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Received: 20 February 2019 / Accepted: 10 April 2019 / Published: 16 April 2019
The current article studies a coupled system of fractional differential equations with boundary conditions and proves the existence and uniqueness of solutions by applying Leray-Schauder’s alternative and contraction mapping principle. Furthermore, the Hyers-Ulam stability of solutions is discussed and sufficient conditions for the stability are developed. Obtained results are supported by examples and illustrated in the last section.
fractional derivative; fixed point theorem; fractional differential equation
Fractional calculus is undoubtedly one of the very fast-growing fields of modern mathematics, due to its broad range of applications in various fields of science and its unique efficiency in modeling complex phenomena [1,2]. In particular, fractional differential equations with boundary conditions are widely employed to build complex mathematical models for numerous real-life problems such as blood flow problem, underground water flow, population dynamics, and bioengineering. As an example, consider the following equation that describes a thermostat model
where and is a positive constant. Note that solutions of the above equation with the specified integral boundary conditions are in fact solutions of the one-dimensional heat equation describing a heated bar with a controller at point 1, which increases or reduces heat based on the temperature picked by a sensor at η. A few of the relevant studies on coupled systems of fractional differential equations with integral boundary conditions are briefly reviewed below and for further information on this topic, refer to References [3,4].
In Reference , Ntouyas and Obaid used Leray-Schauder’s alternative and Banach’s fixed-point theorem to prove the existence and uniqueness of solutions for the following coupled fractional differential equations with Riemann-Liouville integral boundary conditions:
Here, and are Caputo fractional derivatives, , and .
Similarly, Ahmed and Ntouyas  employed Banach fixed-point theorem and Leray-Schauder’s alternative to prove the existence and uniqueness of solutions for the following coupled fractional differential system:
supplemented with coupled and uncoupled slit-strips-type integral boundary conditions, respectively, given by
Furthermore, Alsulami et al.  investigated the following coupled system of fractional differential equations:
subject to the following non-separated coupled boundary conditions:
Note that and denote Caputo fractional derivatives of order α and β. Moreover, , , are real constants with and are appropriately chosen functions. For further details on this topic, refer to References [8,9,10,11,12,13,14,15,16,17,18,19,20,21].
The current paper studies the following coupled system of nonlinear fractional differential equations:
supplemented with boundary conditions of the form:
Here, denotes Caputo fractional derivative of order and are given continuous functions. Note that are real constants such that .
The rest of this paper is organized in the following manner: In Section 2, we briefly review some of the relevant definitions from fractional calculus and prove an auxiliary lemma that will be used later. Section 3 deals with proving the existence and uniqueness of solutions for the given problem, and Section 4 discusses the Hyers-Ulam stability of solutions and presents sufficient conditions for the stability. The paper concludes with supporting examples and obtained results.
We begin this section by reviewing the definitions of fractional derivative and integral [1,2].
The Riemann-Liouville fractional integral of orderfor a continuous functionis given by
provided that the right-hand side is point-wise defined on
The Caputo fractional derivatives of orderfor—times absolutely continuous functionis defined as
whereis the integer part of real number.
Here we prove the following auxiliary lemma that will be used in the next section.
Letthen the unique solution for the problem
General solutions of the fractional differential equations in (3) are known  as
where are arbitrary constants.
Apply conditions and , and we obtain .
Considering boundary conditions
Hence, by substituting the value of a into c, we obtain the final result for these constants as
Substituting the values of in (6) and (7) we get (4) and (5). The converse follows by direct computation. This completes the proof. □
3. Existence and Uniqueness of Solutions
Consider the space endowed with norm . Consequently, the product space is a Banach Space (endowed with ).
In view of Lemma 1, we define the operator as:
Here we establish the existence of the solutions for the boundary value problem (1) and (2) by using Banach’s contraction mapping principle.
Assumeare jointly continuous functions and there exist constants, such that, we have
then the BVP (1) and (2) has a unique solution on. Here
Define and , and , such that
Firstly, we show that .
By our assumption, for we have
which lead to
In a similar manner:
and we get that is .
Now let .
Then we have
From (11) and (12) we have
Since therefore, the operator is a contraction operator. Hence, by Banach’s fixed-point theorem, the operator has a unique fixed point, which is the unique solution of the BVP (1) and (2). This completes the proof. □
Next we will prove the existence of solutions by applying the Leray-Schauder alternative.
“(Leray-Schauder alternative , p. 4) Letbe a completely continuous operator (i.e., a map restricted to any bounded set inis compact). Let. Then either the setis unbounded orhas at least one fixed point)”.
Assumeare continuous functions and there existwhereare real constants andsuch that, we have
whereare defined in (10), then the problem (1) and (2) has at least one solution.
This proof will be presented in two steps.
Step 1: We will show that is completely continuous. The continuity of the operator holds by the continuity of the functions .
Let be bounded. Then there exists positive constants such that
Then and we have
Thus, from the above inequalities, it follows that the operator is uniformly bounded, since
Next, we will show that operator is equicontinuous. Let with This yields
And we obtain
Hence, we have independent of and as Furthermore, we obtain
which implies that independent of and as
Therefore, operator is equicontinuous, and thus is completely continuous.
Step 2: (Boundedness of operator)
Finally, we will show that is bounded. Let with for any , we have
where This proves that is bounded and hence by Leray-Schauder alternative theorem, operator has at least one fixed point. Therefore, the BVP (1) and (2) has at least one solution on . This completes the proof. □
4. Hyers-Ulam Stability
In this section, we will discuss the Hyers-Ulam stability of the solutions for the BVP (1) and (2) by means of integral representation of its solution given by
where and are defined by (8) and (9).
Define the following nonlinear operators
For some we consider the following inequality:
([8,9]).The coupled system (1) and (2) is said to be Hyers-Ulam stable, if there existsuch that for every solutionof the inequality (13), there exists a unique solutionof problems (1) and (2) with
Let the assumptions of Theorem 1 hold. Then the BVP (1) and (2) is Hyers-Ulam-stable.
Let be the solution of the problems (1) and (2) satisfying (8) and (9). Let be any solution satisfying (13):
It follows that
where are defined in (10).
Therefore, we deduce by the fixed-point property of operator , that is given by (8) and (9), which
From (14) and (15) it follows that
Thus, sufficient conditions for the Hyers-Ulam stability of the solutions are obtained. □
Consider the following coupled system of fractional differential equations
Using the given data, we find that
It is clear that
are jointly continuous functions and Lipschitz function with. Moreover,
Thus, all the conditions of Theorem 1 are satisfied, then problem (16) has a unique solution on, which is Hyers-Ulam-stable.
Consider the following system of fractional differential equation
Using the given data, we find that
It is clear that
Note thatandand hence by Theorem 2, problem (17) has at least one solution on.
In this paper, the existence, uniqueness and the Hyers-Ulam stability of solutions for a coupled system of nonlinear fractional differential equations with boundary conditions were established and discussed.
Future studies may focus on different concepts of stability and existence results to a neutral time-delay system/inclusion, time-delay system/inclusion with finite delay.
The authors have made the same contribution. All authors read and approved the final manuscript.
This research received no external funding.
The authors wish to thank the anonymous reviewers for their valuable comments and suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies; Elsevier: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
Chalishajar, D.; Raja, D.S.; Karthikeyan, K.; Sundararajan, P. Existence results for nonautonomous impulsive fractional evolution equations. Res. Nonlinear Anal.2018, 1, 133–147. [Google Scholar]
Chalishajar, D.; Kumar, A. Existence, uniqueness and Ulam’s stability of solutions for a coupled system of fractional differential equations with integral boundary conditions. Mathematics2018, 6, 96. [Google Scholar] [CrossRef]
Ntouyas, S.K.; Obaid, M. A coupled system of fractional differential equations with nonlocal integral boundary conditions. Adv. Differ. Equ.2012, 2012, 130–139. [Google Scholar] [CrossRef]
Ahmad, B.; Ntouyas, S.K. A Coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips-type integral boundary conditions. J. Math. Sci.2017, 226, 175–196. [Google Scholar] [CrossRef]
Alsulami, H.H.; Ntouyas, S.K.; Agarwal, R.P.; Ahmad, B.; Alsaedi, A. A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions. Bound. Value Probl.2017, 2017, 68–74. [Google Scholar] [CrossRef]
Zhang, Y.; Bai, Z.; Feng, T. Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput. Math. Appl.2011, 61, 1032–1047. [Google Scholar] [CrossRef]
Granas, A.; Dugundji, J. Fixed Point Theory; Springer: New York, NY, USA, 2005. [Google Scholar]
Hyers, D.H. On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA1941, 27, 222. [Google Scholar] [CrossRef] [PubMed]
Rus, I.A. Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math.2010, 103–107. [Google Scholar]
Cabada, A.; Wang, G. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl.2012, 389, 403–411. [Google Scholar] [CrossRef]
Graef, J.R.; Kong, L.; Wang, M. Existence and uniqueness of solutions for a fractional boundary value problem on a graph. Fract. Calc. Appl. Anal.2014, 17, 499–510. [Google Scholar] [CrossRef]
Ahmad, B.; Nieto, J.J. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl.2009, 58, 1838–1843. [Google Scholar] [CrossRef]
Su, X. Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett.2009, 22, 64–69. [Google Scholar] [CrossRef]
Wang, J.; Xiang, H.; Liu, Z. Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ.2010, 10, 12. [Google Scholar] [CrossRef]
Ahmad, B.; Ntouyas, S.K.; Alsaedi, A. On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals2016, 83, 234–241. [Google Scholar] [CrossRef]
Zhai, C.; Xu, L. Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul.2014, 19, 2820–2827. [Google Scholar] [CrossRef]
Ahmad, B.; Ntouyas, S.K. Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Appl. Math. Comput.2015, 266, 615–622. [Google Scholar] [CrossRef]
Tariboon, J.; Ntouyas, S.K.; Sudsutad, W. Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions. J. Nonlinear Sci. Appl.2016, 9, 295–308. [Google Scholar] [CrossRef]
Mahmudov, N.I.; Bawaneh, S.; Al-Khateeb, A. On a coupled system of fractional differential equations with four point integral boundary conditions. Mathematics2019, 7, 279. [Google Scholar] [CrossRef]