# Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

Authors: Qiuping Li; Shurong Sun; Ping Zhao; Zhenlai Han
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We discuss the initial value problem for the nonlinear fractional differential equationL(D)u=f(t,u),t∈(0,1],u(0)=0, where L(D)=Dsn-an-1Dsn-1-⋯-a1Ds1, 0<s1<s2<⋯<sn<1, and aj<0, j=1,2,…,n-1, Dsj is the standard Riemann-Liouville fractional derivative and f:[0,1]×ℝ→ℝ is a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.


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## Body

## 1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order, so fractional differential equations have wider application. Fractional differential equations have gained considerable importance; it can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic.In the recent years, there has been a significant development in fractional calculus and fractional differential equations; see Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Baleanu et al. [4], and so forth. Research on the solutions of fractional differential equations is very extensive, such as numerical solutions, see El-Mesiry et al. [5] and Hashim et al. [6], mild solutions, see Chang et al. [7] and Chen et al. [8], the existence and uniqueness of solutions for initial and boundary value problem, see [9–30], and so on.With the deep study, many papers that studied the fractional equations contained more than one fractional differential operator; see [16–20].Babakhani and Daftardar-Gejji in [16] considered the initial value problem of nonlinear fractional differential equation
(1.1)L(D)u=f(t,u),u(0)=0,0<t<1.
By using Banach fixed point theorem and fixed point theorem on a cone some results of existence and uniqueness of solutions are established.Zhang in [17] studied the singular initial value problem for fractional differential equation by nonlinear alternative of Leray-Schauder theorem:
(1.2)L(D)u=f(t,u),t1-snu(t)|t=0=0,0<t≤1.In above two equations,L(D) is defined L(D):=Dsn-an-1Dsn-1-⋯-a1Ds1, where 0<s1<s2<⋯<sn<1, and aj>0, j=1,2,…,n-1, Dsj is the standard Riemann-Liouville fractional derivative.McRae in [14] studied the initial value problem by the method of upper and lower solutions and monotone iterative technique:
(1.3)Dqu=f(t,u),t∈(t0,T],0<q<1,u(t0)=u0=u(t)(t-t0)1-q|t=t0.In this paper, we use similar method as in [16] to consider the initial value problem:
(1.4)L(D)u=f(t,u),t∈(0,1],u(0)=0,
where L(D)=Dsn-an-1Dsn-1-⋯-a1Ds1, 0<s1<s2<⋯<sn<1, and aj<0, j=1,2,…,n-1, Dsj is the standard Riemann-Liouville fractional derivative and f:[0,1]×ℝ→ℝ is a given continuous function.Sincef is assumed continuous, the IVP (1.4) is equivalent to the following Volterra fractional integral equation:
(1.5)u(t)=∑j=1n-1‍ajΓ(sn-sj)∫0t‍(t-s)sn-sn-1-1u(s)ds+1Γ(sn)∫0t‍(t-s)sn-1f(s,u(s))ds.In Section2, we give some definitions and lemmas that will be useful to our main results. In Section 3, we will use the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to investigate the initial value problem (1.4), and some existence and uniqueness results are established. In Section 4, an example is presented to illustrate the main results.


## 2. Preliminaries

In this section, we need the following definitions and lemmas that will be useful to our main results. These materials can be found in the recent literatures; see [1, 11, 16].Definition 2.1 (see [1]).
LetΩ=[a,b](-∞<a<b<+∞) be a finite interval on the real axis ℝ. The Riemann-Liouville fractional integrals Ia+αf and Ib-αf of order α>0 are defined by
(2.1)Ia+αf(x)=1Γ(α)∫ax‍(x-t)α-1f(t)dt,x>a,Ib-αf(x)=1Γ(α)∫xb‍(x-t)α-1f(t)dt,x<b,
respectively. Here Γ(α) is the Gamma function. These integrals are called the left-sided and the right-sided fractional integrals. We denote I0+αf(x) by Iαf(x) in the following paper.Definition 2.2 (see [1]).
LetΩ=[a,b](-∞<a<b<+∞) be a finite interval on the real axis ℝ. The Riemann-Liouville fractional derivatives Da+αf and Db-αf of order α>0 are defined by
(2.2)Da+αf(x)=(ddx)n(Ia+n-α)f(x)=1Γ(n-α)(ddx)n∫ax‍(x-t)n-α-1f(t)dt,x>a,Db-αf(x)=(-ddx)n(Ib-n-α)f(x)=1Γ(n-α)(-ddx)n∫xb‍(x-t)n-α-1f(t)dt,x<b,
respectively, where n=[α]+1, [α] means the integral part of α. These derivatives are called the left-sided and the right-sided fractional derivatives. We denote D0+αf(x) by Dαf(x) in the following paper.Definition 2.3.
Lettingv,w∈C([0,1],ℝ) be locally Hölder continuous with exponent sn<λ<1, we say that w is an upper solution of (1.4) if
(2.3)L(D)w≥f(t,w),w(0)≥0,
and v is a lower solution of (1.4) if
(2.4)L(D)v≤f(t,v),v(0)≤0.Next, we will list the following lemma from [11] that is useful for our main results.Lemma 2.4 (see [11, Lemma 2.1]).
Letm∈C([0,1],ℝ) be locally Hölder continuous with exponent q<λ<1 such that for any t1∈(0,1], we have
(2.5)m(t1)=0,m(t)≤0for0≤t≤t1.
Then it follows that Dqm(t1)≥0.Corollary 2.5.
Letm∈C([0,1],ℝ) be locally Hölder continuous with exponent sn<λ<1 such that for any t1∈(0,1], we have
(2.6)m(t1)=0,m(t)≤0for0≤t≤t1.
Then it follows that L(D)m(t1)≥0 provided aj<0,j=1,2,…,n-1.Lemma 2.6.
Let{uϵ(t)} be a family of continuous functions on [0,1], for each ϵ>0, where L(D)uϵ(t)=f(t,uϵ(t)), uϵ(0)=0 and |f(t,uϵ(t))|≤M for 0≤t≤1. Then the family {uϵ(t)} is equicontinuous on [0,1].Proof.
Since{uϵ(t)} is a family of continuous functions on [0,1], there exists l>0 such that |uϵ(t)|≤l for 0≤t≤1.
Letδ<min{(∑j=1n-1‍ϵΓ(sn-sj+1)/(4l|aj|))1/(sn-sn-1),(ϵΓ(sn+1)/(4M))1/sn}. For 0≤t1<t2≤1,  t2-t1<δ, we get
(2.7)|uϵ(t2)-uϵ(t1)|=|∑j=1n-1‍Isn-sjaju(t2)-∑j=1n-1‍Isn-sjaju(t1)+Isnf(t2,u(t2))-Isnf(t1,u(t1))|=|∑j=1n-1‍ajΓ(sn-sj)∫0t1‍[(t2-s)sn-sj-1-(t1-s)sn-sj-1]u(s)ds+∑j=1n-1‍ajΓ(sn-sj)∫t1t2‍(t2-s)sn-sj-1u(s)ds+1Γ(sn)∫t1t2‍(t2-s)sn-1f(s,u(s))ds+1Γ(sn)∫0t1‍[(t2-s)sn-1-(t1-s)sn-1]f(s,u(s))ds∑j=1n-1‍|≤∑j=1n-1‍l|aj|Γ(sn-sj)∫0t1‍[(t1-s)sn-sj-1-(t2-s)sn-sj-1]ds+∑j=1n-1‍l|aj|Γ(sn-sj)∫t1t2‍(t2-s)sn-sj-1ds+MΓ(sn)∫t1t2‍(t2-s)sn-1ds+MΓ(sn)∫0t1‍[(t1-s)sn-1-(t2-s)sn-1]ds=∑j=1n-1‍l|aj|Γ(sn-sj+1)(t1sn-sj-t2sn-sj)+∑j=1n-1‍2l|aj|Γ(sn-sj+1)(t2-t1)sn-sj+MΓ(sn+1)(t1sn-t2sn)+2MΓ(sn+1)(t2-t1)sn≤∑j=1n-1‍2l|aj|Γ(sn-sj+1)(t2-t1)sn-sn-1+2MΓ(sn+1)(t2-t1)sn≤ϵ2+ϵ2=ϵ.
Thus, {uϵ(t)} is equicontinuous on [0,1].Lemma 2.7 (see [16, Theorem 4.2]).
Letf:[0,1]×ℝ→ℝ be continuous and Lipschitz with respect to second variable with Lipschitz constant L. Let aj satisfy
(2.8)0<LΓ(sn+1)+∑j=1n-1‍|aj|Γ(sn-sj+1)<1.
Then IVP (1.4) has a unique solution.Lemma 2.8.
Letv,w∈C([0,1],ℝ) be locally Hölder continuous with exponent q<λ<1, f∈C([0,1]×ℝ,ℝ) and
(2.9)L(D)w≥f(t,w),L(D)v≤f(t,v),0<t≤1
one of the nonstrict inequalities being strict. Then v(0)<w(0) implies v(t)<w(t), 0≤t≤1.Proof.
Suppose thatv(t)<w(t), 0≤t≤1 is not true. We suppose the inequality L(D)>f(t,w(t)). Letting m(t)=v(t)-w(t), there exists 0<t1≤1 such that m(t)≤0, 0≤t≤t1, and m(t1)=0. Then by Corollary 2.5, we can obtain L(D)m(t1)≥0. From the conditions and the definition of m(t), we have
(2.10)f(t1,v(t1))≥L(D)v(t1)≥L(D)w(t1)>f(t1,w(t1)).
This is a contradiction to v(t1)=w(t1). The proof is complete.Lemma 2.9.
Assume that the conditions of Lemma2.8 hold with nonstrict inequalities (2.3) and (2.4). Furthermore, suppose that
(2.11)f(t,x)-f(t,y)≤N(x-y),wherex≥y,N>0.
Then v(0)≤w(0) implies v(t)≤w(t), 0≤t≤1 provided N<1/Γ(1-sn)-∑j=1n-1‍aj/Γ(1-sj).Proof.
Letwϵ(t)=w(t)+ϵ. For small ϵ>0, we have
(2.12)wϵ(0)>w(0),wϵ(t)>w(t),0≤t≤1.
Then, from (2.11) and (2.12) we get
(2.13)L(D)wϵ(t)=L(D)w(t)+L(D)ϵ=f(t,w(t))+ϵ[t-snΓ(1-sn)-∑j=1n-1‍ajt-sjΓ(1-sj)]≥f(t,wϵ(t))-Nϵ+ϵ[t-snΓ(1-sn)-∑j=1n-1‍ajt-sjΓ(1-sj)]≥f(t,wϵ(t))-Nϵ+ϵ[1Γ(1-sn)-∑j=1n-1‍ajΓ(1-sj)]>f(t,wϵ(t)),0<t≤1.
Applying Lemma 2.8, we obtain v(t)<wϵ(t),  0≤t≤1. By the arbitrariness of ϵ>0, we can conclude that v(t)≤w(t). The proof is complete.Corollary 2.10.
The functionf(t,u)=σ(t)u, where σ(t)≤N, is admissible in Lemma 2.9 to yield v(t)≤0 on 0≤t≤1.


## 3.   Main Results

In this section, we establish the existence and uniqueness criteria of solutions for initial value problem (1.4).Theorem 3.1.
Assume thatf∈C(R0,ℝ), where R0={(t,u):0≤t≤1,|u(t)|≤b} and |f(t,u)|≤M. Then IVP (1.4) possesses at least one solution u(t) on 0≤t≤α, where α=min{1,(bΓ(1+sn)/(2M))1/sn,(∑j=1n-1‍Γ(sn-sj+1)/(2|aj|))1/(sn-sn-1)}.Proof.
Letu0(t) be a continuous function on [-δ,0], δ>0, such that u0(0)=0, |u0(t)|≤b and |L(D)u0(t)|≤M, where D0-sju0(t), j=1,2,…,n-1 are the continuous fractional derivatives. For 0<ϵ≤δ, we define the function uϵ(t)=u0(t) on [-δ,0] and
(3.1)uϵ(t)=1Γ(sn)∫0t‍(t-s)sn-1f(s,uϵ(s-ϵ))ds+∑j=1n-1‍ajΓ(sn-sj)∫0t‍(t-s)sn-sj-1uϵ(s-ϵ)ds
on [0,α1], where α1=min{ϵ,α}. We observe that Dsjuϵ(t),  j=1,2,…,n exist for t∈[0,α1] and
(3.2)|uϵ(t)|≤1Γ(sn)∫0t‍(t-s)sn-1|f(s,uϵ(s-ϵ))|ds+∑j=1n-1‍|aj|Γ(sn-sj)∫0t‍(t-s)sn-sj-1|uϵ(s-ϵ)|ds≤MΓ(sn)∫0t‍(t-s)sn-1ds+∑j=1n-1‍b|aj|Γ(sn-sj)∫0t‍(t-s)sn-sj-1ds=MΓ(sn+1)tsn+∑j=1n-1‍b|aj|Γ(sn-sj+1)tsn-sj≤MΓ(sn+1)αsn+∑j=1n-1‍b|aj|Γ(sn-sj+1)αsn-sn-1≤b2+b2=b.
If α1<α, we can employ (3.1) to extend uϵ(t) as a continuously fractional differentiable function on [-δ,α2], α2=min{α,2ϵ} such that uϵ(t)≤b holds. Continuing this process, we can define uϵ(t) over [-δ,α] so that uϵ(t)≤b; it has a continuous fractional derivative and satisfies (3.1) on the same interval [-δ,α]. Furthermore, |L(D)uϵ(t)|≤M, since |f(t,uϵ(t-ϵ))|≤M on R0. Therefore, from Lemma 2.6, the family {uϵ(t)} is an equicontinuous and uniformly bounded function. An application of Ascoli-Arzela Theorem shows the existence of a sequence {ϵn} such that ϵ1>ϵ2>⋯>ϵn→0 as n→∞, and u(t)=limn→∞uϵn(t) exists uniformly on [-δ,α]. Due to f being uniformly continuous, we can obtain f(t,uϵn(t-ϵn)) which uniformly tends to f(t,u(t)), and uϵn(t-ϵn) uniformly tends to u(t) as n→∞. Therefore, term by term, integration of (3.1) with ϵ=ϵn, α1=α yields
(3.3)u(t)=1Γ(sn)∫0t‍(t-s)sn-1f(s,u(s))ds+∑j=1n-1‍ajΓ(sn-sj)∫0t‍(t-s)sn-sj-1u(s)ds.
This proves that u(t) is a solution of IVP (1.4) and the proof is complete.Theorem 3.2.
Letv,w∈C([0,1],ℝ) be lower and upper solutions of the IVP (1.4) which are locally Hölder continuous with exponent sn<λ<1 such that v(t)≤w(t), t∈[0,1] and f∈C(Ω,ℝ), where Ω={(t,u):v(t)≤u(t)≤w(t),t∈[0,1]}. Furthermore, suppose that
(3.4)(∑j=1n-1‍Γ(sn-sj+1)2|aj|)1/(sn-sn-1)≥1.
Then there exists a solution u(t) of IVP (1.4) satisfying v(t)≤u(t)≤w(t) on [0,1].Proof.
For the need of proof, we define functionp(t,u):[0,1]×ℝ→ℝ as
(3.5)p(t,u)=max⁡{v(t),min⁡{u,w(t)}}.
Therefore, f(t,p(t,u)) defines a continuous extension of f to [0,1]×R which is also bounded because f is bounded on Ω. Then by Theorems 3.1 and 3.2, we can obtain that the initial value problem
(3.6)L(D)u=f(t,p(t,u)),t∈(0,1],u(0)=0,
has a solution on [0,1].
Clearly, from the definition of functionp(t,u), we know that if IVP (3.6) exits a solution u(t) satisfying v(t)≤u(t)≤w(t) on [0,1], then u(t) is also a solution of IVP (1.4). In the following, we will prove that the solution u(t) of IVP (3.6) satisfies v(t)≤u(t)≤w(t) on [0,1].
For anyϵ>0, we consider
(3.7)wϵ(t)=w(t)+ϵ,vϵ(t)=v(t)-ϵ.
Then, we get
(3.8)wϵ(0)=w(0)+ϵ,vϵ(0)=v(0)-ϵ.
Therefore, it follows that vϵ(0)<u(0)<wϵ(0). Next, we will show that vϵ(t)<u(t)<wϵ(t), t∈[0,1]. Suppose that it is not true. Then there exists t1∈(0,1] such that
(3.9)u(t1)=wϵ(t1),vϵ(t)<u(t)<wϵ(t),0≤t<t1.
Therefore, u(t1)>w(t1), p(t1,u(t1))=w(t1) and v(t1)≤p(t1,u(t1))≤w(t1). Letting m(t)=u(t)-wϵ(t), we have m(t1)=0 and m(t)≤0, 0≤t≤t1. Then from Corollary 2.5, we can obtain L(D)m(t1)≥0 and
(3.10)f(t1,w(t1))=f(t1,p(t1,w(t1)))=L(D)u(t1)≥L(D)wϵ(t1)=L(D)w(t1)+L(D)ϵ(t1)=L(D)w(t1)+ϵ[t1-snΓ(1-sn)-∑j=1n-1‍ajt1-sjΓ(1-sj)]>L(D)w(t1)=f(t1,w(t1)),
which is a contradiction. The other case vϵ(t)<u(t) can be proved similarly.
Hence, we getvϵ(t)<u(t)<wϵ(t) on [0,1]. Letting ϵ→0, we obtain v(t)≤u(t)≤w(t) on [0,1]. The proof is complete.Now, we will give the existence of maximal and minimal solutions of initial value problem (1.4).Theorem 3.3.
Letf∈C([0,1]×ℝ,ℝ), v0, w0 be lower and upper solutions of (1.4) such that v0≤w0 on [0,1]. Furthermore, suppose that
(3.11)f(t,x)-f(t,y)≥-N(x-y),forv0≤y≤x≤w0,N≥0,
and aj satisfy
(3.12)0<NΓ(sn+1)+∑j=1n-1‍|aj|Γ(sn-sj+1)<1.
Then there exist monotone sequences {vn} and {wn} such that vn→ρ, wn→r as n→∞ uniformly on [0,1], where ρ and r are minimal and maximal solutions of IVP (1.4), respectively.Proof.
For anyη∈C([0,1],ℝ) satisfying v0≤η≤w0, we consider the following linear fractional differential equation:
(3.13)L(D)u=f(t,η)-N(u-η),t∈(0,1],u(0)=0.
Obviously, the right hand side of (3.13) satisfies the Lipschitz condition. From (3.11) and Lemma 2.7, it is clear that for every η, there exists a unique solution u of (3.13) on [0,1]. Define the operator T by Tη=u and use it to construct the sequences {vn}, {wn}. We need to prove the following propositions hold:  (i)
v0≤Tv0,  w0≥Tw0;(ii)
T is a monotone operator on the segment
(3.14)〈v0,w0〉={u∈C([0,1],R):v0≤u≤w0}.
To prove (i), letTv0=v1, where v1 is the unique solution of (3.13) with η=v0. Letting p=v0-v1, we have
(3.15)L(D)p=L(D)v0-L(D)v1≤f(t,v0)-[f(t,v0)-N(v1-v0)]=-Np,p(0)=v0(0)-v1(0)≤0.
By Corollary 2.10, we can obtain that p(t)≤0 on [0,1], that is, v0≤Tv0.
Similarly, we can getw0≥Tw0.
To prove (ii), letη1, η2∈〈v0,w0〉 such that η1≤η2. Assume that u1=Tη1 and u2=Tη2. Setting p=u1-u2, then using the condition (3.11), we have
(3.16)L(D)p=L(D)u1-L(D)u2=f(t,η1)-N(u1-η1)-[f(t,η2)-N(u2-η2)]≤-N(η1-η2)-N(u1-η1)+N(u2-η2)=-Np,p(0)=u1(0)-u2(0)=0.
From Corollary 2.10, we can obtain that p(t)≤0 on [0,1], which implies Tη1≤Tη2. And (ii) is proved.
Therefore, we can define the sequencesvn=Tvn-1, wn=Twn-1. From the previous discussion, we can get
(3.17)v0≤v1≤⋯≤vn≤wn≤⋯≤w1≤w0on⁡[0,1].
Clearly, the sequences{vn}, {wn} are uniformly bounded on [0,1]. From (3.13), we have |L(D)vn|, |L(D)wn| which are also uniformly bounded. By Lemma 2.6, we know that {vn}, {wn} are equicontinuous on [0,1]. Then applying Ascoli-Arzela Theorem, there exist subsequences {vnk}, {wnk} that converge uniformly on [0,1]. From (3.17), we can see that the entire sequences {vn}, {wn} converge uniformly and monotonically to ρ, r, respectively, as n→∞. It is now easy to show that ρ, r are solutions of IVP (1.4) by the corresponding Volterra fractional integral equation for (3.13).
In the following, we will prove thatρ and r are the minimal and maximal solutions of IVP (1.4), respectively. We need to show that if u is any solution of IVP (1.4) satisfying v0≤u≤w0 on [0,1], then we have v0≤ρ≤u≤r≤w0 on [0,1].
We assume that for somen, vn≤u≤wn on [0,1] and letting p=vn+1-u, we have
(3.18)L(D)p=L(D)vn+1-L(D)u=f(t,vn)-N(vn+1-vn)-f(t,u)≤-N(vn-u)-N(vn+1-vn)=-Np,p(0)=vn+1(0)-u(0)=0,
which implies vn+1≤u. Similarly, we have u≤wn+1 on [0,1]. Since v0≤u≤w0 on [0,1], this proves vn≤u≤wn for all n by induction. Letting n→∞, we conclude that ρ≤u≤r on [0,1] and the proof is complete.Theorem 3.4.
Suppose that the conditions of Theorem3.3 hold. In addition, we assume
(3.19)f(t,x)-f(t,y)≤N(x-y),v0≤y≤x≤w0,N>0.
Then ρ=r=u is the unique solution of IVP (1.4) provided N<1/Γ(1-sn)-∑j=1n-1‍aj/Γ(1-sj).Proof.
We have provedρ≤r in Theorem 3.3, so we just need to prove ρ≥r. Letting p=r-ρ, we get
(3.20)L(D)p=f(t,r)-f(t,ρ)≤Np,p(0)=r(0)-ρ(0)=0.
From Corollary 2.10, we obtain p≤0 on [0,1], which implies ρ≥r. Hence, ρ=r=u is the unique solution of IVP (1.4).


## 4. Examples

In this paper, we will present an example to illustrate the main results.Example 4.1.
Consider the initial value problem of fractional differential equation(4.1)D0.8u+0.4D0.6u=u2t0.210Γ(0.2)-ut0.42Γ(0.4),t∈(0,1],u(0)=0.
Choosew=5, v=-5; then we can obtain
(4.2)D0.8w+0.4D0.6w≥w2t0.210Γ(0.2)-wt0.42Γ(0.4),D0.8v+0.4D0.6v≤v2t0.210Γ(0.2)-vt0.42Γ(0.4).
That is, v and w are the lower and upper solutions of initial value problem (4.1). Furthermore, v and w are locally continuous with exponent 1>λ>0.8.
Since(4.3)(Γ(0.8-0.6+1)2|-0.4|)1/(0.8-0.6)=1.9914>1,
then by Theorem 3.2, there exists a solution u(t) of initial value problem (4.1) satisfying -5≤u(t)≤5.
Next, we will prove the existence of maximal and minimal solutions for initial value problem (4.1) by using Theorem 3.3.
Letv0=-5 and w0=5 be lower and upper solutions of (4.1). Furthermore, for any -5≤y≤x≤5, we have
(4.4)f(t,x)-f(t,y)=x2t0.210Γ(0.2)-xt0.42Γ(0.4)-y2t0.210Γ(0.2)+yt0.42Γ(0.4)=t0.210Γ(0.2)(x-y)(x+y)-t0.42Γ(0.4)(x-y)≥-12Γ(0.4)(x-y).
Then let N=1/2Γ(0.4)≈0.2254. We get
(4.5)0<NΓ(0.8+1)+|-0.4|Γ(0.8-0.6+1)≈0.6777<1.
Thus, from Theorem 3.3, there exist monotone sequences {vn} and {wn} such that vn→ρ, wn→r as n→∞ uniformly on [0,1], where ρ and r are minimal and maximal solutions of initial value problem (4.1), respectively.
In addition,(4.6)f(t,x)-f(t,y)=x2t0.210Γ(0.2)-xt0.42Γ(0.4)-y2t0.210Γ(0.2)+yt0.42Γ(0.4)=t0.210Γ(0.2)(x-y)(x+y)-t0.42Γ(0.4)(x-y)≤1010Γ(0.2)(x-y)≤N(x-y),0<N<1Γ(1-0.8)--0.4Γ(1-0.6)≈0.3982.
Hence, by Theorem 3.4, initial value problem (4.1) has a unique solution.


## 5. Conclusion

In this paper, we considered the initial value problem of nonlinear fractional differential equation(5.1)L(D)u=f(t,u),t∈(0,1],u(0)=0.
The basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique have been applied for the existence and uniqueness of solutions of the initial value problem. And several results were obtained. Besides, we studied the existence of minimal and maximal solutions. In Section 4, we also give an example to illustrate our results.



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*Source: 615230-2012-07-08.xml*