Existence and stability of the solutions for systems of nonlinear fractional differential equations with deviating arguments
© Nisse and Bouaziz; licensee Springer. 2014
Received: 6 June 2014
Accepted: 17 October 2014
Published: 27 October 2014
In this paper, we give sufficient conditions for the existence and uniqueness of the solution for a class of nonlinear fractional differential systems, with variable delays. Our analysis relies on the Banach fixed point theorem. Furthermore, we prove the uniform stability of the solution. Some examples are given to illustrate our results.
In recent years, many research works have been interested in fractional differential equations. This is due, first, to their widespread applications in diverse fields of engineering and natural sciences, and secondly to the intensive development of the theory of fractional calculus (see [1–12]). Furthermore, fractional differential equations with delays have proven more realistic in the description of natural phenomena than those without delays. Therefore, the study of these equations has drawn much attention (see e.g., [13–17]).
El-Sayed and Gaafar  established sufficient conditions for the existence and uniqueness of a solution to some nonlinear Riemann-Liouville fractional differential systems with constant delays. Also, they proved the stability of the solution. Recently, this study has been extended to another class of nonlinear fractional differential equations with delay in .
where is the Caputo fractional derivative of order , , where ′ denotes the transpose of the vector, and are continuous functions for , are continuous real-valued functions defined on , such that , and is a given vector function defined on with values in .
Our purpose is to establish sufficient conditions for the existence and uniqueness of a solution to the problem (1.1)-(1.2), by applying the Banach contraction principle. Furthermore, we prove the uniform stability of the solution.
While most existing research focuses on constant delays, the considered equations (1.1) contain variable delays. Moreover, it is important to note that our results are valid even in the case where the equations are of mixed type. Namely, equations of mixed type are those that have both retarded and advanced arguments. This makes a net difference with the previous works (see Remark 3.3). Thus, the present work generalizes the results obtained in [16, 17].
This paper is organized as follows. In Section 2, we introduce some basic definitions and notations, which are used in the sequel of the paper. In Section 3 and Section 4, we present our main results. Finally, in Section 5, two examples are given, as applications to illustrate our results.
2 Definitions and notations
Let us start by giving the definition of Riemann-Liouville fractional integral, and Caputo fractional derivatives. Further details of related basic properties used in the text can be found in [3, 5, 8].
where is the gamma function.
holds almost everywhere on . Moreover, if or , then the identity holds everywhere on .
If , and (or ) means the n th derivative of a function f, then we have the following definition.
3 Existence and uniqueness
In this section we prove the existence and uniqueness of the solution of the problem (1.1)-(1.2).
Theorem 3.2 Assume that the following hypotheses are satisfied:
where , .
Then the problem (1.1)-(1.2) has a unique solution.
From hypothesis (H4) we have . So, is a contraction. Hence, it has a unique fixed point which is precisely the unique solution of our problem (1.1)-(1.2). □
Remark 3.3 Note that, if for some j the delay function takes negative values, which is possible under the assumptions (H2) and (H3), then (1.1) are with advanced arguments. Thus, the sign of the delay functions being arbitrary, the equations of the considered system (1.1) may contain both types of deviation of argument i.e. both delay and advance. As far as we know, there are no published studies addressing these issues for such systems of equations. However, concerning boundary value problems of fractional order, some results on the existence of solutions are obtained in . But in , the authors considered problems involving only an advanced argument, and they do not address the question about the uniqueness (and the stability) of the solution.
In this section we study the stability of the solution of the problem (1.1)-(1.2).
Definition 4.1 The solution of the problem (1.1)-(1.2) is stable if for any , there exists such that for any two solutions and with the initial condition (1.2) and for , respectively, one has , implies , where denotes the supremum norm defined by , for all bounded vector function Ψ from to .
Theorem 4.2 Assume that hypotheses (H1)-(H4) in Theorem 3.2 are satisfied, then the solution of the problem (1.1)-(1.2) is uniformly stable.
such that if , then , which shows that the solution of the problem (1.1)-(1.2) is uniformly stable. □
where , for . .
where , , and .
Hence, all hypotheses of Theorem 3.2 are fulfilled. Thus, the problem has a unique solution, and by Theorem 4.2 the solution is uniformly stable. Therefore, as a conclusion, the problem has a unique uniform stable solution.
where , for . .
where and .
Hence, all hypotheses of Theorem 3.2 are satisfied. Thus, the problem has a unique solution. Moreover, by Theorem 4.2 the solution is uniformly stable. Therefore, as a conclusion, the problem has a unique uniform stable solution.
The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions, which helped to improve the quality of the paper.
- Ahmad B, Nieto JJ: Existence of solution for non-local boundary value problems of higher-order nonlinear fractional differential equations. Abstr. Appl. Anal. 2009., 2009: Article ID 494720Google Scholar
- Benchohra M, Hellal M: Perturbed partial functional fractional order differential equations with infinite delay. J. Adv. Res. Dyn. Control Syst. 2013, 5(2):1-15.MathSciNetGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Liu S, Jia M, Tian Y: Existence of positive solutions for boundary-value problems with integral boundary conditions and sign changing nonlinearities. Electron. J. Differ. Equ. 2010., 2010: Article ID 163Google Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.Google Scholar
- Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivatives: Theory and Applications. Gordon & Breach, New York; 1993.Google Scholar
- Tarasov VE: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin; 2010.View ArticleGoogle Scholar
- Tenreiro Machado JA, Kiryakova V, Mainardi F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1140-1153. 10.1016/j.cnsns.2010.05.027MathSciNetView ArticleGoogle Scholar
- Wang G, Liu W: Existence results for a coupled system of nonlinear fractional 2m-point boundary value problems at resonance. Adv. Differ. Equ. 2011. 10.1186/1687-1847-2011-44Google Scholar
- Zhong C, Fan X, Chen W: Nonlinear Functional Analysis and Its Application. Lanzhou University Press, Lanzhou; 1998.Google Scholar
- Ravichandran C, Baleanu D: Existence results for fractional neutral functional integro-differential evolution equations with infinite delay in Banach spaces. Adv. Differ. Equ. 2013. 10.1186/1687-1847-2013-215Google Scholar
- Li F: Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay. Adv. Differ. Equ. 2013. 10.1186/1687-1847-2013-327Google Scholar
- Bolat Y: On the oscillation of fractional-order delay differential equations with constant coefficients. Commun. Nonlinear Sci. Numer. Simul. 2014, 19(11):3988-3993. 10.1016/j.cnsns.2014.01.005MathSciNetView ArticleGoogle Scholar
- El-Sayed AMA, Gaafar FM: Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions. Adv. Differ. Equ. 2011. 10.1186/1687-1847-2011-47Google Scholar
- Gao Z, Yang L, Luo Z: Stability of the solutions for nonlinear fractional differential equations with delays and integral boundary conditions. Adv. Differ. Equ. 2013. 10.1186/1687-1847-2013-43Google Scholar
- Wang G, Ntouyas SK, Zhang L: Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument. Adv. Differ. Equ. 2011. 10.1186/1687-1847-2013-43Google Scholar
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