Asymptotic behavior of solutions of linear multi-order fractional differential systems

Definition 2.1

A continuous function x : [0, T] → ℂ^d is called a solution to the differential equation (2.1) with initial condition x(0) = x₀ if it satisfies the integral equation (2.2).

Remark 2.2

Because we assume the function f to be continuous, we can see that, for every solution x of (2.1) in the sense of Definition 2.1, the function f(⋅, x(⋅)) is continuous, too. Therefore, in view of the fact that the solution x satisfies the integral equation (2.2), it follows for i ∈ {1, 2, …, d} that the i-th component of x can be written as the sum of a constant and the Riemann-Liouville integral of order α_i of a continuous function. Using the arguments of [7, proof of Theorem 3.7], we can then conclude that x_i fulfils the conditions of [23, Theorem 5.2] and thus that D∗αixi exists and is continuous. Therefore, under the continuity assumption on f, a solution to (2.1) in the sense of Definition 2.1 is automatically a strong solution to the differential equation in the classical sense.

Theorem 2.3

Consider the equation(2.1). Assume that the function f is continuous and satisfies the Lipschitz condition(2.3). Then, for any x0=(x10,…,xd0)⊤ ∈ ℂ^d, the differential equation (2.1) has a unique solution in the set C([0, T]; ℂ^d) that satisfies the initial condition x(0) = x₀.

Proof

Let λ > 0 be a constant such that

On the space C([0, T];ℂ^d) we define a new norm ∥⋅∥_λ as

Using standard arguments, it is easy to see that (C([0, T];ℂ^d), ∥⋅∥_λ) is a Banach space. For any x0=(x10,…,xd0)⊤ ∈ ℂ^d, we define an operator 𝓣_x0 : C([0, T]; ℂ^d) → C([0, T];ℂ^d) by

where for 1 ≤ i ≤ d, t ∈ [0, T] and φ ∈ C([0, T];ℂ^d),

We see that for every φ, φ^ ∈ C([0, T];ℂ^d), every t ∈ [0, T] and all 1 ≤ i ≤ d,

It is clear that the operator 𝓣_x0 maps the space (C([0, T];ℂ^d), ∥⋅∥_λ) to itself; moreover, from (2.4) we obtain the estimate

which, by definition of λ, shows that this operator is a contractive mapping on this space. Due to the fact that (C([0, T];ℂ^d), ∥⋅∥_λ) is a Banach space, by Banach’s fixed point theorem, there exists a unique fixed point φ of 𝓣_x0 in this space. This fixed point is the unique solution of the Volterra equation (2.2) and hence, as stated above, also the unique solution to the initial value problem consisting of the differential equation (2.1) and the initial condition x(0) = x₀ in C([0, T]; ℂ^d). The proof is complete.□

Corollary 2.4

Let f : [0, ∞) × ℂ^d → ℂ^dbe continuous and satisfy a Lipschitz condition with respect to the second variable. Moreover, let α ∈ (0, 1]^dand x₀ ∈ ℂ^d. Then, the initial value problem

has a unique solution in C([0, ∞); ℂ^d).

Lemma 2.5

Let the values b_1kℓand b_2kℓbe defined as in (2.10) with arbitraryx10,x20 ∈ ℂ. Then, for j ∈ {1, 2} the series

is convergent for all z ∈ ℂ.

Proof of lemma 2.5

Since the series in question does not have any negative summands, we may rearrange the terms according to powers of z; this yields

It is therefore evident that, in order to investigate the convergence radius of this series, we need to estimate expressions of the form

In fact, we shall demonstrate that for sufficiently large k

where c₁ and c₂ are certain positive constants and

Equation (2.11) tells us that the classical power series for the Mittag-Leffler function E_α^* — that is well known to be convergent on the entire complex plane — evaluated at c₂ |z| is a majorant for the series s₁ and s₂ that we are interested in, and hence the series expansions for s₁(z) and s₂(z) also converge for all z as required.

Thus, it only remains to prove (2.11). The left inequality is clear by definition. To prove the right inequality, we employ the relations (2.10a), (2.10b) and (2.10c) and see, using the notation a : = max_{i,j ∈ {1, 2}} |a_ij|, that we have for k ≥ 2 the following chain of inequalities:

with

Both fractions on the right-hand side have the same numerator but their denominators differ by α₂ – α₁; the well known monotonicity of the Gamma function thus allows us to conclude that, for sufficiently large k, we have

with u := −α₁ + k α₂ + 1. For γ > 0 and z → ∞, Stirling’s formula yields the asymptotic relation Γ(z) / Γ(z + γ) = z^−γ (1 + o(1)) which is monotonically decreasing in z. Hence, for sufficiently large k, the quotient on the right-hand side of (2.12) is monotonically decreasing with respect to μ for α₁ ≥ α₂ and monotonically increasing with respect to μ if α₁ < α₂. Therefore, the maximum of this expression over all admissible values of μ is attained at μ = 1 if α₁ ≥ α₂ and at μ = k if α₁ < α₂. These observations may be summarized in the form

and this implies

if k is large enough. Thus, for a sufficiently large and fixed constant N and arbitrary k, by induction, we deduce the estimate

which shows (2.11) and completes the proof of Lemma 2.5. □

The same ideas and methods can be applied if the dimension of the fractional differential equation system is greater than 2. This leads to the following result:

Theorem 2.6

Letα = (α₁, …, α_d) ∈ (0, 1]^dand A ∈ ℂ^{d × d}. Then, for eachx₀ ∈ ℂ^d, the initial value problem

has a uniquely determined solution inC([0,∞); ℂ^d). The components of this solution can be expressed in the form

and the series in eq. (2.14)converges for all t ≥ 0.

Theorem 3.1

1. Every solution of the system(3.1a)converges to zero at infinity if and only if

   |arg(akk)|>αkπ2∀k∈{1,…,d}.(3.4)

2. If there existsk ∈ {1, …, d} such that |arg{(a_kk)| < α_kπ / 2 then there exists some x₀such that the solution to the system(3.1a)that satisfies the initial condition x(0) = x₀is unbounded.

Proof

For the proof of part (i), we will first show that the condition (3.4) is sufficient to assert that every solution of (3.1a) converges to zero at infinity. Indeed, for any initial value x0=(x10,…,xd0)⊤∈Cd, we denote the solution of (3.1a) starting from x₀ by φ (⋅, x₀) = (φ₁(⋅, x₀), …, φ_d(⋅, x₀))^⊤. Our proof will use mathematical induction over the index j of the components of the solution vector in a backward direction. Thus, for our induction basis we consider j = d. Since the d-th equation of the system (3.1a) reads

it follows from Lemma A4(i) that the condition |arg a_dd| > α_dπ / 2 is sufficient to assert that φ_d (t, x₀) → 0 as t → ∞ for all x₀. For the induction step, we assume that we have already shown that the components d, d−1, …, j+1 of the solution tend to 0 as t → ∞ for any choice of the initial values. Then we need to prove that this is also true for the j-th component. To this end we recall that the j-th component of the differential equation system (3.1a) reads

All the terms in the sum are already known and, because of the induction hypothesis, they are continuous and tend to zero as t → ∞. Thus we may apply Lemma A4(i) and immediately deduce that x_j has this property as well.

To conclude the proof of part (i) we now have to demonstrate that (3.4) is also necessary for all solutions of (3.1a) tend to zero as t→∞. To this end we assume that (3.4) does not hold. Then there exists an index k₀ ∈ {1, 2, …, d} which satisfies

i.e. k₀ is the largest index for which (3.4) is violated. Consider the equation

Since (3.4) is true for all i > k₀, the arguments used above imply that f is continuous and tends to zero at infinity. As in the considerations above, we may use the fractional variation-of-constants method [7, Theorem 7.2 and Remark 7.1] to see that the set of all solutions to (3.5) consists of the functions

where xk00 runs through the entire complex plane and where

The well known asymptotic behavior of the Mittag-Leffler functions [13, Proposition 3.6 and Theorem 4.3] then implies that Eαk0(ak0,k0tαk0) does not converge to 0 as t → ∞ because of our assumption on the relation of α_k0 and |arg a_{k0, k0}|. Now assume that there exists some xk00 ∈ ℂ such that φ_k0(t, x₀) → 0 as t → ∞. Then, it follows that for every x~0 ∈ C^d with xk0=x~k0 for k = k₀+1, …, d and xk00≠x~k00, we have

For t → ∞, the last summand on the right-hand side of this equality tends to zero but the other summand does not, and hence we conclude that φ(t, x~0) does not tend to zero as t → ∞ which yields our required contradiction.

For the proof of part (ii), we — much as above — know that there exists an index k₀ ∈ {1, 2, …, d} which satisfies

We may then proceed in the same way as in the second part of the proof of (i). However, now we know that |E_αk0 (a_k0k0t^α_k0)| → ∞ as t → ∞, and therefore we may even conclude that there exists some xk00 ∈ ℂ such that φ_k0(t, x₀) is unbounded. □

Remark 3.2

The same arguments can be used if the coefficient matrix A of the system has a block-upper triangular structure and the differentiation matrix on the left-hand side of the differential equation has a block structure with identical block sizes where each block consists of differential operators of the same order, i.e. if the differential equation has the form

where, using the notation I_μ for the μ-dimensional unit matrix,

A_jk ∈ ℂ^{d_j × d_k} and x = (x₁, …, x_d)^⊤ with d=∑j=1ndj: In this case,

• all solutions of the system (3.7) converge to zero as t → ∞ if and only if, for all j = 1,2,…,n, all eigenvalues λ_jk, k = 1,2,…, d_j, of the matrix A_jj satisfy |arg λ_jk| > α_jπ /2, and

• whenever there exist some j ∈ {1, 2, …, n} and k ∈ {1, 2, …, d_j} with |arg λ_jk| < α_jπ /2, there exists an initial value whose corresponding solution is unbounded.

A close inspection of the proof of Theorem 3.1 reveals that the statement of its part (i) can easily be extended to cover a class of inhomogeneous problems:

Corollary 3.3

Consider the differential equation system

where, for all i = 1,2, …, d, the functionsg_i : [0, ∞) → ℂ are continuous and satisfy

Every solution of the inhomogeneous system(3.8)converges to zero at infinity if and only if all solutions of the associated homogeneous system(3.1a)tend to zero as t→∞, i.e. if and only if condition(3.4)is satisfied.

Proof

Assume first that every solution of (3.8) tends to zero as t → ∞. In order to prove that every solution of the corresponding homogeneous system (3.1a) converges to zero, we choose an arbitrary x₀ ∈ ℂ^d. It is then sufficient to show that the solution of (3.1a) that starts at x₀ converges to zero as t → ∞. To this end, we take the solutions φ(⋅,x₀) and φ(⋅,0) of (3.8) that start at x₀ and at 0, respectively. By assumption, both these functions tend to 0 as t → ∞. Thus, φ(t, x₀) − φ(t, 0) tends to 0 as t → ∞ as well. But clearly, this difference is identical to the solution of the homogeneous system (3.1a) that starts at x₀.

Regarding the proof of the other direction of the equivalence, we assume that the condition (3.4) is satisfied. Under this hypothesis, we may proceed as in the first part of the proof of Theorem 3.1(i). Using the argumentation via Lemma A4(i) employed in the induction step there, we can derive that φ_d(t, x₀) → 0 as t → ∞ for any x₀ ∈ ℂ^d. Then we can proceed inductively as in the first part of the proof of Theorem 3.1(i) and demonstrate that the other components of φ(⋅, x₀) vanish near infinity as well. The proof is complete. □

Remark 3.4

Clearly, the same arguments can be used to extend the statement of Remark 3.2 regarding block triangular systems to the inhomogeneous case as well.