# Existence of Periodic Solution for a Nonlinear Fractional Differential Equation

## Abstract

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.

## 1. Introduction

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, [16].

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 [7].

Fractional calculus appears in rheology, viscoelasticity, electrochemistry, electromagnetism, and so forth. For details, see the monographs of Kilbas et al. [8], Kiryakova [9], Miller and Ross [10], Podlubny [11], Oldham and Spanier [12], and Samko et al. [13], and the papers of Diethelm et al. [1416], Mainardi [17], Metzler et al. [18], Podlubny et al. [19], and the references therein. For some recent advances on fractional calculus and differential equations, see [1, 3, 2024].

In this paper we consider the following nonlinear fractional differential equation of the form

(1.1)

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.

Obviously if

Definition 2.1 (see [11, 13]).

The Riemann-Liouville fractional primitive of order of a function is given by

(2.1)

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

Definition 2.2 (see [11, 13]).

The Riemann-Liouville fractional derivative of order of a continuous function is given by

(2.2)

We have for all .

Lemma 2.3.

Let . If one assumes , then the fractional differential equation

(2.3)

has , as unique solutions.

From this lemma we deduce the following law of composition.

Proposition 2.4.

Assume that is in with a fractional derivative of order that belongs to . Then

(2.4)

for some .

If with and , then .

## 3. Linear Problem

In this section, we will be concerned with the following linear fractional differential equation:

(3.1)

where , and is a continuous function.

Before stating our main results for this section, we study the equation

(3.2)

Then

(3.3)

for some .

Note that and . However, since has a singularity at for

It is easy to show that . Hence we should look for solutions, not in but in . We cannot consider the usual initial condition , but Hence, to study the periodic boundary value problem, one has to consider the following boundary condition of periodic type

(3.4)

From (3.3), we have

(3.5)

that leads to the following.

Theorem 3.1.

The periodic boundary value problem (3.2)–(3.4) has a unique solution if and only if

(3.6)

The previous result remains true even if . In this case, (3.2) is reduced to the ordinary differential equation

(3.7)

with the periodic boundary condition

(3.8)

and the condition (3.6) is reduced to the classical one:

(3.9)

Now, for different from , consider the homogenous linear equation

(3.10)

The solution is given by

(3.11)

Indeed, we have

(3.12)

since the series representing is absolutely convergent.

Using the identities

(3.13)

we get

(3.14)

Then

(3.15)

Note that the solution can be expressed by means of the classical Mittag-Leffler special functions [8]. Indeed

(3.16)

The previous formula remains valid for . In this case,

(3.17)

Then

(3.18)

which is the classical solution to the homogeneous linear differential equation

(3.19)

Now, consider the nonhomogeneous problem (3.1). We seek the particular solution in the following form:

(3.20)

It suffices to show that

(3.21)

Indeed

(3.22)

Using the change of variable

(3.23)

we get

(3.24)

Then,

(3.25)

Hence, the general solution of the nonhomogeneous equation (3.1) takes the form

(3.26)

Now, consider the periodic boundary value problem (3.1)–(3.4). Its unique solution is given by (3.26) for some . Also is in and

(3.27)

From (3.26), we have

(3.28)

(3.29)

since for any , we have

(3.30)

Then the solution of the problem (3.1)–(3.4) is given by

(3.31)

Thus we have the following result.

Theorem 3.2.

The periodic boundary value problem (3.1)–(3.4) has a unique solution given by

(3.32)

where

(3.33)

For , given, is bounded on .

For , (3.1) is

(3.34)

and the boundary condition (3.4) is

(3.35)

In this situation Green's function is

(3.36)

which is precisely Green's function for the periodic boundary value problem considered in [25, 26].

## 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.

Theorem 4.1.

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.

If is a solution of problem (1.1)–(3.4), then it is given by

(4.1)

where is Green's function defined in Theorem 3.2.

Define the operator by

(4.2)

Then the problem (1.1)–(3.4) has solutions if and only if the operator equation has fixed points.

Lemma 4.2.

Suppose that the following hold:

(i)there exists a constant such that

(4.3)

(ii)there exists a constant such that

(4.4)

Then the operator is well defined, continuous, and compact.

Proof.

1. (a)

We check, using hypothesis (4.3), that , for every . Indeed, for any , , we have

(4.5)

From the previous expression, we deduce that, if , then

(4.6)

Indeed, note that the integral is bounded by

(4.7)

A similar argument is useful to study the behavior of the last three terms of the long inequality above. On the other hand, if we denote by the second term in the right-hand side of that inequality, then it is satisfied that

(4.8)

Note that

(4.9)

and, concerning , we distinguish two cases. If is such that , then

(4.10)

and, if is such that , then

(4.11)

In consequence,

(4.12)

The first term in the right-hand side of the previous inequality clearly tends to zero as . On the other hand, denoting by the integer part function, we have

(4.13)

The finite sum obviously has limit zero as . The infinite sum is equal to

(4.14)

and its limit as is zero. Note that is bounded above by .

The previous calculus shows that , for , hence we can define .

1. (b)

Next, we prove that is continuous.

Note that, for and for every , we have, using hypothesis (4.4),

(4.15)

Using the definition of , we get

(4.16)

Moreover,

(4.17)

Using that for , for , and for we obtain

(4.18)

Note that the Beta function, also called the Euler integral of the first kind,

(4.19)

where and , satisfies that . In particular, . On the other hand, using the change of variable , we deduce that

(4.20)

This proves that

(4.21)

Hence,

(4.22)

In consequence,

(4.23)

Finally, we check that is compact. Let be a bounded set in .

1. (i)

First, we check that is a bounded set in .

Indeed,

(4.24)

Hence

(4.25)

and then

(4.26)

which implies that is a bounded set in .

1. (ii)

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 .

Hence, for a sequence in , has a subsequence converging to , that is,

(4.27)

Taking , we get

(4.28)

which means that , which proves that is compact.

Theorem 4.3.

Assume that (4.3) and (4.4) hold. Then the problem (1.1)–(3.4) has at least one solution in

Proof.

Consider the set .

Let be any element of , then for some . Thus for each , we have

(4.29)

As in Lemma 4.2, (i), we have

(4.30)

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.

Remark 4.4.

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.

(i)For each fixed, there exists such that

(4.31)

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).

Theorem 4.5.

Assume that (4.4) holds. Then the problem (1.1)–(3.4) has a unique solution in provided that

(4.32)

Proof.

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).

Remark 4.6.

If , condition (4.32) is reduced to

(4.33)

## References

1. Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. preprint

2. 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.

3. Benchohra M, Cabada A, Seba D: An existence result for nonlinear fractional differential equations on Banach spaces. Boundary Value Problems 2009, 2009:-11.

4. 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.2006.08.105

5. 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.2008.04.065

6. 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.2008.06.052

7. 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.2008.09.011

8. 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.

9. Kiryakova V: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series. Volume 301. Longman Scientific & Technical, Harlow, UK; 1994:x+388.

10. 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.

11. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.

12. Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York, NY, USA; 1974:xiii+234.

13. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.

14. 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.

15. Diethelm K, Ford NJ: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 2002, 265(2):229–248. 10.1006/jmaa.2000.7194

16. Diethelm K, Walz G: Numerical solution of fractional order differential equations by extrapolation. Numerical Algorithms 1997, 16(3–4):231–253.

17. 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.

18. 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. 10.1063/1.470346

19. 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.

20. 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.2007.10.004

21. 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

22. 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.2008.03.014

23. 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. 10.1080/01630560902841146

24. 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.2008.05.008

25. 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.2005.04.005

26. 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)00114-8

## Acknowledgment

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.

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Correspondence to Rosana Rodríguez-López.

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Belmekki, M., Nieto, J. & Rodríguez-López, R. Existence of Periodic Solution for a Nonlinear Fractional Differential Equation. Bound Value Probl 2009, 324561 (2009). https://doi.org/10.1155/2009/324561