Mixed Order Fractional Differential Equations

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

Bonilla et al. [1] studied linear systems of the same order linear FDEs and obtained an explicit representation of the solution.However, there are very few works on the study of mixed order nonlinear fractional differential equations (MOFDEs), which is a natural extension of [1].
This paper is devoted to the study of MOFDEs of the form where D q i 0 + denotes the Caputo derivative with the lower limit at 0, q i ∈ (0, 1], f : R + × R n → R n are continuous, specified below and u i ∈ R n .We may suppose q 1 ≥ • • • ≥ q n .Here R + = [0, ∞).We are interested in the existence of solutions of (1), then their stability and asymptotic properties under reasonable conditions on f i .FDEs with equal order (i.e., q 1 = • • • = q n ) are widely studied, and we refer the reader to the basic books describing FDEs, such as [7,8].On the other hand, there are many MOFDEs with interesting applications-for example, to economic systems in [9].In fact, (1) formulates a model of the national economies in a case of the study of n commonwealth countries, which cannot be simply divided into clear groups of independent and dependent variables.The purpose of this paper is to set a rigorous theoretical background for (1).
The main contributions are stated as follows: We give some existence and uniqueness results for solutions of (1) when the nonlinear term satisfies global and local Lipschitz conditions.
We analyze the upper bound for Lyapunov exponents of solutions of (1).We show that the zero solution of an autonomous version system of (1) is asymptotically stable.

Existence Results
First we prove an existence and uniqueness result for globally Lipschitzian f = ( By C(J, R n ) we denote the Banach space of all continuous functions from a compact interval J ⊂ R to R n with the uniform convergence topology on J. Theorem 1.Let T > 0 and suppose the existence of L > 0 such that (2) Proof of Theorem 1.Note that (2) implies for 1) is equivalent to the fixed point problem Fix α ≥ 0 and set x α = max t∈[0,T] x(t) ∞ e −αt for any x ∈ C(I, R n ).Let x ∈ C(I, R n ) be a solution of (1).For α > 0, (3) and (4) give which implies T q i Γ(q i + 1) . Hence So Similarly, we derive Consequently, assuming (5), we can apply the Banach fixed point theorem to get a unique solution x ∈ C(I, R n ) of (1), which also satisfies (6).The proof is finished.Now we prove an existence and uniqueness result for locally Lipschitzian f .Theorem 2. Suppose that for any r > 0 there is an L r > 0 such that Then (1) has a unique solution x ∈ C(I 0 , R n ), for some L > 0. This extension is given by for a Lipschitz function χ : R + → [0, 1] with χ(r) = 1 for r ∈ [0, 1] and χ(r) = 0 for r ≥ 2.
Applying Theorem 1 to there is a unique solution x ∈ C(I, R n ) of ( 8) which also satisfies ( 6) for α satisfying (5).Note M f = M f .Let us take T ≥ T 0 > 0, α > 0 satisfying (5) and Then the unique solution x ∈ C(I, R n ) of ( 8) satisfies However, this is also a unique solution of ( 1) on I 0 .The proof is finished.
Remark 1.Let us denote by x u the solution from Theorem 1.Then, following the proof of Theorem 1, for any u, v ∈ R n , we derive provided that (5) holds.So, the continuous dependence of the solution of (1) is shown on the initial value u under conditions of Theorem 1 or Theorem 2.

Asymptotic Results
We find the upper bound for Lyapunov exponents of solutions of (1).
Theorem 3. Suppose assumptions of Theorem 2 are satisfied.Moreover, we suppose the existence of a nonnegative n × n-matrix M = {m ij } n i,j≥1 and a nonegative vector Then the Lyapunov exponent where ρ(M) is the spectral radius of M. Note x(0) = u, so we consider as usually Λ : R n → R + (see [10]).
Proof.Clearly (9) implies Then, like in the proof of Theorem 1, for any T > 0 there is a unique solution of (1) on I.However, since T > 0 is arbitrary, we get a unique solution x(t) on R + .Then, we compute .
Consequently, we arrive at for 10) is considered component-wise.From (10), we get for the n × n identity matrix I.By the Neumann lemma, we have which is a positive matrix.Consequently, (11) implies Letting ε → 0, we get The proof is finished.
then the zero solution of (12) is asymptotically stable.
Proof.The proof is motivated by the well-known Geršgoring type method [12,13].Like above, we modify (12) outside of the unit ball B(1) such that the modified system is globally Lipschitz.Then, the solution of the modified (12) has a global unique solution on R + by Theorem 1.This solution of ( 12) has the fixed point form [8] x where E q (t) and E q,q (t) are the classical and two-parametric Mittag-Leffler functions, respectively ([11] p. 56).Fix T > 0 and set |||x||| = max t∈I |x(t)| ∞ (t + 1) q n for x ∈ C(I, R n ).
It is well-known that G : C(I, R n ) → C(I, R n ) is continuous and compact [8].Since B T (r 1 ) is convex and bounded, by the Schauder fixed point theorem, G has a fixed point x ∈ B T (r 1 ).However, this a solution of a modified (12), which has a unique solution on R + .So, this unique solution satisfies x ∈ B T (r 1 ) for any t > 0; i.e., x(t) ∞ ≤ r 1 (t + 1) q n ∀t ≥ 0. ( Certainly (19) gives x(t) ∞ ≤ 1, so x(t) is also a unique solution of the original (12).Moreover, (17) leads to which determines the asymptotic stability of (12) at 0. The proof is finished.Remark 2. Condition (14) is a Geršgoring type assumption [12,13].Even for q 1 = • • • = q n , condition (14) is new, since the general stability assumption (see [5] Theorem 2.2) is difficult to check.The above approach can be extended to FDEs on infinite lattices like in [14].

Examples
In this section, we give examples to demonstrate the validity of our theoretical results.

Conclusions
The existence and uniqueness of solutions of (1) are shown along with their stability and asymptotic properties.Theorems 1 and 2 extended the partial case of q i = q, i = 1, 2, • • • , n which are known in the related literature.Theorems 3 and 4 are original results.Moreover, Theorems 1, 2, and 3 can be directly extended to infinite dimensional cases.The next step should be to derive an exact solution for scalar linear systems (i.e., (12) with g = 0) and to find an explicit variation of constant formula for nonhomogeneous systems.Of course, a more general stability criterion would also be interesting to find by generalizing ([5] Theorem 2.2), which is just for q 1 = • • • = q n .Then, a derivation of a Gronwall-type inequality associated to MOFDEs would be challenging as well.This should extend Theorem 4 to infinite dimensional cases.