DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation14217510.1155/2010/142175142175Research ArticleExistence and Uniqueness of Solutions for theCauchy-TypeProblems of Fractional Differential EquationsKouChunhai1LiuJian1YeYan2ZhangGuang1Department of Applied Mathematics DonghuaUniversityShanghai 201620Chinadhu.edu.cn2College of Information Sciences and TechnologyDonghua UniversityShanghai 201620Chinadhu.edu.cn20101102201020103010200921012010250120102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using the Banach fixed point theorem and step method, we study the existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations. Meanwhile, by citing some counterexamples, it is pointed out that there exist a few defects in the proofs of the known results.

1. Introduction

Recently, fractional differential equations are applied widely in various fields of science and engineering. Regarding applications of fractional differential equations, we refer to  and references cited therein. However, the investigation of basic theory of fractional differential equations is still not complete, and there is a great deal of work which needs to be done. Most of the investigations in this field involve the existence and uniqueness of solutions to fractional differential equations on the finite interval [a,b]. In 1938, Pitcher and Sewell first considered the nonlinear fractional differential equation

(Da+αy)(x)=f[x,y(x)], with the following initial conditions:

(Da+α-ky)(a+)=bk,bk,(k=1,,n,n=-[-α]), where 0<α<1, and Da+α is Riemann-Liouville fractional derivative. Barrett , in 1954, first considered the Cauchy-type problem for the linear fractional differential equation

(Da+αy)(x)-λy(x)=f(x),(n-1R(α)<n,αn-1), with the same initial conditions (1.2). Afterwards, there is a great deal of work about the basic theory . In , Kilbas et al. summarized systematically the main results.

In this paper we consider the cauchy problem (1.1)-(1.2); here Da+α can be Riemann-Liouville fractional derivative, and Hadamard-type fractional derivative. We establish some results about the existence and uniqueness of solution of (1.1)-(1.2). By the way, we will point out that there exist several defects in the proofs of the related theorems of .

This paper is organized as follows: in Section 2, we introduce some preliminaries and notations; main results are proved in Section 3; in Section 4, by citing several counterexamples, we will point out the defects in ; Section 5 is a brief summary of this paper.

2. Preliminaries and Notations

In this section, we introduce some basic definitions and notations about fractional calculus. Meanwhile, several known theorems are given, which are useful in this paper.

Definition 2.1 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let Ω=(a,b](-<a<b<) be a finite interval on the real axis . The Riemann-Liouville left-sided fractional integral Ia+αg of the function g with order α(α>0) is defined by (Ia+αg)(x)=1Γ(α)axg(t)dt(x-t)1-α,(x>a), where the real function g is defined on the interval Ω and the right-side integral of the above equality is assumed to make sense.

Definition 2.2 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let Ω=(a,b](-<a<b<) be a finite interval on the real axis . The Riemann-Liouville left-sided fractional derivative Da+αg of the function g with order α(α0) is defined by (Da+αg)(x)=(ddx)n(Ia+n-αg)(x)=1Γ(n-α)(ddx)naxg(t)dt(x-t)α-n+1,(x>a;n=-[-α]), where the real function g is defined on the interval Ω and the right side of the above equality is assumed to make sense.

Definition 2.3.

Assume that f[x,y] is defined on the set (a,b]×G(G). f[x,y] is said to satisfy Lipschitzian condition with respect to the second variable, if for all x(a,b] and for any y1,y2G one has |f[x,y1]-f[x,y2]|A|y1-y2|, where A>0 does not depend on x(a,b].

Definition 2.4 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let n-1<αn(nN), then the space Cn-αα[a,b] is defined by Cn-αα[a,b]={y(x)Cn-α[a,b]:(Da+αy)(x)Cn-α[a,b]}. Here Cn-α[a,b] is a weighted space of continuous functions Cn-α[a,b]={g:(a,b]R:(x-a)n-αg(x)C[a,b]}, and Da+α is the Riemann-Liouville fractional derivative.

In the space Cn-α[a,b], we define the norm gCn-α=(x-a)n-αg(x)C.

Definition 2.5 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let (a,b)(0<a<b) be a finite or infinite interval of the half-axis +. The Hadamard type left-sided fractional integral 𝒥a+αh of the function h with order α(α>0) is defined by (𝒥a+αh)(x)=1Γ(α)ax(lnxt)α-1h(t)dtt,(a<x<b), where h:(a,b) and the right-side integral of the above equality is assumed to make sense.

Definition 2.6 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let δ=xD(D=d/dx) be the δ-derivative. The Hadamard left-sided fractional derivative 𝒟a+αy of the function y on (a,b) with order α(α0) is defined by (𝒟a+αy)(x)=δn(𝒥a+n-αy)(x)=1Γ(n-α)(xddx)nax(lnxt)n-α-1y(t)dtt,(a<x<b;n=-[-α]), where y:(a,b), δn=δδn, and the right side of the above equality is assumed to make sense.

Definition 2.7 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let n-1<αn(nN),0<a<b<+, and 0γ<1. The space Cδ;n-α,γα[a,b] is defined by Cδ;n-α,γα[a,b]={y(x)Cn-α,ln[a,b]:(𝒟a+αy)Cγ,ln[a,b]}, where 𝒟a+α is a Hadamard left-sided fractional derivative, and Cγ,ln[a,b] is a weighted space of continuous functions Cγ,ln[a,b]={g:(a,b]:(lnxa)γg(x)C[a,b]}.

In the space Cγ,ln[a,b], we define the norm gCγ,ln=(ln(x/a))γg(x)C.

Theorem 2.8 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let α>0, n=-[-α]. Let f:(a,b]× be a function such that f[x,y(x)]Cn-α[a,b] for any y(x)Cn-α[a,b]. If y(x)Cn-α[a,b], then y(x) satisfies the relations: (Da+αy)(x)=f[x,y(x)],(α>0),(Da+α-ky)(a+)=bk,bk,(k=1,,n=-[-α]), if and only if y(x) satisfies the Volterra integral equation y(x)=y0(x)+1Γ(α)axf[t,y(t)]dt(x-t)1-α,(x>a), where y0(x)=j=1nbjΓ(α-j+1)(x-a)α-j, where Da+α is a Riemann-Liouville left-sided fractional derivative.

Theorem 2.9 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

(Banach Fixed Point Theorem) Let (U,d) be a nonempty complete metric space, let 0ω<1, and let T:UU be a map such that, for every u,vU, the relation d(Tu,Tv)ωd(u,v),(0ω<1) holds. Then the operator T has a unique fixed point u*U.

Theorem 2.10 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let 0<a<b<,α>0, n=-[-α], and 0γ<1. Let f:(a,b]× be a function such that f[x,y(x)]Cγ,ln[a,b] for any y(x)Cγ,ln[a,b]. If y(x)Cn-α,ln[a,b], then y(x) satisfies (𝒟a+αy)(x)=f[x,y(x)],(x>a),(𝒟a+α-ky)(a+)=bk,bk,(k=1,,n;n=-[-α]), if and only if y(x) satisfies the Volterra integral equation y(x)=j=1nbjΓ(n-j+1)(lnxa)α-j+1Γ(α)ax(lnxt)α-1f[t,y(t)]dtt,(x>a), where 𝒟a+α is a Hadamard-type left-sided fractional derivative.

Remark 2.11.

It should be worthy noting that the conditions in Theorems 2.8 and 2.10 are a little different from the ones in [28, pages 163, 213]. In , G is an open set in R and f is assumed to be a function such that f[x,y]Cn-α[a,b](Cγ,ln[a,b]) for any yG. In fact, we think that such assumption is not complete for the proof of the related conclusion.

3. Main Results

In this section, we will establish several useful lemmas. It should be pointed out that, in , some analogous lemmas play important roles in the proofs of the related results. However, we have found out that there exist a few defects in these lemmas of , which means that the proofs of the related results in  are not complete. Several counterexamples will be given in Section 4. In a sense, our lemmas are to mend these cracks. Furthermore, several theorems about the existence and uniqueness of solution for the cauchy-type problem (2.10)-(2.11) will be given; then, in the sense of Hadamard fractional derivative, we have the similar result.

Lemma 3.1.

Let γ[0,),a<c<b,gCγ[a,c], and gC[c,b]. Then gCγ[a,b] and gCγ[a,b]max{gCγ[a,c],(b-a)γgC[c,b]}.

Proof.

Since gCγ[a,c] and gC[c,b], then gC(a,b] and gCγ[a,b]. Now we prove the estimate. Because gCγ[a,b], there exists x0[a,b] such that gCγ[a,b]=|(x0-a)γg(x0)|.

If x0[a,c], then

gCγ[a,b]gCγ[a,c].

If x0[c,b], then

gCγ[a,b](b-a)γgC[c,b].

Hence we have

gCγ[a,b]max{gCγ[a,c],(b-a)γgC[c,b]}.

This completes the proof of Lemma 3.1.

Lemma 3.2 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

If γ(0γ<1), then the fractional integration operator Ia+α with order α(α>0) is a mapping from Cγ[a,b] to Cγ[a,b], and Ia+αgCγ(b-a)αΓ(1-γ)Γ(1+α-γ)gCγ, here Ia+α is a Riemann-Liouville fractional integral operator and gCγ[a,b].

Furthermore, we have the following conclusion.

Lemma 3.3.

The fractional integration operator Ia+α with order α(α>0) is a mapping from C[a,b] to C[a,b], and Ia+αgC(b-a)ααΓ(α)gC, where Ia+α is a Riemann-Liouville fractional integral operator and gC[a,b].

Proof.

Firstly we prove that if gC[a,b], then (Ia+αg)(x)C[a,b]. For any x[a,b] and Δx>0,x+Δxb, we have |(Ia+αg)(x+Δx)-(Ia+αg)(x)|=|1Γ(α)ax+Δxg(t)dt(x+Δx-t)1-α-1Γ(α)axg(t)dt(x-t)1-α|1Γ(α){|axg(t)[1(x+Δx-t)1-α-1(x-t)1-α]dt|+|xx+Δxg(t)(x+Δx-t)1-αdt|}gC[a,b]αΓ(α){[(x+Δx-a)α-(x-a)α]+(Δx)α+(Δx)α}.

It is easy to see that as Δx0+, we have

|(Ia+αg)(x+Δx)-(Ia+αg)(x)|0.

Similarly, we can prove that as Δx0-, we have

|(Ia+αg)(x+Δx)-(Ia+αg)(x)|0. Thus Ia+αgC[a,b].

Now we prove the estimate. In fact

Ia+αgC[a,b]=maxx[a,b]|1Γ(α)axg(t)dt(x-t)1-α|gC[a,b]Γ(α)ax(x-t)α-1dt(b-a)ααΓ(α)gC.

This completes the proof of Lemma 3.3.

Lemma 3.4.

Let γ[0,), 0<a<c<b<, gCγ,ln[a,c] and gC[c,b]. Then gCγ,ln[a,b] and gCγ,ln[a,b]max{gCγ,ln[a,c],(lnba)γgC[c,b]}.

Proof.

The proof is similar to the proof of Lemma 3.1. Since gCγ,ln[a,c] and gC[c,b], we have gC(a,b], that is, gCγ,ln[a,b].

Next we give the estimate. Because gCγ,ln[a,b], there exists at least x*[a,b] such that

gCγ,ln[a,b]=|(lnx*a)γg(x*)|.

If x*[a,c], then

gCγ,ln[a,b]gCγ,ln[a,c].

If x*[c,b], then

gCγ,ln[a,b](lnba)γgC[c,b].

Hence we have

gCγ,ln[a,b]max{gCγ,ln[a,c],(lnba)γgC[c,b]}.

This completes the proof of Lemma 3.4.

Next, on the basis of above lemmas, we establish the results about the existence and uniqueness of solution for the cauchy-type problem (2.10)-(2.11) in the sense of Riemann-Liouville fractional derivative and Hadamard fractional derivative.

Theorem 3.5.

Let α>0 and n=-[-α]. Let f:(a,b]× be a function such that f[x,y(x)]Cn-α[a,b] for any y(x)Cn-α[a,b] and the Lipschitzian condition holds with respect to the second variable y. Then there exists a unique solution y(x)Cn-αα[a,b] for the cauchy-type problem (2.10)-(2.11).

Proof.

First we prove the existence of a unique solution y(x)Cn-α[a,b]. According to Theorem 2.8, it is sufficient to prove the existence of a unique solution y(x)Cn-α[a,b] to the nonlinear Volterra integral equation (2.12). Equation (2.12) makes sense in any interval (a,x1](a,b](a<x1<b). Choose x1 such that A(x1-a)αΓ(α-n+1)Γ(2α-n+1)<1, where A>0 is the Lipschitzian coefficient. Next we prove the existence of a unique solution y(x)Cn-α[a,x1] to (2.12) on the interval (a,x1]. For this, we use the Banach fixed point theorem for the space Cn-α[a,x1], which is a complete metric space with the distance given by d(y1,y2)=y1-y2Cn-α[a,x1]=maxx[a,x1]|(x-a)n-α[y1(x)-y2(x)]|.

We rewrite the integral (2.12) in the form

y(x)=(Ty)(x), where (Ty)(x)=y0(x)+1Γ(α)axf[t,y(t)]dt(x-t)1-α.

To apply Theorem 2.9, we have to prove the following: (1) if y(x)Cn-α[a,x1], then (Ty)(x)Cn-α[a,x1]; (2) for any y1,y2Cn-α[a,x1] the following estimate holds:

Ty1-Ty2Cn-α[a,x1]ωy1-y2Cn-α[a,x1],ω=A(x1-a)αΓ(α-n+1)Γ(2α-n+1).

It follows from (2.13) that y0(x)Cn-α[a,x1]. Since f[x,y(x)]Cn-α[a,x1] for any y(x)Cn-α[a,x1], then, by Lemma 3.2  (with γ=n-α,b=x1, and g(x)=f[x,y(x)]), the integral in the right-hand side of (3.19) also belongs to Cn-α[a,x1], and hence (Ty)(x)Cn-α[a,x1]. Now we prove the estimate in (3.21). By (3.20), using the Lipschitzian condition and applying the relation (3.6) (with γ=n-α, b=x1, and g(x)=f[x,y1(x)]-f[x,y2(x)]), we have

Ty1-Ty2Cn-α[a,x1]Ia+α[|f[t,y1(t)]-f[t,y2(t)]|]Cn-α[a,x1]AIa+α[|y1(t)-y2(t)|]Cn-α[a,x1]A(x1-a)αΓ(α-n+1)Γ(2α-n+1)y1-y2Cn-α[a,x1], which yields the estimate (3.21). In accordance with (3.17), 0<ω<1, and hence, by Theorem 2.9, there exists a unique solution y*(x)Cn-α[a,x1] to (2.12) on the interval [a,x1].

By Theorem 2.9, this solution y*(x) is a limit of a convergent sequence (Tmy0*)(x):

limmTmy0*-y*Cn-α[a,x1]=0, where y0*(x) is any function in Cn-α[a,x1]. If there is at least one bk0 in the initial condition (2.11), then we can take y0*(x)=y0(x) with y0(x) defined by (2.13). The last relation can be rewritten into the form limmym-y*Cn-α[a,x1]=0, where ym(x)=(Tmy0*)(x)=y0(x)+1Γ(α)axf[t,(Tm-1y0*)(t)]dt(x-t)1-α,(mN).

Next we consider the interval [x1,b]. Rewrite (2.12) in the form

y(x)=y01(x)+1Γ(α)x1xf[t,y(t)]dt(x-t)1-α, where y01(x) is defined by y01(x)=j=1nbjΓ(α-j+1)(x-a)α-j+1Γ(α)ax1f[t,y(t)]dt(x-t)1-α. We obtain y01(x)C[x1,b]. Next we prove the existence of a unique solution y(x)C[x1,b] to (2.12) on the interval [x1,b]. For this, we also use Banach fixed point theorem for the space C[x1,x2], where x2 satisfies A(x2-x1)ααΓ(α)<1.C[x1,x2] is a complete metric space with the distance given by d(y1-y2)=y1-y2C[x1,x2]=maxx[x1,x2]|y1(x)-y2(x)|.

We rewrite the integral equation (3.26) into the form

y(x)=(Ty)(x), where (Ty)(x)=y01(x)+1Γ(α)x1xf[t,y(t)]dt(x-t)1-α.

To apply Theorem 2.9, we have to prove the following: (1) if y(x)C[x1,x2], then (Ty)(x)C[x1,x2]; (2) for any y1,y2C[x1,x2], the following estimate holds:

Ty1-Ty2C[x1,x2]ωy1-y2C[x1,x2],ω=A(x2-x1)ααΓ(α).

Since f[x,y(x)]Cn-α[a,b] for any y(x)Cn-α[a,b], then, by Lemma 3.3, the integral in the right-hand side of (3.31) also belongs to C[x1,x2], and hence (Ty)(x)C[x1,x2]. Now we prove the estimate in (3.32) as follows:

Ty1-Ty2C[x1,x2]Ia+α[|f[t,y1(t)]-f[t,y2(t)]|]C[x1,x2]A(x2-x1)ααΓ(α)y1-y2C[x1,x2], which yields the estimate (3.32). In accordance with (3.28), then 0<ω<1, and hence by Theorem 2.9, there exists a unique solution y1*(x)C[x1,x2] to (2.12) on the interval [x1,x2]. By Theorem 2.9, this solution in C[x1,x2] is a limit of a convergent sequence (Tmy01*)(x): limmTmy01*-y1*C[x1,x2]=0, where y01*(x) is any function in C[x1,x2]. If y0(x)0 on [x1,x2], then we can take y01*(x)=y0(x) with y0(x) defined by (2.13). The last relation can be rewritten in the form limmym-y1*C[x1,x2]=0, where ym(x)=(Tmy01*)(x)=y01(x)+1Γ(α)x1xf[t,(Tm-1y01*)]dt(x-t)1-α,(mN).

Next we consider the interval [x2,x3], where x3=x2+h2 such that x3b and (A(x3-x2)α/αΓ(α))<1. Using the same arguments as the above, we derive that there exists a unique solution y2*(x)C[x2,x3] to (2.12) on the interval [x2,x3]. If x3b, then take the next interval [x3,x4], where x4=x3+h3 and h3>0 such that x4b and (A(x4-x3)α/αΓ(α))<1. If x4<b, repeating the above process, then we find that there exists a unique solution y(x) to (2.12), y(x)=yk*(x), and yk*(x)C[xk,xk+1](k=1,,L), where x0<x1<<xL+1=b and (A(xk+1-xk)α/αΓ(α))<1, and we take y0(x)=y0k(x), and y0*(x)=y0k*(x)(k=1,,L) on each interval [xk,xk+1]. By (A(xk+1-xk)α/αΓ(α))<1, we know that by finite steps we can arrive at xL+1=b.

Then there exists a unique solution y(x)C[x1,b] to (2.12) on the interval [x1,b]. By Lemma 3.1, we obtain that there exists a unique solution y(x)Cn-α[a,b] to the Volterra integral equation (2.12) on the whole interval [a,b], and hence y(x)Cn-α[a,b] is the unique solution to the cauchy-type problem (2.10)-(2.11).

To complete the proof of Theorem 3.5, we must show that such a unique solution y(x)Cn-α[a,b] belongs to the the space Cn-αα[a,b]; it is sufficient to prove that (Da+αy)(x)Cn-α[a,b]. By the above proof, the solution y(x)Cn-α[a,b] is a limit of the sequence ym(x), where ym(x)=(Tmy0*)Cn-α[a,b]:

limmym-yCn-α[a,b]=0, with the choice of certain y0*(x) on each [a,x1],,[xL,b].

If y0(x)0, then we can take y0*(x)=y0(x).

By (2.10) and the Lipschitzian-condition, we have

Da+αym-Da+αyCn-α=f[x,ym]-f[x,y]Cn-αAym-yCn-α.

Thus

limmDa+αym-Da+αyCn-α=0.

By (Da+αym)(x)=f[x,ym-1(x)] and f[x,y(x)]Cn-α[a,b] for any y(x)Cn-α[a,b], we have f[x,ym-1(x)]Cn-α[a,b], that is, (Da+αym)(x)Cn-α[a,b]. Hence (Da+αy)(x)Cn-α[a,b].

This completes the proof of Theorem 3.5.

Corollary 3.6.

Let α>0 and n=-[-α]. Let f:(a,b]× be a function such that f(x,y)Cn-α[a,b] for any y; the Lipschitzian condition holds with respect to y and limxa(x-a)n-αf(x,y(x)), and limxa(x-a)n-αf(x,y(x)) exist for any y(x)Cn-α[a,b]. Then there exists a unique solution y(x)Cn-αα[a,b] for the cauchy-type problem (2.10)-(2.11).

Remark 3.7.

It should be pointed out that the conditions in Theorem 3.5 are different from the ones in [28, Theorem 3.11, page 165]. In , G is an open set in R and f is assumed to be a function such that f[x,y]Cn-α[a,b] for any yG. In fact, such assumptions are not complete for the proof of the related conclusion. A counterexample will be given in Section 4. There exists the similar problem in [28, Theorem 3.29, page 213]. By applying Lemma 3.4, modifying the conditions in [28, Theorem 3.29, page 213] and using the similar arguments to the proof of Theorem 3.5, we arrive at the following result.

Theorem 3.8.

Let α>0,n=-[-α], and 0γ<1 such that γn-α. Let f:(a,b]×(a>0) be a function such that f[x,y(x)]Cγ,ln[a,b] for any y(x)Cγ,ln[a,b] and the Lipschitzian condition holds with respect to y. Then there exists a unique solution y(x) for the Cauchy-type problem (𝒟a+αy)(x)=f[x,y(x)],α>0;x>a,(𝒟a+α-ky)(a+)=bk,bk(k=1,,n,n=-[-α]). in the space Cδ;n-α,γα[a,b], where 𝒟a+α is a Hadamard fractional derivative.

4. Counterexamples

In this section, by citing some counterexamples we would like to point out that, in [28, Lemmas 3.4, 3.9, and 3.10, pages 165, 202, and 213] are not complete.

Example 4.1.

Let one consider the function g(x)={μ(x-a)γx(a,c],μ(x-c)γx(c,b], where μ0 is a constant number and γ(0,).

From the above definition of g(x), we know that g(x)Cγ[a,c] and g(x)Cγ[c,b], but we cannot get the conclusion g(x)Cγ[a,b] and gCγ[a,b]max{gCγ[a,c],gCγ[c,b]}. Hence the conclusion of [28, Lemma 3.4, page 165] does not hold. We cannot apply it to prove [28, Theorem 3.11, page 165]. Furthermore, there also exists a problem about f in [28, Theorem 3.11, page 165]. For example, choosing f(x,y)=ysin(1/(x-a)), we know that f(x,y)Cγ[a,b] for any y, f(x,y) satisfies Lipschitz condition with respect to the second variable y. However, choosing y(x)=1/(x-a)γ, we can not arrive at (x-a)γf(x,y(x))=sin(1/(x-a))C[a,b]. Hence the condition of f in [28, Theorem 3.11, page 165] is not proper.

The next example illustrates that there also exists a problem in [28, Lemma 3.9, page 202].

Example 4.2.

Consider the function ϕ(x)=lnx,x[a,b],1<a<b.

It is evident that ϕ(x) belongs to the space C1[a,b] = {ψ(x):ψ(1)(x)C[a,b], ψC1[a,b] = ψC[a,b]+ψ(1)C[a,b]}.

Setting c(a,b), that is, a<c<b, then

ϕ(x)C1[a,b],ϕ(x)C1[a,c],ϕ(x)C1[c,b]. We could not conclude that ϕC1[a,b]max{ϕC1[a,c],ϕC1[c,b]}, because ϕC1[a,b]=lnb+1a,ϕC1[a,c]=lnc+1a,ϕC1[c,b]=lnb+1c. However, we have ϕC1[a,b]ϕC1[a,c]+ϕC1[c,b].

The following example is for [28, Lemma 3.10, page 213].

Example 4.3.

Let one consider the function h(x)={μ(lnx/a)γx(a,c],μ(lnx/c)γx(c,b], where μ0 is a constant number and γ(0,).

The same problem exists in [28, Lemma 3.10, page 213]. From the definition of h(x), we have h(x)Cγ,ln[a,c] and h(x)Cγ,ln[c,b]. However, the conclusion that h(x)Cγ,ln[a,b] and hCγ[a,b]max{hCγ[a,c],hCγ[c,b]} is still not correct. This defect means that [28, Lemma 3.10] could not be applied to prove [28, Theorem 3.29].

In a sense, our lemmas and main results have remedied these defects.

5. Conclusion

In this paper, we first get several useful lemmas, especially Lemmas 3.1 and 3.4, which have improved the corresponding lemmas in . By modifying the conditions on f and improving the method used in , we have established the results of existence and uniqueness of solution for the cauchy-type problems involving the Riemann-Liouville fractional derivative and the Hadamard fractional derivative in the weight space of continuous functions. Meanwhile, we have given some counterexamples to prove that [28, Lemmas 3.4, 3.9, and 3.10, pages 165, 203, and 213] are not complete, which means that there exist some defects in the proofs of the related results in .

Acknowledgments

The authors would like to thank the anonymous referee for his/her remarks about the evaluation of the original version of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant no. 10701023 and no. 10971221), the Natural Science Foundation of Shanghai (no. 10ZR1400100), and Chinese Universities Scientific Fund (B08-1).

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