On a fractional differential equation with infinitely many solutions

We present a set of restrictions on the fractional differential equation $x^{(\alpha)}(t)=g(x(t))$, $t\geq0$, where $\alpha\in(0,1)$ and $g(0)=0$, that leads to the existence of an infinity of solutions starting from $x(0)=0$. The operator $x^{(\alpha)}$ is the Caputo differential operator.


Introduction
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 [1], 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 x · χ (0,+∞) (x) has an infinity of solutions x T (t) = (t−T ) 2 4 · χ (T,+∞) (t) defined on the nonnegative half-line which start from x(0) = 0. Here, by χ we denote the characteristic function of a Lebesgue-measurable set.
An interesting classical result [9,1], which generalizes the example, asserts that the initial value problem where the continuous function g : R → R has a zero at x 0 and is positive everywhere else, possesses an infinity of solutions if and only if x 0 + du g(u) < +∞.
Recently, variants of this result have been employed in establishing various facts regarding some mathematical models [12,13]. In particular, if the function g is allowed to have two zeros x 0 < x 1 while remaining positive everywhere else and then the problem (1) has an infinity of solutions (x T ) T >0 such that lim 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 [4].
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 [2,3,5,8,11,15,10,6,7].
Let us consider a function h ∈ C 1 (I, for some α ∈ (0, 1), where I = (0, +∞). The Caputo derivative of order α of h is defined as where Γ is Euler's function Gamma, cf. [ The initial value problem we investigate in this paper is where the function g : R → R is continuous, g(0) = 0 and g(u) > 0 when u ∈ (0, 1]. Further restrictions will be imposed on g to ensure that 0+ du g(u) < +∞.
By means of (2), we deduce that and so the problem (3) can be recast as where y = x (α) , β = 1 − α and the (general) function g has absorbed the constant 1 Γ(α) . In the next section, we look for a family (y T ) T >0 , with y T ∈ C([0, 1], R), of (non-trivial) solutions to (4).
We are now ready to state and prove our main result.
Proof. The operator O : Y → Y with the formula O(y)(t) = g t T y(s) (t − s) β ds , y ∈ Y, t ∈ [T, 1], is well defined.
The operator O being thus a contraction, its fixed point y T in Y is the solution we are looking for. Notice that y T is identically null in [0, T ].