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On a fractional differential equation with infinitely many solutions
Advances in Difference Equations volume 2012, Article number: 145 (2012)
We present a set of restrictions on the fractional differential equation , , where and , that leads to the existence of an infinity of solutions (a continuum of solutions) starting from . The operator is the Caputo differential operator.
The issue of multiplicity for solutions of an initial value problem that is associated to some nonlinear differential equation is essential in the modeling of complex phenomena.
Typically, when the nonlinearity of an equation is not of Lipschitz type , there are only a few techniques to help us decide whether an initial value problem has more than one solution. As an example, the equation has an infinity of solutions (a continuum of solutions , p.15]) defined on the nonnegative half-line which start from . Here, by χ we denote the characteristic function of a Lebesgue-measurable set.
where the continuous function has a zero at and is positive everywhere else, possesses an infinity of solutions if and only if .
Recently, variants of this result have been employed in establishing various facts regarding some mathematical models [13, 14]. In particular, if the function g is allowed to have two zeros while remaining positive everywhere else and
then the problem (1) has an infinity of solutions such that .
Our intention in the following is to discuss a particular case of the above non-uniqueness theorem in the framework of fractional differential equations. To the best of our knowledge, the result has not been established in its full generality yet for any generalized differential equation. We mention at this point the closely connected investigation .
In the last number of years, it became evident that differential equations of non-integer order, also called fractionals (FDE’s), can capture better in models many of the relevant features of complex phenomena from engineering, physics or chemistry, see the references in [2–4, 6, 7, 9, 11, 12, 16].
Let us consider a function with for some , where . The Caputo derivative of order α of h is defined as
where Γ is Euler’s function Gamma, cf. , p.79]. To have an idea about the eventual smoothness of this quantity, we mention that by letting the function be at least absolutely continuous , Chapter 7] the (usual) derivative of will exist almost everywhere with respect to the Lebesgue measure m on , see , p.35, Lemma 2.2]. Further, we have that
provided that is in .
The initial value problem we investigate in this paper is
where the function is continuous, and when . Further restrictions will be imposed on g to ensure that .
By means of (2), we deduce that
and so the problem (3) can be recast as
where , and the (general) function g has absorbed the constant .
In the next section, we look for a family , with , of (non-trivial) solutions to (4).
2 Infinitely many solutions to (4)
We start by noticing that the function with the formula is increasing. Introduce now the numbers and such that
Set in and assume that
As a by-product, . Further, suppose that there exists such that
The latter condition has been inspired by the analysis in .
Introduce now the numbers and with
These will be used in the following for describing the solution .
Several simple estimates, of much help in the proof of our result, are established next. Notice first that, via the change of variables , we get
where and B represents Euler’s function Beta . Also,
Now, returning to (5), there exist , with , such that
In particular, and, by means of (13),
Taking into account (12) and (9), we deduce that
which leads to
and, via (13),
which implies that
Let the set be given by the double inequality
Observe that, by means of (16), (11), (6),
Similarly, via (17),
In conclusion, the mapping is a member of whenever . Also, taking into account (15), we deduce that the quantities from (20) and from (21) belong to as imposed in (6).
We are now ready to state and prove our main result.
Theorem 1 Assume that the nonlinearity g of (4) satisfies the restrictions (5), (6), (7). Given the numbers , , T subject to (8), (9), (10) and the set from (18), the problem (4) has a unique solution in .
Proof The operator with the formula
is well defined.
The typical sup-metric provides the set with the structure of a complete metric space.
Taking into account (7), (19) and (10), we get
The operator being thus a contraction, its fixed point in is the solution we are looking for. Notice that is identically null in . □
As an easily computed example, consider the initial value problem
where and . Then, introducing such that , the problem (22) has the solution throughout which can be extended as a -function downward to 0.
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The work of the second author has been supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0075. The authors would like to thank the referees for their comments and remarks in order to improve the manuscript.
The authors declare that they have no competing interests.
All authors have equal contributions in preparing this manuscript. All authors read and approved the final manuscript.