Abstract
In this paper, we study a coupled fully hybrid system of –Hilfer fractional differential equations equipped with non-symmetric –Riemann-Liouville () integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space (). Moreover, the Ulam–Hyers () stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results.
Keywords:
(k, Φ)–Hilfer fractional derivative; existence; nonlinear analysis; Ulam stability; generalized Banach spaces; Lipschitzian matrix MSC:
34A08; 26A33; 34A34
1. Introduction
Fractional differential equations (FDEs) are equations that include fractional-order derivatives instead of classical integer-order derivatives. There are several types of fractional derivative definitions that have appeared in area of fractional calculus, for example, the Riemann-Liouville (), Hadamard, Grunwald-Letnikov, Caputo, Caputo–Fabrizo, and Atangana–Baleanu–Caputo derivatives [,,,,]. Indeed, FDEs have a large flood of applications in different scopes such as chemistry, physics, finance, engineering, and infectious disease. The combination of FDEs and other analytical and numerical methods can be found in many works such as impulsive FDEs [,], implicit hybrid FDEs [,,], mathematical modelings with the help of FDEs [,,,,], neutral FDEs [,], p-Laplacian FDEs [], variable order time-fractional FDEs [], random and fuzzy FDEs [,], integro-differential inclusions [,], and references therein.
In 2012, -fractional integral was extended by Mubben et al. [] to k--fractional integral. Later, in 2018, Kwun et al. [] introduced the – definition for these operators; then, Kucche and Mali presented the most generalized operator named the –Hilfer fractional operator [], which attracted the attention of many authors such as Samadi et al. [] who studied the existence of solutions for the coupled –Hilfer nonlinear FDEs with – integral conditions. Additionally, Tariboon et al. [] employed the Krasnoselskii, Banach, and Leray-Schauder theorems to study the qualitative properties of –Hilfer FDEs and inclusions with multi-point boundary conditions. Recently, in [], Kamsrisuk et al. investigated the existence and uniqueness results of multi-point non-local –Hilfer FDEs via the fixed-point method.
Over the course of many years, stability was utilized to examine the behavior of solutions for FDEs, and it can be discussed by employing fixed-point methods or by comparing the distance between the solutions of the primary equation and the so-called linearization equation, which relates to the primary equation. We also study this notion for solutions to our proposed system with a special technique. We mention some papers devoted to the study of stability [,,,].
Inspired by the aforementioned works, in this paper, we study the following coupled fully hybrid system of –Hilfer FDEs:
with –fractional integrals conditions
where
is the –Hilfer fractional derivative of orders and types , is the –-fractional integrals of order and , and are continuous functions.
This research is the first paper in which we analyze the uniqueness and existence properties in connection to solutions of a coupled fully hybrid system of –Hilfer-fractional BVPs of FDEs with newly defined –Hilfer-fractional operators. In view of the nature of these operators, our results will cover all of the previous studies in special cases. It is sufficient that we take and ; then, we obtain the classical standard Hilfer fractional derivative. The main technique of this paper for the existence property is to use of Lipschitzian matrices and the Perov theorem. Additionally, another contribution of this paper is that the criterion of Urs is used for studying the stability in combination with –Hilfer-fractional operators. These items constitute the novelty of this paper in comparison to other studies.
This paper is organized as follows: several definitions and preliminaries in connection to these new operators are given in Section 2. The existence and uniqueness of the solutions of the –Hilfer-fractional fully hybrid BVPs of FDEs (1)–(2) are proved in by employing fixed-point theorem techniques in Section 3. In addition, the stability of the solution is established. In Section 4, an application of the main results is illustrated and examined by an example.
2. Background Notions
In this section, we present some notions and definitions that will be used to investigate the desired results.
Assume that is used for the description of the Banach space of each continuous function with the norm Let with . Then, means , and if , then means . Set Moreover, we take
Definition 1
([]). Consider a (increasing) function Φ from into s.t. . Then, the --fractional integral of order for the function is
where the k-Gamma function is formulated by
with some properties such as
Definition 2
([]). Suppose that . Then, the -Hilfer derivative of order with the type for the function is given as
Remark 1.
For s.t. and s.t. and , the -Hilfer fractional derivative can be reformultaed in the sense of --fractional derivative as the following form:
In the next lemmas, we provide some properties of -fractional operators.
Lemma 1
([]). Let . With the same assumptions given in the above remark, we have
Lemma 2
([]). For and with the above assumptions, let and Then,
Lemma 3
([]). Let and s.t. . We have
- (i)
- .
- (ii)
- .
Definition 3
([]). A real square matrix convergent to zero iff its spectral radius is precise less than 1; this means that with for each and represents the unit matrix of .
Theorem 1
([]). Let be a non-negative square matrix. Then, the following items are equivalent:
- (i)
- As , ;
- (ii)
- The spectral radius ;
- (iii)
- is non-singular and
- (iv)
- The matrices and are non-singular and non-negative, respectively.
Definition 4
([,]). Let the generalized metric space be denoted by . If there is a matrix converging to zero, then the mapping is contractive, where
Now, we recall two fixed-point theorems that will be used in the next sections.
Theorem 2
([,]). Let complete generalized metric space be If the mapping is a contractive with Lipschitz’s matrix , then Π possesses one and only one fixed point , and , we obtain
Theorem 3
([]). Assume that Ψ be a convex, closed, non-empty subset of a Ξ. Let Π and Υ map Ψ into Ξ such that
- (i)
- ;
- (ii)
- The mapping Π is continuous and compact;
- (iii)
- The mapping Υ is an -contraction.
Then, possesses at least one solution on Ψ.
3. Qualitative Results
Throughout this section, we prove the existence, uniqueness, and stability of solutions for the coupled fully hybrid system of –Hilfer BVPs (1)–(2).
Now, in order to establish qualitative results of the mentioned system (1)–(2), we need to provide the following lemma. In this lemma, we derive the main structure of solution in form of an integral equation.
Lemma 4.
If the solution of the fully hybrid –Hilfer BVP given by
exists, then it is equivalent to the integral equation
Proof.
Let be a solution of the problem (3). Integrating on (3) and then, using the properties of the fractional operators, we obtain
and (4) can be obtained. Secondly, let be a solution of the integral Equation (4). Then,
and
This completes the proof. □
In view of Lemma 4, we need to present the following lemma, which plays a key role in the main theorems. In fact, this lemma shows the solution of the given system via two integral equations.
Lemma 5.
Now, the product space is a with the following norm:
Additionally, let the operator define
with
and
For computational convenience, we introduce the following notations:
Let us list the following hypotheses:
- (HP1)
- For , the functions and are bounded on the subject to bounds and , respectively.
- (HP2)
- and , , exist, where
- (HP3)
- where
Next, we are in a position to investigate and prove the uniqueness result by using Perov’s fixed-point theorem.
Theorem 4.
Proof.
In order to show that has exactly one fixed point, we will use Perov’s fixed-point theorem. Indeed, we prove that the mapping is an -contraction on . For given , and , using and , we can obtain
Hence,
By the same technique, we can also obtain
The following result is achieved based on the Krasnoselskii’s Theorem 3. In fact, here we prove the existence result with the help of the Krasnoselskii’s fixed-point theorem in a generalized Banach space.
Theorem 5.
Proof.
In order to use Theorem (3), we need to take a set such that is closed, convex, bounded, and defined as
with such that
where , and are non-negative real numbers that will be specified later.
Now, consider the mappings and on as
and
It is obvious that both and are well-defined. Moreover, by Lemma 5 the mappings form the system (5) as
Our purpose is to confirm this fact that and fulfill all properties of Theorem 3. For better clarity, the proof is broken down into three steps.
Step 1:, .
In fact, from , for , we can obtain
Hence,
By similar procedure, we obtain
In a similar way, we obtain
where
Therefore, we check for such that . Regarding this, in view of (14), it is sufficient to verify that
where and
Equivalently,
Since the spectral radius of is <1, according to Theorem 1, we have the matrix is non-singular and has positive elements. So, (15) is equal to
In addition, if we take
thus, we find
Therefore, .
Step 2: The mapping is -contraction on .
Indeed, and for any by a similar procedure in the proof of Theorem 4, it is not difficult to verify that
Since the spectral radius of is <1, the mapping is an -contraction on .
Step 3: The mapping is continuous and compact.
By the continuity of and , we deduce that is continuous. Moreover, we show that is uniformly bounded on . From (12), and we find that
This means that the mapping is uniformly bounded on .
At the last step, we are going to prove that is equicontinuous. From the hypotheses and , for and for any , we obtain
Similarly,
Therefore,
Thus, we deduce that is equicontinuous. Due to Arzelà–Ascoli’s theorem, we conclude that the mapping is compact. Hence, the requirements of Theorem 3 are fulfilled. Thus, in view of the Krasnoselskii’s FPT, we derive that the mapping defined by (9) possesses at least one fixed point , which is the solution of the coupled fully hybrid system of –Hilfer BVPs (1)–(2). □
Now, we end this section by discussing the stability of the coupled fully hybrid system of –Hilfer BVPs (1)–(2) by utilizing its solution in the sense of integral form given as
such that and are given in (6) and (7).
Let us define the following mappings as:
In addition, we assume that the next inequalities
for some are to be held.
Definition 5
In this part of the paper, we aim to prove that the given coupled fully hybrid system (1)–(2) is -stable. To do this, we use Urs’s technique.
Theorem 6.
Proof.
Let be the solution of the coupled fully hybrid system of –Hilfer BVPs (1)–(2) satisfying (6) and (7). Assume that is any solution verifying (16):
So,
and
Hence, we obtain
Similarly, we have
Inequalities (21) and(22) can be rewritten in a matrix form as
where is the matrix given by (8). Since the spectral radius of is by Theorem 1, we deduce that is non-singular and possesses positive elements. Hence, (23) is equivalent to the form
which yields that
where are the elements of . Consequently, the coupled fully hybrid system of –Hilfer BVPs (1)–(2) is -stable. □
4. Applications
We provide an example in this part to investigate and guarantee the validity of the results.
Example 1.
Consider the following coupled fully hybrid system of –Hilfer BVPs:
with –fractional integrals conditions
where
Here, and the functions
Obviously, the functions are continuous. Furthermore, for all and each we have satisfied as follows:
where
Additionally, is satisfied for
and we can calculate that
Thus, we obtain
5. Conclusions
This research paper was devoted to studying a coupled fully hybrid system of quadratic differential equations in the sense of the –Hilfer fractional derivative with subject to the – fractional integral conditions. The existence and uniqueness of solutions for such a system were established by utilizing the Perov and Krasnoselskii fixed-point theorems in . Moreover, stability was proved by Urs’s technique. Finally, an example was provided for checking the validity of our results. For the next research projects, we would like to extend our methods and techniques in the context of post-quantum calculus. One can combine these methods with numerical techniques for studying the dynamics of the solutions based on -operators.
Author Contributions
Conceptualization, A.B. and S.E.; formal analysis, A.B., S.E., S.T.M.T., S.K.N., S.R. and J.T.; funding acquisition, J.T.; methodology, A.B., S.E., S.T.M.T., S.K.N., S.R. and J.T.; and software, S.E. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by the National Science, Research, and Innovation Fund (NSRF) and King Mongkut’s University of Technology North Bangkok with Contract no. KMUTNB-FF-66-11.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Acknowledgments
The second and fifth authors would like to thank Azarbaijan Shahid Madani University. Additionally, the authors would like to thank the respected and dear reviewers for providing useful comments to improve the quality of the paper.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Podlubny, I. Fractional Differential Equations, Mathematics in Science and Engineering; 198; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1, 73–85. [Google Scholar]
- Atangana, A.; Baleanu, D. New fractional derivative without nonlocal and nonsingular kernel: Theory and application to heat transfer model. Therm. Sci. 2016, 20, 763–769. [Google Scholar] [CrossRef]
- Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 2017, 44, 460–481. [Google Scholar] [CrossRef]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific Publishing Co.: River Edge, NJ, USA, 2000. [Google Scholar]
- Rodríguez-López, R.; Tersian, S. Multiple solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 2014, 17, 1016–1038. [Google Scholar] [CrossRef]
- Anguraj, A.; Karthikeyan, P. Anti-periodic boundary value problem for impulsive fractional integro differential equations. Acta Math. Hung. 2010, 13, 281–293. [Google Scholar]
- Boutiara, A.; Abdo, M.S.; Almalahi, M.A.; Ahmad, H.; Ishan, A. Implicit hybrid fractional boundary value problem via generalized Hilfer derivative. Symmetry 2021, 13, 1937. [Google Scholar] [CrossRef]
- Amara, A. Existence results for hybrid fractional differential equations with three-point boundary conditions. AIMS Math. 2020, 5, 1074–1088. [Google Scholar] [CrossRef]
- Khan, R.A.; Gul, S.; Jarad, F.; Khan, H. Existence results for a general class of sequential hybrid fractional differential equations. Adv. Differ. Equ. 2021, 2021, 284. [Google Scholar] [CrossRef]
- Thabet, S.T.M.; Abdo, M.S.; Shah, K. Theoretical and numerical analysis for transmission dynamics of COVID-19 mathematical model involving Caputo-–Fabrizio derivative. Adv. Differ. Equ. 2021, 2021, 184. [Google Scholar] [CrossRef]
- Thabet, S.T.M.; Abdo, M.S.; Shah, K.; Abdeljawad, T. Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative. Res. Phys. 2020, 19, 103507. [Google Scholar] [CrossRef]
- Baleanu, D.; Etemad, S.; Mohammadi, H.; Rezapour, S. A novel modeling of boundary value problems on the glucose graph. Commun. Nonlinear Sci. Numer. Simul. 2021, 100, 105844. [Google Scholar] [CrossRef]
- Etemad, S.; Rezapour, S. On the existence of solutions for fractional boundary value problems on the ethane graph. Adv. Differ. Equ. 2020, 2020, 276. [Google Scholar] [CrossRef]
- Amara, A.; Etemad, S.; Rezapour, S. Approximate solutions for a fractional hybrid initial value problem via the Caputo conformable derivative. Adv. Differ. Equ. 2020, 2020, 608. [Google Scholar] [CrossRef]
- Adjimi, N.; Boutiara, A.; Abdo, M.S.; Benbachir, M. Existence results for nonlinear neutral generalized Caputo fractional differential equations. J. Pseudo Differ. Oper. Appl. 2021, 12, 1–17. [Google Scholar] [CrossRef]
- Boutiara, A.; Matar, M.M.; Kaabar, M.K.A.; Martínez, F.; Etemad, S.; Rezapour, S. Some qualitative analyses of neutral functional delay differential equation with generalized Caputo operator. J. Funct. Spaces 2021, 2021, 9993177. [Google Scholar] [CrossRef]
- Tang, X.; Yan, C.; Liu, Q. Existence of solutions of two point boundary value problems for fractional p-Laplace differential equations at resonance. J. Appl. Math. Comput. 2013, 41, 119–131. [Google Scholar] [CrossRef]
- Abuasbeh, K.; Kanwal, A.; Shafqat, R.; Taufeeq, B.; Almulla, M.A.; Awadalla, M. A method for solving time-fractional initial boundary value problems of variable order. Symmetry 2023, 15, 519. [Google Scholar] [CrossRef]
- Abuasbeh, K.; Shafqat, R. Fractional Brownian motion for a system of fuzzy fractional stochastic differential equation. J. Math. 2022, 2022, 3559035. [Google Scholar] [CrossRef]
- Abuasbeh, K.; Shafqat, R.; Alsinai, A.; Awadalla, M. Analysis of controllability of fractional functional random integroevolution equations with delay. Symmetry 2023, 15, 290. [Google Scholar] [CrossRef]
- El-Sayed, A.; Hashem, H.; Al-Issa, S. Analysis of a hybrid integro-differential inclusion. Bound. Value Probl. 2022, 2022, 68. [Google Scholar] [CrossRef]
- Abbas, M.I.; Ghaderi, M.; Rezapour, S.; Thabet, S.T.M. On a coupled system of fractional differential equations via the generalized proportional fractional derivatives. J. Funct. Spaces 2022, 2022, 4779213. [Google Scholar] [CrossRef]
- Mubben, S.; Habibullah, G.M. k-fractional integrals and application. Int. J. Contemp. Math. Sci. 2012, 7, 89–94. [Google Scholar]
- Kwun, Y.C.; Farid, G.; Nazeer, W.; Ullah, S.; Kang, S.M. Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities. IEEE Access 2018, 6, 64946–64953. [Google Scholar] [CrossRef]
- Kucche, K.D.; Mali, A.D. On the nonlinear (k,Φ)-Hilfer fractional differential equations. Chaos Solitons Fractals 2021, 152, 111335. [Google Scholar] [CrossRef]
- Samadi, A.; Ntouyas, S.K.; Ahmad, B.; Tariboon, J. Investigation of a nonlinear coupled (k,Φ)–Hilfer fractional differential system with coupled (k,Φ)–Riemann–Liouville fractional integral boundary conditions. Foundations 2022, 2, 918–933. [Google Scholar] [CrossRef]
- Tariboon, J.; Samadi, A.; Ntouyas, S.K. Multi-point boundary value problems for (k,ϕ)-Hilfer fractional differential equations and inclusions. Axioms 2022, 11, 110. [Google Scholar] [CrossRef]
- Kamsrisuk, N.; Ntouyas, S.K.; Ahmad, B.; Samadi, A.; Tariboon, J. Existence results for a coupled system of (k,φ)-Hilfer fractional differential equations with nonlocal integro-multi-point boundary conditions. AIMS Math. 2023, 8, 4079–4097. [Google Scholar] [CrossRef]
- Awadalla, M.; Subramanian, M.; Abuasbeh, K. Existence and Ulam–Hyers stability results for a system of coupled generalized Liouville–Caputo fractional Langevin equations with multipoint boundary conditions. Symmetry 2023, 15, 198. [Google Scholar] [CrossRef]
- Baitiche, Z.; Derbazi, C.; Benchohra, M.; Zhou, Y. A new class of coupled systems of nonlinear hyperbolic partial fractional differential equations in generalized Banach spaces involving the ψ–Caputo fractional derivative. Symmetry 2021, 13, 2412. [Google Scholar] [CrossRef]
- Rezapour, S.; Thabet, S.T.M.; Matar, M.M.; Alzabut, J.; Etemad, S. Some existence and stability criteria to a generalized FBVP having fractional composite p-Laplacian operator. J. Funct. Spaces 2021, 2021, 9554076. [Google Scholar] [CrossRef]
- Boutiara, A.; Etemad, S.; Hussain, A.; Rezapour, S. The generalized U–H and U–H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving ϕ-Caputo fractional operators. Adv. Differ. Equ. 2021, 2021, 95. [Google Scholar] [CrossRef]
- Varga, R.S. Matrix Iterative Analysis, 2nd revised and expanded ed.; Springer Series in Computational Mathematics, 27; Springer: Berlin, Germany, 2000. [Google Scholar]
- Precup, R.; Viorel, A. Existence results for systems of nonlinear evolution equations. Int. J. Pure Appl. Math. 2008, 47, 199–206. [Google Scholar]
- Rus, I.A. Generalized Contractions and Applications; Cluj University Press: Cluj, Romania, 2001. [Google Scholar]
- Perov, A.I. On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uvavn. Vyp. 1964, 2, 115–134. [Google Scholar]
- Petre, I.R.; Petruşel, A. Krasnoselskii’s theorem in generalized Banach spaces and applications. Electron. J. Qual. Theory Differ. Equ. 2012, 85, 20. [Google Scholar] [CrossRef]
- Urs, C. Coupled fixed-point theorems and applications to periodic boundary value problems. Miskolc Math. Notes 2013, 14, 323–333. [Google Scholar] [CrossRef]
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