Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 142175, 15 pages
doi:10.1155/2010/142175
Research Article

Existence and Uniqueness of Solutions for theCauchy-TypeProblems of Fractional Differential Equations

Chunhai Kou,1 Jian Liu,1 and Yan Ye2

1Department of Applied Mathematics, DonghuaUniversity, Shanghai 201620, China
2College of Information Sciences and Technology, Donghua University, Shanghai 201620, China

Received 30 October 2009; Revised 21 January 2010; Accepted 25 January 2010

Academic Editor: Guang Zhang

Copyright © 2010 Chunhai Kou et al. This 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.

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 ) − 𝜆 𝑦 ( 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 + 𝑔 ( 𝑥 ) = 1 Γ ( 𝛼 ) 𝑥 𝑎 𝑔 ( 𝑡 ) 𝑑 𝑡 ( 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 + 𝑔 ( 𝑥 ) = 𝑑 𝑑 𝑥 𝑛 𝐼 𝑛 − 𝛼 𝑎 + 𝑔 ( 𝑥 ) = 1 Γ ( 𝑛 − 𝛼 ) 𝑑 𝑑 𝑥 𝑛 𝑥 𝑎 𝑔 ( 𝑡 ) 𝑑 𝑡 ( 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 𝑦 1 , 𝑦 2 ∈ 𝐺 one has | | 𝑓 𝑥 , 𝑦 1 − 𝑓 𝑥 , 𝑦 2 | | ≤ 𝐴 | | 𝑦 1 − 𝑦 2 | | , ( 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 − 𝛼 𝑔 ( 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 ‖ 𝐶 𝑛 − 𝛼 = ‖ ( 𝑥 − 𝑎 ) 𝑛 − 𝛼 𝑔 ( 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 + ℎ ( 𝑥 ) = 1 Γ ( 𝛼 ) 𝑥 𝑎 l n 𝑥 𝑡 𝛼 − 1 ℎ ( 𝑡 ) 𝑑 𝑡 𝑡 , ( 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 = 𝑥 𝐷 ( 𝐷 = 𝑑 / 𝑑 𝑥 ) 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 + 𝑦 ( 𝑥 ) = 1 Γ ( 𝑛 − 𝛼 ) 𝑥 𝑑 𝑑 𝑥 𝑛 𝑥 𝑎 l n 𝑥 𝑡 𝑛 − 𝛼 − 1 𝑦 ( 𝑡 ) 𝑑 𝑡 𝑡 , ( 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 , l n [ 𝑎 , 𝑏 ] ∶ 𝒟 𝛼 𝑎 + 𝑦 ∈ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] , ( 2 . 8 ) where 𝒟 𝛼 𝑎 + is a Hadamard left-sided fractional derivative, and 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] is a weighted space of continuous functions 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] = 𝑔 ∶ ( 𝑎 , 𝑏 ] ⟶ ℝ ∶ l n 𝑥 𝑎 𝛾 𝑔 ( 𝑥 ) ∈ 𝐶 [ 𝑎 , 𝑏 ] . ( 2 . 9 )
In the space 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] , we define the norm ‖ 𝑔 ‖ 𝐶 𝛾 , l n = ‖ ( l n ( 𝑥 / 𝑎 ) ) 𝛾 𝑔 ( 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 ) , ( 2 . 1 0 ) 𝐷 𝛼 − 𝑘 𝑎 + 𝑦 ( 𝑎 + ) = 𝑏 𝑘 , 𝑏 𝑘 ∈ ℝ , ( 𝑘 = 1 , … , 𝑛 = − [ − 𝛼 ] ) , ( 2 . 1 1 ) if and only if 𝑦 ( 𝑥 ) satisfies the Volterra integral equation 𝑦 ( 𝑥 ) = 𝑦 0 ( 𝑥 ) + 1 Γ ( 𝛼 ) 𝑥 𝑎 𝑓 [ 𝑡 , 𝑦 ( 𝑡 ) ] 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 , ( 𝑥 > 𝑎 ) , ( 2 . 1 2 ) where 𝑦 0 ( 𝑥 ) = 𝑛 𝑗 = 1 𝑏 𝑗 Γ ( 𝛼 − 𝑗 + 1 ) ( 𝑥 − 𝑎 ) 𝛼 − 𝑗 , ( 2 . 1 3 ) 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 ( 𝑇 𝑢 , 𝑇 𝑣 ) ≤ 𝜔 𝑑 ( 𝑢 , 𝑣 ) , ( 0 ≤ 𝜔 < 1 ) ( 2 . 1 4 ) 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 , l n [ 𝑎 , 𝑏 ] for any 𝑦 ( 𝑥 ) ∈ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] . If 𝑦 ( 𝑥 ) ∈ 𝐶 𝑛 − 𝛼 , l n [ 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 . 1 5 ) if and only if 𝑦 ( 𝑥 ) satisfies the Volterra integral equation 𝑦 ( 𝑥 ) = 𝑛 𝑗 = 1 𝑏 𝑗 Γ ( 𝑛 − 𝑗 + 1 ) l n 𝑥 𝑎 𝛼 − 𝑗 + 1 Γ ( 𝛼 ) 𝑥 𝑎 l n 𝑥 𝑡 𝛼 − 1 𝑓 [ 𝑡 , 𝑦 ( 𝑡 ) ] 𝑑 𝑡 𝑡 , ( 𝑥 > 𝑎 ) , ( 2 . 1 6 ) 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 , l n [ 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 ‖ 𝐶 𝛾 [ 𝑎 , 𝑏 ] ≤ m a x ‖ 𝑔 ‖ 𝐶 𝛾 [ 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 𝑥 0 ∈ [ 𝑎 , 𝑏 ] such that ‖ 𝑔 ‖ 𝐶 𝛾 [ 𝑎 , 𝑏 ] = | | 𝑥 0 − 𝑎 𝛾 𝑔 𝑥 0 | | . ( 3 . 2 )
If 𝑥 0 ∈ [ 𝑎 , 𝑐 ] , then
‖ 𝑔 ‖ 𝐶 𝛾 [ 𝑎 , 𝑏 ] ≤ ‖ 𝑔 ‖ 𝐶 𝛾 [ 𝑎 , 𝑐 ] . ( 3 . 3 )
If 𝑥 0 ∈ [ 𝑐 , 𝑏 ] , then
‖ 𝑔 ‖ 𝐶 𝛾 [ 𝑎 , 𝑏 ] ≤ ( 𝑏 − 𝑎 ) 𝛾 ‖ 𝑔 ‖ 𝐶 [ 𝑐 , 𝑏 ] . ( 3 . 4 )
Hence we have
‖ 𝑔 ‖ 𝐶 𝛾 [ 𝑎 , 𝑏 ] ≤ m a x ‖ 𝑔 ‖ 𝐶 𝛾 [ 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 ‖ ‖ 𝐶 𝛾 ≤ ( 𝑏 − 𝑎 ) 𝛼 Γ ( 1 − 𝛾 ) Γ ( 1 + 𝛼 − 𝛾 ) ‖ 𝑔 ‖ 𝐶 𝛾 , ( 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 ) 𝛼 𝛼 Γ ( 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 Δ 𝑥 > 0 , 𝑥 + Δ 𝑥 ≤ 𝑏 , we have | | 𝐼 𝛼 𝑎 + 𝑔 ( 𝑥 + Δ 𝑥 ) − 𝐼 𝛼 𝑎 + 𝑔 ( 𝑥 ) | | = | | | | 1 Γ ( 𝛼 ) 𝑥 + Δ 𝑥 𝑎 𝑔 ( 𝑡 ) 𝑑 𝑡 ( 𝑥 + Δ 𝑥 − 𝑡 ) 1 − 𝛼 − 1 Γ ( 𝛼 ) 𝑥 𝑎 𝑔 ( 𝑡 ) 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 | | | | ≤ 1 Γ ( 𝛼 ) ⎧ ⎨ ⎩ | | | | 𝑥 𝑎 𝑔 ( 𝑡 ) 1 ( 𝑥 + Δ 𝑥 − 𝑡 ) 1 − 𝛼 − 1 ( 𝑥 − 𝑡 ) 1 − 𝛼 𝑑 𝑡 | | | | + | | | | 𝑥 + Δ 𝑥 𝑥 𝑔 ( 𝑡 ) ( 𝑥 + Δ 𝑥 − 𝑡 ) 1 − 𝛼 𝑑 𝑡 | | | | ⎫ ⎬ ⎭ ≤ ‖ 𝑔 ‖ 𝐶 [ 𝑎 , 𝑏 ] 𝛼 Γ ( 𝛼 ) ( 𝑥 + Δ 𝑥 − 𝑎 ) 𝛼 − ( 𝑥 − 𝑎 ) 𝛼 + ( Δ 𝑥 ) 𝛼 + ( Δ 𝑥 ) 𝛼 . ( 3 . 8 )
It is easy to see that as Δ 𝑥 → 0 + , we have
| | 𝐼 𝛼 𝑎 + 𝑔 ( 𝑥 + Δ 𝑥 ) − 𝐼 𝛼 𝑎 + 𝑔 ( 𝑥 ) | | ⟶ 0 . ( 3 . 9 )
Similarly, we can prove that as Δ 𝑥 → 0 − , we have
| | 𝐼 𝛼 𝑎 + 𝑔 ( 𝑥 + Δ 𝑥 ) − 𝐼 𝛼 𝑎 + 𝑔 ( 𝑥 ) | | ⟶ 0 . ( 3 . 1 0 ) Thus 𝐼 𝛼 𝑎 + 𝑔 ∈ 𝐶 [ 𝑎 , 𝑏 ] .
Now we prove the estimate. In fact
‖ ‖ 𝐼 𝛼 𝑎 + 𝑔 ‖ ‖ 𝐶 [ 𝑎 , 𝑏 ] = m a x 𝑥 ∈ [ 𝑎 , 𝑏 ] | | | | 1 Γ ( 𝛼 ) 𝑥 𝑎 𝑔 ( 𝑡 ) 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 | | | | ≤ ‖ 𝑔 ‖ 𝐶 [ 𝑎 , 𝑏 ] Γ ( 𝛼 ) 𝑥 𝑎 ( 𝑥 − 𝑡 ) 𝛼 − 1 𝑑 𝑡 ≤ ( 𝑏 − 𝑎 ) 𝛼 𝛼 Γ ( 𝛼 ) ‖ 𝑔 ‖ 𝐶 . ( 3 . 1 1 )
This completes the proof of Lemma 3.3.

Lemma 3.4. Let 𝛾 ∈ [ 0 , ∞ ) , 0 < 𝑎 < 𝑐 < 𝑏 < ∞ , 𝑔 ∈ 𝐶 𝛾 , l n [ 𝑎 , 𝑐 ] and 𝑔 ∈ 𝐶 [ 𝑐 , 𝑏 ] . Then 𝑔 ∈ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] and ‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] ≤ m a x ‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑐 ] , l n 𝑏 𝑎 𝛾 ‖ 𝑔 ‖ 𝐶 [ 𝑐 , 𝑏 ] . ( 3 . 1 2 )

Proof. The proof is similar to the proof of Lemma 3.1. Since 𝑔 ∈ 𝐶 𝛾 , l n [ 𝑎 , 𝑐 ] and 𝑔 ∈ 𝐶 [ 𝑐 , 𝑏 ] , we have 𝑔 ∈ 𝐶 ( 𝑎 , 𝑏 ] , that is, 𝑔 ∈ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] .
Next we give the estimate. Because 𝑔 ∈ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] , there exists at least 𝑥 ∗ ∈ [ 𝑎 , 𝑏 ] such that
‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] = | | | | l n 𝑥 ∗ 𝑎 𝛾 𝑔 𝑥 ∗ | | | | . ( 3 . 1 3 )
If 𝑥 ∗ ∈ [ 𝑎 , 𝑐 ] , then
‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] ≤ ‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑐 ] . ( 3 . 1 4 )
If 𝑥 ∗ ∈ [ 𝑐 , 𝑏 ] , then
‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] ≤ l n 𝑏 𝑎 𝛾 ‖ 𝑔 ‖ 𝐶 [ 𝑐 , 𝑏 ] . ( 3 . 1 5 )
Hence we have
‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑏 ] ≤ m a x ‖ 𝑔 ‖ 𝐶 𝛾 , l n [ 𝑎 , 𝑐 ] , l n 𝑏 𝑎 𝛾 ‖ 𝑔 ‖ 𝐶 [ 𝑐 , 𝑏 ] . ( 3 . 1 6 )
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 , 𝑥 1 ] ⊂ ( 𝑎 , 𝑏 ] ( 𝑎 < 𝑥 1 < 𝑏 ) . Choose 𝑥 1 such that 𝐴 𝑥 1 − 𝑎 𝛼 Γ ( 𝛼 − 𝑛 + 1 ) Γ ( 2 𝛼 − 𝑛 + 1 ) < 1 , ( 3 . 1 7 ) where 𝐴 > 0 is the Lipschitzian coefficient. Next we prove the existence of a unique solution 𝑦 ( 𝑥 ) ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] to (2.12) on the interval ( 𝑎 , 𝑥 1 ] . For this, we use the Banach fixed point theorem for the space 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] , which is a complete metric space with the distance given by 𝑑 𝑦 1 , 𝑦 2 = ‖ ‖ 𝑦 1 − 𝑦 2 ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] = m a x 𝑥 ∈ 𝑎 , 𝑥 1 | | ( 𝑥 − 𝑎 ) 𝑛 − 𝛼 𝑦 1 ( 𝑥 ) − 𝑦 2 ( 𝑥 ) | | . ( 3 . 1 8 )
We rewrite the integral (2.12) in the form
𝑦 ( 𝑥 ) = ( 𝑇 𝑦 ) ( 𝑥 ) , ( 3 . 1 9 ) where ( 𝑇 𝑦 ) ( 𝑥 ) = 𝑦 0 ( 𝑥 ) + 1 Γ ( 𝛼 ) 𝑥 𝑎 𝑓 [ 𝑡 , 𝑦 ( 𝑡 ) ] 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 . ( 3 . 2 0 )
To apply Theorem 2.9, we have to prove the following: ( 1 ) if 𝑦 ( 𝑥 ) ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] , then ( 𝑇 𝑦 ) ( 𝑥 ) ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] ; ( 2 ) for any 𝑦 1 , 𝑦 2 ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] the following estimate holds:
‖ ‖ 𝑇 𝑦 1 − 𝑇 𝑦 2 ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] ≤ 𝜔 ‖ ‖ 𝑦 1 − 𝑦 2 ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] , 𝜔 = 𝐴 𝑥 1 − 𝑎 𝛼 Γ ( 𝛼 − 𝑛 + 1 ) Γ ( 2 𝛼 − 𝑛 + 1 ) . ( 3 . 2 1 )
It follows from (2.13) that 𝑦 0 ( 𝑥 ) ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] . Since 𝑓 [ 𝑥 , 𝑦 ( 𝑥 ) ] ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] for any 𝑦 ( 𝑥 ) ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] , then, by Lemma 3.2 [28] (with 𝛾 = 𝑛 − 𝛼 , 𝑏 = 𝑥 1 , and 𝑔 ( 𝑥 ) = 𝑓 [ 𝑥 , 𝑦 ( 𝑥 ) ] ) , the integral in the right-hand side of (3.19) also belongs to 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] , and hence ( 𝑇 𝑦 ) ( 𝑥 ) ∈ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] . 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 = 𝑥 1 , and 𝑔 ( 𝑥 ) = 𝑓 [ 𝑥 , 𝑦 1 ( 𝑥 ) ] − 𝑓 [ 𝑥 , 𝑦 2 ( 𝑥 ) ] ) , we have
‖ ‖ 𝑇 𝑦 1 − 𝑇 𝑦 2 ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] ≤ ‖ ‖ 𝐼 𝛼 𝑎 + | | 𝑓 𝑡 , 𝑦 1 ( 𝑡 ) − 𝑓 𝑡 , 𝑦 2 ( 𝑡 ) | | ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] ≤ 𝐴 ‖ ‖ 𝐼 𝛼 𝑎 + | | 𝑦 1 ( 𝑡 ) − 𝑦 2 ( 𝑡 ) | | ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] ≤ 𝐴 𝑥 1 − 𝑎 𝛼 Γ ( 𝛼 − 𝑛 + 1 ) Γ ( 2 𝛼 − 𝑛 + 1 ) ‖ ‖ 𝑦 1 − 𝑦 2 ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] , ( 3 . 2 2 ) 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 , 𝑥 1 ] to (2.12) on the interval [ 𝑎 , 𝑥 1 ] .
By Theorem 2.9, this solution 𝑦 ∗ ( 𝑥 ) is a limit of a convergent sequence ( 𝑇 𝑚 𝑦 ∗ 0 ) ( 𝑥 ) :
l i m 𝑚 → ∞ ‖ ‖ 𝑇 𝑚 𝑦 ∗ 0 − 𝑦 ∗ ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] = 0 , ( 3 . 2 3 ) where 𝑦 ∗ 0 ( 𝑥 ) is any function in 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] . If there is at least one 𝑏 𝑘 ≠ 0 in the initial condition (2.11), then we can take 𝑦 ∗ 0 ( 𝑥 ) = 𝑦 0 ( 𝑥 ) with 𝑦 0 ( 𝑥 ) defined by (2.13). The last relation can be rewritten into the form l i m 𝑚 → ∞ ‖ ‖ 𝑦 𝑚 − 𝑦 ∗ ‖ ‖ 𝐶 𝑛 − 𝛼 [ 𝑎 , 𝑥 1 ] = 0 , ( 3 . 2 4 ) where 𝑦 𝑚 ( 𝑥 ) = 𝑇 𝑚 𝑦 ∗ 0 ( 𝑥 ) = 𝑦 0 ( 𝑥 ) + 1 Γ ( 𝛼 ) 𝑥 𝑎 𝑓 𝑡 , 𝑇 𝑚 − 1 𝑦 ∗ 0 ( 𝑡 ) 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 , ( 𝑚 ∈ 𝑁 ) . ( 3 . 2 5 )
Next we consider the interval [ 𝑥 1 , 𝑏 ] . Rewrite (2.12) in the form
𝑦 ( 𝑥 ) = 𝑦 0 1 ( 𝑥 ) + 1 Γ ( 𝛼 ) 𝑥 𝑥 1 𝑓 [ 𝑡 , 𝑦 ( 𝑡 ) ] 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 , ( 3 . 2 6 ) where 𝑦 0 1 ( 𝑥 ) is defined by 𝑦 0 1 ( 𝑥 ) = 𝑛 𝑗 = 1 𝑏 𝑗 Γ ( 𝛼 − 𝑗 + 1 ) ( 𝑥 − 𝑎 ) 𝛼 − 𝑗 + 1 Γ ( 𝛼 ) 𝑥 1 𝑎 𝑓 [ 𝑡 , 𝑦 ( 𝑡 ) ] 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 . ( 3 . 2 7 ) We obtain 𝑦 0 1 ( 𝑥 ) ∈ 𝐶 [ 𝑥 1 , 𝑏 ] . Next we prove the existence of a unique solution 𝑦 ( 𝑥 ) ∈ 𝐶 [ 𝑥 1 , 𝑏 ] to (2.12) on the interval [ 𝑥 1 , 𝑏 ] . For this, we also use Banach fixed point theorem for the space 𝐶 [ 𝑥 1 , 𝑥 2 ] , where 𝑥 2 satisfies 𝐴 𝑥 2 − 𝑥 1 𝛼 𝛼 Γ ( 𝛼 ) < 1 . ( 3 . 2 8 ) 𝐶 [ 𝑥 1 , 𝑥 2 ] is a complete metric space with the distance given by 𝑑 𝑦 1 − 𝑦 2 = ‖ ‖ 𝑦 1 − 𝑦 2 ‖ ‖ 𝐶 [ 𝑥 1 , 𝑥 2 ] = m a x 𝑥 ∈ 𝑥 1 , 𝑥 2 | | 𝑦 1 ( 𝑥 ) − 𝑦 2 ( 𝑥 ) | | . ( 3 . 2 9 )
We rewrite the integral equation (3.26) into the form
𝑦 ( 𝑥 ) = ( 𝑇 𝑦 ) ( 𝑥 ) , ( 3 . 3 0 ) where ( 𝑇 𝑦 ) ( 𝑥 ) = 𝑦 0 1 ( 𝑥 ) + 1 Γ ( 𝛼 ) 𝑥 𝑥 1 𝑓 [ 𝑡 , 𝑦 ( 𝑡 ) ] 𝑑 𝑡 ( 𝑥 − 𝑡 ) 1 − 𝛼 . ( 3 . 3 1 )
To apply Theorem 2.9, we have to prove the following: ( 1 ) if 𝑦 ( 𝑥 ) ∈ 𝐶 [ 𝑥 1 , 𝑥