New conditions for the exponential stability of fractionally perturbed ODEs

The aim of this paper is to present some results on the exponential stability of the zero solution for a class of fractionally perturbed ordinary differential equations, whose right-hand sides involve the Riemann–Liouville substantial fractional integrals of different orders and we assume that they are polynomially bounded. In their proofs we apply a method recently developed by Rigoberto Medina. We also prove an existence result for this type of equations.


Introduction
It is well known that the system of linear fractional differential equations where D α x(t) is the Riemann-Liouville or the Caputo derivative of x(t) of the order α ∈ (0, 1) and A is a constant matrix, do not have exponentially stable solutions, but asymptotically stable only.The equilibrium x = 0 of this equation is asymptotically stable if and only if | arg(λ)| > απ 2 for all eigenvalues λ of the matrix A. In this case all components of x(t) decay towards 0 like t −α (see e.g.[8]).
In the paper [3] a sufficient condition for the exponential stability of the zero solution of nonlinear fractional systems of equations of the following class ẋ(t) = Ax(t) + g t, x(t), RL I α 1 x(t), . . ., RL I α m x(t) , x(t) ∈ R N , ( is proved.Here A is a constant matrix and Corresponding author.Email: Milan.Medved@fmph.uniba.sk is the Riemann-Liouville fractional integral of order α of the function x(t).The aim of this paper is to prove a result of that type for the following class of fractional system ẋ(t) = A(t)x(t) + F t, x(t) + f t, I (α 1 ,β 1 ) x(t), . . ., I (α m ,β m ) x(t) , t ≥ 0, x(t) ∈ R N , x(t 0 ) = x 0 , (1.4) where is the so-called fractional substantial integral of the function x(t) of order α > 0 with a parameter β > 0 (see e.g.[5]).This integral is more general than integrals defining the following fractional derivations: (1.6) ) ( We remark that the substantial fractional derivative, corresponding to the substantial fractional integral is defined as (1.9) Definition 1.1.We say that x(t) is a solution of the initial value problem (2.1), defined on the interval [t 0 , T) it it is C 1 -differentiable, the fractional integrals in this equation exists, x(t) fulfils the equality (2.1) for all t ∈ (0, T) with x(0) = x 0 .It is called maximal, if there no its proper continuation, i.e. there is no > 0, such that there exists a solution y(t) of this problem, defined on the interval [t 0 , T + ) with y(t) = x(t) for all t ∈ [t 0 , T).If T = ∞, the this solution is called global.
In the paper [4] the problem of exponential stability of fractional differential equations of the type (1.2), where instead of the Riemann-Liouville fractional integrals there are Caputo-Fabrizio fractional integrals, is studied.
The aim of this paper is to prove a result on the exponential stability of the zero solution of equations of the form (1.2), where instead of the constant matrix A there is a time-dependent matrix A(t) and instead the Riemann-Liouville fractional integrals there are the Riemann-Liouville substantial fractional integrals.These integrals have some better properties, convenient for the study asymptotic properties of solutions, than the Riemann-Liouville fractional integrals.
In the papers [7] a sufficient condition for the asymptotic stability of the zero solution of the equation where f , Φ are continuous functions, are proved.In this case solutions decay toward 0 as t → ∞ like t −α .It is proven in the paper [13] that solutions of the equation have the same asymptotic properties.This equation can be written in the form of the system (1.4) and this means that there is a chance to obtain some conditions for the exponential stability of the zero solution of a fractional perturbation of the equation (1.12), or the corresponding system, only if we consider time dependent coefficients a, b.We consider this type of equations in [3,4] with the Riemann-Liouville and Caputo-Fabrizio fractional integrals and in this paper we study equations of this type with the Riemann-Liouville substantial fractional integrals.

Existence result
In this section, we prove a local existence and uniqueness result concerning the initial value problem where A(t) is a continuous matrix function and ) be a continuous locally Lipschitz mapping.Then for any (t 0 , x 0 ) ∈ G, t 0 ≥ 0, there exists a δ > 0 such that the initial value problem (2.1) has a unique solution x(t) on the interval I δ = [t 0 , t 0 + δ). Proof.Let for some a > 0, b > 0. Let and the mapping F satisfies the condition where c = min 1≤i≤m Γ(α i )α i be the Banach space of continuous mappings from I δ into R N endowed with the metrics d(h, g) and so, we have (2.8) Hence, the first approximation x 1 (t) is well defined and This yields the inequality for all t ∈ I δ .Now, similarly as in the proof of the existence theorem in [3] we find using the Lipschitz condition (2.4) and the inequality (2.7) that where k = M 3 + L 0 + ∑ m i=1 L i and one can show by induction that we obtain (2.15) From the definition of δ it follows that kδ < 1, and so the series x 0 + ∑ ∞ i=1 (kδ) i is convergent.This yields the uniform convergence of the sequence {x n (t)} ∞ i=0 on the interval I δ to a continuous mapping x ∈ C δ , which is a unique solution of the equation (2.1).
Corollary 2.2.For any x 0 ∈ R N and any t 0 ≥ 0 there exists a maximal solution of the initial value problem (2.1).This corollary is a consequence of Theorem 2.1.

Exponential stability of fractionally perturbed ODEs with linearly bounded right-hand sides
The results described in this section, is based upon a method developed by Rigoberto Medina in the paper [9] for systems of the form (1.4) without the fractional part.We extend his results to the fractional system (1.4).We will work with the logarithmic norm µ(B), of a square N × N matrix B = (b ij ) defined by where I is the unit matrix and • is a norm on R N .For example, with respect to the 1-norm Lemma 5]).We will apply the following Coppel's inequality: To established the main results we make the following assumptions: (H1) There are positive numbers Θ, q such that where • denotes a norm in R N .
where G(A(•), F, f ) is given by (3.8).Since the right-hand side of (3.8) is independent of T this inequality holds for all t ∈ [0, ∞).
Hence the condition x(0 ensure the exponential stability of the solution x(t) with respect to the ball Ω(λ).
If r < ∞, then using the Uryson's lemma [2, Lemma 10.2], we get the exponential stability in this case.

Example 1
Let us illustrate Theorem 3.1 by the following example, which is a fractional perturbation of the [9, Example 9, p. 4]: where where a 1 , a 2 , γ 1 , γ 2 are positive constants, d 1 (t), d 2 (t) are continuous nonnegative and bounded functions.
Then the zero solution of the equation (4.1) is exponentially stable with respect to the ball Ω(λ 0 ) with λ 0 = r(1 − S 0 ).
Proof.One can check that the condition (C1) yields the inequality i.e. the condition (H1) of Theorem 3.1 is fulfilled.By the formula [9, (44)] µ(A(t)) = −ρ, t 0, where ρ is defined in (C2), the condition (H2) of Theorem 3.1 is also fulfilled.Since ) is satisfied, then the condition formulated in Theorem 3.1 is satisfied and hence we have proved that the assertion of Theorem 4.1 is a consequence of Theorem 3.1.

Exponential stability of fractionally perturbed ODEs with several power nonlinearities
In this section we consider the equation (1.4) under the following assumptions: (G1) There are positive numbers Θ, q such that (G2) For any logarithmic norm µ, the matrix A(t) satisfies (G3) For a positive r < ∞, there are constants γ = γ(r), i = i (r) such that where (G4) There are positive constants η i , ξ i , i = 1, 2, . . ., m and µ i , where 1 < ω 1 < ω 2 < • • • < ω m are constants, independent of r with the additional property: Theorem 5.1.Let the conditions (G1)-(G4) be satisfied.In addition, let G(A(•), F, f ) := q Γ(Θ + 1) Then the solution x(t) of the initial value problem (2.1) with t 0 = 0 is global and where ) where (5.10) Proof.Let x(t) be a solution of the initial value problem (2.1) with x(0) = x 0 .Then x(t) ≤ x(0) e −ρt + q t 0 e −ρ(t−s) |s − τ| Θ x(s) ds + γ t 0 e −ρ(t−s) x(s) ds (5.12) The first three integrals are the same as in the linear case studied in Section 4. Therefore we can apply the same procedure as in the proof of Theorem 3.1.Denote by Φ(t) the right-hand side of the inequality (5.12).Hence, if K = 1 − G(A(•), F, f ) −1 , then from this inequality we have (5.13) Now, let us apply the desingularization method suggested in the paper [10] (see also [11,12]).Using the Hölder inequality with ω i and κ i = ω i ω i −1 we obtain the estimate: (5.14) We have the following estimate: where (5.17) Using this inequality we obtain from the inequality (5.13): (5.18) From this inequality it follows the following inequality for v(t) = Φ(t)e ρt : where (5.20) From Pinto's inequality [14], which is a generalization of the Bihari inequality [1], it follows an integral inequality, corresponding to several power nonlinearities, formulated and proved in [3] (see [3, Lemma 3.1]), we obtain the inequality: where where and since the function H(z) is continuous on Ω(r) and H(0) = 0, from the inequality (5.26) it follows that the maximal solution x(t) is global and that if x(0) ∈ Ω(r), then lim t→∞ x(t) = 0.

7 ) 1 ξKG 1 = 8 )
The function H(z) is obviously defined for all z ∈ Ω(r) with r = sup z : |z| < If x(t) is a solution of the initial value problem (4.1), then by Theorem 5.1