Variation of Constant Formulas of Linear Autonomous Grünwald-Letnikov-type Fractional Difference

AIP Conference Proceedings 2116, 450082 (2019); https://doi.org/10.1063/1.5114549 2116, 450082

© 2019 Author(s).

Variation of constant formulas of linear autonomous Grünwald-Letnikov-type fractional difference equations

Cite as: AIP Conference Proceedings 2116, 450082 (2019); https://doi.org/10.1063/1.5114549 Published Online: 24 July 2019

Anh Pham The, Piotr Jurgás, Michał Niezabitowski, and Stefan Siegmund

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Variation of Constant Formulas of Linear Autonomous Grünwald-Letnikov-type Fractional Difference

Equations

Anh Pham The

^1,b)

, 3LRWU-XUJiV

²

, Michał Niezabitowski

^2,D)

and Stefan Siegmund

^3,c)

1Faculty of Information Technology, Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha noi, Viet nam

2InstituteofAutomaticControl,SilesianUniversityofTechnology,Akademicka16,44-100Gliwice,Poland

3Technische Universit¨at Dresden, Faculty of Mathematics, Center for Dynamics, Zellescher Weg 12-14, 01069 Dresden, Germany

a)Corresponding author: michal.niezabitowski@polsl.pl

b)anhpt@lqdtu.edu.vn

c)stefan.siegmund@tu-dresden.de

Abstract.In this paper, the variation of constant formulas for Gr¨unwald-Letnikov-type fractional difference equations are estab- lished. As an application, we prove a stability result for solutions of a class of autonomous scalar fractional difference equations.

Introduction

It is well-known that the Laplace transform method can be utilized to derive a variation of constants formula for linear fractional differential equations [1]. In this paper we use theZ-transform to establish variation of constant formulas for Gr¨unwald-Letnikov-type fractional difference equations in Section 2. In Section 3 we prove some stability results for solutions of a class of autonomous scalar fractional difference equations.

Letα∈(0,1),h>0. Consider a fractional difference equation of the form

(0Δ^α_hx)(t+h)=A(t)x(t)+f(t) (t∈hN), (1) wherex: hN→R^d,0Δ^α_h is the Gr¨unwald-Letnikov-type fractionalh-difference operator of orderα, f:hN→R^dand A:hN→R^d×d. For an initial valuex₀∈Rⁿ, (1) has a unique solutionx:hN→R^dwhich satisfies the initial condition x(0)=x₀. We denotexbyϕG-L(·,x₀).

Remark 1(Solving linear homogeneous fractional difference equations (1)). The linear homogeneous initial value problem,(0Δ^α_hx)(t+h) =Ax(t),x(0) =x₀ ∈R^d, t ∈hN, A∈ R^d×d, has a unique solution given by the formula (see e.g. [2, Proposition 32])

ϕG-L(t,x0)=E(α,α)

Ah^α,_h^t

x0 (t∈hN).

A reader who is familiar with fractional difference equations may very well skip the remainder of the introduction in which we recall notation to keep the paper self-contained. Denote byRthe set of real numbers, byZthe set of integers, byN Z_≥0 the set{0,1,2, . . .}of natural numbers including 0, and byZ_≤0 {0,−1,−2, . . .}the set of non-positive integers. Fora∈R,h>0, we denote by (hN)athe sethN+a={a,a+h, . . .}. We writehNinstead of (hN)₀. ByΓ:R\Z≤0→Rwe denote the Euler Gamma function defined by

Γ(α) lim

n→∞

n^αn!

α(α+1)· · ·(α+n) (α∈R\Z≤0). (2) Note that

Γ(α)=⎧⎪⎪⎨

⎪⎪⎩

∞

0 x^α−1e^−xdx ifα >0,

Γ(α+1)

α ifα <0 andα∈R\Z_≤0. (3)

Binomial coefficients_r

m

can be defined for anyr,m∈Cas described in [3, Section 5.5, formula (5.90)]. Forr ∈R andm∈Zthe binomial coefficient satisfies [3, Section 5.1, formula (5.1)]

r m

=⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

r(r−1)···(r−m+1)

m! ifm∈Z≥1,

1 ifm=0,

0 ifm∈Z_≤−1.

Definition 2. [2, Definition 27] Letα,a∈R. The Gr¨unwald-Letnikov-type h-difference operatoraΔ^α_hof orderαfor a function x: (hN)_a→Ris defined by

(aΔ^α_hx)(t)=

t−a

h

k=0

a^(α)_k x(t−kh) (t∈(hN)a),

where a^(α)_k =(−1)^k_α

k

1 h^α.

Forβ∈C, we define a discrete-time Mittag-Leffler functionE_(α,β)(A,·) by E_(α,β)(A,n)=^∞

k=0

A^k

n−k+kα+β−1 n−k

(n∈Z), (4)

see [2, 4, 5] for this and similar definitions.

Proposition 3. [2, Proposition 9] Letα ∈ (0,1],λ, β ∈ Randβ < α+1. If all solutions z ∈ Cof the equation (z−1)^α=λz^α−1satisfy|z|<1, thenlim_n→∞E_(α,β)(λ,n)=0.

Corollary 4. [2, Corollary 10] Letα∈(0,1],λ, β∈Randβ < α+1. All solutions z∈Cof the equation(z−1)^α = λz^α−1satisfy|z|<1if and only if−2^α< λ <0.

Variation of constants formulas

Theorem 5(Variation of constant formulas for Gr¨unwald-Letnikov-type fractional difference equations). Letα ∈ (0,1), A∈R^d×dand f:hN→R^d. The solution of (1),(0Δ^α_hx)(t+h)=Ax(t)+f(t), x(0)=x0∈Rⁿis given by

ϕG-L(t,x₀)=E_(α,α)(Ah^α,_h^t)x₀+

t

h

k=0

E_(α,α)(Ah^α,_h^t −k)f(k) (t∈hN). (5)

Remark 6 (Variation of constants formula applied to nonlinear equation). Let α ∈ (0,1) and x: hN → R^d be a solution of the nonlinear fractional difference equation (0Δ^α_hx)(t+h)=Ax(t)+g(x(t)), where0Δ^α_h is the Gr¨unwald- Letnikov-type difference operator of orderα, g:R^d → R^d and A∈ R^d×d. Then x is also a solution of the (nonau- tonomous) linear fractional difference equation(1)with f: hN → R^d, t → g(x(t)). By Theorem 5, x satisfies the implicit equation

x(t)=E_(α,α)(Ah^α,_h^t)x₀+

t

h

k=0

E_(α,α)(A,_h^t −k)g(x(k)) (t∈hN).

To prepare the proof of Theorem 5, we summarize some results about theZ-transform of a sequence x: N → R, which is defined by Z[x](z)=^∞

i=0

x(i)z⁻ⁱforz∈C,|z|>Rlim sup_i→∞|x(i)|^1/i, see e.g. [6, Chapter 6] and [7]. The Z-transform ofR^dorR^d×dvalued sequences is defined component-wise. The proof of the following lemma follows from [8, Lemma 3] and [2, Proposition 28].

Lemma 7(Z-transform of Mittag-Leffler functions and fractional differences). Let a∈R,α∈(0,1).

(i) Let A∈R^d×d,β∈R. ThenZ

E(α,β)(A,·) (z)=

z z−1

_β I−¹_z

z z−1

_α A₋₁

.

(ii) Let x: (hN)a → R^d and y(n) (aΔ^α_hx)(t) for t ∈ (hN)_α and n ∈ N0 such that t = a+nh. ThenZ y

(z) = zh

z−1

_−α

Z[x](z), where X(z)=Z[x](z)and x(n)x(a+nh).

Proof of Theorem 5. By Lemma 7(ii), applying theZ-transform to equation (1) with the Gr¨unwald-Letnikov-type forward difference operator, we get

zh z−1

_−α

ZϕG-L(·,x₀)

(z)=AZϕG-L(·,x₀)

(z)+Z[f](z).

Using Lemma 7(i), we obtain ZϕG-L(·,x0)

(z)=Z

E(α,α)(Ah^α,·)(z)x0+Z

E(α,α)(Ah^α,·)(z)x0Z[f](z).

Applying the inverse of theZ-transform yields

ϕG-L(·,x₀)=E_(α,α)(Ah^α,_h^t)x₀+Z⁻¹Z

E_(α,α)(Ah^α,·)(z)x0Z[f](z). Hence, we get

ϕG-L(t,x₀)=E_(α,α)(Ah^α,_h^t)x₀+

t

h

k=0

E_(α,α)(Ah^α,_h^t −k)f(k) (t∈hN).

Boundedness of scalar linear Gr ¨unwald-Letnikov-type difference equations

Theorem 8(Stability of linear homogeneous Gr¨unwald-Letnikov-type scalar difference equation). Let α ∈ (0,1), λ∈R. If all solutions z∈Cof

1−1 z

z z−1

_α

λh^α=0 (6)

satisfy|z|<1, then

(0Δ^α_hx)(t+h)=λx(t) (t∈hN) (7)

is asymptotically stable.

Proof. From Theorem 5, we haveϕG-L(t,x₀)=E_(α,α)(λh^α,_h^t)x₀.From (8), we have (z−1)^α=λh^αz^α−1.Hence, from Proposition 3, we have lim^t

h→∞E_(α,α)(λ^α,_h^t)=0.

Theorem 9(Boundedness of nonautonomous linear Gr¨unwald-Letnikov-type scalar difference equation). Let α ∈ (0,1),λ∈Rand(λn)n∈Na sequence in l¹. If all solutions z∈Cof

1−1 z

z z−1

_α

λh^α=0. (8)

satisfy|z|<1, then the solution of

(0Δ^α_hx)(t+h)=λx(t)+λt/h (t∈hN), (9)

with initial condition x(0)=x0∈Rⁿ, is bounded.

Proof. By Theorem 5, the solution of (9),x(0)=x₀∈Rⁿ, is given by ϕG-L(t,x₀)=E_(α,α)(λh^α,_h^t)x₀+

t

h

k=0

E_(α,α)(λh^α,_h^t −k)λk (t∈hN).

By Theorem (8), there existsM≥0 such that|E_(α,α)(λ^α,_h^t)| ≤Mfort∈hN. Then,

t

h

k=0

E(α,α)(λh^α, t h −k)λk

≤M

t

h

k=0

|λk| (t∈hN),

proving that|ϕG-L(t,x₀)| ≤M|x₀|+M_∞

k=0|λk|.

Using Corollary 4, the following Corollary follows from Theorem 9.

Corollary 10. Letα∈(0,1),λ∈R,(λn)n∈Na sequence in l¹. Then the solution of(9)is bounded if−₂

h

_α

< λ <0.

ACKNOWLEDGMENTS

The research of the second and third author was funded by the National Science Centre in Poland granted according to decisions DEC-2015/19/D/ST7/03679, DEC-2017/25/B/ST7/02888, respectively. The research of the last author was partially supported by an Alexander von Humboldt Polish Honorary Research Fellowship.

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