# Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay

Authors: Azizollah Babakhani
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.


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## Body

## 1. Introduction

Fractional differential equations have gained considerable importance due to their varied applications [1–5] in viscoelasticity, electroanalytical chemistry, and many physical problems [1]. So far there have been several fundamental works on the fractional derivative and fractional differential equations, written by Oldham and Spanier [3], Miller and Ross [2], Podlubny [1], and others. These works are an introduction to the theory of the fractional derivative and fractional differential equations and provide a systematic understanding of the fractional calculus such as the existence and the uniqueness, some analytic methods for solving fractional differential equations, namely, Green`s function method, the Mellin transform method, and the power series.Control systems subject to delays have been extensively studied [6] and the delay differential equations are large and important class of dynamic systems. They often arise in either natural or technological control problems. Equations with discontinuity often appear in various control theory models [7, 8]. Time delay, always existing in real systems, usually results in oscillations around the discontinuity surface. Shustin [9] has studied various aspects of oscillations for the system of differential equations with the delay(1.1)ẋ(t)=Fi(x1(t),…,xn(t),t)-signxi(t-hi),i=1,…,n,
in dynamics of oscillations in a multidimension, where F1,…,Fn∈C1(ℝn+1) and h1,…,hn are positive constants.Roy et al. developed a control methodology for linear time-invariant plants that uses multiple delayed observations in feedback [10]. Fractional differential systems have proved to be useful in control processing for the last two decades [11, 12].As explained in the above text regarding the ordinary differential equations in control theory and others, we can modify the applications of ordinary differential equations to ordinary fractional differential equations.The existence and uniqueness of solution for the system of fractional differential equation have been studied in the papers [13, 14]. As a pursuit in this paper, we discuss the existence and uniqueness of positive solutions for system of nonlinear fractional differential equations in two-dimensional with the delay(1.2)Dα[x(t)-x(0)]=f(t,xt,yt),t∈(0,T],Dβ[y(t)-y(0)]=g(t,xt,yt),t∈(0,T],(x(t),y(t))=(ϕ(t),ψ(t)),t∈[-τ,0],
where (i)
0<α,β<1 and Dα,Dβ are the Riemann-Liouville fractional derivatives,(ii)
f,g:(0,T]×C2 → [0,+∞) are given continuous functions so that limt→0+f(t,xt,yt) = limt→0+g(t,xt,yt)=+∞ (i.e., f and g are singular at t=0) and C=C([-τ,0],ℝ≥0),(iii)
xt,yt∈C, where xt(s)=x(t+s),yt(s)=y(t+s),s∈[-τ,0],t∈(0,T].The paper is organized as follows. In Section2, we give definitions of the fractional derivative and fractional integral with some basic properties. Required topics of functional analysis were also introduced. Section 3 deals with existence of positive solution theorem and gives an illustrative example. The unique positive solution theorem with an example has been discussed in Section 4.


## 2. Basic Definitions and Preliminaries

We recall some standard definitions and results [1–4, 15, 16]. In the following Π and Σ are real Banach spaces and W is an operator (not necessarily linear) with domain in Π and range in Σ.Definition 2.1.
A subsetK of Π is called a cone if the following conditions hold well:
(i)  the setK is closed,
(ii)  iff,g∈K,α>0,  β>0 implies that αf+βg∈K,
(iii)  iff∈K,-f∈K implies that f=θ which is the zero element of Π.Note:
Every coneK induces a semiordering in Π, namely,   f≺g if g-f∈K.Definition 2.2.
Forf,g∈E the order interval 〈f,g〉 is defined as [17]
(2.1)〈f,g〉={h∈E:f≤h≤g}.Definition 2.3.
A subsetE⊂Π is called order bounded if E is contained in some order interval.Definition 2.4.
A coneK is called normal if there exists a positive constant μ such that f,g∈V and θ≺f≺g implies that ∥f∥≤μ∥g∥.We state the two fixed point results which will be needed in this paper. Our first result is a nonlinear alternative of Leray-Schauder type in a cone whereas our second is Krasnoselskii`s fixed point theorem.Theorem 2.5 (Leray-Schauder Theorem).
LetE be a Banach space with C⊆E closed and convex. Assume that U is relatively open subset of C with 0∈U and W:U¯→C is a continuous, compact map. Then either
(i)W has fixed point in U¯ or
(ii)  there exist u∈∂U and γ∈(0,1) with u=γWu.Theorem 2.6 (Krasnoselskii`s fixed point theorem [16]).
LetE=(E,∥·∥) be a Banach space and let K⊂E be a cone in E. Assume that Ω1 and Ω2 are open subsets of E with 0∈Ω1 and Ω̅1⊂Ω2 and let W:K∩(Ω¯2∖Ω1)→K be continuous and completely continuous. In addition suppose that either
(i)∥Wu∥≤∥u∥ for u∈K∩∂Ω1 and ∥Wu∥≥∥u∥ for u∈K∩Ω2 or
(ii)∥Wu∥≤∥u∥ for u∈K∩∂Ω1 and ∥Wu∥≤∥u∥ for u∈K∩Ω2.
Then,W has a fixed point in K∩(Ω̅2∖Ω1).In this paper the Beta functionB(α,β) is used also. B(α,β) is closely related to the Gamma function [1]. If α,β>0, then it has the integral representation (2.2)B(α,β)=∫01tα-1(1-t)β-1dt.
It may be written in terms of the Gamma function as B(α,β)=Γ(α)Γ(β)/Γ(α+β).Definitions of Riemann-Liouville fractional derivative/integral and their properties are given bellow [1–4].Definition 2.7.
Letx : [a,b]→ℝ and x∈L1[a,b] then the expression
(2.3)Ia+αx(t)=1Γ(α)∫at(t-s)α-1x(t)dt,x>a
is called a left-sided fractional integral of order α.Definition 2.8.
Letn be a positive integer number and α∈(n-1,n]. The left-sided fractional derivative of a function x:[a,b]→ℝ is defined as
(2.4)Da+αx(t)=Dn(In-αx(t)),t∈[a,b].
We denote Da+αx(t) as Daαx(t) and Ia+αx(t) as Iaαx(t). Further D0+αx(t) and I0+αx(t) are referred to as Dαx(t) and Iαx(t), respectively.If the fractional derivativeDaβx(t) is integrable then(2.5)Iaα(Daβx(t))=Iaα-βx(t)-{Ian-βx(t)}t=a(t-a)α-nΓ(α-n+1)-∑j=1n-1{Daβ-jx(t)}t=a(t-a)α-jΓ(α-j+1),
where n-1≤β≤α<n and n∈ℕ [1]. Further, if x∈C[a,b], then {Ian-βx(t)}t=a=0 and then (2.5) implies that(2.6)Iaα(Daαx(t))=x(t),0<α<1.


## 3. Existence Theorem

In this section we discuss the system of nonlinear fractional differential equation (1.2) which has at least one positive solution.Lemma 3.1 (see [18]).
Leth:(0,T]→ℝ be a continuous function and limt→0+h(t)=+∞. If there exits σ∈(0,1) such that 0<σ<α and tσh(t) is a continuous function on [0,T], then H(t)=Iαtσh(t) is continuous [0,T].In the following theorem we want to prove that (1.2) is equivalent to a system of integral equations.Theorem 3.2.
Suppose thatf,g : (0,T]×C2→[0,+∞) are given continuous functions with limt→0f(t,xt,yt)=+∞=limt→0g(t,xt,yt)=+∞. If there exist ρ,σ∈(0,1) such that 0<ρ,σ<α and tρf(t,xt,yt),tσg(t,xt,yt) are continuous functions on [0,T], then (1.2) is equivalent to the system of integral equations
(3.1)x(t)=x(0)+Iαf(t,xt,yt),t∈(0,T],y(t)=y(0)+Iβg(t,xt,yt),t∈(0,T],x(t)=ϕ(t)≥0,y(t)=ψ(t)≥0,t∈[-τ,0].Proof.
It is sufficient to prove that the first equation of system equation (1.2) is equivalent to the first equation of system equation (3.1). So,
(3.2)Iα{Dα[x(t)-x(0)]}=Iαf(t,xt,yt),t∈(0,T].
Using (2.6) we get Iα{Dα[x(t)-x(0)}=x(t)-x(0). Hence (3.2) implies that x(t)=x(0)+Iαf(t,xt,yt),t∈(0,T].
Conversely, first we note thatIαf(t,x(t),y(t)) = Iα{tρt-ρf(t,xt,yt)} exists by Lemma 3.1. DαIαf(t,xt,yt)= f(t,xt,yt) as t-ρf(t,xt,yt) is continuous and Iαf(t,xt,yt)∈C[0,T]. Therefore, the system of fractional integral equation (3.1) is a solution of (1.2).
Letω : [-τ,T]→[0,+∞)×[0,+∞) be a function defined by
(3.3)ω(t)={(ϕ(0),ψ(0)),t∈[0,T],(ϕ(t),ψ(t)),t∈[-τ,0],
for each z(t)=(η(t),γ(t)) with z(0)=(0,0) where η and γ belong to C([0,T],ℝ). We introduce the notation z¯ defined by
(3.4)z¯(t)={(η(t),γ(t)),t∈[0,T],(0,0),t∈[-τ,0].
We can decompose (x(·),y(·)) as (x(t),y(t))=z¯(t)+ω(t),t∈[-τ,T] which implies that (xt,yt)=z¯t+ωt,t∈[0,T]. Hence, by Theorem 3.2, (1.2) is equivalent to the system of integral equations
(3.5)η(t)=Iαf(t,z¯t+ωt),γ(t)=Iβg(t,z¯t+ωt),
where t∈[0,T]. Set Ω={z=(η(t),γ(t)):η,γ∈C([0,T],ℝ),η(0)=γ(0))=0}. For each z∈Ω, let ∥z∥T be the seminorm in Ω defined by
(3.6)∥z∥T=∥z(0)∥+∥z∥=∥z∥=sup{|η(t)|,|γ(t)|:t∈[0,T]}.Ω is a Banach space with norm ∥·∥T. Let K be a cone of Ω,(3.7)K={z=(η(t),γ(t))∈Ω:η(t)≥0,γ(t)≥0,t∈[0,T]},K*={((x,y):x,y∈C([-τ,T],ℝ+)),(x(t),y(t))=(ϕ(t)≥0,ψ(t)≥0),t∈[-τ,0]}.
For each t∈[0,T], we define the operator W : K→K by
(3.8)Wz(t)=(Iαf(t,z¯t+ωt),Iβg(t,z¯t+ωt))=(Fη(t),Gγ(t)),
where
(3.9)Fη(t)=Iαf(t,z¯t+ωt),Gη(t)=Iβg(t,z¯t+ωt).Lemma 3.3.
Letf(t,xt,yt),g(t,xt,yt),and t∈(0,T] be nonnegative continuous functions, where xt,yt∈C. Then, the operator W with the following conditions is maps-bounded set into bounded sets in K.
(H1)0<limt→0+f(t,xt,yt)=limt→0+g(t,xt,yt)=+∞(H2) if there exist ρ,σ∈(0,1)suchthatρ,σ<α,  tρf(t,xt,yt),andtσg(t,xt,yt) are continuous on [0,T].Proof.
By assumptions of Lemma3.3 and by using Lemma 3.1, it is clear that Wz(t)=(Fη(t),γ(t))∈K.  There exist positive constants Λ1,Λ2 such that |tρf(t,xt,yt)|≤Λ1 and |tσg(t,xt,yt)|≤Λ2, as tρf(t,xt,yt) and tσg(t,xt,yt) are continuous on [0,T]×C2. Hence,
(3.10)Iαf(t,xt,yt)=Iα(t-ρtρf(t,xt,yt))≤Λ1Iαt-ρ=Λ1Γ(α)∫0t(t-s)α-1s-ρds=Λ1tα-ρΓ(α)B(1-ρ,α)≤Λ1Tα-ρΓ(α)B(1-ρ,α).
Similarly,
(3.11)Iβg(t,xt,yt)≤Λ2Tβ-σΓ(β)B(1-σ,β).
LetH⊂K be bounded, that is, for each z=(η,γ)∈H there exist positive constants L such that ∥z∥T≤L. In view of (3.10), (3.11) we have
(3.12)|Fη(t)|≤|Iαf(t,z̅t+ωt)|≤Λ1Tα-ρΓ(α)B(1-ρ,α),|Gγ(t)|≤|Iαf(t,z̅t+ωt)|≤Λ2Tβ-σΓ(α)B(1-σ,β).
Therefore, ∥Wz(t)∥≤Λ where
(3.13)Λ=max{Λ1Tα-ρΓ(α)B(1-ρ,α),Λ2Tβ-σΓ(β)B(1-σ,β)}.
Hence, W(K) is bounded.Lemma 3.4.
Suppose thatf(t,xt,yt),g(t,xt,yt),xt,yt∈C,t∈(0,T] are continuous functions and further two conditions (H1) and (H2) are satisfied. Then, the operator W is continuous and completely continuous.Proof.
In the following we want to prove thatW is continuous. It is sufficient to prove that F and G are continuous. Let z0=(u0,v0)∈K with ∥u0∥=c0 and ∥v0∥=d0. If z=(u,v)∈K with ∥u-u0∥<l1 and ∥v-v0∥<l2, then ∥u∥<l1+c0 and ∥v∥<l2+d0. By the continuity of tρf(t,z¯t+ωt) and tσg(t,z¯t+ωt) on [0,T], we conclude that tρf(t,z¯t+ωt) and tσg(t,z¯t+ωt) are uniformly continuous on [0,T]×[0,c0]×[0,do]. Therefore, |Fu(t)-Fu0(t)|,|Gv(t)-Gv0(t)|,t∈[0,T] are continuous and hence,
(3.14)∥Wz(t)-Wz0(t)∥=max{|Fu(t)-Fu0(t)|,|Gv(t)-Gv0(t)|}.
is continuous.
Finally, we want to prove that the operatorW is equicontinuous. Let H⊂K be bounded. Suppose that, z=(u,v)∈K and t,r∈[0,T] such that t<r. For given ϵ>0, there exists δ1>0, so that if |t-r|<δ1, then |tρf(t,z¯t+ωt)-tρf(t,z¯r+ωr)|<ϵ:(3.15)|Fu(t)-Fu(r)|=|Iα{f(t,z¯t+ωt)-f(r,z¯r+ωr)}|≤1Γ(α)|∫0t(t-s)α-1s-ρsρf(s,z¯s+ωs)ds-∫0r(r-s)α-1s-ρsρf(s,z¯s+ωs)ds|≤1Γ(α)|∫0r(t-s)α-1s-ρsρf(s,z¯s+ωs)-∫0r(r-s)α-1s-ρsρf(s,z¯s+ωs)ds|+1Γ(α)|∫tr(t-s)α-1s-ρsρf(s,z¯s+ωs)ds|≤1Γ(α)∫0r|(t-s)α-1s-ρ-(r-s)α-1s-ρ|×|sρf(s,z¯s+ωs)-sρf(s,z¯s+ωs)|ds+1Γ(α)∫tr|(t-s)α-1s-ρ||sρf(s,z¯s+ωs)|ds<ϵΓ(α)∫0r{(r-s)α-1s-ρ+(r-s)α-1s-ρ}ds+Λ1Γ(α)∫tr(s-t)α-1s-ρds=2ϵΓ(α)B(1-ρ,α)+Λ1Γ(α)∫tr(s-t)α-1s-ρds.
Similarly, for given ϵ>0, there exists δ2>0, so that if |t-r|<δ2, then
(3.16)|Gu(t)-Gu(r)|≤2ϵΓ(β)B(1-σ,β)+Λ2Γ(β)∫tr(s-t)β-1s-σdsCase 1.
Ift=0 and 0<r<δi,i=1,2, then
(3.17)∫tr(s-t)α-1s-ρds≤max0≤r≤T∫0rsα-ρ-1ds≤Tα-ρα-ρ.
Set
(3.18)λ1=max{12Γ(α)B(1-ρ,α),2Λ1ϵ(α-ρ)Tα-ρ},λ2=max{12Γ(β)B(1-σ,β),2Λ2ϵ(β-σ)Tβ-σ}.
Hence, |Fu(t)-Fu(r)|<λ1ϵ and |Gv(t)-Gv(r)|<λ2ϵ.Case 2.
If0<t<r<δi≤T,i=1,2, then
(3.19)m1=Λ1Γ(α)∫tr(s-t)α-1s-ρds,m2=Λ2Γ(β)∫tr(s-t)β-1s-σds
are finite. Set
(3.20)μ1=max{12Γ(α)B(1-ρ,α)},μ2=max{12Γ(β)B(1-σ,β)}.
Hence, |Fu(t)-Fu(r)|<μ1ϵ and |Gv(t)-Gv(r)|<μ2ϵ. Thus, for any given ϵ>0, for all z=(u,v)∈K and for all t,r∈[0,T] with |r-t|<δ where δ=min{δ1,δ2}, for the Euclidian distance d on ℝ2 we have
(3.21)d(Wz(t),Wz(r))={[Fu(r)-Fu(t)]2}1/2+{[Gv(r)-Gv(t)]2}1/2<η2ϵ,
where η=max{μ1,μ2,λ1,λ2}. Therefore, W(H) is equicontinuous and the Ascoli-Arzela theorem implies that W(H)¯ is compact and hence W:K→K is completely continuous.Theorem 3.5.
Iff(t,xt,yt) and g(t,xt,yt) on (0,T]×C2 are nonnegative continuous functions, then the system of nonlinear fractional differential equation (1.2) with the conditions (H1) and (H2) has at least one positive solution x*∈K*, satisfying ∥x*∥≤max{∥ϕ∥,∥ψ∥,h}, where h is observed in the proof of the theorem.Proof.
By Lemma3.3, the operator W:K→K is continuous and completely continuous. We prove that there exits an open set U⊆K, with z≠γW(z) for γ∈(0,1) and z∈∂U. Let z=(u,v)∈K be any solution of z=γW(z),γ∈(0,1). In view of Theorem 3.5, we have
(3.22)z(t)=γWz(t)=γ(Fu(t),Gv(t)),t∈[0,T].
Hence,
(3.23)z(t)=γWz(t)=(γIαt-ρtρf(t,z¯t+ωt),γIβt-σtσg(t,z¯t+ωt)).
There exist positive constants N1,N2 such that |tρf(t,z¯t+ωt)|≤N1 and |tρg(t,z¯t+ωt)|≤N2, as tρf(t,z¯t+ωt) and tσg(t,z¯t+ωt) are continuous on [0,T]×(C[-τ,T])2. Hence,
(3.24)|u(t)|≤N1Γ(α)max0≤t≤Ttα-ρB(1-ρ,α)≤N1Tα-ρΓ(α)B(1-ρ,α),|v(t)|≤N2Γ(β)max0≤t≤Ttβ-σB(1-σ,β)≤N2Tβ-σΓ(β)B(1-σ,β).
If we consider
(3.25)h=1+max{N1Tα-ρΓ(α)B(1-α,α),N2Tβ-σΓ(β)B(1-β,β)},
then any solution z=γW(z) satisfies ∥z∥≠h. Set U={z∈K:∥z∥<h}. Theorem 2.5 guarantees that W has fixed point z∈U¯. Theorem 3.2 gives that (1.2) has a positive solution x*∈K* satisfying ∥x*∥≤max{∥ϕ∥,∥ψ∥,h}.Example 3.6.
Consider the system of nonlinear fractional differential equation(3.26)D4/5[x(t)-x(0)]=e-tt(xt+yt),t∈(0,1],D3/4[y(t)-y(0)]=e-tt4(xt+yt),t∈(0,1],(x(t),y(t))=(-sinπt,-sin(πt2)),t∈[-1,0],
where
(3.27)xt=∫-1-tds1+x2(t+s)=∫t-10ds1+x2(s),t∈[-1,0],yt=∫-1-ty2(t+s)1+y2(t+s)ds=∫t-10y2(s)1+y2(s)ds,t∈[-1,0].
Here, α=4/5,β=3/4,f(t,xt,yt)=e-tt-1/2(xt+yt), and g(t,xt,yt)=e-tt-1/4(xt+yt) with limt→0+f(t,·,·)=+∞=limt→0+g(t,·,·)=+∞. If we select ρ=3/4 and σ=1/2, then tρf(t,·,·) and tσg(t,·,·) are continuous on [0,1].
By Theorem3.2, the system of nonlinear fractional differential equation (3.26) is equivalent to the system of integral equation
(3.28)x(t)=x(0)+I4/5(e-tt-1/2(xt+yt)),t∈(0,1],y(t)=y(0)+I3/4(e-tt-1/4(xt+yt)),t∈(0,1],(x(t),y(t))=(-sinπt,-sinπ2t),t∈[-1,0].
Therefore, by Theorem 3.5, (3.26) has at least one positive solution x*∈K* satisfying ∥x*∥≤max{∥ϕ∥,ψ,h} where
(3.29)∥ϕ∥=max{max-1≤t≤0|-sinπt|,max-1≤t≤0|-sinπ2t|}=1,h=1+max{N1Γ(4/5)B(14,45),N2Γ(3/4)B(12,34)},N1=max0≤t≤1{tρ(e-tt-1/2(xt+yt))}=max0≤t≤1{e-tt1/4(xt+yt)}≤44e4,N2=max0≤t≤1{tσ(e-tt-1/4(xt+yt))}=max0≤t≤1{e-tt1/2(xt+yt)}≤62e.
Note that, for each t∈[0,1], we have 0≤xt,yt≤2 and
(3.30)max0≤t≤1{t1/4e-t}=14e4,max0≤t≤1{t1/2e-t}=12e.Theorem 3.7.
Suppose thatf,g : (0,T]×C2→[0,∞) are nonnegative continuous with limt→+∞f(t,xt,yt)=limt→+∞g(t,xt,yt)=+∞. If there exist ρ,σ∈(0,1) such that 0<ρ<α,0<σ<β and tρf(t,xt,yt) and tσg(t,xt,yt) are continuous on [0,T]×C2, then (1.2) has at least one positive solution with the following conditions:
(H3)
tρf(t,xt,yt)≤ρ1Tρ-α(Γ(1-ρ+α)/Γ(1-ρ)),tσg(t,xt,yt)≤ρ2Tσ-β(Γ(1-σ+β)/Γ(1-σ)),
(H4) tρf(t,xt,yt)≥μ1Tρ-α(Γ(1-ρ+α)/Γ(1-ρ)),tσg(t,xt,yt)≥μ2Tσ-β(Γ(1-σ+β)/Γ(1-σ)),
where(t,xt,yt)∈[0,T]×[0,ρ]2,(t,xt,yt)∈[0,T]×[0,μ]2,ρ=max{ρ1,ρ2}, μ=min{μ1,μ2}, and 0<μ<ρ.Proof.
For proving this theorem we provide the conditions required in Theorem2.6. Set
(3.31)Ω1={z=(u,v)∈K:∥z∥<μ},Ω2={z=(u,v)∈K:∥z∥<ρ}.
For each z=(u,v)∈K∩∂Ω2 and t∈[0,T] we have 0≤u(t)≤ρ and 0≤v(t)≤ρ. Condition (H3) implies that
(3.32)Fu(t)=1Γ(α)∫0t(t-s)α-1s-ρsρf(s,z¯s+ωs)ds≤ρiTρ1-αΓ(1-ρ+α)Γ(1-ρ)1Γ(α)∫0t(t-s)α-1s-ρds=ρ1Tρ-αtα-ρΓ(1-ρ+α)Γ(1-ρ)Γ(1-ρ)Γ(1-ρ+α)=ρ1Tρ-αtα-ρ.
Hence ∥Fu∥=max0≤t≤T|Fu(t)|≤ρ. Similarly ∥Gv∥=max0≤t≤T|Gv(t)|≤ρ. Thus, ∥Wz∥≤ρ=∥z∥ for z=(u,v)∈K∩∂Ω2. By using condition (H4) and the above-mentioned proof, we have ∥Wz∥≥μ=∥z∥ for z∈K∩∂Ω1. Therefore, by Theorem 2.6, (ii), and Theorem 3.2, the proof is completed.Example 3.8.
Consider the system of nonlinear fractional differential equation(3.33)D1/4[x(t)-x(0)]=1t5zt,t∈(0,1],D1/2[y(t)-y(0)]=1t4xt,t∈(0,1],D3/4[z(t)-z(0)]=1tyt,t∈(0,1],x(t)=(-sinπt,-sin(πt4),-sin(πt2)),t∈[-1,0],
where xt,yt,zt are continuous functions from [-1,0] to [0,+∞) so that
(3.34)xt=Γ(5/4)Γ(3/4){1+∫-1-t2x2(t+s)+11+x2(t+s)ds},yt=Γ(5/4)Γ(1/4){2+∫-1-t3y2(t+s)+21+y2(t+s)ds},zt=Γ(21/20)Γ(4/5){3+∫-1-t4z2(t+s)+31+z2(t+s)ds}.
Here α = 1/4,β = 1/2,γ = 3/4,f(t,xt,yt,zt) = t-1/5zt, g(t,xt,yt,zt) = t-1/4xt,h(t,xt,yt,zt) = t-1/2yt  such that limt→0+f(t,·,·) = limt→0+g(t,·,·) = limt→0+h(t,·,·) = +∞. Also tρf(t,·,·),tσg(t,·,·), and tυh(t,·,·) are continuous, where ρ = 1/5,σ = 1/4, and υ = 1/2. In the following calculations, we review conditions (H3) and (H4) by using (3.34):
(3.35)t1/5f(t,xt,yt,zt)=Γ(21/20)Γ(4/5){3+∫-1-t2z2(t+s)+11+z2(t+s)ds}=Γ(21/20)Γ(4/5){∫t-102z2(s)+11+z(s)2ds}≤Γ(21/20)Γ(4/5){3+4∫t-10ds}≤7Γ(21/20)Γ(4/5)=7Γ(1-ρ+α)Γ(1-ρ),t1/4g(t,xt,yt,zt)=Γ(5/4)Γ(3/4)∫-1-t{1+2x2(t+s)+11+x2(t+s)ds}=Γ(5/4)Γ(3/4)∫t-10{1+2x12(s)+11+[x1(s)]2ds}≤Γ(5/4)Γ(3/4){1+2∫t-10ds}≤3Γ(5/4)Γ(3/4)=3Γ(1-σ+β)Γ(1-σ),t1/2h(t,xt,yt,zt)=Γ(5/4)Γ(1/4){2+∫-1-t3y2(t+s)+11+y2(t+s)ds}=Γ(5/4)Γ(1/4){2+∫t-103x22(s)+11+[x2(s)]2ds}≤Γ(5/4)Γ(1/4){2+3∫t-10ds}≤5Γ(5/4)Γ(1/4)=5Γ(1-υ+γ)Γ(1-υ).
Hence, ρ = max{3,5,7} = 7 and the above calculations satisfy condition (H3). On the other hand,
(3.36)t1/5f(t,xt,yt,zt)=Γ(21/20)Γ(4/5){3+∫-1-t4z2(t+s)+11+z2(t+s)ds}=Γ(21/20)Γ(4/5){3+∫t-104z2(s)+11+z2(s)ds}≥Γ(21/20)Γ(4/5){3+3∫t-10ds}≥3Γ(21/20)Γ(4/5)=3Γ(1-ρ+α)Γ(1-ρ),t1/4g(t,xt,yt,zt)=Γ(5/4)Γ(3/4){1+∫-1-t2x2(t+s)+11+x2(t+s)ds}=Γ(5/4)Γ(3/4){1+∫t-102x2(s)+11+x2(s)ds}≥Γ(5/4)Γ(3/4){1+∫t-10ds}≥Γ(5/4)Γ(3/4)=Γ(1-σ+β)Γ(1-σ),t1/2h(t,xt,yt,zt)=Γ(5/4)Γ(1/4){2+∫-1-t3y2(t+s)+11+y2(t+s)ds}=Γ(5/4)Γ(1/4){2+∫t-103y2(s)+11+y2(s)ds}≥Γ(5/4)Γ(1/4){2+2∫t-10ds}≥2Γ(5/4)Γ(1/4)=2Γ(1-υ+γ)Γ(1-υ).
Hence μ=min{1,2,3}=1 and the above calculations satisfy condition (H4). Then, (3.33) has at least one positive solution.


## 4. Unique Existence of Solution

In this section we give conditions onf and g which render unique positive solution to (1.2).Theorem 4.1.
Letf,g : (0,T]×C2→[0,∞) be continuous and limt→0+f(t,xt,yt)=limt→0+g(t,xt,yt)=+∞. If there exist ρ,σ∈(0,1) such that 0<ρ<α<1,0<σ<β<1 and tρf(t,xt,yt),tσg(t,xt,yt) are continuous functions on [0,T]×C2, then (1.2) has unique positive solution with the following conditions:
(H5)
|tρf(t,u)-tρf(t,v)|<L1∥u-v∥,|tσg(t,u)-tρg(t,v)|<L2∥u-v∥;(H6)
L=max{(L1/Γ(α))tα-ρB(1-ρ,α),(L2/Γ(β))tβ-σB(1-σ,β)}<1,
where u,v∈C2, and L1,L2 are positive constants.Proof.
As pointed out in the proceeding section, (1.2) is equivalent to the integral equation (3.1) and the solution of (3.1) is equivalent to the fixed point of operator W. Thus for u=(u1,v1),v=(u2,v2)∈K(4.1)|Fu1(t)-Fu2(t)|≤Iα{t-ρ|tρfi(t,u¯t+ωt)-tρf(t,v¯t+ωt)|}≤L1∥u-v∥Iα{t-ρ}=∥u-v∥L1Γ(α)∫0t(t-s)α-1s-ρ=L1Γ(α)∥u-v∥B(1-ρ,α)tα-ρ.
Similarly,
(4.2)|Gv1(t)-Fv2(t)|≤L2Γ(β)∥u-v∥B(1-σ,β)tβ-σ.
Hence, for each u,v∈K, we have
(4.3)∥Wu(t)-Wv(t)∥=max{max0≤t≤T|Fu1(t)-Fv1(t)|,|Fu1(t)-Fv1(t)|}=L∥u-v∥.
Hence, by application of the Banach fixed point theorem, W has unique fixed point in K, which is the unique positive solution of (1.2).Example 4.2.
Consider the system of nonlinear fractional differential equation(4.4)D4/5[x(t)-x(0)]=1(1+8t2)t(xt+yt),t∈(0,1],D3/4[y(t)-y(0)]=1(1+4t2)t4(xt+yt),t∈(0,1],(x(t),y(t))=(-sinπt,-sin(πt2)),t∈[-1,0],
where
(4.5)xt=∫-1-tds1+x2(t+s),yt=∫-1-t2y(t+s)1+y2(t+s)ds.
Here, α = 4/5,β = 3/4,f(t,xt,yt) = e-tt-1/2(xt+yt), and g(t,xt,yt)  = e-tt-1/4(xt+yt) such that limt→0+f(t,·,·) = limt→0+g(t,·,·) = +∞. We select ρ = 3/4 and σ = 1/2. Hence tρf(t,·,·) and tσg(t,·,·) are continuous on [0,1]:
(4.6)|tρf(t,xt,yt)-tσg(t,xt,yt)|≤max{18Γ(1/2)B(14,45),15Γ(3/4)B(12,34)}∥u-v∥<0.4∥u-v∥,
where u=(xt,yt),v=(xt′,yt′). Equation (4.4) satisfies the conditions required in Theorem 4.1 and this theorem implies that the nonlinear equation (4.4) has unique solution in K. Using Theorem 3.2, (4.4) is equivalent to the system of integral equations
(4.7)x(t)=x(0)+I4/5{1(1+8t2)t(xt+yt)},t∈(0,1],y(t)=y(0)+I3/4{1(1+4t2)t4(xt+yt)},t∈(0,1],(x(t),y(t))=(-sinπt,-sinπ2t),t∈[-1,0],
and the solution of (4.4) is (x(t),y(t))=limn→+∞(xn(t),yn(t)) where
(4.8)(xn+1(t),yn+1(t))︸n=0,1,2,…=(Fxn(t),Gyn(t)),x0(t)=x(0),y0(0)=y(0).


## 5. Conclusions

In this paper the existence of positive solutions for system of nonlinear fractional differential equations in two dimensions with the delay comprising of standard Riemann-Liouville fractional derivatives has been discussed in Banach space. The unique solution under Lipschitz condition is also derived. For getting our results in this paper Leray-Schauder`s theorem, Krasnoselskii`s fixed point theorem, and Banach contraction principle had played important roles. Our research work in this paper can be generalized to system of nonlinear fractional differential equations in multiple dimensions with the finite delay or infinite delay. The present work provides insights in the equations encountered in fractional order dynamic systems and the control systems which further may be explored.



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*Source: 536317-2010-05-31.xml*