# Remark on Existence and Uniqueness of Solutions for a Coupled System of Multiterm Nonlinear Fractional Differential Equations

Authors: Huichao Zou; Yonghong Fan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The aim of this paper is to extend the work of Sun et al. (2012) to a more general case for a wider range of function classes off and g. Our results include the case of the previous work, which are essential improvement of the work of Sun et al. (2012), especially.


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## Body

## 1. Introduction

Fractional calculus can give a more vivid and accurate description of problems in various fields of sciences than the traditional calculus [1–3]. Recently many complicated dynamic phenomena were modeled by fractional order calculus system and have received more and more attention; see [4–16].In recent work [12], Sun et al. studied the existence and uniqueness of solutions for a coupled system of multiterm nonlinear fractional differential equations with an initial value condition
(1)-Dαx(t)=f(t,y(t),Dβ1y(t),…,DβNy(t)),hhhhhhhhhhhhhhDα-ix(0)=0,i=1,2,…,n1,-Dσy(t)=g(t,x(t),Dρ1x(t),…,DρNx(t)),hhhhhhhhhhhhhDσ-jy(0)=0,j=1,2,…,n2,
where t∈(0,1], α>β1>β2>⋯>βN>0, σ>ρ1>ρ2>⋯>ρN>0, n1=[α]+1, n2=[σ]+1 for α,σ∉ℕ and n1=α, n2=σ for α,σ∈ℕ, βq,ρq<1 for any q∈{1,2,…,N}, D is the standard Riemann-Liouville derivative, and f,g:[0,1]×ℝN+1→ℝ are given functions. In order to obtain the existence and uniqueness of solutions of (1), the following growth conditions are introduced in [12].(H1)
There exist two nonnegative functionsa(t),b(t)∈L1[0,1] such that
(2)|f(t,x0,x1,…,xN)|≤a(t)+c0|x0|γ0+c1|x1|γ1+⋯+cN|xN|γN,|G(t,x0,x1,…,xN)|≤b(t)+d0|x0|θ0+d1|x1|θ1+⋯+dN|xN|θN,whereci,di≥0, 0<γi, θi<1 for i=0,1,2,…,N.(H2)
The functionsf and g satisfy
(3)|f(t,x0,x1,…,xN)|≤c0|x0|γ0+c1|x1|γ1+⋯+cN|xN|γN,|g(t,x0,x1,…,xN)|≤d0|x0|θ0+d1|x1|θ1+⋯+dN|xN|θN,whereci,di≥0, γi,θi>1 for i=0,1,2,…,N.However, there are many functions which cannot satisfy conditions (H1) and (H2); for example,(4)g(t,x0,x1)=t6.08+125.26[x0+ex1].
Hence the results of [12] are limited only to a small subset of functions which satisfy (H1) and (H2). This paper thus aims to extend the work of Sun et al. [12] to a more general case with more general conditions on f and g. Our major contributions of this paper include three aspects.(1)
We extend the function classes to more general case; that is, the power growth assumptions (H1) and (H2) are replaced by a very general assumption where the functionsϕ(|x0|,|x1|,…,|xN|) and ψ(|x0|,|x1|,…,|xN|) are only required to be nondecreasing function classes (see (A1)), which implies that the function classes are extended to more general case and also include the case of [12] as a special case. In mathematics and applied science, this generalization is important and interesting.(2)
In [12], the weight functions considered constants c0,c1,…,cN. But in physics, the influence of weight functions for the whole system is important, so in this work, we improve the weight functions to general Lebesgue integral functions b(t),d(t)∈L1[0,1], which is also an essential improvement.(3)
In this paper, the nonlinearitiesf and g are allowed to be exponential growth. However, in [12], the nonlinearities f and g are only allowed to be power growth. It is known that in most cases exponential growth is faster than power growth. From this aspect, this is also a major contribution of this paper.The remaining part of the paper is organized as follows. In Section2, some preliminary results including definitions, notations, and lemmas are given. Section 3 presents the main results and the proof of the results. In addition, an example is given to illustrate the application of the main results.


## 2. Preliminaries and Lemmas

Definition 1 (see [1–3]).
The fractional integral of orderα>0 of a function x:(a,+∞)→R is given by
(5)Iαx(t)=1Γ(α)∫at‍(t-s)α-1x(s)ds,
provided that the right-hand side is pointwisely defined on (a,+∞).Definition 2 (see [1–3]).
The Riemann-Liouville fractional derivative of orderα>0 of a function x: (a,+∞)→R is given by
(6)Dαx(t)=1Γ(n-α)(ddt)n∫at‍(t-s)n-α-1x(s)ds,
where n=[α]+1, [α] denotes the integer part of the number α, and t>a, provided that the right-hand side is defined on (a,+∞).Lemma 3 (see [1]).
Assume thatx∈L1[0,1] with a fractional derivative of order α>0; then
(7)IαDαx(t)=x(t)+c1tα-1+c2tα-2+⋯+cntα-n,
where ci∈R, i=1,2,…,n, n=[α]+1.Lemma 4 (see [12]).
Suppose thath∈L1[0,1]. Then the initial value problem
(8)Dαx(t)=h(t),α>0,t∈[a,b],Dαx(a)=bk,k=1,2,…n,
has a unique solution
(9)x(t)=∑j=1n‍bjΓ(α-j+1)(t-a)α-j+1Γ(α)∫at‍h(s)(t-s)1-αds,
where n=[α]+1 for α∉ℕ and α=n for α∈ℕ.LetI=[0,1] and let C(I) be the space of all continuous functions defined on I. We define the space
(10)X×Y={(Dρjx(t),Dβjy(t))(x,y)∣(x,y)∈C(I)×C(I),fffffff(Dρjx(t),Dβjy(t))∈C(I)fffffff×C(I),j=1,2,…,N(Dρjx(t),Dβjy(t))}
endowed with the norm ∥(x,y)∥X×Y=max{∥x∥X,∥y∥Y}, where
(11)∥x∥X=maxt∈I|x(t)|+∑j=1N‍maxt∈I|Dρjx(t)|,∥y∥Y=maxt∈I|y(t)|+∑j=1N‍maxt∈I|Dβjy(t)|.
Then X×Y is a Banach space with norm ∥(x,y)∥X×Y.By Lemma4, system (1) is equivalent to the following coupled system of integral equations:
(12)x(t)=1Γ(α)∫0t‍(t-s)α-1f(s,y(s),Dβ1y(s),…,DβNy(s))ds,y(t)=1Γ(σ)∫0t‍(t-s)σ-1g(s,x(s),Dρ1x(s),…,DρNx(s))ds.
Define an operator T:X×Y→X×Y(13)T(x,y)(t)=(T1x(t),T2y(t))=(1Γ(α)∫0t‍(t-s)α-1fffffffff×f(s,y(s),Dβ1y(s),…,DβNy(s))ds,dddd1Γ(σ)∫0t‍(t-s)σ-1ddgggggdd×g(s,x(s),Dρ1x(s),…,DρNx(s))ds1Γ(α)∫0t‍).
It is obvious that a fixed point of operator T is the solution of coupled system (1).


## 3. Main Result

Theorem 5.
Letf,g:[0,1]×ℝN+1→ℝ be continuous. Assume that(A1)
there exist nonnegative functionsa,b,c,d∈L1[0,1] and nonnegative nondecreasing functions ϕ,ψ with respect to each variable xi,  i=0,1,2,…,N, such that
(14)|f(t,x0,x1,…,xN)|≤a(t)+b(t)ϕ(|x0|,|x1|,…,|xN|),|g(t,x0,x1,…,xN)|≤c(t)+d(t)ψ(|x0|,|x1|,…,|xN|);(A2)
there exists a constantR0>max{k1,l1} such that
(15)ϕ(R0,R0,…,R0)≤R0-k1k2,ψ(R0,R0,…,R0)≤R0-l1l2,where(16)k1=maxt∈I(∑j=1N‍1Γ(α)∫0t‍(t-s)α-1|a(s)|dsggggg+∑j=1N‍1Γ(α-ρj)∫0t‍(t-s)α-ρj-1|a(s)|ds),k2=maxt∈I(∑j=1N‍1Γ(α)∫0t‍(t-s)α-1|b(s)|dsffffff+∑j=1N‍1Γ(α-ρj)∫0t‍(t-s)α-ρj-1|b(s)|ds),l1=maxt∈I(∑j=1N‍1Γ(σ)∫0t‍(t-s)σ-1|c(s)|dsgggggg+∑j=1N‍1Γ(σ-βj)∫0t‍(t-s)σ-βj-1|c(s)|ds),l2=maxt∈I(∑j=1N‍1Γ(σ)∫0t‍(t-s)σ-1|d(s)|dsffFFFF+∑j=1N‍1Γ(σ-βj)∫0t‍(t-s)σ-βj-1|d(s)|ds).Then the coupled system (1) has a solution.Proof.
Define a closed ball of Banach spaceX×Y(17)B={(x,y)∈X×Y:∥(x,y)∥X×Y≤R0}.
We will prove that T:B→B. In fact, for any (x,y)∈B, by (A1), we have
(18)|T1x(t)|=|1Γ(α)∫0t‍(t-s)α-1f(s,y(s),Dβ1y(s),…,DβNy(s))ds|≤1Γ(α)∫0t‍(t-s)α-1|a(s)|ds+ϕ(R0,R0,…,R0)Γ(α)∫0t‍(t-s)α-1|b(s)|ds,|DρjT1x(t)|=|DρjIαf(t,y(t),Dβ1y(t),…,DβNy(t))|=|Iα-ρjf(t,y(t),Dβ1y(t),…,DβNy(t))|=1Γ(α-ρj)×∫0t‍(t-s)α-ρj-1f(s,y(s),Dβ1y(s),…,DβNy(s))ds≤1Γ(α-ρj)×∫0t‍(t-s)α-ρj-1|a(s)|ds+ϕ(R0,R0,…,R0)Γ(α-ρj)∫0t‍(t-s)α-ρj-1|b(s)|ds.
Thus it follows from (18) and (A2) that
(19)∥T1x∥X=maxt∈I|T1x(t)|+∑j=1N‍maxt∈I|DρjT1x(t)|≤k1+k2ϕ(R0,R0,…,R0)≤R0.
In the same way, we also have(20)∥T2y∥Y=maxt∈I|T2y(t)|+∑j=1N‍maxt∈I|DρjT2y(t)|≤l1+l2ψ(R0,R0,…,R0)≤R0.
Consequently, ∥T1x∥X≤R0 and ∥T2y∥Y≤R0, and then ∥T∥X×Y≤R0 for any (x,y)∈B; that is, T:B→B.
By [12], we know that the operator T is completely continuous. Therefore, the Schauder fixed point theorem implies that coupled system (1) has a solution in B. The proof is completed.From Theorem5, we easily obtain the following corollaries.Corollary 6.
Letf,g:[0,1]×ℝN+1→ℝ be continuous. Assume that(A1)
there exist nonnegative functionsc,d∈L1[0,1] and nonnegative nondecreasing functions ϕ,ψ with respect to each variable xi,  i=0,1,2,…,N, such that
(21)|f(t,x0,x1,…,xN)|≤b(t)ϕ(|x0|,|x1|,…,|xN|),|g(t,x0,x1,…,xN)|≤d(t)ψ(|x0|,|x1|,…,|xN|);(A2)
there exists a positive constantR0 such that
(22)ϕ(R0,R0,…,R0)≤R0k2,ψ(R0,R0,…,R0)≤R0l2,where(23)k2=maxt∈I(∑j=1N‍1Γ(α)∫0t‍(t-s)α-1|b(s)|dsdddddd+∑j=1N‍1Γ(α-ρj)∫0t‍(t-s)α-ρj-1|b(s)|ds),l2=maxt∈I(∑j=1N‍1Γ(σ)∫0t‍(t-s)σ-1|d(s)|dsggfgggg+∑j=1N‍1Γ(σ-βj)∫0t‍(t-s)σ-βj-1|d(s)|ds).Then the coupled system (1) has a solution.Corollary 7.
Letf,g:[0,1]×ℝN+1→ℝ be continuous. Assume that(A*1)
there exist nonnegative functionsa,c∈L1[0,1] such that(24)|f(t,x0,x1,…,xN)|≤a(t),|g(t,x0,x1,…,xN)|≤c(t).
Then the coupled system (1) has a solution.Proof.
In fact, let us chooseR0=max{k1,l1}, where
(25)k1=maxt∈I(∑j=1N‍1Γ(α)∫0t‍(t-s)α-1|a(s)|dsgggggGg+∑j=1N‍1Γ(α-ρj)∫0t‍(t-s)α-ρj-1|a(s)|ds),l1=maxt∈I(∑j=1N‍1Γ(σ)∫0t‍(t-s)σ-1|c(s)|dsffffff+∑j=1N‍1Γ(σ-βj)∫0t‍(t-s)σ-βj-1|c(s)|ds),
and construct a closed ball of Banach space X×Y(26)B={(x,y)∈X×Y:∥(x,y)∥X×Y≤R0}.
The rest of proof is similar to Theorem 5.Remark 8.
The condition (A1) is weaker than (H1) and (H2). Clearly,ϕ(|x0|,|x1|,…,|xN|) and ψ(|x0|,|x1|,…,|xN|) include c0|x0|γ0+c1|x1|γ1+⋯+cN|xN|γN and d0|x0|θ0+d1|x1|θ1+⋯+dN|xN|θN, θi,γi≠1 as special cases. Moreover (A1) also includes the case θi=1 or/and γi=1, but (H1) and (H2) do not be allowed.Remark 9.
In Corollary7, for the special case a,c∈C[0,1], clearly f,g:[0,1]×ℝN+1→ℝ are continuous and bounded. This leads to the Corollary 3.1 of [12]. Therefore, Corollary 3.1 of [12] is only a special case of Corollary 7.In the following, we focus on the uniqueness of the solution of the system (1).Theorem 10.
Letf,g:[0,1]×ℝN+1→ℝ be continuous. Assume that(B1)
there exist nonnegative functionsa,c∈L1[0,1] and nonnegative nondecreasing functions ϕ,ψ with respect to each variable xi,  i=0,1,2,…,N, such that
(27)|f(t,u0,u1,…,uN)-f(t,v0,v1,…,vN)|≤a(t)ϕ(|u0-v0|,|u1-v1|,…,|uN-vN|),|g(t,u0,u1,…,uN)-g(t,v0,v1,…,vN)|≤b(t)ψ(|u0-v0|,|u1-v1|,…,|uN-vN|);(B2)
for anys>0,
(28)ϕ(s,s,…,s)≤s,ψ(s,s,…,s)≤s,andmax{k12,l12}<1, where
(29)k1=maxt∈I(∑j=1N‍1Γ(α)∫0t‍(t-s)α-1a(s)dsfffff+∑j=1N‍1Γ(α-ρj)∫0t‍(t-s)α-ρj-1a(s)ds),l1=maxt∈I(∑j=1N‍1Γ(σ)∫0t‍(t-s)σ-1c(s)dsfffff+∑j=1N‍1Γ(σ-βj)∫0t‍(t-s)σ-βj-1c(s)ds).Then coupled system (1) has a unique solution.Proof.
We prove that the operatorT:X×Y→X×Y is contraction. To do this, let (x1,y1), (x2,y2)∈X×Y; we have
(30)|T1x2(t)-T1x1(t)|=|1Γ(α)∫0t‍(t-s)α-1ddddddddd×f(s,y2(s),Dβ1y2(s),…,DβNy2(s))ds=-1Γ(α)∫0t‍(t-s)α-1ddfffffffd×f(s,y1(s),Dβ1y1(s),…,DβNy1(s))ds1Γ(α)∫0t‍|≤1Γ(α)∫0t‍(t-s)α-1a(s)fffffffff×ϕ(|y2(s)-y1(s)|,|Dβ1y2(s)-Dβ1y1(s)|,…,fffffffffffff|DβNy2(s)-DβNy1(s)|)ds≤1Γ(α)∫0t‍(t-s)α-1a(s)ffffffffff×ϕ(∥y2-y1∥,∥Dβ1y2-Dβ1y1∥,…,ffffffffffffff∥DβNy2-DβNy1∥)ds≤1Γ(α)∫0t‍(t-s)α-1a(s)ffffffffff×ϕ(∥y2-y1∥Y,∥y2-y1∥Y,…,∥y2-y1∥Y)ds≤1Γ(α)∫0t‍(t-s)α-1a(s)ds∥y2-y1∥Y,|DρjT1x2(t)-DρjT1x1(t)|=|Iα-ρj(f(t,y2(t),Dβ1y2(t),…,DβNy2(t))fffffff-f(t,y1(t),Dβ1y1(t),…,DβNy1(t)))|≤1Γ(α-ρj)×∫0t‍(t-s)α-ρj-1fffffff×|f(t,y2(t),Dβ1y2(t),…,DβNy2(t))ffffffffff-f(t,y1(t),Dβ1y1(t),…,DβNy1(t))|ds≤1Γ(α-ρj)×∫0t‍(t-s)α-ρj-1a(s)fffffff×ϕ(∥y2-y1∥,∥Dβ1y2-Dβ1y1∥,…,ffffffffffff∥DβNy2-DβNy1∥)ds≤1Γ(α-ρj)×∫0t‍(t-s)α-ρj-1a(s)gggggg×ϕ(∥y2-y1∥Y,∥y2-y1∥Y,…,∥y2-y1∥Y)ds≤1Γ(α-ρj)∫0t‍(t-s)α-ρj-1a(s)ds∥y2-y1∥Y.
Thus it follows from (30) and (B2) that
(31)∥T1x2-T1x1∥X=maxt∈I|T1x2(t)-T1x1(t)|+∑j=1N‍maxt∈I|Dρj(T1x2(t)-T1x1(t))|≤(∑j=1N‍1Γ(α)∫0t‍(t-s)α-1a(s)dsffff+∑j=1N‍1Γ(α-ρj)∫0t‍(t-s)α-ρj-1a(s)ds)∥y2-y1∥Y≤k1∥y2-y1∥Y.
Similarly, we can get
(32)|T1y2(t)-T1y1(t)|≤1Γ(σ)∫0t‍(t-s)σ-1c(s)ds∥x2-x1∥X,|DβjT1y2(t)-DβjT1y1(t)|≤1Γ(σ-βj)∫0t‍(t-s)σ-βj-1c(s)ds∥x2-x1∥X,∥T1y2-T1y1∥Y=maxt∈I|T1y2(t)-T1y1(t)|+∑j=1N‍maxt∈I|𝒟ρj(T1y2(t)-T1y1(t))|≤(∑j=1N‍1Γ(σ)∫0t‍(t-s)σ-1c(s)dsggggg+∑j=1N‍1Γ(σ-βj)∫0t‍(t-s)σ-βj-1c(s)ds)∥x2-x1∥X≤l1∥x2-x1∥X.
Hence, for the Euclidean distanced on ℝ2, we get
(33)d(T(x2,y2),T(x1,y1))=∥T1x2-T1x1∥X2+∥T1y2-T1y1∥Y2≤k12∥x2-x1∥X2+l12∥y2-T1∥Y2≤max{k12,l12}∥x2-x1∥X2+∥y2-T1∥Y2=max{k12,l12}d((x2,y2),(x1,y1)).
Thus T is a contraction since max{k12,l12}<1.
By Banach contraction principle,T has a unique fixed point, which is a solution of the coupled system (1). The proof is completed.An Example. Consider the existence of solutions for the following coupled system of multiterm nonlinear fractional differential equations:
(34)-D3.5x(t)=t6.08+125.26[y(t)+e(D0.8y(t))],hhhhhhhFFhhhhD3.5x(0)=0,i=1,2,…,4,-D4.2y(t)=100005501[t-1/2x0.2(t)+t2(D0.5x(t))0.5],hhhhhhhhhhhhhhhhhD4.2-jy(0)=0,j=1,2,…,5,
where t∈(0,1].Let(35)f(t,x0,x1)=t6.08+125.26[x0+ex1],g(t,x0,x1)=t-1/2x00.2+t2x10.5,
and choose
(36)a(t)=t6.08,b(t)=125.26,ϕ(x0,x1)=x0+ex1,c(t)=0,d(t)=100005501[t-1/2+t2],ψ(x0,x1)=x00.2+x10.5.
Then
(37)f(t,x0,x1)≤a(t)+b(t)ϕ(x0,x1),g(t,x0,x1)≤c(t)+d(t)ψ(x0,x1);
consequently, (A1) holds.In the following, we check the condition (A1). Since(38)k1=max(1Γ(3.5)∫0t‍(t-s)2.5s6.08ds+1Γ(3)∫0t‍(t-s)2s6.08ds)=0.01,k2=max(1Γ(3.5)∫0t‍(t-s)2.525.26ds+1Γ(3)∫0t‍(t-s)225.26ds)k2=0.01,l1=0,l2=100005501×max(1Γ(4.2)∫0t‍(t-s)3.2(s-1/2+s2)dsfgggggfff+1Γ(3.4)∫0t‍(t-s)2.4(s-1/2+s2)ds)=1,
take R0=5; we have
(39)ϕ(R0,R0)=R0+eR0=5+e5=153.44<R0-k1k2=5-0.010.01=499,ψ(R0,R0)=R00.2+R00.5=50.2+50.5=3.6158<R0-l1l2=5,
which implies that (A2) is satisfied. Hence, by Theorem 5, the coupled system of fractional differential equation (34) has a solution.Remark 11.
In the coupled system of fractional differential equation (34), the nonlinear function f involves exponential growth, but the results of [12] are only allowed to be power growth; that is, (34) cannot be solved by using the results of [12]. So the results obtained in this paper give a significant improvement of the previous work in [12].



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