Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 16, 1-9; http://www.math.u-szeged.hu/ejqtde/

Stability of a monotonic solution of a non-autonomous

multidimensional delay differential equation of arbitrary

(fractional) order

A. M. A. El-Sayed Faculty of Science, Alexandria University, Alexandria, Egypt e.mail: amasayed@hotmail.com E. M. El-Maghrabi Faculty of Science, Benha University, Benha 13518, Egypt e.mail: esam mh@yahoo.com

Abstract We are concerned here with the existence of monotonic and uniformly asymptotically stable solution of an initial-value problem of non-autonomous delay differential equations of arbitrary (fractional) orders.

Key words: Fractional calculus, delay differential equations, monotonic solutions, asymptotically stable solution.

1 Introduction

In a number of papers [1, 2, 3, 7] stability and existence of solution of some equations of fractional order has been investigated. In [2] the authors proved the stability (and some other properties concerning the existence and uniqueness) of solution of the problem of non-autonomous system

D_{t_{0}}^{α} X(t) = A(t) X(t) + f(t), α ∈ (0, 1], X(t_{0}) = X^{0} ,

where the coefficients of the matrix functions A(t) and f(t) are absolutely continuous functions. And the problem

\frac{d}{dt}X(t) = A(t) \frac{d}{dt} I_{t_{0}}^{α} X(t) + f(t), α ∈ (0, 1], X(t_{0}) = X^{0}

where the coefficients of the matrix A(t) are bounded and measurable and the function f(t) is integrable.

Let α_{i} ∈ (0, 1], β_{j} ∈ [0, 1), i = 1, 2, …, n, j = 1, 2, …, m, and r_{i}, σ_{j} are positive constants. Consider the initial-value problem

\begin{matrix}\frac{d}{dt}x(t) − f(t, x(t), & D_{r_{1}}^{α_{1}}x(t − r_{1}), …, D_{r_{n}}^{α_{n}}x(t − r_{n})) \\ = g(t, x(t), D_{σ_{1}}^{β_{1}}x(t − σ_{1}), …, D_{σ_{m}}^{β_{m}}x(t − σ_{m})),  t > 0, \end{matrix} (1.1)
x(t) = x_{0},  t ≤ 0,  (1.2)

We prove here the existence of uniformly asymptotically stable and monotonic increasing solution x ∈ C[0, T] of (1.1) with \frac{dx}{dt} ∈ C[0, T]. As an application the special (linear) case

\begin{aligned}\frac{d}{dt} x(t) − a(t) x(t) & = ∑_{i=1}^{n} a_{i}(t)D_{r_{i}}^{α_{i}}x(t − r_{i}) + ∑_{j=1}^{m} b_{j}(t)D_{σ_{j}}^{β_{j}}x(t − σ_{j}) \\ x(t) & = x_{0} t ≤ 0 \end{aligned}  (1.3)

and the initial value problem of the multidimensional neutral delay differential equation

\begin{aligned}\frac{d}{dt} (x(t) − ∑_{i=0}^{n} a_{i}(t) x(t − r_{i})) & = ∑_{j=1}^{m} b_{j}(t) x(t − σ_{j}) \\ x(t) & = 0 t ≤ 0 \end{aligned}  (1.4)

will be studied

2 Preliminaries

Let L^{1}[a, b] denote the space of all Lebesgue integrable functions on the interval [a, b], 0 ≤ a < b < ∞.

Definition 2.1. The fractional (arbitrary) order integral of the function f ∈ L^{1}[a, b] of order β ∈ R^{+} is defined by (see [4] - [6])

I_{a}^{β} f(t) = ∫_{a}^{t} \frac{(t − s)^{β − 1}}{Γ(β)} f(s) ds,

where Γ(.) is the gamma function.

Definition 2.2. The (Caputo) fractional-order derivative D^{α} of order α ∈ (0, 1] of the function g(t) is defined as (see [4] - [6])

D_{a}^{α} g(t) = I_{a}^{1 − α} \frac{d}{dt} g(t), t ∈ [a, b].

The following properties are some of the main ones of the fractional derivatives and integrals. Let β, γ ∈ R^{+} and α ∈ (0, 1). Then we have

(i) I_{a}^{β} : L^{1} → L^{1}, and if f(x) ∈ L^{1}, then I_{a}^{γ} I_{a}^{β} f(x) = I_{a}^{γ+β} f(x). (ii) lim_{β→n} I_{a}^{β} f(x) = I_{a}^{n} f(x) uniformly on [a,b], n = 1, 2, 3, ⋯, where I_{a}^{1} f(x) = ∫_{a}^{x} f(s) ds. (iii) lim_{β→0} I_{a}^{β} f(x) = f(x) weakly. (iv) If f(x) is absolutely continuous on [a, b], then lim_{α→1} D_{a}^{α} f(x) = \frac{df(x)}{dx}.

(v) If f(x) = k ≠ 0, k is a constant, then D_{a}^{α} k = 0.

3 Existence of solution

Let C[0, T] be the class of continuous functions. For x ∈ C[0, T] we use the norm ∥x∥_{1} = sup_{t} e^{−Nt}|x(t)| , N > 0. Let f : [0, T]×ℝ^{n+1} → ℝ_{+}, g : [0, T]×ℝ^{m+1} → ℝ_{+} and f, g satisfy the Lipschitz condition

\begin{aligned}|f(t, x_{0}, x_{1}, …, x_{n}) − f(t, y_{0}, y_{1}, …, y_{n})| & ≤ k_{1} ∑_{i=0}^{n} |x_{i} − y_{i}| , k_{1} > 0 \\ |g(t, x_{0}, x_{1}, …, x_{m}) − g(t, y_{0}, y_{1}, …, y_{m})| & ≤ k_{2} ∑_{j=0}^{m} |x_{j} − y_{j}| , k_{2} > 0 \end{aligned}  (3.1)

Theorem 3.1. If f and g, satisfy the Lipschitz condition (3.1), then the Cauchy problem (1.1) - (1.2) has a monotonic increasing solution x ∈ C[0, T] and \frac{dx}{dt} ∈ C[0, T].

Proof. Let y(t) = \frac{d}{dt} x(t) then x(t) = x_{0} + I y(t), which implies

x(t − r_{i}) = x_{0} + ∫_{r_{i}}^{t} y(θ − r_{i})dθ, i = 0, 1, 2, …, n

i.e.

x(t − r_{i}) = x_{0} + I_{r_{i}} y(t − r_{i}),

from which we obtain

I_{r_{i}}^{1−α_{i}} \frac{d}{dt} x(t − r_{i}) = I_{r_{i}}^{1−α_{i}} y(t − r_{i}), i = 0, 1, …, n .

Similarly we obtain

I_{σ_{j}}^{1−β_{j}} \frac{d}{dt} x(t − σ_{j}) = I_{σ_{j}}^{1−β_{j}} y(t − σ_{j}), j =, 0, 1, …, m .

Now equation (1.1) can be written as

\begin{aligned}y(t) & = f(t, x_{0} + Iy(t), I_{r_{1}}^{1−α_{1}} y(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} y(t − r_{n})) \\ & + g(t, x_{0} + I y(t), I_{σ_{1}}^{1−β_{1}} y(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} y(t − σ_{n})), \end{aligned}  (3.2)

Define the operator F : C[0, T] → C[0, T] by

\begin{aligned}F y(t) & = f(t, x_{0} + Iy(t), I_{r_{1}}^{1−α_{1}} y(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} y(t − r_{n})) \\ & + g(t, x_{0} + I y(t), I_{σ_{1}}^{1−β_{1}} y(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} y(t − σ_{n})), \end{aligned}

then

F y(t) − F z(t) = f (t, x_{0} + Iy(t), I_{r_{1}}^{1−α_{1}} y(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} y(t − r_{n}))
− f (t, x_{0} + Iz(t), I_{r_{1}}^{1−α_{1}} z(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} z(t − r_{n}))
+ g (t, x_{0} + I y(t), I_{σ_{1}}^{1−β_{1}} y(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} y(t − σ_{n}))
− g (t, x_{0} + I z(t), I_{σ_{1}}^{1−β_{1}} z(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} z(t − σ_{n})) ,

and

e^{−Nt} |F y(t) − F z(t)| = e^{−Nt} |f (t, x_{0} + Iy(t), I_{r_{1}}^{1−α_{1}} y(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} y(t − r_{n}))

− f (t, x_{0} + Iz(t), I_{r_{1}}^{1−α_{1}} z(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} z(t − r_{n}))|

+ e^{−Nt} |g (t, x_{0} + I y(t), I_{σ_{1}}^{1−β_{1}} y(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} y(t − σ_{n}))

− g (t, x_{0} + I z(t), I_{σ_{1}}^{1−β_{1}} z(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} z(t − σ_{n}))| ,

≤ (k_{1} + k_{2}) ∫_{0}^{t} e^{−N(t)} |y(s) − z(s)|ds

+ k_{1} ∑_{i=1}^{n} ∫_{r_{i}}^{t} e^{−N(t)} \frac{(t − s)^{−α_{i}}}{Γ(1 − α_{i})} |y(s − r_{i}) − z(s − r_{i})|ds

+ k_{2} ∑_{j=1}^{m} ∫_{σ_{j}}^{t} e^{−N(t)} \frac{(t − s)^{−β_{j}}}{Γ(1 − β_{j})} |y(s − σ_{j}) − z(s − σ_{j})|ds

which implies that

∥F y − F z∥_{1} ≤ (k_{1} + k_{2}) ∥y − z∥_{1} ∫_{0}^{t} e^{−N(t−θ)} dθ

+ k_{1} ∥y − z∥_{1} ∑_{i=1}^{n} e^{−Nr_{i}} ∫_{0}^{t−r_{i}} e^{−N(t−r_{i}−θ)} \frac{(t − r_{i} − θ)^{−α_{i}}}{Γ(1 − α_{i})} dθ

+ k_{2} ∥y − z∥_{1} ∑_{j=1}^{m} e^{−Nσ_{j}} ∫_{0}^{t−σ_{j}} e^{−N(t−σ_{j}−θ)} \frac{(t − σ_{j} − θ)^{−β_{j}}}{Γ(1 − β_{j})} dθ

≤ \frac{k_{1} + k_{2}}{N} ∥y − z∥_{1}

+ k_{1} ∥y − z∥_{1} ∑_{i=1}^{n} ∫_{0}^{N(t−r_{i})} \frac{e^{−u} u^{−α_{i}}}{N^{−α_{i}} Γ(1 − α_{i})} \frac{du}{N}

+ k_{2} ∥y − z∥_{1} ∑_{j=1}^{m} ∫_{0}^{N(t−σ_{j})} \frac{e^{−u} u^{−β_{j}}}{N^{−β_{j}} Γ(1 − β_{j})} \frac{du}{N}

≤ (\frac{k_{1} + k_{2}}{N} + k_{1} ∑_{i=1}^{n} \frac{1}{N^{1−α_{i}}} + k_{2} ∑_{j=1}^{m} \frac{1}{N^{1−β_{j}}} ) ∥y − z∥_{1} < K̃ ∥y − z∥_{1} ,

where

K̃ = (\frac{k_{1} + k_{2}}{N} + k_{1} ∑_{i=1}^{n} \frac{1}{N^{1−α_{i}}} + k_{2} ∑_{j=1}^{n} \frac{1}{N^{1−β_{j}}} )

Now we choose N large enough such that K̃ < 1, then

∥F y − F z∥_{1} < ∥y − z∥_{1} ,

and the map F is contraction and has a unique positive fixed point y ∈ C[0, T] which proves the existence of a unique solution x ∈ C[0, T] of the Cauchy problem (1.1) - (1.2). Now the solution of (1.1) - (1.2) can be written as x(t) = x_{0} + I y(t), this implies that \frac{dx}{dt} = y > 0 which prove that \frac{dx}{dt} ∈ C[0, T] and x is monotonic increasing.

4 Stability

In this section we study the stability of the solution of the initial value problem (1.1) - (1.2).

Theorem 4.1. If the solution of the initial value problem (1.1) - (1.2) exists, then this solution is uniformly asymptotically stable.

Proof. Let y(t) be a solution of problem (1.1) - (1.2) and y^{∗}(t) be a solution of problem (1.1) with the initial value x^{∗}(t) = x_{0}^{∗} t ≤ 0 then

e^{−Nt} |y(t) − y^{∗}(t)| ≤ e^{−Nt} |f(t, x_{0} + Iy(t), I_{r_{1}}^{1−α_{1}} y(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} y(t − r_{n}))

− f(t, x_{0}^{∗} + Iy^{∗}(t), I_{r_{1}}^{1−α_{1}} y^{∗}(t − r_{1}), …, I_{r_{n}}^{1−α_{n}} y^{∗}(t − r_{n}))| + e^{−Nt} |g(t, x_{0} + I y(t), I_{σ_{1}}^{1−β_{1}} y(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} y(t − σ_{n})) , − g(t, x_{0}^{∗} + I y^{∗}(t), I_{σ_{1}}^{1−β_{1}} y^{∗}(t − σ_{1}), …, I_{σ_{n}}^{1−β_{n}} y^{∗}(t − σ_{n}))| ,

≤ (k_{1} + k_{2}) {e^{−Nt}|x_{0} − x_{0}^{∗}| + ∫_{0}^{t} e^{−N(t−s)} e^{−Ns} |y(s) − y^{∗}(s)|ds}

+k_{1}e^{−Nt} ∑_{i=1}^{n} ∫_{r_{i}}^{t} \frac{(t − s)^{−α_{i}}}{Γ(1 − α_{i})} |y(s − r_{i}) − y^{∗}(s − r_{i})| ds

+k_{2}e^{−Nt} ∑_{j=1}^{n} ∫_{σ_{j}}^{t} \frac{(t − s)^{−β_{j}}}{Γ(1 − β_{j})} |y(s − σ_{i}) − y^{∗}(s − σ_{i})| ds

≤ (k_{1} + k_{2}) {e^{−Nt}|x_{0} − x_{0}^{∗}| + ∫_{0}^{t} e^{−N(t−s)} e^{−Ns} |y(s) − y^{∗}(s)|ds}

+ k_{1} ∑_{i=1}^{n} ∫_{0}^{t−r_{i}} e^{−N(t−θ)} \frac{(t − r_{i} − θ)^{−α_{i}}}{Γ(1 − α_{i})} e^{−Nθ} |y(θ) − y^{∗}(θ)|dθ

+ k_{2} ∑_{j=1}^{n} ∫_{0}^{t−σ_{j}} e^{−N(t−θ)} \frac{(t − σ_{j} − θ)^{−β_{j}}}{Γ(1 − β_{j})} e^{−Nθ} |y(θ) − y^{∗}(θ)|dθ

and

∥y − y^{∗}∥_{1} ≤ (k_{1} + k_{2}) e^{−Nt} ∥x_{0} − x_{0}^{∗}∥_{1} + \frac{(k_{1} + k_{2})}{N} ∥y − y^{∗}∥_{1}

+ k_{1} ∥y − y^{∗}∥_{1} ∑_{i=1}^{n} ∫_{0}^{N(t−r_{i})} \frac{e^{−u} u^{−α_{i}}}{N^{−α_{i}} Γ(1 − α_{i})} \frac{du}{N}

+k_{2} ∥y − y^{∗}∥_{1} ∑_{j=1}^{n} ∫_{0}^{N(t−σ_{j})} \frac{e^{−u} u^{−β_{j}}}{N^{−β_{j}} Γ(1 − β_{j})} \frac{du}{N}

≤ (k_{1} + k_{2}) e^{−Nt} ∥x_{0} − x_{0}^{∗}∥_{1}

+ (\frac{k_{1} + k_{2}}{N} + k_{1} ∑_{i=1}^{n} \frac{1}{N^{1−α_{i}}} + k_{2} ∑_{j=1}^{n} \frac{1}{N^{1−β_{j}}} ) ∥y − y^{∗}∥_{1}

this implies that

∥y − y^{∗}∥_{1} ≤ (1 − K̃)^{−1} (k_{1} + k_{2}) e^{−Nt} ∥x_{0} − x_{0}^{∗}∥_{1}

where

K̃ = (\frac{k_{1} + k_{2}}{N} + k_{1} ∑_{i=1}^{n} \frac{1}{N^{1−α_{i}}} + k_{2} ∑_{i=1}^{n} \frac{1}{N^{1−β_{i}}} ) < 1.

Now

x(t) − x^{∗}(t) = x_{0} − x_{0}^{∗} + ∫_{0}^{t}(y(s) − y^{∗}(s))ds

this implies that

e^{−Nt}|x(t) − x^{∗}(t)| ≤ e^{−Nt}|x_{0} − x_{0}^{∗}| + ∫_{0}^{t} e^{−N(t−s)} e^{−Ns}|(y(s) − y^{∗}(s))|ds

then

∥x − x^{∗}∥_{1} ≤ e^{−Nt} ∥x_{0} − x_{0}^{∗}∥_{1} + \frac{1}{N} ∥y − y^{∗}∥_{1}
≤ e^{−Nt} ∥x_{0} − x_{0}^{∗}∥_{1} + (1 − K̃)^{−1} (\frac{k_{1} + k_{2}}{N}) e^{−Nt} ∥x_{0} − x_{0}^{∗}∥_{1}
≤ (1 + \frac{k_{1} + k_{2}}{N(1 − K̃)}) e^{−Nt} ∥x_{0} − x_{0}^{∗}∥_{1} ,

therefore lim_{t→∞} ∥x − x^{∗}∥_{1} = 0, then the solution of the system (1.1) is uniformly asymptotically stable.

5 Application

(1) Consider now the initial value problem (1.3). Letting

f(t, x(t), D_{r_{1}}^{α_{1}}x(t − r_{1}), …, D_{r_{n}}^{α_{n}}x(t − r_{n})) = a(t) x(t) + ∑_{i=1}^{n} a_{i}(t)D_{r_{i}}^{α_{i}}x(t − r_{i})

and

g(t, x(t), D_{σ_{1}}^{β_{1}}x(t − σ_{1}), …, D_{σ_{m}}^{β_{m}}x(t − σ_{m})) = ∑_{j=1}^{m} b_{j}(t)D_{σ_{j}}^{β}x(t − σ_{j})

where a_{i} and b_{j}, i, j = 0, 1, 2 ⋯ are positive continuous functions on [0, T] with |a_{i}(t)| ≤ a and |b_{j}(t)| ≤ b. Then our results here can be applied to the initial value problem (1.3) and prove the existence of positive monotonic and uniformly asymptotically stable solution for the initial value problem (1.3).

(2) Consider now the initial value problem (1.4). Letting α_{i} → 1 and β → 0 in (1.3) and use the proberties of the fractional derivative ([1]) we obtain (1.4).

References

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EJQTDE, 2008 No. 16, p. 8 [7] Bai C., Positive solutions for nonlinear fractional differential equations with coefficient

that changes sign, Nonlinear Analysis 64 (2006) 677–685

(Received January 3, 2008)