# Some Properties for Solutions of Riemann-Liouville Fractional Differential Systems with a Delay

Authors: Jing Zhao; Fanwei Meng
Journal: Journal of Function Spaces
 (2018)

---

## Abstract

In this paper, we study properties for solutions of Riemann-Liouville (R-L) fractional differential systems with a delay. Some results on integral inequalities are first presented by Ho¨lder inequality. Then we investigate properties on solutions for R-L fractional systems with a delay by using the obtained inequalities and obtain upper bound of solutions. Finally, an illustrative example is considered to support our new results.


---

## Body

## 1. Introduction

Fractional differential equations have been studied for several centuries. At first the researches were only on the pure theoretical aspect. In the last few years, more and more fractional differential equations have been applied to described some actual researches, such as mechanics, aerodynamics, chemistry, and the electrodynamics of complex mediums [1–8].Integral inequalities play an important role in researches not only on properties of solutions for various differential and integral equations [9–12], but also on some fractional differential equations. Recently, some results are obtained on properties of solutions for a fractional differential equation with or without delays. For example, Ma [13] obtained upper bounds for solutions of a class of nonlinear fractional differential systems by a result of two dimensional linear integral inequalities. Ye [14] studied the Henry-Gronwall type retarded integral inequalities and then obtained a certain properties of fractional differential equations with delay. This paper studies some properties for solutions of R-L fractional differential systems with a delay. First, we obtain some results on the integral inequalities by Ho¨lder inequality. Then, using the obtained inequalities, properties are investigated on solutions for R-L fractional systems with a delay, and upper bound of solutions is obtained. Moreover, an illustrative example is studied to show that new results presented in this paper work very well.


## 2. Main Results

This section is devoted to studying properties of solutions for R-L fractional differential systems with a delay and presents the main result of this paper. First, we give some lemmas on integral inequalities.Lemma 1 (letI=[t0,+∞) and R+=[0,+∞)).
Supposeai(t),bi(t), and ci(t)∈C(I,R+), ϕi(t)∈C([t0-τ,t0],R+), ai(t0)=ϕi(t0), i=1,2, and τ>0 is a contant. If ui(t)∈C([t0-τ,+∞),R+) and(1)u1t≤a1t+∫t0tb1su1s-τ+c1su2s-τds,t∈t0,+∞,u2t≤a2t+∫t0tb2su1s-τ+c2su2s-τds,t∈t0,+∞,u1t≤ϕ1t,t∈t0-τ,t0,u2t≤ϕ2t,t∈t0-τ,t0,then (2)u1tu2t≤At+Ft,t∈I,where A(t)=a1(t)a2(t),(3)Ft=exp∫t0+τtHsds∫t0t0+τHsΦs-τds+∫t0+τtexp∫ξtHsdsHξAξ-τdξ,t∈t0+τ,+∞,∫t0tHsΦs-τds,t∈t0,t0+τ,(4)Ht=b1tc1tb2tc2t,Φt=ϕ1tϕ2t.Proof.
Setv1(t)=∫t0t[b1(s)u1(s-τ)+c1(s)u2(s-τ)]ds, v2(t)=∫t0t[b2(s)u1(s-τ)+c2(s)u2(s-τ)]ds. Then, ui(t)≤ai(t)+vi(t), vi(t)≥0(i=1,2), are nondecreasing for t∈[t0,+∞). Hence, for t∈[t0+τ,+∞), we have (5)v1′t=b1tu1t-τ+c1tu2t-τ≤b1ta1t-τ+v1t-τ+c1ta2t-τ+v2t-τ=b1ta1t-τ+c1ta2t-τ+b1tv1t-τ+c1tv2t-τ≤b1ta1t-τ+c1ta2t-τ+b1tv1t+c1tv2t,(6)v2′t=b2tu1t-τ+c2tu2t-τ≤b2ta1t-τ+v1t-τ+c2ta2t-τ+v2t-τ=b2ta1t-τ+c2ta2t-τ+b2tv1t-τ+c2tv2t-τ≤b2ta1t-τ+c2ta2t-τ+b2tv1t+c2tv2t.
SetWt=v1tv2t. (5) and (6) can be rewritten as a matrix form (7)W′t≤HtAt-τ+HtWt.Then we have (8)exp⁡-∫t0+τtHsdsWt′≤exp-∫t0+τtHsdsHtAt-τ.
Hence,(9)∫t0+τtexp-∫t0+τξHsdsWξ′dξ≤∫t0+τtexp-∫t0+τξHsdsHξAξ-τdξ. That is,(10)exp-∫t0+τtHsdsWt-Wt0+τ≤∫t0+τtexp-∫t0+τξHsdsHξAξ-τdξ. Then(11)Wt≤exp∫t0+τtHsdsWt0+τ+∫t0+τtexp∫ξtHsdsHξAξ-τdξ.
Whent∈[t0,t0+τ], from (1),(12)Wt=v1tv2t≤∫t0tb1sϕ1s-τ+c1sϕ2s-τds∫t0tb2sϕ1s-τ+c2sϕ2s-τds=∫t0tHsΦs-τds.
Let(13)Ft=exp∫t0+τtHsds∫t0t0+τHsΦs-τds+∫t0+τtexp∫ξtHsdsHξAξ-τdξ,t∈t0+τ,+∞,∫t0tHsΦs-τds,t∈t0,t0+τ,
Thus,(14)u1tu2t≤At+Wt≤At+Ft,t∈I.Therefore, (2) is satisfied and the proof is completed.Lemma 2.
Supposeai(t),bi(t), and ci(t)∈C(I,R+), ϕi(t)∈C([t0-τ,t0],R+), ai(t0)=ϕi(t0), i=1,2, and τ>0 and 0<β<1 are constants. If ui(t)∈C([t0-τ,+∞),R+) and (15)u1t≤a1t+∫t0tt-sβ-1b1su1s-τ+c1su2s-τds,t∈t0,+∞,u2t≤a2t+∫t0tt-sβ-1b2su1s-τ+c2su2s-τds,t∈t0,+∞,u1t≤ϕ1t,t∈t0-τ,t0,u2t≤ϕ2t,t∈t0-τ,t0,then,
(i) when1/2<β<1, set(16)u1tu2t=etw11/2tetw21/2tand we have(17)Wt=w1tw2t≤At+Qt,where(18)Qt=exp∫t0+τtHsds∫t0t0+τHsΨs-τds+∫t0+τtexp∫ξtHsdsHξAξ-τdξ,t∈t0+τ,+∞,∫t0tHsΨs-τds,t∈t0,t0+τ,(19)At=α1tα2t,Ht=l1tm1tl2tm2t,Ψtψ1tψ2t=e-2tϕ12te-2tϕ22t,and αi(t)=2e-2tai2(t), li(t)=Γ(2β-1)/22β-3e-2τbi2(t), mi(t)=Γ(2β-1)/22β-3e-2τci2(t), i=1,2;
(ii) when0<β≤1/2, let p=1+β, q=1+1/β(20)u1tu2t=etw~11/qtetw~21/qtand we have(21)W~t=w~1tw~2t≤A~t+Q~t,where(22)Q~t=exp∫t0+τtH~sds∫t0t0+τH~sΨ~s-τds+∫t0+τtexp∫ξtH~sdsH~ξA~ξ-τdξ,t∈t0+τ,+∞,∫t0tH~sΨ~s-τds,t∈t0,t0+τ,(23)A~t=α~1tα~2t,H~t=l~1tm~1tl~2tm~2t,Ψ~t=ψ~1tψ~2t=e-qtϕ1qte-qtϕ2qt and α~i(t)=2q-1[e-tai(t)]q, l~i(t)=41/βΓ1/β(β2)/pβe-qτbiq(t), m~i(t)=41/βΓ1/β(β2)/pβe-qτciq(t), i=1,2.Proof.
(i) Supposeβ>1/2. By Ho¨lder inequality and (a+b)2≤2(a2+b2), for t∈[t0+τ,+∞) and i=1,2, we have(24)
u
i
t
≤
a
i
t
+
∫
t
0
t
t
-
s
β
-
1
e
s
e
-
s
b
i
s
u
1
s
-
τ
+
c
i
s
u
2
s
-
τ
d
s
≤
a
i
t
+
∫
t
0
t
t
-
s
2
β
-
1
e
2
s
d
s
1
/
2
∫
t
0
t
e
-
2
s
b
i
s
u
1
s
-
τ
+
c
i
s
u
2
s
-
τ
2
d
s
1
/
2
≤
a
i
t
+
2
1
/
2
∫
t
0
t
t
-
s
2
β
-
1
e
2
s
d
s
1
/
2
∫
t
0
t
b
i
2
s
e
-
2
s
u
1
2
s
-
τ
+
c
i
2
s
e
-
2
s
u
2
2
s
-
τ
d
s
1
/
2
=
a
i
t
+
2
1
/
2
-
∫
t
-
t
0
0
s
2
β
-
1
e
2
t
-
s
d
s
1
/
2
∫
t
0
t
b
i
2
s
e
-
2
s
u
1
2
s
-
τ
+
c
i
2
s
e
-
2
s
u
2
2
s
-
τ
d
s
1
/
2
≤
a
i
t
+
e
t
2
β
-
1
∫
0
+
∞
2
s
2
β
-
1
-
1
e
-
2
s
d
2
s
1
/
2
∫
t
0
t
b
i
2
s
e
-
2
s
u
1
2
s
-
τ
+
c
i
2
s
e
-
2
s
u
2
2
s
-
τ
d
s
1
/
2
.
Thus,(25)ui2t≤2ai2t+e2t22β-3Γ2β-1∫t0tbi2se-2su12s-τ+ci2se-2su22s-τds,i=1,2.Let wi(t)=[e-tui(t)]2, αi(t)=2e-2tai2(t), li(t)=Γ(2β-1)/22β-3e-2τbi2(t), mi(t)=Γ(2β-1)/22β-3e-2τci2(t),i=1,2. Then, for t∈[t0,+∞), we have(26)wit≤αit+∫t0tlitw1s-τ+mitw2s-τds.Set ψi(t)=e-2tϕi2(t)(i=1,2). For t∈[t0-τ,t0](27)wit≤ψiti=1,2.By (26), (27), and Lemma 1, (17) is satisfied.
(ii) Suppose0<β≤1/2. Set p=β+1 and q=1+1/β. Then 1/p+1/q=1. By Ho¨lder inequality and (a+b)l≤2l-1(al+bl)(0<l<1), for t∈[t0+τ,+∞) and i=1,2, we have(28)uit≤ait+∫t0tt-sβ-1ese-sbisu1s-τ+cisu2s-τds≤ait+∫t0tt-spβ-1epsds1/p∫t0te-qsbisu1s-τ+cisu2s-τqds1/q≤ait+21/pet∫0∞sβ2-1e-psds1/p∫t0tbiqse-qsu1qs-τ+ciqse-qsu2qs-τds1/q≤ait+21/petΓ1/pβ2pβ2/p∫t0tbiqse-qsu1qs-τ+ciqse-qsu2qs-τds1/q.Thus,(29)e-tuitq≤2q-1e-taitq+22/βΓ1/ββ2pβ∫t0tbiqse-qsu1qs-τ+ciqse-qsu2qs-τds,i=1,2. Let w~i(t)=[e-tui(t)]q, α~i(t)=2q-1[e-tai(t)]q, l~i(t)=41/βΓ1/β(β2)/pβe-qτbiq(t), mi~(t)=41/βΓ1/β(β2)/pβe-qτbiq(t), i=1,2. Then, we have(30)w~it≤αi~t+∫t0tli~sw~1s-τ+mi~sw~2s-τds,t∈t0,+∞.Set ψi~(t)=e-qtϕiq(t)(i=1,2). For t∈[t0-τ,t0],(31)w~it≤ψi~ti=1,2.By (30), (31), and Lemma 1, (21) is satisfied and the proof is completed.Now, we introduce some definitions on Riemann-Liouville fractional derivative and fractional primitive.Definition 3 (see [2]).
The fractional derivative of order0<α<1 of a function x(t)∈C(R,R) is given by(32)Dα=1Γ1-αddx∫ats-t-αxsds.Definition 4 (see [2]).
The fractional derivative of order0<α<1 of a function x(t)∈C(R,R) is given by(33)Iα=1Γα∫ats-tα-1xsds.Consider the following Riemann-Liouville fractional differential system with a delay:(34)Dαxt=ft,xt-τ,yt-τ,Dαyt=gt,xt-τ,yt-τ,t∈t0,+∞,Dα-1xt=ξ,(35)Dα-1yt=η,t∈t0-τ,t0,where 0<α<1, τ>0,ξ,η are constants, and f, g:[t0,+∞)×R2→R are continuous functions. Then, a result can be obtained on the solutions of system (34).Theorem 5.
Consider system (34). f(t,x,y) and g(t,x,y)∈C([t0,+∞)×R2,R) and satisfy the following condition:(36)ft,x,y≤b1tx+c1ty,gt,x,y≤b2tx+c2ty,where bi(t) and ci(t)∈C([t0,+∞),R+). Then, solutions of system (34) satisfy that
(i) when1/2<α<1,(37)xtyt=etw11/2tetw21/2t,(38)w1tw2t≤At+Rt,where(39)Rt=exp∫t0+τtHsds∫t0t0+τHsΨs-τds+∫t0+τtexp∫ξtHsdsHξAξ-τdξ,t∈t0+τ,+∞,∫t0tHsΨs-τds,t∈t0,t0+τ,(40)At=α1tα2t,Ht=l1tm1tl2tm2t,Ψt=e-2ta12te-2ta22t,a1(t)=ξ/Γ(α)tα-1, a2(t)=η/Γ(α)tα-1 and αi(t)=2e-2tai2(t), li(t)=Γ(2α-1)/22α-3e-2τbi2(t), mi(t)=Γ(2α-1)/22α-3e-2τci2(t), i=1,2;
(ii) when0<α≤1/2,(41)xtyt=etw~11/qtetw~21/qt,(42)w~1tw~2t≤A~t+R~t, where(43)R~t=exp∫t0+τtH~sds∫t0t0+τH~sΨ~s-τds+∫t0+τtexp∫ξtH~sdsH~ξA~ξ-τdξ,t∈t0+τ,+∞,∫t0tH~sΨ~s-τds,t∈t0,t0+τ,(44)A~t=α~1tα~2t,H~t=l~1tm~1tl~2tm~2t,Ψ~t=ψ~1tψ~2t=e-qtϕ1qte-qtϕ2qtp=1+α, q=1+1/α and α~i(t)=2q-1[e-tai(t)]q, l~i(t)=41/αΓ1/α(α2)/pαe-qτbiq(t), m~i(t)=41/αΓ1/α(α2)/pαe-qτciq(t), i=1,2.Proof.
Fractional differential system (34) with a delay can be converted to the following integral equation:(45)xt=ξΓαtα-1+1Γα∫t0tt-sα-1fs,xs-τ,ys-τds,yt=ηΓαtα-1+1Γα∫t0tt-sα-1gs,xs-τ,ys-τds,t∈t0,+∞,xt=ξΓαtα-1,yt=ηΓαtα-1,t∈t0-τ,t0.By condition (36), we have(46)xt≤ξΓαtα-1+1Γα∫t0tt-sα-1b1sxs-τ+c1sys-τds,yt≤ηΓαtα-1+1Γα∫t0tt-sα-1b2sxs-τ+c2sys-τds,t∈t0,+∞,xt=ξΓαtα-1,yt=ηΓαtα-1,t∈t0-τ,t0.
Seta1(t)=|ξ|/Γ(α)tα-1 and a2(t)=|η|/Γ(α)tα-1. Then, according to Lemma 2, the result is obtained and the proof is completed.


## 3. An Illustrative Example

In this section, we give an illustrative example to show effectiveness of results obtained in this paper.Example 1.
Consider the following fractional differential equation:(47)D3/4xt=ft,xt-1,yt-1,D3/4yt=gt,xt-1,yt-1,t∈1,+∞,D-1/4xt=D-1/4yt=Γ34,t∈0,1,where f(t,x,y)=g(t,x,y)=e/Γ(1/2)(t-1)1/4[x+y], t∈[1,+∞).It is obvious that|f(t,x,y)|=|g(t,x,y)|≤e/Γ(1/2)(t-1)1/4[|x|+|y|], t∈[1,+∞). From (47) and Theorem 5, we obtain a1(t)=a2(t)=t-1/4, α1(t)=α2(t)=2t-1/2e-2t, li(t)=mi(t)=2t-1, i=1,2. Thus, A(t)=2t-1/2e-2t11, H(t)=2t-11111, Ψ(t)=t-1/2e-2t11.Set(48)xtyt=etw11/2tetw21/2t.Therefore,(49)w1tw2t≤At+Rt,where for t∈[1,2], (50)Rt=∫1tHsΨs-1ds=211∫1te-2s-1ds=1-e-2t-111, and for t∈[2,+∞), (51)Rt=exp∫2tHsds∫12HsΨs-1ds+∫2texp∫ξtHsdsHξAξ-1dξ≤2exp⁡43t-13/221-e-2+16e4/3t-13/2∫1te-2ξdξ11≤4e4/3t-13/2+4e8/3t-13/2-411.



---

*Source: 4151089-2018-10-21.xml*