputo fractional derivative of variable-order via a linear feedback controller. Chaos Solitons Fract.2021, 153, 111525. [Google Scholar] [CrossRef]
Zhang, X. The general solution of differential equations with Caputo–Hadamard fractional derivatives and impulsive effect. Adv. Differ. Equ.2015, 2015, 215. [Google Scholar] [CrossRef][Green Version]
Zhao, K.; Ma, S. Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses. AIMS Math.2021, 7, 3169–3185. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely
those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or
the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas,
methods, instructions or products referred to in the content.