Existence and Uniqueness of Solutions for the Cauchy-Type Problems of Fractional Differential Equations

Abstract

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 [1–15] 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[16] first considered the nonlinear fractional differential equation

(𝐷^𝛼_𝑎+𝑦)(𝑥)=𝑓[𝑥,𝑦(𝑥)], (1.1) with the following initial conditions:

(𝐷^𝛼−𝑘_𝑎+𝑦)(𝑎+)=𝑏_𝑘,𝑏_𝑘∈ℝ,(𝑘=1,…,𝑛,𝑛=−[−𝛼]), (1.2) where 0<𝛼<1, and 𝐷^𝛼_𝑎+ is Riemann-Liouville fractional derivative. Barrett [17], in 1954, first considered the Cauchy-type problem for the linear fractional differential equation

(𝐷^𝛼_𝑎+𝑦)(𝑥)−𝜆𝑦(𝑥)=𝑓(𝑥),(𝑛−1≤𝑅(𝛼)<𝑛,𝛼≠𝑛−1), (1.3) with the same initial conditions (1.2). Afterwards, there is a great deal of work about the basic theory [18–27]. In [28], Kilbas et al. summarized systematically the main results.

In this paper we consider the cauchy problem (1.1)-(1.2); here 𝐷^𝛼_𝑎+ 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 [28].

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 [28]; 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 [28]). Let Ω=(𝑎,𝑏](−∞<𝑎<𝑏<∞) be a finite interval on the real axis ℝ. The Riemann-Liouville left-sided fractional integral 𝐼^𝛼_𝑎+𝑔 of the function 𝑔 with order 𝛼∈ℝ(𝛼>0) is defined by (𝐼^𝛼_𝑎+1𝑔)(𝑥)=Γ(𝛼)^𝑥_𝑎𝑔(𝑡)𝑑𝑡(𝑥−𝑡)^1−𝛼,(𝑥>𝑎), (2.1) 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 [28]). Let Ω=(𝑎,𝑏](−∞<𝑎<𝑏<∞) be a finite interval on the real axis ℝ. The Riemann-Liouville left-sided fractional derivative 𝐷^𝛼_𝑎+𝑔 of the function 𝑔 with order 𝛼∈ℝ(𝛼≥0) is defined by (𝐷^𝛼_𝑎+𝑑𝑔)(𝑥)=()𝑑𝑥^𝑛(𝐼^𝑛−𝛼_𝑎+=1𝑔)(𝑥)(𝑑Γ(𝑛−𝛼))𝑑𝑥^𝑛𝑥_𝑎𝑔(𝑡)𝑑𝑡(𝑥−𝑡)^{𝛼−𝑛+1},(𝑥>𝑎;𝑛=−[−𝛼]), (2.2) 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⊂ℝ). 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,𝑦₂∈𝐺 one has |𝑓[𝑥,𝑦₁]−𝑓[𝑥,𝑦₂]|≤𝐴|𝑦₁−𝑦₂|, (2.3) where 𝐴>0 does not depend on 𝑥∈(𝑎,𝑏].

Definition 2.4 (see [28]). Let 𝑛−1<𝛼≤𝑛(𝑛∈𝑁), then the space 𝐶^𝛼_𝑛−𝛼[𝑎,𝑏] is defined by 𝐶^𝛼_𝑛−𝛼[𝑎,𝑏]={𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏]∶(𝐷^𝛼_𝑎+𝑦)(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏]}. (2.4) Here 𝐶_𝑛−𝛼[𝑎,𝑏] is a weighted space of continuous functions 𝐶_𝑛−𝛼[𝑎,𝑏]={𝑔∶(𝑎,𝑏]→𝑅∶(𝑥−𝑎)^𝑛−𝛼𝑔(𝑥)∈𝐶[𝑎,𝑏]}, (2.5) and 𝐷^𝛼_𝑎+ is the Riemann-Liouville fractional derivative.
In the space 𝐶_𝑛−𝛼[𝑎,𝑏], we define the norm ‖𝑔‖_{𝐶𝑛−𝛼}=‖(𝑥−𝑎)^𝑛−𝛼𝑔(𝑥)‖_𝐶.

Definition 2.5 (see [28]). Let (𝑎,𝑏)(0<𝑎<𝑏≤∞) be a finite or infinite interval of the half-axis ℝ⁺. The Hadamard type left-sided fractional integral 𝒥^𝛼_𝑎+ℎ of the function ℎ with order 𝛼∈ℝ(𝛼>0) is defined by (𝒥^𝛼_𝑎+1ℎ)(𝑥)=Γ(𝛼)^𝑥_𝑎𝑥(ln𝑡)^𝛼−1ℎ(𝑡)𝑑𝑡𝑡,(𝑎<𝑥<𝑏), (2.6) where ℎ∶(𝑎,𝑏)→ℝ and the right-side integral of the above equality is assumed to make sense.

Definition 2.6 (see [28]). Let 𝛿=𝑥𝐷(𝐷=𝑑/𝑑𝑥) be the 𝛿-derivative. The Hadamard left-sided fractional derivative 𝒟^𝛼_𝑎+𝑦 of the function 𝑦 on (𝑎,𝑏) with order 𝛼∈ℝ(𝛼≥0) is defined by (𝒟^𝛼_𝑎+𝑦)(𝑥)=𝛿^𝑛(𝒥^𝑛−𝛼_𝑎+=1𝑦)(𝑥)𝑑Γ(𝑛−𝛼)(𝑥)𝑑𝑥^𝑛𝑥_𝑎𝑥(ln𝑡)^{𝑛−𝛼−1}𝑦(𝑡)𝑑𝑡𝑡,(𝑎<𝑥<𝑏;𝑛=−[−𝛼]), (2.7) where 𝑦∶(𝑎,𝑏)→ℝ, 𝛿^𝑛=𝛿⋯𝛿_𝑛, and the right side of the above equality is assumed to make sense.

Definition 2.7 (see [28]). Let 𝑛−1<𝛼≤𝑛(𝑛∈𝑁),0<𝑎<𝑏<+∞, and 0≤𝛾<1. The space 𝐶^𝛼_{𝛿;𝑛−𝛼,𝛾}[𝑎,𝑏] is defined by 𝐶^𝛼_{𝛿;𝑛−𝛼,𝛾}[𝑎,𝑏]={𝑦(𝑥)∈𝐶_{𝑛−𝛼,ln}[𝑎,𝑏]∶(𝒟^𝛼_𝑎+𝑦)∈𝐶_𝛾,ln[𝑎,𝑏]}, (2.8) where 𝒟^𝛼_𝑎+ is a Hadamard left-sided fractional derivative, and 𝐶_𝛾,ln[𝑎,𝑏] is a weighted space of continuous functions 𝐶_𝛾,ln𝑥[𝑎,𝑏]={𝑔∶(𝑎,𝑏]→ℝ∶(ln𝑎)^𝛾𝑔(𝑥)∈𝐶[𝑎,𝑏]}. (2.9)
In the space 𝐶_𝛾,ln[𝑎,𝑏], we define the norm ‖𝑔‖_𝐶𝛾,ln=‖(ln(𝑥/𝑎))^𝛾𝑔(𝑥)‖_𝐶.

Theorem 2.8 (see [28]). Let 𝛼>0, 𝑛=−[−𝛼]. Let 𝑓∶(𝑎,𝑏]×ℝ→ℝ be a function such that 𝑓[𝑥,𝑦(𝑥)]∈𝐶_𝑛−𝛼[𝑎,𝑏] for any 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏]. If 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏], then 𝑦(𝑥) satisfies the relations: (𝐷^𝛼_𝑎+𝑦)(𝑥)=𝑓[𝑥,𝑦(𝑥)],(𝛼>0),(𝐷^𝛼−𝑘_𝑎+𝑦)(𝑎+)=𝑏_𝑘,𝑏_𝑘∈ℝ,(𝑘=1,…,𝑛=−[−𝛼]), (2.10)(2.11) if and only if 𝑦(𝑥) satisfies the Volterra integral equation 𝑦(𝑥)=𝑦₀1(𝑥)+Γ(𝛼)^𝑥_𝑎𝑓[𝑡,𝑦(𝑡)]𝑑𝑡(𝑥−𝑡)^1−𝛼,(𝑥>𝑎), (2.12) where 𝑦₀(𝑥)=^𝑛_𝑗=1𝑏_𝑗Γ(𝛼−𝑗+1)(𝑥−𝑎)^𝛼−𝑗, (2.13) where 𝐷^𝛼_𝑎+ is a Riemann-Liouville left-sided fractional derivative.

Theorem 2.9 (see [28]). (Banach Fixed Point Theorem) Let (U,d) be a nonempty complete metric space, let 0≤𝜔<1, and let 𝑇∶𝑈→𝑈 be a map such that, for every 𝑢,𝑣∈𝑈, the relation 𝑑(𝑇𝑢,𝑇𝑣)≤𝜔𝑑(𝑢,𝑣),(0≤𝜔<1) (2.14) holds. Then the operator T has a unique fixed point 𝑢^∗∈𝑈.

Theorem 2.10 (see [28]). Let 0<𝑎<𝑏<∞,𝛼>0, 𝑛=−[−𝛼], and 0≤𝛾<1. Let 𝑓∶(𝑎,𝑏]×ℝ→ℝ be a function such that 𝑓[𝑥,𝑦(𝑥)]∈𝐶_𝛾,ln[𝑎,𝑏] for any 𝑦(𝑥)∈𝐶_𝛾,ln[𝑎,𝑏]. If 𝑦(𝑥)∈𝐶_{𝑛−𝛼,ln}[𝑎,𝑏], then 𝑦(𝑥) satisfies (𝒟^𝛼_𝑎+𝑦)(𝑥)=𝑓[𝑥,𝑦(𝑥)],(𝑥>𝑎),(𝒟^𝛼−𝑘_𝑎+𝑦)(𝑎+)=𝑏_𝑘,𝑏_𝑘∈ℝ,(𝑘=1,…,𝑛;𝑛=−[−𝛼]), (2.15) if and only if 𝑦(𝑥) satisfies the Volterra integral equation 𝑦(𝑥)=^𝑛_𝑗=1𝑏_𝑗𝑥Γ(𝑛−𝑗+1)(ln𝑎)^𝛼−𝑗+1Γ(𝛼)^𝑥_𝑎𝑥(ln𝑡)^𝛼−1𝑓[𝑡,𝑦(𝑡)]𝑑𝑡𝑡,(𝑥>𝑎), (2.16) where 𝒟^𝛼_𝑎+ 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 [28], 𝐺 is an open set in 𝑅 and 𝑓 is assumed to be a function such that 𝑓[𝑥,𝑦]∈𝐶_𝑛−𝛼[𝑎,𝑏](𝐶_𝛾,ln[𝑎,𝑏]) for any 𝑦∈𝐺. 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 [28], 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 [28], which means that the proofs of the related results in [28] 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,𝑔∈𝐶_𝛾[𝑎,𝑐], and 𝑔∈𝐶[𝑐,𝑏]. Then 𝑔∈𝐶_𝛾[𝑎,𝑏] and ‖𝑔‖_{𝐶𝛾[𝑎,𝑏]}≤max{‖𝑔‖_{𝐶𝛾[𝑎,𝑐]},(𝑏−𝑎)^𝛾‖𝑔‖_{𝐶[𝑐,𝑏]}}. (3.1)

Proof. Since 𝑔∈𝐶_𝛾[𝑎,𝑐] and 𝑔∈𝐶[𝑐,𝑏], then 𝑔∈𝐶(𝑎,𝑏] and 𝑔∈𝐶_𝛾[𝑎,𝑏]. Now we prove the estimate. Because 𝑔∈𝐶_𝛾[𝑎,𝑏], there exists 𝑥₀∈[𝑎,𝑏] such that ‖𝑔‖_{𝐶𝛾[𝑎,𝑏]}=|(𝑥₀−𝑎)^𝛾𝑔(𝑥₀)|. (3.2)
If 𝑥₀∈[𝑎,𝑐], then
‖𝑔‖_{𝐶𝛾[𝑎,𝑏]}≤‖𝑔‖_{𝐶𝛾[𝑎,𝑐]}. (3.3)
If 𝑥₀∈[𝑐,𝑏], then
‖𝑔‖_{𝐶𝛾[𝑎,𝑏]}≤(𝑏−𝑎)^𝛾‖𝑔‖_{𝐶[𝑐,𝑏]}. (3.4)
Hence we have
‖𝑔‖_{𝐶𝛾[𝑎,𝑏]}≤max{‖𝑔‖_{𝐶𝛾[𝑎,𝑐]},(𝑏−𝑎)^𝛾‖𝑔‖_{𝐶[𝑐,𝑏]}}. (3.5)
This completes the proof of Lemma 3.1.

Lemma 3.2 (see [28]). If 𝛾∈ℝ(0≤𝛾<1), then the fractional integration operator 𝐼^𝛼_𝑎+ with order 𝛼∈ℝ(𝛼>0) is a mapping from 𝐶_𝛾[𝑎,𝑏] to 𝐶_𝛾[𝑎,𝑏], and ‖𝐼^𝛼_𝑎+𝑔‖_𝐶𝛾≤(𝑏−𝑎)^𝛼Γ(1−𝛾)Γ(1+𝛼−𝛾)‖𝑔‖_𝐶𝛾, (3.6) here 𝐼^𝛼_𝑎+ is a Riemann-Liouville fractional integral operator and 𝑔∈𝐶_𝛾[𝑎,𝑏].

Furthermore, we have the following conclusion.

Lemma 3.3. The fractional integration operator 𝐼^𝛼_𝑎+ with order 𝛼∈ℝ(𝛼>0) is a mapping from 𝐶[𝑎,𝑏] to 𝐶[𝑎,𝑏], and ‖𝐼^𝛼_𝑎+𝑔‖_𝐶≤(𝑏−𝑎)^𝛼𝛼Γ(𝛼)‖𝑔‖_𝐶, (3.7) where 𝐼^𝛼_𝑎+ is a Riemann-Liouville fractional integral operator and 𝑔∈𝐶[𝑎,𝑏].

Proof. Firstly we prove that if 𝑔∈𝐶[𝑎,𝑏], then (𝐼^𝛼_𝑎+𝑔)(𝑥)∈𝐶[𝑎,𝑏]. For any 𝑥∈[𝑎,𝑏] and Δ𝑥>0,𝑥+Δ𝑥≤𝑏, we have |(𝐼^𝛼_𝑎+𝑔)(𝑥+Δ𝑥)−(𝐼^𝛼_𝑎+1𝑔)(𝑥)|=|Γ(𝛼)_𝑎^𝑥+Δ𝑥𝑔(𝑡)𝑑𝑡(𝑥+Δ𝑥−𝑡)^1−𝛼−1Γ(𝛼)^𝑥_𝑎𝑔(𝑡)𝑑𝑡(𝑥−𝑡)^1−𝛼|≤1Γ(𝛼){|^𝑥_𝑎1𝑔(𝑡)[(𝑥+Δ𝑥−𝑡)^1−𝛼−1(𝑥−𝑡)^1−𝛼]𝑑𝑡|+|_𝑥^𝑥+Δ𝑥𝑔(𝑡)(𝑥+Δ𝑥−𝑡)^1−𝛼≤𝑑𝑡|}‖𝑔‖_{𝐶[𝑎,𝑏]}𝛼Γ(𝛼){[(𝑥+Δ𝑥−𝑎)^𝛼−(𝑥−𝑎)^𝛼]+(Δ𝑥)^𝛼+(Δ𝑥)^𝛼}. (3.8)
It is easy to see that as Δ𝑥→0⁺, we have
|(𝐼^𝛼_𝑎+𝑔)(𝑥+Δ𝑥)−(𝐼^𝛼_𝑎+𝑔)(𝑥)|→0. (3.9)
Similarly, we can prove that as Δ𝑥→0⁻, we have
|(𝐼^𝛼_𝑎+𝑔)(𝑥+Δ𝑥)−(𝐼^𝛼_𝑎+𝑔)(𝑥)|→0. (3.10) Thus 𝐼^𝛼_𝑎+𝑔∈𝐶[𝑎,𝑏].
Now we prove the estimate. In fact
‖𝐼^𝛼_𝑎+𝑔‖_{𝐶[𝑎,𝑏]}=max_{𝑥∈[𝑎,𝑏]}|1Γ(𝛼)^𝑥_𝑎𝑔(𝑡)𝑑𝑡(𝑥−𝑡)^1−𝛼|≤‖𝑔‖_{𝐶[𝑎,𝑏]}Γ(𝛼)^𝑥_𝑎(𝑥−𝑡)^𝛼−1≤𝑑𝑡(𝑏−𝑎)^𝛼𝛼Γ(𝛼)‖𝑔‖_𝐶. (3.11)
This completes the proof of Lemma 3.3.

Lemma 3.4. Let 𝛾∈[0,∞), 0<𝑎<𝑐<𝑏<∞, 𝑔∈𝐶_𝛾,ln[𝑎,𝑐] and 𝑔∈𝐶[𝑐,𝑏]. Then 𝑔∈𝐶_𝛾,ln[𝑎,𝑏] and ‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑏]}≤max{‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑐]}𝑏,(ln𝑎)^𝛾‖𝑔‖_{𝐶[𝑐,𝑏]}}. (3.12)

Proof. The proof is similar to the proof of Lemma 3.1. Since 𝑔∈𝐶_𝛾,ln[𝑎,𝑐] and 𝑔∈𝐶[𝑐,𝑏], we have 𝑔∈𝐶(𝑎,𝑏], that is, 𝑔∈𝐶_𝛾,ln[𝑎,𝑏].
Next we give the estimate. Because 𝑔∈𝐶_𝛾,ln[𝑎,𝑏], there exists at least 𝑥^∗∈[𝑎,𝑏] such that
‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑏]}𝑥=|(ln^∗𝑎)^𝛾𝑔(𝑥^∗)|. (3.13)
If 𝑥^∗∈[𝑎,𝑐], then
‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑏]}≤‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑐]}. (3.14)
If 𝑥^∗∈[𝑐,𝑏], then
‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑏]}𝑏≤(ln𝑎)^𝛾‖𝑔‖_{𝐶[𝑐,𝑏]}. (3.15)
Hence we have
‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑏]}≤max{‖𝑔‖_{𝐶𝛾,ln[𝑎,𝑐]}𝑏,(ln𝑎)^𝛾‖𝑔‖_{𝐶[𝑐,𝑏]}}. (3.16)
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]×ℝ→ℝ be a function such that 𝑓[𝑥,𝑦(𝑥)]∈𝐶_𝑛−𝛼[𝑎,𝑏] for any 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏] and the Lipschitzian condition holds with respect to the second variable 𝑦. Then there exists a unique solution 𝑦(𝑥)∈𝐶^𝛼_𝑛−𝛼[𝑎,𝑏] for the cauchy-type problem (2.10)-(2.11).

Proof. First we prove the existence of a unique solution 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏]. According to Theorem 2.8, it is sufficient to prove the existence of a unique solution 𝑦(𝑥)∈𝐶_𝑛−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, (3.17) where 𝐴>0 is the Lipschitzian coefficient. Next we prove the existence of a unique solution 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑥₁] to (2.12) on the interval (𝑎,𝑥₁]. For this, we use the Banach fixed point theorem for the space 𝐶_𝑛−𝛼[𝑎,𝑥₁], which is a complete metric space with the distance given by 𝑑(𝑦₁,𝑦₂)=‖𝑦₁−𝑦₂‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}=max_{𝑥∈[𝑎,𝑥1]}|(𝑥−𝑎)^𝑛−𝛼[𝑦₁(𝑥)−𝑦₂(𝑥)]|. (3.18)
We rewrite the integral (2.12) in the form
𝑦(𝑥)=(𝑇𝑦)(𝑥), (3.19) where (𝑇𝑦)(𝑥)=𝑦₀1(𝑥)+Γ(𝛼)^𝑥_𝑎𝑓[𝑡,𝑦(𝑡)]𝑑𝑡(𝑥−𝑡)^1−𝛼. (3.20)
To apply Theorem 2.9, we have to prove the following: (1) if 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑥₁], then (𝑇𝑦)(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑥₁]; (2) for any 𝑦₁,𝑦₂∈𝐶_𝑛−𝛼[𝑎,𝑥₁] the following estimate holds:
‖𝑇𝑦₁−𝑇𝑦₂‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}≤𝜔‖𝑦₁−𝑦₂‖_{𝐶𝑛−𝛼[𝑎,𝑥1]},𝜔=𝐴(𝑥₁−𝑎)^𝛼Γ(𝛼−𝑛+1)Γ(2𝛼−𝑛+1). (3.21)
It follows from (2.13) that 𝑦₀(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑥₁]. Since 𝑓[𝑥,𝑦(𝑥)]∈𝐶_𝑛−𝛼[𝑎,𝑥₁] for any 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑥₁], then, by Lemma 3.2 [28] (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 𝐶_𝑛−𝛼[𝑎,𝑥₁], and hence (𝑇𝑦)(𝑥)∈𝐶_𝑛−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−𝑇𝑦₂‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}≤‖𝐼^𝛼_𝑎+[|𝑓[𝑡,𝑦₁(𝑡)]−𝑓[𝑡,𝑦₂(𝑡)]|]‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}≤𝐴‖𝐼^𝛼_𝑎+[|𝑦₁(𝑡)−𝑦₂(𝑡)|]‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}≤𝐴(𝑥₁−𝑎)^𝛼Γ(𝛼−𝑛+1)Γ(2𝛼−𝑛+1)‖𝑦₁−𝑦₂‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}, (3.22) 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)∈𝐶_𝑛−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 (𝑇^𝑚𝑦^∗₀)(𝑥):
lim_𝑚→∞‖𝑇^𝑚𝑦^∗₀−𝑦^∗‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}=0, (3.23) where 𝑦^∗₀(𝑥) is any function in 𝐶_𝑛−𝛼[𝑎,𝑥₁]. If there is at least one 𝑏_𝑘≠0 in the initial condition (2.11), then we can take 𝑦^∗₀(𝑥)=𝑦₀(𝑥) with 𝑦₀(𝑥) defined by (2.13). The last relation can be rewritten into the form lim_𝑚→∞‖𝑦_𝑚−𝑦^∗‖_{𝐶𝑛−𝛼[𝑎,𝑥1]}=0, (3.24) where 𝑦_𝑚(𝑥)=(𝑇^𝑚𝑦^∗₀)(𝑥)=𝑦₀1(𝑥)+Γ(𝛼)^𝑥_𝑎𝑓[𝑡,(𝑇^𝑚−1𝑦^∗₀)(𝑡)]𝑑𝑡(𝑥−𝑡)^1−𝛼,(𝑚∈𝑁). (3.25)
Next we consider the interval [𝑥₁,𝑏]. Rewrite (2.12) in the form
𝑦(𝑥)=𝑦₀₁1(𝑥)+Γ(𝛼)^𝑥_𝑥1𝑓[𝑡,𝑦(𝑡)]𝑑𝑡(𝑥−𝑡)^1−𝛼, (3.26) where 𝑦₀₁(𝑥) is defined by 𝑦₀₁(𝑥)=^𝑛_𝑗=1𝑏_𝑗Γ(𝛼−𝑗+1)(𝑥−𝑎)^𝛼−𝑗+1Γ(𝛼)^𝑥1_𝑎𝑓[𝑡,𝑦(𝑡)]𝑑𝑡(𝑥−𝑡)^1−𝛼. (3.27) 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. (3.28)𝐶[𝑥₁,𝑥₂] is a complete metric space with the distance given by 𝑑(𝑦₁−𝑦₂)=‖𝑦₁−𝑦₂‖_{𝐶[𝑥1,𝑥2]}=max_{𝑥∈[𝑥1,𝑥2]}|𝑦₁(𝑥)−𝑦₂(𝑥)|. (3.29)
We rewrite the integral equation (3.26) into the form
𝑦(𝑥)=(𝑇𝑦)(𝑥), (3.30) where (𝑇𝑦)(𝑥)=𝑦₀₁1(𝑥)+Γ(𝛼)^𝑥_𝑥1𝑓[𝑡,𝑦(𝑡)]𝑑𝑡(𝑥−𝑡)^1−𝛼. (3.31)
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,𝑦₂∈𝐶[𝑥₁,𝑥₂], the following estimate holds:
‖𝑇𝑦₁−𝑇𝑦₂‖_{𝐶[𝑥1,𝑥2]}≤𝜔‖𝑦₁−𝑦₂‖_{𝐶[𝑥1,𝑥2]},𝜔=𝐴(𝑥₂−𝑥₁)^𝛼.𝛼Γ(𝛼) (3.32)
Since 𝑓[𝑥,𝑦(𝑥)]∈𝐶_𝑛−𝛼[𝑎,𝑏] for any 𝑦(𝑥)∈𝐶_𝑛−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−𝑇𝑦₂‖_{𝐶[𝑥1,𝑥2]}≤‖𝐼^𝛼_𝑎+[|𝑓[𝑡,𝑦₁(𝑡)]−𝑓[𝑡,𝑦₂(𝑡)]|]‖_{𝐶[𝑥1,𝑥2]}≤𝐴(𝑥₂−𝑥₁)^𝛼𝛼Γ(𝛼)‖𝑦₁−𝑦₂‖_{𝐶[𝑥1,𝑥2]}, (3.33) 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 𝑦^∗₁(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 (𝑇^𝑚𝑦^∗₀₁)(𝑥): lim_𝑚→∞‖𝑇^𝑚𝑦^∗₀₁−𝑦^∗₁‖_{𝐶[𝑥1,𝑥2]}=0, (3.34) where 𝑦^∗₀₁(𝑥) is any function in 𝐶[𝑥₁,𝑥₂]. If 𝑦₀(𝑥)≠0 on [𝑥₁,𝑥₂], then we can take 𝑦^∗₀₁(𝑥)=𝑦₀(𝑥) with 𝑦₀(𝑥) defined by (2.13). The last relation can be rewritten in the form lim_𝑚→∞‖𝑦_𝑚−𝑦^∗₁‖_{𝐶[𝑥1,𝑥2]}=0, (3.35) where 𝑦_𝑚(𝑥)=(𝑇^𝑚𝑦^∗₀₁)(𝑥)=𝑦₀₁1(𝑥)+Γ(𝛼)^𝑥_𝑥1𝑓[𝑡,(𝑇^𝑚−1𝑦^∗₀₁)]𝑑𝑡(𝑥−𝑡)^1−𝛼,(𝑚∈𝑁). (3.36)
Next we consider the interval [𝑥₂,𝑥₃], where 𝑥₃=𝑥₂+ℎ₂ such that 𝑥₃≤𝑏 and (𝐴(𝑥₃−𝑥₂)^𝛼/𝛼Γ(𝛼))<1. Using the same arguments as the above, we derive that there exists a unique solution 𝑦^∗₂(𝑥)∈𝐶[𝑥₂,𝑥₃] to (2.12) on the interval [𝑥₂,𝑥₃]. If 𝑥₃≠𝑏, then take the next interval [𝑥₃,𝑥₄], where 𝑥₄=𝑥₃+ℎ₃ and ℎ₃>0 such that 𝑥₄≤𝑏 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)=𝑦^∗_𝑘(𝑥), and 𝑦^∗_𝑘(𝑥)∈𝐶[𝑥_𝑘,𝑥_𝑘+1](𝑘=1,…,𝐿), where 𝑥₀<𝑥₁<⋯<𝑥_𝐿+1=𝑏 and (𝐴(𝑥_𝑘+1−𝑥_𝑘)^𝛼/𝛼Γ(𝛼))<1, and we take 𝑦₀(𝑥)=𝑦_0𝑘(𝑥), and 𝑦^∗₀(𝑥)=𝑦^∗_0𝑘(𝑥)(𝑘=1,…,𝐿) on each interval [𝑥_𝑘,𝑥_𝑘+1]. By (𝐴(𝑥_𝑘+1−𝑥_𝑘)^𝛼/𝛼Γ(𝛼))<1, we know that by finite steps we can arrive at 𝑥_𝐿+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)∈𝐶_𝑛−𝛼[𝑎,𝑏] to the Volterra integral equation (2.12) on the whole interval [𝑎,𝑏], and hence 𝑦(𝑥)∈𝐶_𝑛−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)∈𝐶_𝑛−𝛼[𝑎,𝑏] belongs to the the space 𝐶^𝛼_𝑛−𝛼[𝑎,𝑏]; it is sufficient to prove that (𝐷^𝛼_𝑎+𝑦)(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏]. By the above proof, the solution 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏] is a limit of the sequence 𝑦_𝑚(𝑥), where 𝑦_𝑚(𝑥)=(𝑇^𝑚𝑦^∗₀)∈𝐶_𝑛−𝛼[𝑎,𝑏]:
lim_𝑚→∞‖𝑦_𝑚−𝑦‖_{𝐶𝑛−𝛼[𝑎,𝑏]}=0, (3.37) with the choice of certain 𝑦^∗₀(𝑥) on each [𝑎,𝑥₁],…,[𝑥_𝐿,𝑏].
If 𝑦₀(𝑥)≠0, then we can take 𝑦^∗₀(𝑥)=𝑦₀(𝑥).
By (2.10) and the Lipschitzian-condition, we have
‖𝐷^𝛼_𝑎+𝑦_𝑚−𝐷^𝛼_𝑎+𝑦‖_{𝐶𝑛−𝛼}=‖𝑓[𝑥,𝑦_𝑚]−𝑓[𝑥,𝑦]‖_{𝐶𝑛−𝛼}≤𝐴‖𝑦_𝑚−𝑦‖_{𝐶𝑛−𝛼}. (3.38)
Thus
lim_𝑚→∞‖𝐷^𝛼_𝑎+𝑦_𝑚−𝐷^𝛼_𝑎+𝑦‖_{𝐶𝑛−𝛼}=0. (3.39)
By (𝐷^𝛼_𝑎+𝑦_𝑚)(𝑥)=𝑓[𝑥,𝑦_𝑚−1(𝑥)] and 𝑓[𝑥,𝑦(𝑥)]∈𝐶_𝑛−𝛼[𝑎,𝑏] for any 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏], we have 𝑓[𝑥,𝑦_𝑚−1(𝑥)]∈𝐶_𝑛−𝛼[𝑎,𝑏], that is, (𝐷^𝛼_𝑎+𝑦_𝑚)(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏]. Hence (𝐷^𝛼_𝑎+𝑦)(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏].
This completes the proof of Theorem 3.5.

Corollary 3.6. Let 𝛼>0 and 𝑛=−[−𝛼]. Let 𝑓∶(𝑎,𝑏]×ℝ→ℝ be a function such that 𝑓(𝑥,𝑦)∈𝐶_𝑛−𝛼[𝑎,𝑏] for any 𝑦∈ℝ; the Lipschitzian condition holds with respect to 𝑦 and lim_𝑥→𝑎(𝑥−𝑎)^𝑛−𝛼𝑓(𝑥,𝑦(𝑥)), and lim_𝑥→𝑎(𝑥−𝑎)^𝑛−𝛼𝑓(𝑥,𝑦(𝑥)) exist for any 𝑦(𝑥)∈𝐶_𝑛−𝛼[𝑎,𝑏]. Then there exists a unique solution 𝑦(𝑥)∈𝐶^𝛼_𝑛−𝛼[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 [28], 𝐺 is an open set in 𝑅 and 𝑓 is assumed to be a function such that 𝑓[𝑥,𝑦]∈𝐶_𝑛−𝛼[a,𝑏] for any 𝑦∈𝐺. 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 0≤𝛾<1 such that 𝛾≥𝑛−𝛼. Let 𝑓∶(𝑎,𝑏]×ℝ→ℝ(𝑎>0) be a function such that 𝑓[𝑥,𝑦(𝑥)]∈𝐶_𝛾,ln[𝑎,𝑏] for any 𝑦(𝑥)∈𝐶_𝛾,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 (𝒟^𝛼_𝑎+𝑦)(𝑥)=𝑓[𝑥,𝑦(𝑥)],𝛼>0;𝑥>𝑎,(𝒟^𝛼−𝑘_𝑎+𝑦)(𝑎+)=𝑏_𝑘,𝑏_𝑘∈ℝ(𝑘=1,…,𝑛,𝑛=−[−𝛼]). (3.40) in the space 𝐶^𝛼_{𝛿;𝑛−𝛼,𝛾}[𝑎,𝑏], where 𝒟^𝛼_𝑎+ 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 𝜇𝑔(𝑥)=(𝑥−𝜇𝑎)^𝛾𝑥∈(𝑎,𝑐],(𝑥−𝑐)^𝛾𝑥∈(𝑐,𝑏], (4.1) where 𝜇≠0 is a constant number and 𝛾∈(0,∞).

From the above definition of 𝑔(𝑥), we know that 𝑔(𝑥)∈𝐶_𝛾[𝑎,𝑐] and 𝑔(𝑥)∈𝐶_𝛾[𝑐,𝑏], but we cannot get the conclusion 𝑔(𝑥)∈𝐶_𝛾[𝑎,𝑏] and ‖𝑔‖_{𝐶𝛾[𝑎,𝑏]}≤max{‖𝑔‖_{𝐶𝛾[𝑎,𝑐]},‖𝑔‖_{𝐶𝛾[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)∈𝐶_𝛾[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. (4.2)
It is evident that 𝜙(𝑥) belongs to the space 𝐶¹[𝑎,𝑏] = {𝜓(𝑥)∶𝜓⁽¹⁾(𝑥)∈𝐶[𝑎,𝑏], ‖𝜓‖_{𝐶1[𝑎,𝑏]} = ‖𝜓‖_{𝐶[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]. (4.3) We could not conclude that ‖𝜙‖_{𝐶1[𝑎,𝑏]}≤max{‖𝜙‖_{𝐶1[𝑎,𝑐]},‖𝜙‖_{𝐶1[𝑐,𝑏]}}, (4.4) because ‖𝜙‖_{𝐶1[𝑎,𝑏]}1=ln𝑏+𝑎,‖𝜙‖_{𝐶1[𝑎,𝑐]}1=ln𝑐+𝑎,‖𝜙‖_{𝐶1[𝑐,𝑏]}1=ln𝑏+𝑐. (4.5) However, we have ‖𝜙‖_{𝐶1[𝑎,𝑏]}≤‖𝜙‖_{𝐶1[𝑎,𝑐]}+‖𝜙‖_{𝐶1[𝑐,𝑏]}. (4.6)

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

Example 4.3. Let one consider the function 𝜇ℎ(𝑥)=(ln𝑥𝜇/𝑎)^𝛾𝑥∈(𝑎,𝑐],(ln𝑥/𝑐)^𝛾𝑥∈(𝑐,𝑏], (4.7) where 𝜇≠0 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)∈𝐶_𝛾,ln[𝑎,𝑐] and ℎ(𝑥)∈𝐶_𝛾,ln[𝑐,𝑏]. However, the conclusion that ℎ(𝑥)∈𝐶_𝛾,ln[𝑎,𝑏] and ‖ℎ‖_{𝐶𝛾[𝑎,𝑏]}≤max{‖ℎ‖_{𝐶𝛾[𝑎,𝑐]},‖ℎ‖_{𝐶𝛾[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 [28]. By modifying the conditions on 𝑓 and improving the method used in [28], 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 [28].

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).