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.
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