Existence of Periodic Solution for a Nonlinear Fractional Differential Equation
© Mohammed Belmekki et al. 2009
Received: 2 February 2009
Accepted: 4 June 2009
Published: 14 July 2009
We study the existence of solutions for a class of fractional differential equations. Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green's function and give some existence results for the linear case and then we study the nonlinear problem.
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The subject is as old as the differential calculus, and goes back to time when Leibnitz and Newton invented differential calculus. The idea of fractional calculus has been a subject of interest not only among mathematicians but also among physicists and engineers. See, for instance, [1–6].
Fractional-order models are more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a "memory" term in a model. This memory term insures the history and its impact to the present and future. For more details, see .
Fractional calculus appears in rheology, viscoelasticity, electrochemistry, electromagnetism, and so forth. For details, see the monographs of Kilbas et al. , Kiryakova , Miller and Ross , Podlubny , Oldham and Spanier , and Samko et al. , and the papers of Diethelm et al. [14–16], Mainardi , Metzler et al. , Podlubny et al. , and the references therein. For some recent advances on fractional calculus and differential equations, see [1, 3, 20–24].
where is the standard Riemann-Liouville fractional derivative, is continuous, and .
This paper is organized as follows. in Section 2 we recall some definitions of fractional integral and derivative and related basic properties which will be used in the sequel. In Section 3, we deal with the linear case where is a continuous function. Section 4 is devoted to the nonlinear case.
2. Preliminary Results
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Let the Banach space of all continuous real functions defined on with the norm Define for , . Let , be the space of all functions such that which turn out to be a Banach space when endowed with the norm
By we denote the space of all real functions defined on which are Lebesgue integrable.
provided the right side is pointwise defined on , and where is the gamma function.
For instance, exists for all , when ; note also that when , then and moreover
Let , if with , then , with If , then is bounded at the origin, whereas if with , then we may expect to be unbounded at the origin.
Recall that the law of composition holds for all
We have for all .
has , as unique solutions.
From this lemma we deduce the following law of composition.
for some .
If with and , then .
3. Linear Problem
where , and is a continuous function.
for some .
Note that and . However, since has a singularity at for
that leads to the following.
since the series representing is absolutely convergent.
Using the change of variable
Thus we have the following result.
For , given, is bounded on .
4. Nonlinear Problem
In this section we will be concerned with the existence and uniqueness of solution to the nonlinear problem (1.1)–(3.4). To this end, we need the following fixed point theorem of Schaeffer.
Assume to be a normed linear space, and let operator be compact. Then either
(i)the operator has a fixed point in , or
(ii)the set is unbounded.
where is Green's function defined in Theorem 3.2.
Then the problem (1.1)–(3.4) has solutions if and only if the operator equation has fixed points.
Suppose that the following hold:
Then the operator is well defined, continuous, and compact.
- (a)We check, using hypothesis (4.3), that , for every . Indeed, for any , , we have(4.5)
and its limit as is zero. Note that is bounded above by .
Next, we prove that is continuous.
First, we check that is a bounded set in .
Now, we prove that is an equicontinuous set in . Following the calculus in (a), we show that tends to zero as .
Then is equicontinuous in the space , where , for .
As a consequence of (i) and (ii), is a bounded and equicontinuous set in the space .
which means that , which proves that is compact.
Assume that (4.3) and (4.4) hold. Then the problem (1.1)–(3.4) has at least one solution in
Consider the set .
which implies that the set is bounded independently of . Using Lemma 4.2 and Theorem 4.1, we obtain that the operator has at least a fixed point.
In Lemma 4.2, condition (4.3) is used to prove that the operator is continuous. Hence, in Lemma 4.2 and, in consequence, in Theorem 4.3, we can assume the weaker condition.
instead of (4.3).
However, to prove the existence and uniqueness of solution given in the following theorem, we need to assume the Lipschitzian character of (condition (4.3).
We use the Banach contraction principle to prove that the operator has a unique fixed point.
Using the calculus in (b) Lemma 4.2, is a contraction by condition (4.32). As a consequence of Banach fixed point theorem, we deduce that has a unique fixed point which gives rise to a unique solution of problem (1.1)–(3.4).
The research of J. J. Nieto and R. Rodríguez-López has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.
- Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. preprint
- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
- Benchohra M, Cabada A, Seba D: An existence result for nonlinear fractional differential equations on Banach spaces. Boundary Value Problems 2009, 2009:-11.Google Scholar
- Bonilla B, Rivero M, Rodríguez-Germá L, Trujillo JJ: Fractional differential equations as alternative models to nonlinear differential equations. Applied Mathematics and Computation 2007, 187(1):79–88. 10.1016/j.amc.2006.08.105MATHMathSciNetView ArticleGoogle Scholar
- Daftardar-Gejji V, Bhalekar S: Boundary value problems for multi-term fractional differential equations. Journal of Mathematical Analysis and Applications 2008, 345(2):754–765. 10.1016/j.jmaa.2008.04.065MATHMathSciNetView ArticleGoogle Scholar
- Varlamov V: Differential and integral relations involving fractional derivatives of Airy functions and applications. Journal of Mathematical Analysis and Applications 2008, 348(1):101–115. 10.1016/j.jmaa.2008.06.052MATHMathSciNetView ArticleGoogle Scholar
- Lazarević MP, Spasić AM: Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach. Mathematical and Computer Modelling 2009, 49(3–4):475–481. 10.1016/j.mcm.2008.09.011MATHMathSciNetView ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
- Kiryakova V: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series. Volume 301. Longman Scientific & Technical, Harlow, UK; 1994:x+388.Google Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar
- Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York, NY, USA; 1974:xiii+234.MATHGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
- Diethelm K, Freed AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. Edited by: Keil F, Mackens W, Voss H, Werther J. Springer, Heidelberg, Germany; 1999:217–224.Google Scholar
- Diethelm K, Ford NJ: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 2002, 265(2):229–248. 10.1006/jmaa.2000.7194MATHMathSciNetView ArticleGoogle Scholar
- Diethelm K, Walz G: Numerical solution of fractional order differential equations by extrapolation. Numerical Algorithms 1997, 16(3–4):231–253.MATHMathSciNetView ArticleGoogle Scholar
- Mainardi F: Fractional calculus: some basic problems in continuum and statistical mechanis. In Fractals and Fractional Calculus in Continuum Mechanics. Edited by: Carpinteri A, Mainardi F. Springer, Wien, Austria; 1997:291–348.View ArticleGoogle Scholar
- Metzler R, Schick W, Kilian H-G, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. The Journal of Chemical Physics 1995, 103(16):7180–7186. 10.1063/1.470346View ArticleGoogle Scholar
- Podlubny I, Petráš I, Vinagre BM, O'Leary P, Dorčák L: Analogue realizations of fractional-order controllers. Nonlinear Dynamics 2002, 29(1–4):281–296.MATHMathSciNetView ArticleGoogle Scholar
- Araya D, Lizama C: Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):3692–3705. 10.1016/j.na.2007.10.004MATHMathSciNetView ArticleGoogle Scholar
- Benchohra M, Hamani S, Nieto JJ, Slimani BA: Existence results for fractional differential inclusions with fractional order and impulses. to appear in Computers & Mathematics with Applications
- Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009, 49(3–4):605–609. 10.1016/j.mcm.2008.03.014MATHMathSciNetView ArticleGoogle Scholar
- Chang Y-K, Nieto JJ: Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators. Numerical Functional Analysis and Optimization 2009, 30: 227–244. 10.1080/01630560902841146MATHMathSciNetView ArticleGoogle Scholar
- Jafari H, Seifi S: Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation 2009, 14(5):2006–2012. 10.1016/j.cnsns.2008.05.008MATHMathSciNetView ArticleGoogle Scholar
- Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. Journal of Mathematical Analysis and Applications 2007, 325(1):226–236. 10.1016/j.jmaa.2005.04.005MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ: Differential inequalities for functional perturbations of first-order ordinary differential equations. Applied Mathematics Letters 2002, 15(2):173–179. 10.1016/S0893-9659(01)00114-8MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.