International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 2 (2017), 216-230 http://www.etamaths.com BASIC THEORY FOR DIFFERENTIAL EQUATIONS WITH UNIFIED REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES BAŞAK KARPUZ1,∗, UMUT M. ÖZKAN2, TUĞBA YALÇIN2 AND MUSTAFA K. YILDIZ2 Abstract. In this paper, we extend the definition of the fractional integral and derivative intro- duced in [Appl. Math. Comput. 218 (2011)] by Katugampola, which exhibits nice properties only for numbers whose real parts lie in [0, 1]. We prove some interesting properties of the fractional inte- grals and derivatives. Based on these properties, the following concepts for the new type fractional differential equations are explored: Existence and uniqueness of solutions; Solutions of autonomous fractional differential equations; Dependence on the initial conditions; Green’s function; Variation of parameters formula. 1. Introduction The history of fractional calculus was originated in the seventeenth century, when the half-order derivative was discussed by Leibnitz in 1695. Since then, this theory became one of the interesting subjects to mathematicians as well as biologists, chemists, economists, engineers and physicists. There are several books written on this subject, for instance [3,9–11,13]. [13] is one of the most comprehensive main tools of the subject, where several types of derivatives (such as Riemann-Liouville, Hadamard, Grünwald-Letnikov, Riesz and Caputo) were introduced. Derivatives of fractional order are defined by integrals with a fractional order kernel. Reimann- Liouville ([3,9–11,13]) and Hadamard ([1,2,7,8,12]) type fractional integrals are two of the most studied forms of fractional integrals. The Riemann–Liouville fractional integral of order α for a function f is defined by ∫ t s [t−η]α−1 Γ(α) f(η)dη for t > s and α > 0, (1.1) which is motivated by the Cauchy integral formula∫ t s ∫ η1 s · · · ∫ ηn−1 s f(ηn)dηn · · ·dη2dη1 = ∫ t s [t−η]n−1 Γ(n) f(η)dη for t > s and n ∈ N. Another one is the Hadamard fractional integral introduced in [4], which reads as ∫ t s 1 η1 ∫ η1 s 1 η2 · · · ∫ ηn−1 s f(ηn) ηn dηn · · ·dη2dη1 = 1 Γ(n) ∫ t s [ ln ( t η )]n−1 f(η) η dη for t > s and n ∈ N, from which the following fractional integral of f is deduced by∫ t s 1 Γ(α) [ ln ( t η )]α−1 f(η) η dη for t > s and α > 0. (1.2) In [5], Katugampola unified the Reimann-Liouville fractional integral and the Hadamard fractional integral by ∫ t s [tρ −ηρ]α−1ηρ−1 ρα−1Γ(α) f(η)dη for t > s and α > 0, (1.3) Received 25th October, 2016; accepted 10th January, 2017; published 1st March, 2017. 2010 Mathematics Subject Classification. Primary 26A33; Secondary 33E12, 34A08, 34K37. Key words and phrases. Reimann-Liouville fractional calculus; Hadamard fractional calculus; existence and unique- ness; dependence on initial conditions; Green’s function; variation of parameters formula. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 216 REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES 217 where ρ > 0, which we will call as the Reimann-Liouville-Hadamard (RLH) fractional integral. As limρ→0+ tρ−sρ ρ = ln( t s ), we see that (1.3) with ρ = 1 and ρ → 0+ contains (1.1) and (1.2), respectively. This fractional integral has also been extended to fractional derivative in [6], which holds “nicely” for α with Re(α) ∈ (0, 1) (see [6, § 3]). Motivated by the definition of fractional order derivatives given in [6], we will give a new extended the definition for arbitrary positive numbers. Based on this fractional derivative, we will study important properties of the fractional differential equations (FDE) defined with this new type of derivatives. The paper covers the following concepts: • Existence and uniqueness of solutions to FDEs • Solutions of autonomous FDEs • Dependence of solutions on the initial conditions • Green’s function for RLH FDEs • Variation of parameters formula The paper is organized as follows. In § 2, we give the basic definitions and related auxiliary results. § 3 includes the fundamental properties of the fractional integral/derivative, which will be required in the latter sections. In § 4, we will provide existence and uniqueness for solutions of differential equations of the new type of fractional derivative. By using direct substitution technique and the Picard iterates, we will consider autonomous type fractional differential equations in § 5. In § 6, we will provide a result on dependence of the initial conditions. § 7 is dedicated to the concept of Green’s function and the variation of parameters formula for the new type fractional differential equations. Finally, in § 8, we present some directions for future research and make our final discussion to conclude the paper. 2. Definitions and Auxiliary Results Let us first introduce the kernel function Kαρ : R×R → C, where α ∈ C\Z − 0 and ρ ∈ R +, defined by Kαρ (t,s) := [tρ −sρ]α−1sρ−1 ρα−1Γ(α) for s,t ∈ R. (2.1) We assume for convenience Kαρ (t,s) ≡ 0 for α ∈ Z − 0 . Also, for n ∈ N, k ∈ {1, 2, · · · ,n} and ρ ∈ R +, we let An,k(ρ) :=   [1 − (n− 1)ρ]An−1,1(ρ), k = 1 An−1,k−1(ρ) + [k − (n− 1)ρ]An−1,k(ρ), k = 2, 3, · · · ,n− 1 1, k = n. (2.2) Definition 2.1 (Cf. [5]). Let α ∈ R, ρ ∈ R+ and f : (0,∞) → R. We define the α-order fractional integration of f by [ Jαρ f ] (t) :=   ∫ t 0 Kαρ (t,η)f(η)dη, α ∈ R\Z − 0 f(t), α = 0 (−α)∑ i=1 A(−α),i(ρ) t(−α)ρ−i ( d dt )i f(t), α ∈ Z− (2.3) for t > 0. Remark 2.1. One can show that An,k(1) = δn,k, where δ is Kronecker’s delta. Hence, Jαρ f = f(−α) for α ∈ Z−. Example 2.1. For α ∈ R+0 , ν ∈ (−1,∞) and ρ ∈ R +, we have [ Jαρ ∗ ρν ] (t) = Γ(ν + 1) ραΓ(ν + α + 1) tρ(ν+α) for t > 0. 218 KARPUZ, ÖZKAN, YALÇIN AND YILDIZ The proof is trivial for α = 0. We let α ∈ R+ and compute for t > 0 that [ Jα∗ρν ] (t) = ∫ t 0 Kαρ (t,η)η ρνdη = ∫ t 0 [tρ −ηρ]α−1ηρ−1 ρα−1Γ(α) ηρνdη = tρ(α+ν) ραΓ(α) ∫ 1 0 [1 − ζ]α−1ζνdζ = tρ(α+ν) ραΓ(α) B(α,ν + 1) = Γ(ν + 1) ραΓ(α + ν + 1) tρ(α+ν). We proceed by recalling some important properties of the kernel K. Lemma 2.1. The following basic properties of the kernel K are true. (i) ∫ t s Kαρ (t,η)K β ρ (η,s)dη = K α+β ρ (t,s) for t ≥ s ≥ 0 and α,β ∈ R+. (ii) tρ−1Kαρ (t,s) = (−1)αsρ−1Kαρ (s,t) for s,t ∈ R and α ∈ C. (iii) ∂ ∂t Kα+1ρ (t,s) = tρ−1Kαρ (t,s) for s,t ∈ R and α ∈ C\Z − 0 . (iv) ∂ ∂s Kα+1ρ (t,s) sρ−1 = −Kαρ (t,s) for s ∈ R\{0}, t ∈ R and α ∈ C\Z − 0 . Proof. (i) Then, we compute for t ≥ s ≥ 0 that ∫ t s Kαρ (t,η)K β ρ (η,s)dη = ∫ t s [tρ −ηρ]α−1ηρ−1 ρα−1Γ(α) [ηρ −sρ]β−1sρ−1 ρβ−1Γ(β) dη = 1 ρα+β−2Γ(α)Γ(β) ∫ t s [tρ −ηρ]α−1ηρ−1[ηρ −sρ]β−1sρ−1dη = [tρ −sρ]α+β−1sρ−1 ρα+β−1Γ(α)Γ(β) ∫ 1 0 [1 − ζ]α−1ζβ−1dζ = [tρ −sρ]α+β−1sρ−1 ρα+β−1Γ(α)Γ(β) B(α,β) = [tρ −sρ]α+β−1sρ−1 ρα+β−1Γ(α + β) = Kα+βρ (t,s). (ii) The proof is trivial and thus we omit it here. (iii) For t ≥ s ≥ 0, we have ∂ ∂t Kα+1ρ (t,s) = ∂ ∂t [tρ −sρ]αsρ−1 ραΓ(α + 1) = αρtρ−1[tρ −sρ]α−1sρ−1 ραΓ(α + 1) = tρ−1[tρ −sρ]α−1sρ−1 ρα−1Γ(α) = tρ−1Kαρ (t,s). (iv) The proof can be given similar to that of (iii). � Lemma 2.2. Note that for α ∈ R, we have [ Jαρ f ] (t) = 1 tρ−1 d dt [Jα+1ρ f](t) for t > 0. (2.4) Proof. We proceed with the following three distinct cases. REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES 219 • Let α ∈ R\Z−. Then, we have for t > 0 that d dt [ Jα+1ρ f ] (t) = d dt ∫ t 0 Kα+1ρ (t,η)f(η)dη = d dt ∫ t 0 [tρ −ηρ]αηρ−1 ραΓ(α + 1) f(η)dη = ∫ t 0 d dt [tρ −ηρ]αηρ−1 ραΓ(α + 1) f(η)dη + [tρ − tρ]αηρ−1 ραΓ(α + 1) f(t) =αρtρ−1 ∫ t 0 [tρ −ηρ]α−1ηρ−1 ραΓ(α + 1) f(η)dη =tρ−1 ∫ t 0 [tρ −ηρ]α−1ηρ−1 ρα−1Γ(α) f(η)dη =tρ−1 [ Jαρ f ] (t). • Let α = −1. Then, d dt [J 0ρ f](t) = d dt f(t) = tρ−1 1 tρ−1 d dt f(t) = tρ−1 [ J−1ρ f ] (t) for t > 0. • Let α ∈{·· · ,−3,−2}. Then, putting n := −α for simplicity, we compute for t > 0 that d dt [ Jα+1ρ f ] (t) = d dt n−1∑ i=1 An−1,i t(n−1)ρ−i ( d dt )i f(t) = n−1∑ i=1 d dt An−1,i t(n−1)ρ−i ( d dt )i f(t) = n−1∑ i=1 [ An−1,i t(n−1)ρ−i ( d dt )i+1 − [(n− 1)ρ− i] An−1,i t(n−1)ρ−i+1 ( d dt )i] f(t) =tρ−1 n−1∑ i=1 [ An−1,i tnρ−(i+1) ( d dt )i+1 − [(n− 1)ρ− i] An−1,i tnρ−i ( d dt )i] f(t) =tρ−1 [ n−1∑ i=1 An−1,i tnρ−(i+1) ( d dt )i+1 f(t) − n−1∑ i=1 [(n− 1)ρ− i] An−1,i tnρ−i ( d dt )i f(t) ] =tρ−1 [ n∑ i=2 An−1,i−1 tnρ−i ( d dt )i f(t) − n−1∑ i=1 [(n− 1)ρ− i] An−1,i tnρ−i ( d dt )i f(t) ] =tρ−1 [ An−1,n−1 tn(ρ−1) ( d dt )n f(t) + n−1∑ i=2 An−1,i−1 tnρ−i ( d dt )i f(t) − n−1∑ i=2 ( (n− 1)ρ− i )An−1,i tnρ−i ( d dt )i f(t) − [(n− 1)ρ− 1] An−1,1 tnρ−1 d dt f(t) ] =tρ−1 [ An−1,n−1 tn(ρ−1) ( d dt )n f(t) + n−1∑ i=2 [ An−1,i−1 − [(n− 1)ρ− i]An−1,i ] 1 tnρ−i ( d dt )i f(t) − [(n− 1)ρ− 1] An−1,1 tnρ−1 d dt f(t) ] =tρ−1 [ An,n tn(ρ−1) ( d dt )n f(t) + n−1∑ i=2 An,i tnρ−i ( d dt )i f(t) + An,1 tnρ−1 d dt f(t) ] =tρ−1 n∑ i=1 An,i tnρ−i ( d dt )i f(t) = tρ−1 [ Jαρ f ] (t). The proof is completed by considering the three cases above. � Motivated by Lemma 2.2, we suggest the following definition for the fractional derivative of a function. 220 KARPUZ, ÖZKAN, YALÇIN AND YILDIZ Definition 2.2. Let α ∈ R, ρ ∈ R+ and f : (0,∞) → R. We define the α-order fractional derivative of f iteratedly by [ Dαρ f ] (t) :=   [ J−αρ f ] (t), α ∈ R−0 1 tρ−1 d dt [ Dα−1ρ f ] (t), α ∈ R+. Example 2.2. For α,ν ∈ R+0 and ρ ∈ R +, we have[ Dαρ∗ ρν ] (t) = ραΓ(ν + 1) Γ(ν −α + 1) tρ(ν−α) for t > 0. (2.5) We will prove this by applying induction on n ∈ Z+0 for α ∈ [n,n + 1). First, let α ∈ [0, 1), then we have [ Dαρ∗ ρν ] (t) = 1 tρ−1 d dt [ Dα−1ρ ∗ ρν ] (t) = 1 tρ−1 d dt [ J 1−αρ ∗ ρν ] (t) = 1 tρ−1 d dt Γ(ν + 1) ρ1−αΓ ( ν + (1 −α) + 1 )tρ(ν+(1−α)) = ραΓ(ν + 1) Γ(ν −α + 1) tρ(ν−α) for t > 0, where we have applied Example 2.1 in the second line above. This proves validity of (2.5) for all α ∈ [0, 1). Let n ∈ Z+0 , and assume now for all α ∈ [n,n + 1) that (2.5) is true. By Definition 2.2, we have for any α ∈ [n + 1,n + 2) that[ Dαρ∗ ρν ] (t) = 1 tρ−1 d dt [ Dα−1ρ ∗ ρν ] (t) = 1 tρ−1 d dt ρα−1Γ(ν + 1) Γ ( ν − (α− 1) + 1 )tρ(ν−(α−1)) = ραΓ(ν + 1) Γ(ν −α + 1) tρ(ν−α) for t > 0, which completes the proof. In the following lemma, we provide a direct form for the definition of the fractional derivative in terms of the coefficients defined in (2.2). Lemma 2.3. For α,ρ ∈ R+, we have [ Dαρ f ] (t) = dαe∑ i=1 Adαe,i(ρ) tdαeρ−i ( d dt )i[ J dαe−αρ f ] (t) for t > 0. Proof. We compute that[ Dαρ f ] (t) = 1 tρ−1 d dt [ Dα−1ρ f ] (t) = 1 tρ−1 d dt [ 1 tρ−1 d dt [ Dα−2ρ f ] (t) ] = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ Dα−dαeρ f ] (t) · · · ]] = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ J dαe−αρ f ] (t) · · · ]] , where we have for a total of dαe usual derivatives above. Let us denote g := J dαe−αρ f and use (2.4) repeatedly inside to outside, then [ Dαρ f ] (t) = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ J 0ρ g ] (t) · · · ]] = · · · = 1 tρ−1 d dt [ J−dαe+1ρ g ] (t) = [ J−dαeρ g ] (t), which completes the proof by using (2.3). � REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES 221 3. Properties of the Operators J and D The main result of this section is the following theorem. Theorem 3.1. The following properties hold. (i) Dαρ = J−αρ for α ∈ R. (ii) Jαρ Jβρ = Jα+βρ for α,β ∈ R + 0 . (iii) DαρDβρ = Dα+βρ for α ∈ Z + 0 and β ∈ R + 0 or for α,β ∈ R +\Z+ with (α + β) 6∈ Z+. (iv) DαρJαρ = I for α ∈ R, where I is the identity operator. (v) [ Jαρ Dαρ f ] (t) = f(t) − dαe∑ i=1 tρ(α−i) ρα−iΓ(α− i + 1) [ Dα−iρ f ] (0+) for α ∈ R+. Proof. (i) For α ∈ R+0 , the proof is similar to that of Lemma 2.3, and for α ∈ R −, the proof follows from Definition 2.2. (ii) The proof is trivial for α = 0 or β = 0. Hence, we consider below the case where α,β ∈ R+. Then, [ Jαρ J β ρ f ] (t) = ∫ t 0 Kαρ (t,η) ∫ η 0 Kβρ (η,ζ)f(ζ)dζdη = ∫ t 0 ∫ η 0 Kαρ (t,η)K β ρ (η,ζ)f(ζ)dζdη = ∫ t 0 ∫ t ζ Kαρ (t,η)K β ρ (η,ζ)f(ζ)dηdζ = ∫ t 0 [∫ t ζ Kαρ (t,η)K β ρ (η,ζ)dη ] f(ζ)dζ = ∫ t 0 Kα+βρ (t,ζ)f(ζ)dζ = [ Jα+βρ f ] (t), where we have applied Lemma 2.1 (i) for the last line. (iii) The proof is trivial for α = 0 or β = 0. Below, we consider the case where α 6= 0 and β 6= 0. (a) For α ∈ Z+ and β ∈ R+, then [ DαρD β ρf ] (t) = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ Dβρf ] (t) · · · ]] = · · · = 1 tρ−1 d dt [ Dβ+(α−1)ρ f ] (t) = [ Dβ+αρ f ] (t). (b) For α,β ∈ R+\Z+ with (α + β) 6∈ Z+, then DαρDβρ = Dα+βρ as in (ii). (iv) This follows from (i) by using the steps in the proof of (ii) and (iii). (v) Performing integration by parts, for α ∈ R+, we obtain [ Jα+1ρ D α ρ f ] (t) = [ Jα+1ρ 1 ∗ρ−1 [ Dα−1ρ f ]′] (t) = ∫ t 0 Kα+1ρ (t,η) ηρ−1 [ Dα−1ρ f ]′ (η)dη = Kα+1ρ (t,η) ηρ−1 [ Dα−1ρ f ] (η) ∣∣∣∣η=t η=0+ − ∫ t 0 ∂ ∂η ( Kα+1ρ (t,η) ηρ−1 )[ Dα−1ρ f ] (η)dη = − [tρ −ηρ]α ραΓ(α + 1) [ Dα−1ρ f ] (η) ∣∣∣∣η=t η=0+ + ∫ t 0 Kαρ (t,η) [ Dα−1ρ f ] (η)dη = [ Jαρ D α−1 ρ f ] (t) − tρα ραΓ(α + 1) [ Dα−1ρ f ] (0+) where ∗′ in the first line stands for the usual derivative. Using (ii), we get[ J 1ρJ α ρ D α ρ f ] (t) = [ J 1ρJ α−1 ρ D α−1 ρ f ] (t) − tρα ραΓ(α + 1) [ Dα−1ρ f ] (0+). 222 KARPUZ, ÖZKAN, YALÇIN AND YILDIZ An application of D1ρ on both sides yields by using (iv) that[ Jαρ D α ρ f ] (t) = [ Jα−1ρ D α−1 ρ f ] (t) − tρ(α−1) ρα−1Γ(α) [ Dα−1ρ f ] (0+). Repeating this procedure for a total of dαe times, we get [ Jαρ D α ρ f ] (t) = [ Jα−dαeρ D α−dαe ρ f ] (t) − dαe∑ i=1 tρ(α−i) ρα−iΓ(α− i + 1) [ Dα−iρ f ] (0+) (3.1) for all t > 0. By Definition 2.1, Definition 2.2 and (ii), we have Jα−dαeρ D α−dαe ρ = { J 0ρD0ρ = I, α ∈ N Jα−dαeρ J dαe−α ρ = J 0ρ = I, α ∈ R+\N, which completes the proof by using this in (3.1). Thus, we have justified the validity of each of the properties above, and completed the proof. � 4. Existence and Uniqueness for RLH FDEs Let us consider the initial-value problem{[ Dαρ y ] (t) = f ( t,y(t) ) for t > 0[ Dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe, (4.1) where α ∈ R+ and y0,y1, · · · ,ydαe−1 ∈ R. Suppose that f is defined in a domain Ω of a plane (t,y), and define a region R(h,K) ⊂ Ω as a set of points (t,y) ∈ Ω, which satisfy the inequality∣∣∣∣∣y(t) − dαe∑ i=1 tρ(α−i) ρα−iΓ(α− i + 1) ∣∣∣∣∣ ≤ K for all t ∈ (0,h), where h and K are constants. Theorem 4.1. Let f : Ω → R satisfy the Lipschitz condition with respect to its second component, i.e., |f(t,y1) −f(t,y2)| ≤ L|y1 −y2| for all (t,y1), (t,y2) ∈ Ω, where L ∈ R+, and f be bounded on Ω, i.e., |f(t,y)| ≤ M for all (t,y) ∈ Ω, where M ∈ R+. Further, assume that there exist h,K ∈ R+ such that Mhρα ραΓ(α + 1) ≤ K. Then, there exists a unique and continuous solution of the problem (4.1) in the region R(h,K) ⊂ Ω. Proof. The method proof based on the ideas in [11, Theorem 3.4]. First, consider Theorem 3.1 (v) and reduce the problem (4.1) to an equivalent fractional integral equation y(t) = dαe∑ i=1 ydαe−i ρα−iΓ(α− i + 1) tρ(α−i) + ∫ t 0 Kαρ (t,η)f ( η,y(η) ) dη for t ∈ (0,h]. (4.2) If y satisfies (4.1), then it also satisfies the equation (4.2). On the other hand, if y is a solution of (4.2), then it is satisfies (4.1) initial-value problem. Therefore, the equation (4.2) is equivalent to the initial value problem (4.1). Now, let us define the sequence of functions {ym}m∈N0 by ym(t) =   dαe∑ i=1 ydαe−i ρα−iΓ(α− i + 1) tρ(α−i), m = 0 y0(t) + [ Jαρ f ( ∗,ym−1(∗) )] (t), m ∈ N (4.3) REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES 223 for t ∈ (0,h]. We will show that limm→∞ym exists and gives the required solution y of the integral equation (4.2). First, it can be shown by induction that ym(t) ∈ R(h,K) for all t ∈ (0,h] and m ∈ N0. Indeed, for all t ∈ (0,h] and all m ∈ N0, we obtain |ym(t) −y0(t)| = ∣∣∣∣ ∫ t 0 Kαρ (t,η)f ( η,ym−1(η) ) dη ∣∣∣∣ ≤ ∫ t 0 Kαρ (t,η) ∣∣f(η,ym−1(η))∣∣dη ≤M ∫ t 0 Kαρ (t,η)dη = Mtρα ραΓ(α + 1) ≤ Mhρα ραΓ(α + 1) ≤ K and thus |y1(t) −y0(t)| ≤ Mhρα ραΓ(α + 1) ≤ K for all t ∈ (0,h]. (4.4) Let us show by induction that |ym(t) −ym−1(t)| ≤ MLm−1tmρα ρmαΓ(mα + 1) for all t ∈ (0,h] and all m ∈ N. (4.5) It follows from (4.4) that (4.5) holds for m = 1. Suppose for some m ∈ N that |ym(t) −ym−1(t)| ≤ MLm−1tmρα ρmαΓ(mα + 1) for all t ∈ (0,h]. (4.6) Then, using (4.3) and (4.6), we have |ym+1(t) −ym(t)| = ∣∣∣∣ ∫ t 0 Kαρ (t,η) [ f ( η,ym(η) ) −f ( η,ym−1(η) )] dη ∣∣∣∣ ≤ ∫ t 0 Kαρ (t,η) ∣∣f(η,ym(η))−f(η,ym−1(η))∣∣dη ≤L ∫ t 0 Kαρ (t,η)|ym(η) −ym−1(η)|dη ≤ MLm ρmαΓ(mα + 1) ∫ t 0 Kαρ (t,η)η mραdη = MLm ρ(m+1)αt (m+1)ρα Γ(α)Γ(mα + 1) ∫ 1 0 [1 − ζ]α−1ζmαdζ = MLmt(m+1)ρα ρ(m+1)αΓ(α)Γ(mα + 1) ∫ 1 0 [1 − ζ]α−1ζmαdζ = MLmt(m+1)ρα ρ(m+1)αΓ(α)Γ(mα + 1) B(α,mα + 1) = MLmt(m+1)ρα ρ(m+1)αΓ ( (m + 1)α + 1 ) for all t ∈ (0,h]. This means that (4.5) is true. Let us consider the limiting function y(t) := lim m→∞ ym(t) = y0(t) + ∞∑ j=1 [yj(t) −yj−1(t)] for t ∈ (0,h]. (4.7) According to the estimate (4.5), for t ∈ (0,h], the absolute value of its terms is less than the corre- sponding terms of the convergent numeric series ∞∑ j=1 |yj(t) −yj−1(t)| ≤ ∞∑ j=1 MLj−1hjρα ρjαΓ(jα + 1) = M L ∞∑ j=1 Ljhjρα ρjαΓ(jα + 1) = M L [ Eα,1 ( Lhρα ρα ) − 1 ] , 224 KARPUZ, ÖZKAN, YALÇIN AND YILDIZ where E is the two-parameter Mittag-Leffler function defined by Eα,β(z) := ∞∑ j=0 zj Γ(αj + β) for z ∈ C and α,β ∈ C, (4.8) which converges for all values of z (i.e., it is an entire function). This means that the series (4.7) converges uniformly. Letting m →∞ in (4.3) and using (4.7), we get y(t) = y0(t) + ∫ t 0 Kαρ (t,η)f ( η,y(η) ) dη for all t ∈ (0,h]. Therefore, y defined by (4.7) is a solution of (4.2), and thus (4.1). What follows next is to prove the uniqueness of the solution. Let us suppose that z is another solution of the equation (4.2), which is continuous in the interval (0,h]. Then w(t) := y(t) − z(t) for t ∈ (0,h], then satisfies the equation w(t) = ∫ t 0 Kαρ (t,η) [ f ( η,y(η) ) −f ( η,z(η) )] dη (4.9) from which it follows that w(0+) = 0. Therefore, w extends continuously to [0,h]. Then, |w(t)| ≤ C for all t ∈ (0,h], where C ∈ R+, and we obtain from (4.9) that |w(t)| ≤ CLtρα ραΓ(α + 1) for all t ∈ (0,h]. Repeating this procedure for a total of m ∈ N times, we obtain |w(t)| ≤ CLmtmρα ρmαΓ(mα + 1) for all t ∈ (0,h]. In the right-hand side, we recognize the general term of the series for the Mittag-Leffler function Eα,1( Ltρα ρα ), and therefore lim m→∞ Lmtmρα ρmαΓ(mα + 1) = 0 for all t ∈ (0,h]. Then, we have w(t) ≡ 0 for all t ∈ (0,h], and thus y(t) ≡ z(t) for all t ∈ (0,h]. This ends the proof. � 5. The Autonomous Equation of RL Type Let us consider initial-value problem{[ Dαρ y ] (t) = λy(t) for t > 0[ Dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe, (5.1) where λ ∈ R. In this case, when compared to (4.1), we have f(t,y) = λy. Now, we will introduce two techniques for obtaining the unique solution of (5.1). 5.1. Direct Substitution. Let α ∈ R and β ∈ R and define yα,β(t) := tρβ ρβ Eα,β+1 ( λtρα ρα ) for t > 0, where E is the two-parameter Mittag-Leffler function defined in (4.8). REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES 225 Then, Dαρ y = λy. Indeed, we have [ Dαρ yα,β ] (t) = [ Dαρ ∗ρβ ρβ Eα,β+1 ( λ∗ρα ρα )] (t) = [ Dαρ ∞∑ j=0 λj∗ρ(αj+β) ραj+βΓ(αj + β + 1) ] (t) = ∞∑ j=0 λj [ Dαρ∗ρ(αj+β) ] (t) ραj+βΓ(αj + β + 1) = ∞∑ j=0 λjtρ(α(j−1)+β) ρα(j−1)+βΓ ( α(j − 1) + β + 1 ) = tρ(β−α) ρβ−αΓ(β −α + 1) + ∞∑ j=1 λjtρ(α(j−1)+β) ρα(j−1)+βΓ ( α(j − 1) + β + 1 ) = tρ(β−α) ρβ−αΓ(β −α + 1) + λ tρβ ρβ ∞∑ j=0 λjtραj ραjΓ(αj + β + 1) = tρ(β−α) ρβ−αΓ(β −α + 1) + λyα,β(t) for all t > 0. Here, we see that yα,β solves Dαρ y = λy provided that (β−α) is a negative integer. That is, yα,α−i, where i = 1, 2, · · · ,dαe, satisfies Dαρ y = λy. Moreover, we compute that [ Dα−kρ yα,α−i ] (t) = [ Dα−kρ ∗ρ(α−i) ρα−i Eα,α−i+1 ( λ∗ρα ρα )] (t) = [ Dα−kρ ∞∑ j=0 λj∗ρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 )](t) = ∞∑ j=0 λj [ Dα−kρ ∗ρ(α(j+1)−i) ] (t) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) = ∞∑ j=0 λjtρ(αj−i+k) ραj−i+kΓ(αj − i + k + 1) for t > 0 and k = 1, 2, · · · ,dαe. Using the properties of the Gamma function and considering the positive powers of t, we find that[ Dα−kρ yα,α−i ] (0+) = δi,k for k = 1, 2, · · · ,dαe, where δ is Kronecker’s delta. Therefore, {yα,α−i} dαe i=1 is the set of normalized fundamental solutions of Dαρ y = λy. Moreover, the following linear combination of functions y(t) := dαe∑ i=1 ydαe−i tρ(α−i) ρα−i Eα,α−i+1 ( λtρα ρα ) for t > 0 forms the solution desired of (5.1). 5.2. The Picard Iterates. In accordance with the proof of Theorem 4.1, let us take ym(t) =   dαe∑ i=1 tρ(α−i) ρα−iΓ(α− i + 1) [Dα−iρ y](0 +), m = 0 y0(t) + λ [ Jαρ ym−1 ] (t), m ∈ N (5.2) for t ∈ (0,h]. We will show by induction that ym(t) = dαe∑ i=1 ydαe−i m∑ j=0 tρ(α(j+1)−i)λj ρ(α(j+1)−i)Γ ( α(j + 1) − i + 1 ) (5.3) 226 KARPUZ, ÖZKAN, YALÇIN AND YILDIZ for all t ∈ (0,h] and m ∈ N0. The claim holds for m = 0 by (5.2). Assume for some m ∈ N0 that ym(t) = dαe∑ i=1 ydαe−i m∑ j=0 tρ(α(j+1)−i)λj ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) for all t ∈ (0,h], which together with Example 2.1 and (5.2) yields ym+1(t) =y0(t) + λ [ Jαρ ym ] (t) = y0(t) + λ dαe∑ i=1 ydαe−i m∑ j=0 λj [ Jαρ ∗ρ(α(j+1)−i) ] (t) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ydαe−i tρ(α−i) ρα−iΓ(α− i + 1) + dαe∑ i=1 ydαe−i m∑ j=0 λj+1tρ((j+2)α−i) ρ(j+2)α−iΓ ( (j + 2)α− i + 1 ) = dαe∑ i=1 ydαe−i m+1∑ j=0 λjtρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) for all t ∈ (0,h]. This justifies (5.3). Letting m →∞ in (5.3), we obtain the solution of the problem (5.2) as y(t) = lim m→∞ ym(t) = dαe∑ i=1 ydαe−i ∞∑ j=0 λjtρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ydαe−i tρ(α−i) ρα−i Eα,α−i+1 ( λtρα ρα ) for t > 0, where E is the two-parameter Mittag-Leffler function defined in (4.8). 6. Dependence on Initial Conditions Let us introduce small changes in the initial conditions of (4.1) and consider{[ Dαρ y ] (t) = f ( t,y(t) ) for t > 0[ Dα−kρ y ] (0+) = ydαe−k + εdαe−k for k = 1, 2, · · · ,dαe, (6.1) where εdαe−k are arbitrary constants. Theorem 6.1. Assume that conditions of Theorem 4.1 hold. Let y and z be respective solutions of the initial value problems (4.1) and (6.1). Then, |y(t) −z(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ ρi Atρi Eα,1−i ( Atρα ρα ) for t ∈ (0,h]. Proof. In conformity with Theorem 4.1, we have y(t) = lim m→∞ ym(t) for t ∈ (0,h], where the sequence of functions {ym}m∈N0 is defined by (4.3) for t ∈ (0,h]. Similarly, z(t) = lim m→∞ zm(t) for t ∈ (0,h], where zm(t) =   dαe∑ i=1 tρ(α−i) ρα−iΓ(α− i + 1) (ydαe−i + εdαe−i), m = 0 z0(t) + [ Jαρ f ( ∗,zm−1(∗) )] (t), m ∈ N (6.2) for t ∈ (0,h]. Let us prove by induction that |ym(t) −zm(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 Ajtρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) (6.3) REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES 227 for all t ∈ (0,h] and all m ∈ N0. From (4.3) and (6.2), it directly follows that |y0(t) −z0(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iΓ(α− i + 1) for all t ∈ (0,h]. Assume now for some m ∈ N that |ym(t) −zm(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 Ajtρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) (6.4) for all t ∈ (0,h]. Then, using (4.3) and (6.2), the Lipschitz condition for the function f together with the inequality (6.4), we obtain |ym+1(t) −zm+1(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iΓ(α− i + 1) + A ∫ t 0 Kαρ (t,η)|ym(η) −zm(η)|dη ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iΓ(α− i + 1) + A ∫ t 0 Kαρ (t,η) dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 Ajηρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 )dη = dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iΓ(α− i + 1) + dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 Aj+1tρ((j+2)α−i) ρ(j+2)α−iΓ ( (j + 2)α− i + 1 ) = dαe∑ i=1 ∣∣εdαe−i∣∣m+1∑ j=0 Ajtρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) for all t ∈ (0,h]. This proves (6.3). Taking the limit of (6.4) as m →∞, we obtain |y(t) −z(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ ∞∑ j=0 Ajtρ(α(j+1)−i) ρα(j+1)−iΓ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ∣∣εdαe−i∣∣ ρi Atρi ∞∑ j=0 Aj+1t(j+1)ρα ρα(j+1)Γ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ∣∣εdαe−i∣∣ ρi Atρi Eα,1−i ( Atρα ρα ) for all t ∈ (0,h], which completes the proof. � 7. The Green’s Function for Linear Equations The objective of this section is to define the Green’s function notion for the initial value problem{[ Dαρ y ] (t) = p(t)y(t) + f(t) for t > 0[ Dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe, (7.1) where p,f : [0,∞) → R are continuous functions, and then present its role in obtaining the solution of the equation. Let ∆ := {(t,s) : t > s ≥ 0} and denote by sDαρ f and sJαρ f the fractional derivative and the fractional integral of a function f centered at s ∈ [0,∞), respectively. Definition 7.1 (Green’s function). Let the continuous function Gρ : ∆ → R satisfy the following properties. (i) [sDαρGρ(∗,s)](t) = p(t)Gρ(t,s) for all (t,s) ∈ ∆. (ii) lim s→t− [sDα−kρ Gρ(∗,s)](t) = δk,1 for t > 0 and k = 1, 2, · · · ,dαe. 228 KARPUZ, ÖZKAN, YALÇIN AND YILDIZ (iii) lim s→t− t→0+ [sDα−kρ Gρ(∗,s)](t) = 0 for k = 1, 2, · · · ,dαe− 1. Then, Gρ is called the Green’s function for the initial value problem (7.1). Theorem 7.1. Let Gρ be the Green’s function for the initial value problem (7.1). Then, y(t) := ∫ t 0 Gρ(t,η)ηρ−1f(η)dη for t > 0 (7.2) is the (unique) solution of the initial value problem{[ Dαρ y ] (t) = p(t)y(t) + f(t) for t > 0[ Dα−kρ y ] (0+) = 0 for k = 1, 2, · · · ,dαe. (7.3) Proof. First, we will show that y defined by (7.2) solves the fractional differential equation in (7.3). To this end, we let β = α−dαe + 1, then β ∈ (0, 1]. From (7.2), we have for t > 0 that[ Dβρy ] (t) = 1 tρ−1 d dt [ Dβ−1ρ y ] (t) = 1 tρ−1 d dt [ J 1−βρ y ] (t) = 1 tρ−1 d dt [∫ t 0 K1−βρ (t,η) ∫ η 0 Gρ(η,ζ)ζρ−1f(ζ)dζdη ] = 1 tρ−1 d dt [∫ t 0 ∫ η 0 K1−βρ (t,η)Gρ(η,ζ)ζ ρ−1f(ζ)dζdη ] = 1 tρ−1 d dt [∫ t 0 ∫ t ζ K1−βρ (t,η)Gρ(η,ζ)ζ ρ−1f(ζ)dηdζ ] = 1 tρ−1 d dt [∫ t 0 [∫ t ζ K1−βρ (t,η)Gρ(η,ζ)dη ] ζρ−1f(ζ)dζ ] = 1 tρ−1 d dt [∫ t 0 [ ζJ 1−βρ Gρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ ] = ∫ t 0 1 tρ−1 d dt [ ζJ 1−βρ Gρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + 1 tρ−1 lim ζ→t− [[ ζJ 1−βρ Gρ(∗,ζ)(t) ] ζρ−1f(ζ) ] , where we have applied the well-known Leibnitz rule. Using the definition of the fractional derivative, we obtain [ Dβρy ] (t) = ∫ t 0 [ ζDβρGρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + lim ζ→t− [ ζDβ−1ρ Gρ(∗,ζ) ] (t)f(t) for all t > 0. Applying D1 repeatedly for a total of (dαe− 1) times and using Definition 7.1 (ii), we find for all t > 0 that[ Dαρ y ] (t) = ∫ t 0 [ ζDαρGρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + lim ζ→t− [ ζDα−1ρ Gρ(∗,ζ) ] (t)f(t) (7.4) = ∫ t 0 [ ζDαρG(∗,ζ) ] (t)ζρ−1f(ζ)dζ + f(t) =p(t) ∫ t 0 Gρ(t,ζ)ζρ−1f(ζ)dζ + f(t) =p(t)y(t) + f(t) for all t > 0 by Definition 7.1 (i). Thus, y is a solution of the fractional differential equation in (7.3). Next, we justify the initial conditions. As in (7.4), we compute that[ Dα−kρ y ] (t) = ∫ t 0 [ ζDα−kρ Gρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + lim ζ→t− [ ζDα−k+1ρ Gρ(∗,ζ) ] (t)f(t) for all t > 0 and k = 1, 2, · · · ,dαe . By Definition 7.1 (iii) and letting t → 0+, we obtain [ Dα−kρ y ] (0+) = 0 for k = 1, 2, · · · ,dαe. We have therefore justified that y defined in (7.2) solves the initial value problem (7.3). � REIMANN-LIOUVILLE AND HADAMARD TYPE FRACTIONAL DERIVATIVES 229 Corollary 7.1. If {Hi} dαe i=1 forms the set of normalized fundamental solutions of the homogeneous initial value problem associated with (7.1), i.e.,{ [Dαρ Hi](t) = p(t)Hi(t) for t > 0 [Dα−kρ Hi](0+) = δi,k for k = 1, 2, · · · ,dαe, and G is the Green’s function for the initial value problem (7.1), then the solution of the initial value problem (7.1) is given by y(t) = dαe∑ k=1 ydαe−iHi(t) + ∫ t 0 Gρ(t,η)ηρ−1f(η)dη for t > 0. 7.1. Representation of Solutions for the Autonomous Equation. In this section, we confine our attention to the linear autonomous initial value problem{[ Dαρ y ] (t) = λy(t) + f(t) for t > 0[ Dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe. (7.5) For linear autonomous equations, we can easily verify that Gρ(t,s) = Gρ ( ρ √ tρ −sρ, 0 ) for all (t,s) ∈ ∆. (7.6) Moreover, Gρ(∗, 0) is the solution of the associated homogeneous equation{[ Dαρ y ] (t) = λy(t) for t > 0[ Dα−kρ y ] (0+) = δk,1 for k = 1, 2, · · · ,dαe. Due to the discussion made in § 5.1, we see that the Green’s function of (7.5) is given by Gρ(t,s) = [tρ −sρ]α−1 ρα−1 Eα,α ( λ[tρ −sρ]α ρα ) for (t,s) ∈ ∆. Further, the variation of parameters formula for the initial value problem (7.5) is given by y(t) = dαe∑ i=1 ydαe−i tρ(α−i) ρα−i Eα,α−i+1 ( λtρα ρα ) + ∫ t 0 [tρ −ηρ]α−1 ρα−1 Eα,α ( λ[tρ −ηρ]α ρα ) ηρ−1f(η)dη (7.7) for t > 0 (cf. [9, Equation (3.1.11)]). 8. Final Comments The following two examples can be easily verified. Example 8.1. For α ∈ R+0 , ν ∈ (−1,∞) and ρ ∈ R +, we have[ sJαρ [∗ ρ −sρ]ν ] (t) = Γ(ν + 1) ραΓ(ν + α + 1) [tρ −sρ]ν+α for t > s ≥ 0. Example 8.2. For α,ν ∈ R+0 and ρ ∈ R +, we have[ sDαρ [∗ ρ −sρ]ν ] (t) = ραΓ(ν + 1) Γ(ν −α + 1) [tρ −sρ]ν−α for t > s ≥ 0. As our first remark, we would like to say that one can justify (7.6) similar to that in the third part of [11, § 5.1.2]. As an another note, we would like to emphasis that one can justify the variation of parameters formula given in (7.7) by using Example 8.2 and applying the Picard iterates technique (used in § 5.2) to (7.5). As to some directions for future research, we would like to mention that the study of numerical solutions to FDEs and numerical integration techniques would very important. Next, we note that the extension of any of the results in this paper to the case FDEs with Miller-Ross type sequential derivatives would be of significant interest too. Finally, obtaining the solutions of FDEs of the type (7.5) by the Laplace transform would also deserve attention for the sake of completeness. 230 KARPUZ, ÖZKAN, YALÇIN AND YILDIZ References [1] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), no. 2, 387–400. [2] P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), no. 1, 1–27. [3] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010. [4] J. Hadamard, Essai sur l’étude des fonctions données par leur développement de Taylor, Journal de Mathématiques Pures et Appliquées 4 (1892), no. 8, 101–186. [5] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), no. 3, 860–865. [6] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 4 (2014), no. 6, 1–15. [7] A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191–1204. [8] A. A. Kilbas and J. J. Trujillo, Hadamard-type integrals as G-transforms, Integral Transforms Spec. Funct. 14 (2003), no. 5, 413–427. [9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier Science B.V., Amsterdam, 2006. [10] K. B. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974. [11] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [12] S. Pooseh, P. Almeida and D. F. M. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. 33 (2012), no. 3, 301–319. [13] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. 1Dokuz Eylül University, Tınaztepe Campus, Faculty of Science, Department of Mathematics, Buca, 35160 İzmir, Turkey. 2Afyon Kocatepe University, ANS Campus, Faculty of Science and Arts, Department of Mathematics, 03200 Afyonkarahisar, Turkey. ∗Corresponding author: bkarpuz@gmail.com 1. Introduction 2. Definitions and Auxiliary Results 3. Properties of the Operators J and D 4. Existence and Uniqueness for RLH FDEs 5. The Autonomous Equation of RL Type 5.1. Direct Substitution 5.2. The Picard Iterates 6. Dependence on Initial Conditions 7. The Green's Function for Linear Equations 7.1. Representation of Solutions for the Autonomous Equation 8. Final Comments References