# Uniform Asymptotic Stability of Solutions of Fractional Functional Differential Equations

Authors: Yajing Li; Yejuan Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Some global existence and uniform asymptotic stability results for fractional functional differential equations are proved. It is worth mentioning that whenα=1 the initial value problem Dαyteβt=ft,yteβt,t∈t0,∞,t0⩾0,0<α<1,yt=ϕt,t0-h⩽t⩽t0, reduces to a classical dissipative differential equation with delays as in Caraballo et al.'s work (2005).


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## Body

## 1. Introduction

Consider the initial value problem (IVP for short) of the following fractional functional differential equation:(1)Dα[y(t)eβt]=f(t,yt)eβt,t∈[t0,∞),t0⩾0,ddddddddffddssdd0<α<1,y(t)=ϕ(t),t0-h⩽t⩽t0,
where Dα is the Caputo fractional derivative, β>0, f:J×C([-h,0],ℝ)→ℝ, where J=[t0,∞), is a given function satisfying some assumptions that will be specified later, h>0, and ϕ∈C([t0-h,t0],ℝ). If y∈C([t0-h,∞),ℝ), then for any t∈[t0,∞), define yt by
(2)yt(θ)=y(t+θ),θ∈[-h,0].The study of retarded differential equations is an important area of applied mathematics due to physical reasons, noninstant transmission phenomena, memory processes, and specially biological motivations (see, e.g., [1–4]). Fractional differential equations have attracted much attention recently (see, e.g., [5–11] and the references cited therein for the applications in various sciences such as physics, mechanics, chemistry, and engineering).Some attractive results for fractional functional differential equations and nonlinear functional integral equations are obtained by using the fixed point theory; see [12–16] and references therein. Global asymptotic stability of solutions of a functional integral equation is discussed in [17]; however, there is no work on uniform asymptotic stability of solutions of fractional functional differential equation. It is our intention here to show the global existence and uniform asymptotic stability of the fractional functional differential equation (1).We organize the paper as follows. In Section2, we recall some necessary concepts and results. In Section 3 we give the global existence and uniform asymptotic stability of fractional functional differential equations. Finally, two examples are given to illustrate our main results.


## 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.We considerBC:=BC([t0-h,∞),ℝ) the Banach space of all bounded and continuous functions from [t0-h,∞) into ℝ with the norm
(3)∥y∥∞:=sup{|y(t)|:t∈[t0-h,∞)}.
Let ∥yt∥=sup-h⩽θ⩽0|y(t+θ)| for t∈J.Throughout this paper, we always assume thatf(t,xt) satisfies the following condition:(H0)
f(t,xt) is Lebesgue measurable with respect to t on [t0,∞), and f(t,φ) is continuous with respect to φ on C([-h,0],ℝ). By condition (H0) and the technique used in [7], we get the equivalent form of IVP (1) as
(4)y(t)={y(t0)e-β(t-t0)+1Γ(α)∫t0t(t-s)α-1e-β(t-s)ssssssss×f(s,ys)ds,t⩾t0,ϕ(t),t∈[t0-h,t0],
where Γ(·) is the gamma function.Definition 1.
We say that the solutionsy and x of IVP (1) are uniformly asymptotically stable if for any bounded subset B of C([-h,0],ℝ) and ε>0, there exists a T>0 such that
(5)|y(t,t0,ϕ)-x(t,t0,ψ)|⩽ε∀t⩾T,sssssssssssssssddddddddϕ,ψ∈B.
We recall the following generalization of Gronwall’s lemma for singular kernels [18], which will be used in the sequel.Lemma 2.
Letv:[t0,b]→[0,+∞) be a real function and w(·) is a nonnegative, locally integrable function on [t0,b] and there are constants a>0 and 0<α<1 such that
(6)v(t)⩽w(t)+a∫t0t‍v(s)(t-s)αds.
Then there exists a constant K=K(α) such that
(7)v(t)⩽w(t)+Ka∫t0t‍w(s)(t-s)αds,
for every t∈[t0,b].Theorem 3 (Leray-Schauder fixed-point theorem).
LetP be a continuous and compact mapping of a Banach space X into itself, such that the set
(8){x∈X:x=λPxforsome0⩽λ⩽1},
is bounded. Then, P has a fixed point.


## 3. FDEs of Fractional Order

In this section, we will investigate the IVP (1). Our first global existence and uniform asymptotic stability result for the IVP (1) is based on the Banach contradiction principle and Lemma 2.Theorem 4.
Assume thatf(t,yt) satisfies conditions (H0) and(H1)
there existsl>0 such that
(9)|f(t,ut)-f(t,vt)|⩽l∥ut-vt∥,for t∈J and every ut,vt∈C([-h,0],ℝ). Moreover, the function t↦f(t,0) is bounded with f0=supt⩾t0|f(t,0)|.
If(10)l((t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1))<1,
then the IVP (1) has a unique solution in the space BC. Moreover, solutions of IVP (1) are uniformly asymptotically stable.Proof.
We divide the proof into two steps.
Step
1. We define the operator P:C([t0-h,∞),ℝ)→C([t0-h,∞),ℝ) by
(11)(Py)(t)={y(t0)e-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)ssddss×f(s,ys)ds,t⩾t0,ϕ(t),t∈[t0-h,t0].
The operator P maps BC into itself. Indeed for each y∈BC, and for each t⩾2t0+h, it follows from (H1) that
(12)|(Py)(t)|⩽|y(t0)|e-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)(f0+l∥ys∥)ds⩽∥ϕ∥e-β(t-t0)+f0+l∥y∥∞Γ(α)×(∫t0t-(t0+h)‍(t0+h)α-1e-β(t-s)ds+∫t-(t0+h)t‍(t-s)α-1ds)⩽∥ϕ∥+(f0+l∥y∥∞)((t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1)).
For each t∈[t0,2t0+h], we have
(13)|(Py)(t)|⩽∥ϕ∥+(f0+l∥y∥∞)(t0+h)αΓ(α+1),
and consequently P(y)∈BC.
SinceBC:=BC([t0-h,∞),ℝ) is a Banach space with norm ∥·∥∞, we will show that P:BC→BC is a contraction map. Let y1,y2∈BC. Then, we have for each t⩾t0,
(14)|(Py1)(t)-(Py2)(t)|⩽1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)ffffff×|f(s,y1s)-f(s,y2s)|ds⩽lΓ(α)∫t0t‍(t-s)α-1e-β(t-s)∥y1s-y2s∥ds.
Therefore, for anyt⩾2t0+h,
(15)|(Py1)(t)-(Py2)(t)|⩽lΓ(α)∥y1(·)-y2(·)∥∞×(∫t0t-(t0+h)‍(t0+h)α-1e-β(t-s)dsdd+∫t-(t0+h)t‍(t-s)α-1ds)⩽l((t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1))×∥y1(·)-y2(·)∥∞,
and for t0-h⩽t⩽2t0+h,
(16)|(Py1)(t)-(Py2)(t)|⩽lΓ(α)∥y1(·)-y2(·)∥∞∫t0t‍(t-s)α-1ds⩽l(t0+h)αΓ(α+1)∥y1(·)-y2(·)∥∞,
and thus
(17)∥(Py1)(·)-(Py2)(·)∥∞⩽l((t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1))×∥y1(·)-y2(·)∥∞.
Hence, (10) and (17) imply that the operator P is a contraction. Therefore, P has a unique fixed point by Banach’s contraction principle.
Step
2. For any two solutions x=x(t) and y=y(t) of IVP (1) corresponding to initial values ψ and ϕ, by (4) we can deduce that for all t⩾t0+h and all θ∈[-h,0],
(18)|x(t+θ)-y(t+θ)|⩽|x(t0)-y(t0)|e-β(t+θ-t0)+1Γ(α)∫t0t+θ‍(t+θ-s)α-1e-β(t+θ-s)ssssddssss×|f(s,xs)-f(s,ys)|ds⩽|x(t0)-y(t0)|e-β(t+θ-t0)+lΓ(α)∫t0t+θ‍(t+θ-s)α-1e-β(t+θ-s)∥xs-ys∥ds.
Then, it follows that
(19)eβt∥xt-yt∥⩽|x(t0)-y(t0)|eβ(h+t0)+leβhΓ(α)∫t0t‍(t-s)α-1eβs∥xs-ys∥ds.
Let w(t)=eβt∥xt-yt∥. Then, we have
(20)w(t)⩽|x(t0)-y(t0)|eβ(h+t0)+leβhΓ(α)∫t0t‍(t-s)α-1w(s)ds.
Applying Lemma 2, one can see that there exists a constant K such that
(21)w(t)⩽|x(t0)-y(t0)|eβ(h+t0)+KleβhΓ(α)∫t0t‍(t-s)α-1|x(t0)-y(t0)|eβ(h+t0)ds⩽|x(t0)-y(t0)|eβ(h+t0)(1+KleβhΓ(α+1)(t-t0)α).
Hence, we obtain
(22)eβt∥xt-yt∥=w(t)⩽|x(t0)-y(t0)|eβ(h+t0)×(1+KleβhΓ(α+1)(t-t0)α),
and thus for all t⩾t0+h,
(23)|x(t)-y(t)|⩽|x(t0)-y(t0)|e-β(t-h-t0)(1+KleβhΓ(α+1)(t-t0)α),
which implies that the solutions of IVP (1) are uniformly asymptotically stable.Now we give global existence and uniform asymptotic stability results based on the nonlinear alternative of Leray-Schauder type.Theorem 5.
Assume that the following hypotheses hold:(H2)
f is a continuous function;(H3)
there exist positive functionsk1, k2∈BC([t0,∞),ℝ+) such that
(24)|f(t,ut)|⩽k1(t)+k2(t)∥ut∥for t∈J and every ut∈C([-h,0],ℝ);(H4)
moreover, assume that(25)limt→∞∫t0t‍(t-s)α-1e-β(t-s)k1(s)ds=0,limt→∞∫t0t‍(t-s)α-1e-β(t-s)k2(s)ds=0.Then the IVP (1) admits a solution in the space BC. Moreover, solutions of IVP (1) are uniformly asymptotically stable.Proof.
LetP:C([t0-h,∞),ℝ)→C([t0-h,∞),ℝ) be defined as in (11). First, we show that P maps BC into itself. Let K1=supt⩾t0k1(t), K2=supt⩾t0k2(t). Indeed, the map P(y) is continuous on [t0-h,+∞) for each y∈BC, and for each t⩾2t0+h, (H2) implies that
(26)|(Py)(t)|⩽|y(t0)|e-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)(K1+K2∥ys∥)ds⩽∥ϕ∥e-β(t-t0)+K1+K2∥y∥∞Γ(α)×(∫t0t-(t0+h)‍(t0+h)α-1e-β(t-s)dsssss+∫t-(t0+h)t‍(t-s)α-1ds)⩽∥ϕ∥+(K1+K2∥y∥∞)×((t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1)),
for each t∈[t0,2t0+h], we have
(27)|(Py)(t)|⩽∥ϕ∥+(K1+K2∥y∥∞)(t0+h)αΓ(α+1),
and for any t∈[t0-h,t0],
(28)|(Py)(t)|⩽∥ϕ∥.
Thus,
(29)∥P(y)∥∞⩽∥ϕ∥+(K1+K2∥y∥∞)×((t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1)),
and consequently P(y)∈BC.
Next, we show that the operatorP is continuous and completely continuous, and there exists an open set U⊂BC with y≠λP(y) for λ∈(0,1) and y∈∂U.
Step
1 (
P
is continuous). Let {yn} be a sequence such that yn→y in BC. Then, there exist R>0 and N>0 such that
(30)∥yn∥∞+∥y∥∞<R,∀n⩾N.
Let ε>0 be given. Since (H4) holds, there is a real number T>0 such that
(31)2Γ(α)∫t0t‍(t-s)α-1e-β(t-s)(k1(s)+k2(s)R)ds<ε,
for all t⩾T. Now we consider the following two cases.
Case
1. If t⩾T, then it follows from (H3) and (30)-(31) that for n sufficiently large
(32)|Pyn(t)-Py(t)|⩽|yn(t0)-y(t0)|e-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)|f(s,yns)-f(s,ys)|ds⩽|yn(t0)-y(t0)|+2Γ(α)∫t0t‍(t-s)α-1e-β(t-s)dddddddd×(k1(s)+k2(s)R)ds<2ε.
Case2. If t0⩽t⩽T, since f is a continuous function, one has
(33)|Pyn(t)-Py(t)|⩽|yn(t0)-y(t0)|+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)|f(s,yns)-f(s,ys)|ds⩽|yn(t0)-y(t0)|+(T-t0)αΓ(α+1)sups∈[t0,T]|f(s,yns)-f(s,ys)|.
Note that yn→y in BC. Hence, (32) and (33) imply that
(34)∥P(yn)-P(y)∥∞⟶0asn⟶∞.
Step
2 (P maps bounded sets into bounded sets in
B
C
). Indeed, it is enough to show that for any η>0, there exists a positive constant ℓ such that for each y∈Bη={y∈BC:∥y∥∞⩽η} one has ∥P(y)∥∞⩽ℓ. Let y∈Bη. Then, we have for each t⩾2t0+h,
(35)|(Py)(t)|⩽|y(t0)|e-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)|f(s,ys)|ds⩽η+K1+K2∥y∥∞Γ(α)∫t0t‍(t-s)α-1e-β(t-s)ds⩽η+(K1+K2η)×((t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1))=:ℓ,
and for each t with t0⩽t⩽2t0+h,
(36)|(Py)(t)|⩽η+(K1+K2η)(t0+h)αΓ(α+1).
Hence,∥P(y)∥∞⩽ℓ.
Step
3 (Pmaps bounded sets into equicontinuous sets on every compact subset[t0-h,b]of[t0-h,∞)). Let t1,t2∈[t0,b], t1<t2, and let Bη be a bounded set of BC as in Step 2. Let y∈Bη. Then, we have
(37)|(Py)(t2)-(Py)(t1)|⩽|y(t0)e-β(t2-t0)-y(t0)e-β(t1-t0)|+1Γ(α)∫t0t1‍|((t2-s)α-1e-β(t2-s)ddddddddddd-(t1-s)α-1e-β(t1-s))f(s,ys)|ds+1Γ(α)∫t1t2‍|(t2-s)α-1e-β(t2-s)f(s,ys)|ds⩽|y(t0)|eβt0|e-βt2-e-βt1|+K1+K2ηΓ(α+1)(t2-t1)α+K1+K2ηΓ(α)∫t0t1‍((t1-s)α-1e-β(t1-s)ddddddddddddd-(t2-s)α-1e-β(t2-s))ds⩽|y(t0)|eβt0|e-βt2-e-βt1|+K1+K2ηΓ(α+1)(t2-t1)α+K1+K2ηΓ(α)∫t0t1‍((t1-s)α-1e-β(t1-s)ddddddddddddd-(t2-s)α-1e-β(t1-s))ds+K1+K2ηΓ(α)∫t0t1‍((t2-s)α-1e-β(t1-s)ssssssssssshsssdh-(t2-s)α-1e-β(t2-s))ds.
Observing that(38)K1+K2ηΓ(α)×∫t0t1‍((t1-s)α-1e-β(t1-s)-(t2-s)α-1e-β(t1-s))ds⩽K1+K2ηΓ(α)∫t0t1‍((t1-s)α-1-(t2-s)α-1)ds⩽K1+K2ηΓ(α+1)((t1-t0)α-(t2-t0)α+(t2-t1)α)⩽K1+K2ηΓ(α+1)(t2-t1)α,
from Taylor’s theorem, we obtain
(39)K1+K2ηΓ(α)×∫t0t1‍((t2-s)α-1e-β(t1-s)-(t2-s)α-1e-β(t2-s))ds⩽K1+K2ηΓ(α)(t2-t1)α-1∫t0t1‍(e-β(t1-s)-e-β(t2-s))ds⩽K1+K2ηβΓ(α)(t2-t1)α-1(1-e-β(t2-t1))=K1+K2ηΓ(α)((t2-t1)α+o(t2-t1)t2-t1(t2-t1)α),
where limt2-t1→0(o(t2-t1)/(t2-t1))=0. By (37)–(39), we can conclude that
(40)|(Py)(t2)-(Py)(t1)|⩽ηeβt0|e-βt2-e-βt1|+2(K1+K2η)Γ(α+1)(t2-t1)α+K1+K2ηΓ(α)((t2-t1)α+o(t2-t1)t2-t1(t2-t1)α).
As t1→t2, the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t1<t2⩽t0 and t1⩽t0⩽t2 is obvious.
Step
4 (Pmaps bounded sets into equiconvergent sets). Let y∈Bη. Then
(41)|(Py)(t)|⩽|y(t0)|e-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)|f(s,ys)|ds⩽ηe-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)(k1(s)+k2(s)η)ds.
Therefore, (H4) implies that |(Py)(t)| uniformly (with respect to y∈B(η)) converges to 0 as t→∞. As a consequence of Steps 1–4, we can conclude that P:BC→BC is continuous and completely continuous.
Step
5 (a priori bounds). We now show that there exists an open set U⊆BC with y≠λP(y) for λ∈(0,1) and y∈∂U.
Lety∈BC and y=λP(y) for some 0<λ<1. Then, for each t∈[t0,∞), we obtain
(42)y(t)=λ[1Γ(α)∫t0t‍y(t0)e-β(t-t0)dddd+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)f(s,ys)ds].
By (H3), we have that for all θ∈[-h,0] and t⩾t0+h,
(43)|y(t+θ)|⩽|y(t0)|e-β(t+θ-t0)+1Γ(α)∫t0t+θ‍(t+θ-s)α-1e-β(t+θ-s)|f(s,ys)|ds⩽|y(t0)|e-β(t+θ-t0)+1Γ(α)∫t0t+θ‍(t+θ-s)α-1e-β(t+θ-s)(K1+K2∥ys∥)ds,
and thus
(44)∥yt∥⩽|y(t0)|e-β(t-h-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-h-s)(K1+K2∥ys∥)ds.
From the arguments in (26)-(27), we can conclude that for each t∈[t0,∞),
(45)1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)ds⩽(t0+h)α-1e-β(t0+h)βΓ(α)+(t0+h)αΓ(α+1)=:R1.
Hence,
(46)eβt∥yt∥⩽∥ϕ∥eβ(h+t0)+eβ(h+t)K1R1+K2eβhΓ(α)∫t0t‍(t-s)α-1eβs∥ys∥ds.
Let R2=∥ϕ∥eβ(h+t0)+eβhK1R1. Then, from Lemma 2, there exists K such that we have for all t⩾t0+h,
(47)∥yt∥⩽R2+KK2R2eβhΓ(α+1)(t-t0)αe-βt.
Since limt→∞(t-t0)αe-βt=0, there exists R3>0 such that
(48)∥y∥∞⩽R3.
Set(49)U={y∈BC:∥y∥∞<R3+1}.P:U→BC is continuous and completely continuous. From the choice of U, there is no y∈∂U such that y=λP(y), for λ∈(0,1). As a consequence of Leray-Schauder fixed-point theorem, we deduce that P has a fixed point y in U.
Step
6 (uniform asymptotic stability of solutions). Let B⊂C([-h,0],ℝ) be bounded; that is, there exists d⩾0 such that
(50)∥ψ∥=supθ∈[-h,0]|ψ(θ)|⩽d∀ψ∈B.
From the similar arguments in Step 4, we can deduce that there exists R4>0 such that for all solutions y(t,t0,ϕ) of IVP (1) with initial data ϕ∈B, we have
(51)∥y∥∞⩽R4,∀ϕ∈B.
Now we consider two solutionsx=x(t) and y=y(t) of IVP (1) corresponding to the initial values ψ and ϕ. Note that for all t⩾t0,
(52)|x(t)-y(t)|⩽|x(t0)-y(t0)|e-β(t-t0)+1Γ(α)∫t0t‍(t-s)α-1e-β(t-s)(|f(s,xs)|+|f(s,ys)|)ds⩽2de-β(t-t0)+2Γ(α)∫t0t‍(t-s)α-1e-β(t-s)(k1(s)+k2(s)R4)ds.
Then, the proof of uniform asymptotic stability of solutions can be done by making use of (H4) and (52).
The proof of Theorem5 is completed.


## 4. Examples

Example 1.
Consider the fractional functional differential equation(53)D1/2[y(t)et]=e2t8(et+e-t)sin4(y(t-1))+et,t⩾0,y(t)=ϕ(t),-1⩽t⩽0,
where f(t,yt)=(et/8(et+e-t))sin4(y(t-1))+1. It is clear that condition (H0) holds. Let xt, yt∈C([-1,0],ℝ). Then for all t∈[0,∞), we have
(54)|f(t,xt)-f(t,yt)|=et8(et+e-t)|sin4(x(t-1))-sin4(y(t-1))|⩽et2(et+e-t)|x(t-1)-y(t-1)|⩽12|x(t-1)-y(t-1)|.
On the other hand, note that f(t,0)=1 for each t∈[0,∞) and (1/2)(e-1/Γ(1/2)+1/Γ(3/2))<1. Hence, conditions (H1) and (10) hold. By Theorem 4, we conclude that IVP (53) has a unique solution in the space BC([-1,∞),ℝ), and the solution of IVP (53) is uniformly asymptotically stable.Example 2.
Consider the fractional functional differential equation(55)D1/2[y(t)et]=10et(t+1)-3/4y(t-1)1+|y(t-1)|,t⩾0,y(t)=ϕ(t),-1⩽t⩽0,
where f(t,yt)=10(t+1)-3/4(y(t-1)/(1+|y(t-1)|)). It is easy to see that condition (H2) holds. Let yt∈C([-1,0],ℝ). Then, for all t∈[0,∞), we find that
(56)|f(t,yt)|=|10(t+1)-3/4y(t-1)1+|y(t-1)||⩽10(t+1)-3/4|y(t-1)|,
where 10(t+1)-3/4∈BC([0,∞),ℝ+) with supt⩾010(t+1)-3/4=10 and
(57)1Γ(1/2)∫0t‍(t-s)-1/2e-(t-s)10(s+1)-3/4ds⩽10Γ(1/2)∫0t‍(t-s)-1/2s-3/4ds=10Γ(1/4)Γ(3/4)t-1/4⟶0ast⟶∞.
Thus, conditions (H3) and (H4) hold, and the global existence and the uniform asymptotic stability of solutions of IVP (55) can be obtained by applying Theorem 5.
By using the algorithm given in [19], we numerically simulate Example 1 with the initial conditions ϕ(t)=sin(t),cos(t),-cos(t),1.5, and Example 2 with ϕ(t)=t,cos(t),-cos(t),1.5; see Figures 1 and 2. From the numerical results, it can be noted that both of the solutions of Examples 1 and 2 converge uniformly, and the solutions of Example 1 converge faster than the ones of Example 2. The numerical results confirm the theoretical analysis.Figure 1
The numerical solutions of Example1 with the initial conditions ϕ(t)=sin(t),cos(t),-cos(t), and 1.5, respectively.Figure 2
The numerical solutions of Example2 with the initial conditions ϕ(t)=t,cos(t),-cos(t), and 1.5, respectively.



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*Source: 532589-2013-11-26.xml*