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In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical (α = 1) differential equation in the fractional case (here using fractional Caputo's derivatives) and the numerical scheme which are associated (here with discrete Grünwald-Letnikov derivatives). Our main concerns are positivity, order preserving ,equilibrium points and stability of these points. We formulate explicit conditions under which a fractional system preserves positivity. We deduce also sufficient conditions to ensure order preserving. We deduce from these results a fractional persistence theorem which ensures that positivity, order preserving, equilibrium points and stability is preserved under a Caputo fractional embedding of a given differential equation. At the discrete level, the problem is more complicated. Following a strategy initiated by R. Mickens dealing with non local approximations, we define a non standard finite difference scheme for fractional differential equations based on discrete Grünwald-Letnikov derivatives, which preserves positivity unconditionally on the discretization increment. We deduce a discrete version of the fractional persistence theorem for what concerns positivity and equilibrium points. We then apply our results to study a fractional prey-predator model introduced by Javidi and al.

Starting from the Riemann-Liouville derivative, many authors have built their own notion of fractional derivative in order to avoid some classical difficulties like a non zero derivative for a constant function or a rather complicated analogue of the Leibniz relation. Discussing in full generality the existence of such operator over continuous functions, we derive some obstruction Lemma which can be used to prove the triviality of some operators as long as the linearity and the Leibniz property are preserved. As an application, we discuss some properties of the Jumarie's fractional derivative as well as the local fractional derivative. We also discuss the chain rule property in the same perspective.

Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in [8] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [6]) using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular ([7],Theorem 32). However, the counterexample does not explain why and where the proof given in [8] does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in [7] which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in [8]. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost's method, in particular in the discrete setting. We then derive a fractional Noether's Theorem following this strategy, correcting the initial statement of Frederico and Torres in [7] and obtaining an alternative proof of the main result of Atanackovic and al. [2].

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