Impulsive Multiorders Riemann-Liouville Fractional Differential ...

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 603893, 9 pages http://dx.doi.org/10.1155/2015/603893

Research Article Impulsive Multiorders Riemann-Liouville Fractional Differential Equations Weera Yukunthorn,1,2 Sotiris K. Ntouyas,3,4 and Jessada Tariboon1,2 1

Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand 2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand 3 Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece 4 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Jessada Tariboon; [email protected] Received 9 October 2014; Accepted 6 April 2015 Academic Editor: Seenith Sivasundaram Copyright © 2015 Weera Yukunthorn 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. Impulsive multiorders fractional differential equations are studied. Existence and uniqueness results are obtained for first- and second-order impulsive initial value problems by using Banach’s fixed point theorem in an appropriate weighted space. Examples illustrating the main results are presented.

𝑝

1. Introduction Fractional calculus has become very useful over the last years because of its many applications in almost all applied sciences. By now, almost all fields of research in science and engineering use fractional calculus to better describe them. Fractional differential equations have been of great interest and are caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various science such as physics, mechanics, chemistry, and engineering. For details and some recent results on the subject we refer to the papers [1–3], books [4–7], and references cited therein. Recently in [8], Wang et al. studied existence and uniqueness results for the following impulsive multipoint fractional integral boundary value problem involving multiorders fractional derivatives and deviating argument: 𝑐

𝛼

𝐷𝑡+𝑘 𝑢 (𝑡) = 𝑓 (𝑡, 𝑢 (𝑡) , 𝑢 (𝜃 (𝑡))) , 𝑘

1 < 𝛼𝑘 ≤ 2,

Δ𝑢 (𝑡𝑘 ) = 𝐼𝑘 (𝑢 (𝑡𝑘 )) , Δ𝑢󸀠 (𝑡𝑘 ) = 𝐼𝑘∗ (𝑢 (𝑡𝑘 )) ,

𝑘 = 1, 2, . . . , 𝑝,

𝛽

𝑢 (0) = ∑ 𝜆 𝑘 J𝑡+𝑘 𝑢 (𝜂𝑘 ) , 𝑘=0

𝑘

𝑢󸀠 (0) = 0, 𝑡𝑘 < 𝜂𝑘 < 𝑡𝑘+1 , (1) 𝛼

𝛽

where 𝑐 𝐷𝑡+𝑘 is the Caputo fractional derivative of order 𝛼𝑘 , J𝑡+𝑘 𝑘 𝑘 is fractional Riemann-Liouville integral of order 𝛽𝑘 > 0, 𝑓 ∈ 𝐶(𝐽 × R × R, R), 𝐼𝑘 , 𝐼𝑘∗ ∈ 𝐶(R, R), 𝜃 ∈ 𝐶(𝐽, 𝐽), 𝐽 = [0, 𝑇] (𝑇 > 0), 0 = 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑘 < ⋅ ⋅ ⋅ < 𝑡𝑝 < 𝑡𝑝+1 = 𝑇, Δ𝑢(𝑡𝑘 ) = 𝑢(𝑡𝑘+ )−𝑢(𝑡𝑘− ), and Δ𝑢󸀠 (𝑡𝑘 ) = 𝑢󸀠 (𝑡𝑘+ )−𝑢󸀠 (𝑡𝑘− ) where 𝑢(𝑡𝑘+ ), 𝑢󸀠 (𝑡𝑘+ ) and 𝑢(𝑡𝑘− ), 𝑢󸀠 (𝑡𝑘− ) denote the right and left hand limits of 𝑢(𝑡) and 𝑢󸀠 (𝑡) at 𝑡 = 𝑡𝑘 (𝑘 = 1, 2, . . . , 𝑝). We notice that there are some discussions on the concept of solution for impulsive fractional differential equations for both Riemann-Liouville and Caputo fractional derivatives. We refer the interested reader to some recent papers [9– 11] and the references cited therein. However, we can point out the problems caused by using the definition of Caputo

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fractional derivatives of order 𝛼 with the lower limit 0 for a function 𝑓 as 𝑐

𝛼

𝐷0 𝑓 (𝑡) =

𝑡 1 ∫ (𝑡 − 𝑠)𝑛−𝛼−1 𝑓(𝑛) (𝑠) 𝑑𝑠. Γ (𝑛 − 𝛼) 0

(2)

If there are impulse points 𝑡𝑘 such that 𝑡𝑘 ∈ (0, 𝑡) for some 𝑘 ∈ N, then the 𝑓(𝑛) (𝑡𝑘 ) does not exist, which leads to nonintegrability of the right-hand side of (2). The key idea for solving this problem is to apply the definition of fractional derivative only on an interval (𝑡𝑘 , 𝑡𝑘+1 ) and combining all intervals by using impulsive conditions. In this paper, we study impulsive multiorders RiemannLiouville fractional differential equations. More precisely, in Section 3 we study the existence and uniqueness of solutions for the following initial value problem for impulsive multiorders Riemann-Liouville fractional differential equations of order 0 < 𝛼𝑘 ≤ 1 of the form 𝛼

𝐷𝑡𝑘𝑘 𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡)) , ̃ (𝑡𝑘 ) = 𝜑𝑘 (𝑥 (𝑡𝑘 )) , Δ𝑥

𝑡 ∈ 𝐽, 𝑡 ≠ 𝑡𝑘 , 𝑘 = 1, 2, 3, . . . , 𝑚,

(3)

𝑥 (0) = 0, 𝛼

where 𝐷𝑡𝑘𝑘 is the Riemann-Liouville fractional derivative of order 0 < 𝛼𝑘 ≤ 1 on intervals 𝐽𝑘 , 𝑘 = 0, 1, 2, . . . , 𝑚, 0 = 𝑡0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑘 < ⋅ ⋅ ⋅ < 𝑡𝑚 < 𝑡𝑚+1 = 𝑇, 𝑓 : 𝐽 × R → R is a ̃ 𝑘) continuous function, and 𝜑𝑘 ∈ 𝐶(R, R). The notation Δ𝑥(𝑡 is defined by ̃ (𝑡𝑘 ) = 𝐼𝑡1−𝛼𝑘 𝑥 (𝑡+ ) − 𝐼𝑡1−𝛼𝑘−1 𝑥 (𝑡𝑘 ) , Δ𝑥 𝑘 𝑘 𝑘−1 𝑘 = 1, 2, 3, . . . , 𝑚,

(4)

1−𝛼

where 𝐼𝑡𝑘 𝑘 is the Riemann-Liouville fractional integral of order 1 − 𝛼𝑘 on interval 𝐽𝑘 . It should be noticed that if 𝛼𝑘 = ̃ 𝑘 ) = Δ𝑥(𝑡𝑘 ) = 𝑥(𝑡+ ) − 𝑥(𝑡𝑘 ) for 𝑘 = 1 in (4), then Δ𝑥(𝑡 𝑘 1, 2, 3, . . . , 𝑚. In Section 4, we investigate the initial value problem of impulsive Riemann-Liouville fractional differential equations of the form 𝛼

𝐷𝑡𝑘𝑘 𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡)) ,

𝑡 ∈ 𝐽, 𝑡 ≠ 𝑡𝑘 ,

̃ (𝑡𝑘 ) = 𝜑𝑘 (𝑥 (𝑡𝑘 )) , Δ𝑥

𝑘 = 1, 2, 3, . . . , 𝑚,

Δ∗ 𝑥 (𝑡𝑘 ) = 𝜑𝑘∗ (𝑥 (𝑡𝑘 )) ,

𝑘 = 1, 2, 3, . . . , 𝑚,

(5)

𝐷𝛼0 −1 𝑥 (0) = 𝛽,

In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later. Definition 1. The Riemann-Liouville fractional derivative of order 𝛼 > 0 of a continuous function 𝑓 : (𝑎, 𝑏) → R is defined by 𝐷𝑎𝛼 𝑓 (𝑡) =

1 𝑑 𝑛 𝑡 ( ) ∫ (𝑡 − 𝑠)𝑛−𝛼−1 𝑓 (𝑠) 𝑑𝑠, Γ (𝑛 − 𝛼) 𝑑𝑡 𝑎

(7)

𝑛 − 1 < 𝛼 < 𝑛, 𝑡 ∈ (𝑎, 𝑏) , where 𝑛 = [𝛼] + 1, [𝛼] denotes the integer part of a real number 𝛼, provided the right-hand side is pointwise defined on (𝑎, 𝑏), where Γ is the gamma function defined by Γ(𝛼) = ∞ ∫0 𝑒−𝑠 𝑠𝛼−1 𝑑𝑠. Definition 2. For at least 𝑛-times differentiable function 𝑓 : (𝑎, 𝑏) → R, the Caputo derivative of fractional order 𝛼 is defined as 𝑡 1 𝑐 𝛼 ∫ (𝑡 − 𝑠)𝑛−𝛼−1 𝑓(𝑛) (𝑠) 𝑑𝑠, 𝐷𝑎 𝑓 (𝑡) = Γ (𝑛 − 𝛼) 𝑎 (8) 𝑛 − 1 < 𝛼 < 𝑛, 𝑡 ∈ (𝑎, 𝑏) , where 𝑛 = [𝛼] + 1. Definition 3. The Riemann-Liouville fractional integral of order 𝛽 > 0 of a continuous function 𝑓 : (𝑎, 𝑏) → R is defined by 𝑡 1 ∫ (𝑡 − 𝑠)𝛼−1 𝑓 (𝑠) 𝑑𝑠, Γ (𝛼) 𝑎

𝑡 ∈ (𝑎, 𝑏)

(9)

provided the right-hand side is pointwise defined on (𝑎, 𝑏).

where 𝛽 ∈ R, 0 = 𝑡0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑘 < ⋅ ⋅ ⋅ < 𝑡𝑚 < 𝑡𝑚+1 = 𝑇, 𝑓 : 𝐽 × R → R is a continuous function, 𝜑𝑘 , 𝜑𝑘∗ ∈ 𝛼 𝐶(R, R) for 𝑘 = 1, 2, . . . , 𝑚, and 𝐷𝑡𝑘𝑘 is the Riemann-Liouville fractional derivative of order 1 < 𝛼𝑘 ≤ 2 on intervals 𝐽𝑘 for ̃ 𝑘 ) is defined by (4) and 𝑘 = 0, 1, 2, . . . , 𝑚. The notation Δ𝑥(𝑡 ∗ Δ 𝑥(𝑡𝑘 ) is defined by 2−𝛼𝑘

2. Preliminaries

𝐼𝑎𝛼 𝑓 (𝑡) =

𝑥 (0) = 0,

Δ∗ 𝑥 (𝑡𝑘 ) = 𝐼𝑡𝑘

2−𝛼

where 𝐼𝑡𝑘 𝑘 is the Riemann-Liouville fractional integral of order 2 − 𝛼𝑘 > 0 on 𝐽𝑘 . It should be noticed that if 𝛼𝑘 = 2 ̃ 𝑘 ) = 𝐷𝑡 𝑥(𝑡+ ) − 𝐷𝑡 𝑥(𝑡𝑘 ) and Δ∗ 𝑥(𝑡𝑘 ) = in (6), then Δ𝑥(𝑡 𝑘 𝑘 𝑘−1 + Δ𝑥(𝑡𝑘 ) = 𝑥(𝑡𝑘 ) − 𝑥(𝑡𝑘 ) for 𝑘 = 1, 2, . . . , 𝑚. By using Banach’s fixed point theorem we prove existence and uniqueness results for the problem (3) and (5) in an appropriate weighted space. The paper is organized as follows: Section 2 contains some preliminary notations, definitions, and lemmas that we need in the sequel. In Section 3 we present the main results for problem (3), while in Section 4 we present the main results for problem (5). Examples illustrating the obtained results are also presented.

2−𝛼

𝑥 (𝑡𝑘+ ) − 𝐼𝑡𝑘−1 𝑘−1 𝑥 (𝑡𝑘 ) , 𝑘 = 1, 2, . . . , 𝑚,

(6)

Lemma 4 (see [5]). Let 𝛼 > 0 and 𝑥 ∈ 𝐶(𝑎, 𝑏) ∩ 𝐿(𝑎, 𝑏). Then the fractional differential equation 𝐷𝑎𝛼 𝑥 (𝑡) = 0

(10)

has a unique solution 𝑥 (𝑡) = 𝑘1 (𝑡 − 𝑎)𝛼−1 + 𝑘2 (𝑡 − 𝑎)𝛼−2 + ⋅ ⋅ ⋅ + 𝑘𝑛 (𝑡 − 𝑎)𝛼−𝑛 , where 𝑘𝑖 ∈ R, 𝑖 = 1, 2, . . . , 𝑛, and 𝑛 − 1 < 𝑞 < 𝑛.

(11)

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3

Lemma 5 (see [5]). Let 𝛼 > 0. Then for 𝑥 ∈ 𝐶(𝑎, 𝑏) ∩ 𝐿(𝑎, 𝑏) it holds (12)

3. Impulsive Riemann-Liouville Fractional Differential Equations of Orders 0 < 𝛼𝑘 ≤ 1 Let 𝐽 = [0, 𝑇], 𝐽0 = [𝑡0 , 𝑡1 ], 𝐽𝑘 = (𝑡𝑘 , 𝑡𝑘+1 ] for 𝑘 = 1, 2, 3, . . . , 𝑚. Let 𝑃𝐶(𝐽, R) = {𝑥 : 𝐽 → R, 𝑥(𝑡) is continuous everywhere except for some 𝑡𝑘 at which 𝑥(𝑡𝑘+ ) and 𝑥(𝑡𝑘− ) exist, and 𝑥(𝑡𝑘− ) = 𝑥(𝑡𝑘 ), 𝑘 = 1, 2, 3, . . . , 𝑚}. For 𝛾 ∈ R+ , we introduce the space 𝐶𝛾,𝑘 (𝐽𝑘 , R) = {𝑥 : 𝐽𝑘 → R : (𝑡 − 𝑡𝑘 )𝛾 𝑥(𝑡) ∈ 𝐶(𝐽𝑘 , R)} with the norm ‖𝑥‖𝐶𝛾,𝑘 = sup𝑡∈𝐽𝑘 |(𝑡 − 𝑡𝑘 )𝛾 𝑥(𝑡)| and 𝑃𝐶𝛾 = {𝑥 : 𝐽 → R : for each 𝑡 ∈ 𝐽𝑘 and (𝑡 − 𝑡𝑘 )𝛾 𝑥(𝑡) ∈ 𝐶(𝐽𝑘 , R), 𝑘 = 0, 1, 2, . . . , 𝑚} with the norm ‖𝑥‖𝑃𝐶𝛾 = max{sup𝑡∈𝐽𝑘 |(𝑡 − 𝑡𝑘 )𝛾 𝑥(𝑡)| : 𝑘 = 0, 1, 2, . . . , 𝑚}. Clearly 𝑃𝐶𝛾 is a Banach space. In this section we study problem (3). Lemma 6. If 𝑥 ∈ 𝑃𝐶(𝐽, R) is a solution of (3), then, for any 𝑡 ∈ 𝐽𝑘 , 𝑘 = 0, 1, 2, . . . , 𝑚, 𝛼 −1

[ ∑ (𝐼𝑡1𝑘−1 𝑓 (𝑡𝑘 , 𝑥 (𝑡𝑘 )) 0 1 for 𝑘 = 0, 1, 2, . . . , 𝑚, then the initial value problem (3) has a unique solution on [0, 𝑇].

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Proof. In view of Lemma 6, we define the operator K : 𝑃𝐶 → 𝑃𝐶 as 𝛼 −1

K𝑥 (𝑡) =

(𝑡 − 𝑡𝑘 ) 𝑘 Γ (𝛼𝑘 )

[ ∑ (𝐼𝑡1𝑘−1 𝑓 (𝑡𝑘 , 𝑥 (𝑡𝑘 )) 0 0, ∀𝑥, 𝑦 ∈ R. (46)

𝛼

+ 𝐼𝑡𝑘𝑘 𝑓 (𝑡, 𝑥 (𝑡)) , for 𝑡 ∈ 𝐽𝑘 , 𝑘 = 0, 1, 2, . . . , 𝑚 with ∑𝑏𝑗=𝑎 = 0 for 𝑏 < 𝑎. It is straightforward to show that A𝑥 ∈ 𝑃𝐶𝛾 (𝐽, R); see Theorem 7. Next we will show that A𝐵𝑟 ⊂ 𝐵𝑟 , where a ball 𝐵𝑟 is defined by 𝐵𝑟 = {𝑥 ∈ 𝑃𝐶𝛾 (𝐽, R), ‖𝑥‖𝑃𝐶𝛾 ≤ 𝑟}. Assume that sup𝑡∈𝐽 |𝑓(𝑡, 0)| = 𝑀, max{|𝜑𝑘 (0)| : 𝑘 = 1, 2, 3, . . . , 𝑚} = 𝑁 and max{|𝜑𝑘∗ (0)| : 𝑘 = 1, 2, 3, . . . , 𝑚} = 𝑃. Setting 𝑇2∗ (𝑀 (Φ + 𝑇 + 1) + 𝑁 (2𝑚 − 1) + 𝑃𝑚 Γ2∗ 󵄨 󵄨 + 󵄨󵄨󵄨𝛽󵄨󵄨󵄨 (𝑇 + 1)) ,

Ω2 =

(50)

we choose a constant 𝑟 such that 𝑟≥

Ω2 . 1 − Ω1

Let 𝑥 ∈ 𝐵𝑟 . For any 𝑡 ∈ 𝐽𝑘 , 𝑘 = 0, 1, 2, . . . , 𝑚, we have

(51)

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7

𝛼 −2

|(A𝑥) (𝑡)| ≤

𝑘−1 (𝑡 − 𝑡𝑘 ) 𝑘 [󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 1 󵄨 󵄨󵄨𝛽󵄨󵄨 𝑡𝑘 + ∑ ((𝑡𝑘 − 𝑡𝑗 ) (𝐼𝑡𝑗−1 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠))󵄨󵄨󵄨 (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗 (𝑥 (𝑡𝑗 ))󵄨󵄨󵄨󵄨)) Γ (𝛼𝑘 − 1) 𝑗=1 [ 𝛼 −1

𝑘 (𝑡 − 𝑡𝑘 ) 𝑘 󵄨 󵄨 󵄨 󵄨 + ∑ (𝐼𝑡2𝑗−1 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠))󵄨󵄨󵄨 (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗∗ (𝑥 (𝑡𝑗 ))󵄨󵄨󵄨󵄨)] + Γ (𝛼𝑘 ) 𝑗=1 ]

𝑘 [󵄨󵄨󵄨𝛽󵄨󵄨󵄨 + ∑ (𝐼1 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠))󵄨󵄨󵄨 (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗 (𝑥 (𝑡𝑗 ))󵄨󵄨󵄨󵄨)] 𝑡𝑗−1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑗=1 ] [

𝛼 −2

(𝑡 − 𝑡𝑘 ) 𝑘 [󵄨󵄨 󵄨󵄨 𝛼 󵄨 󵄨 + 𝐼𝑡𝑘𝑘 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠))󵄨󵄨󵄨 (𝑡) ≤ 󵄨𝛽󵄨 𝑡 Γ (𝛼𝑘 − 1) 󵄨 󵄨 𝑘 [ 𝑘−1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + ∑ ((𝑡𝑘 − 𝑡𝑗 ) (𝐼𝑡1𝑗−1 (󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 0)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑠, 0)󵄨󵄨󵄨) (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗 (𝑥 (𝑡𝑗 )) − 𝜑𝑗 (0)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜑𝑗 (𝑥 (0))󵄨󵄨󵄨󵄨)) 𝑗=1 𝑘

+ ∑ (𝐼𝑡2𝑗−1 𝑗=1

𝛼 −1

(𝑡 − 𝑡𝑘 ) 𝑘 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 0)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑠, 0)󵄨󵄨󵄨) (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗∗ (𝑥 (𝑡𝑗 )) − 𝜑𝑗∗ (0)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜑𝑗∗ (0)󵄨󵄨󵄨󵄨)] + Γ (𝛼𝑘 ) ]

[󵄨󵄨󵄨𝛽󵄨󵄨󵄨 󵄨 󵄨 [

𝑘 󵄨 󵄨 󵄨 󵄨 𝛼 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + ∑ (𝐼𝑡1𝑗−1 (󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 0)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑠, 0)󵄨󵄨󵄨) (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗 (𝑥 (𝑡𝑗 )) − 𝜑𝑗 (0)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜑𝑗 (0)󵄨󵄨󵄨󵄨)] + 𝐼𝑡𝑘𝑘 (󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 0)󵄨󵄨󵄨 𝑗=1 ] 𝛼 −2

𝑘−1 (𝑡 − 𝑡𝑘 ) 𝑘 [󵄨󵄨 󵄨󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝑓 (𝑠, 0)󵄨󵄨󵄨) (𝑡) ≤ 󵄨󵄨𝛽󵄨󵄨 𝑡𝑘 + (𝐿 1 𝑟 + 𝑀) ( ∑ (𝑡𝑘 − 𝑡𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 )) + (𝐿 2 𝑟 + 𝑁) (𝑘 − 1) Γ (𝛼𝑘 − 1) 𝑗=1 [ 2

𝑘

+ (𝐿 1 𝑟 + 𝑀) (∑

(𝑡𝑗 − 𝑡𝑗−1 )

) + (𝐿 3 𝑟 + 𝑃) 𝑘] + ]

2

𝑗=1

𝛼 −1

(𝑡 − 𝑡𝑘 ) 𝑘 Γ (𝛼𝑘 )

𝑘

[󵄨󵄨󵄨𝛽󵄨󵄨󵄨 + (𝐿 1 𝑟 + 𝑀) ( ∑ (𝑡𝑗 − 𝑡𝑗−1 )) + (𝐿 2 𝑟 + 𝑁) 𝑘] 󵄨 󵄨 𝑗=1 ] [

𝛼

+ (𝐿 1 𝑟 + 𝑀)

(𝑡 − 𝑡𝑘 ) 𝑘 . Γ (𝛼𝑘 + 1)

Multiplying both sides of the above inequality by (𝑡 − 𝑡𝑘 )𝛾 for 𝑡 ∈ 𝐽𝑘 , we have 𝛾+𝛼 −2

(𝑡 − 𝑡𝑘 ) 𝑘 [󵄨󵄨 󵄨󵄨 (𝑡 − 𝑡𝑘 ) |(A𝑥) (𝑡)| ≤ 󵄨󵄨𝛽󵄨󵄨 𝑡𝑘 Γ (𝛼𝑘 − 1) [ 𝛾

𝑘−1

+ (𝐿 1 𝑟 + 𝑀) ( ∑ (𝑡𝑘 − 𝑡𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 )) 𝑗=1

𝑘

+ (𝐿 1 𝑟 + 𝑀) ( ∑ (𝑡𝑗 − 𝑡𝑗−1 )) + (𝐿 2 𝑟 + 𝑁) 𝑘] 𝑗=1 ] 𝛾+𝛼

+ (𝐿 1 𝑟 + 𝑀)

(𝑡 − 𝑡𝑘 ) 𝑘 𝑇2∗ [󵄨󵄨 󵄨󵄨 ≤ ∗ 󵄨󵄨𝛽󵄨󵄨 𝑇 Γ2 Γ (𝛼𝑘 + 1) [ 𝑚−1

+ (𝐿 1 𝑟 + 𝑀) ( ∑ (𝑡𝑚 − 𝑡𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 )) 𝑗=1

+ (𝐿 2 𝑟 + 𝑁) (𝑘 − 1) 𝑘

+ (𝐿 1 𝑟 + 𝑀) ( ∑

(𝑡𝑗 − 𝑡𝑗−1 )

𝑗=1

𝛾+𝛼 −1

+

(𝑡 − 𝑡𝑘 ) 𝑘 Γ (𝛼𝑘 )

+ (𝐿 2 𝑟 + 𝑁) (𝑚 − 1) 2

[󵄨󵄨󵄨𝛽󵄨󵄨󵄨 󵄨 󵄨 [

2

) + (𝐿 3 𝑟 + 𝑃) 𝑘] ]

𝑚

+ (𝐿 1 𝑟 + 𝑀) (∑

𝑗=1 𝑚

(𝑡𝑗 − 𝑡𝑗−1 ) 2

2

󵄨 󵄨 ) + (𝐿 3 𝑟 + 𝑃) + 󵄨󵄨󵄨𝛽󵄨󵄨󵄨

+ (𝐿 1 𝑟 + 𝑀) ( ∑ (𝑡𝑗 − 𝑡𝑗−1 )) + (𝐿 2 𝑟 + 𝑁) 𝑚 𝑗=1

(52)

8

Discrete Dynamics in Nature and Society 𝑇2∗ (𝑀 (Φ + 𝑇 + 1) + 𝑁 (2𝑚 − 1) + 𝑃𝑚 Γ2∗ 󵄨 󵄨 + 󵄨󵄨󵄨𝛽󵄨󵄨󵄨 (𝑇 + 1)) = Ω1 𝑟 + Ω2 .

+

+ (𝐿 1 𝑟 + 𝑀)] ] = 𝑟[

(53)

𝑇2∗ (𝐿 (Φ + 𝑇 + 1) + 𝐿 2 (2𝑚 − 1) + 𝐿 3 𝑚)] Γ2∗ 1

This implies that A𝐵𝑟 ⊂ 𝐵𝑟 . Finally we will show that A is a contraction mapping on 𝐵𝑟 . For 𝑥, 𝑦 ∈ 𝑃𝐶𝛾 (𝐽, R) and for each 𝑡 ∈ 𝐽 we have

𝛼 −2

𝑘−1 𝑘 󵄨 (𝑡 − 𝑡𝑘 ) 󵄨󵄨 [ ∑ ((𝑡𝑘 − 𝑡𝑗 ) (𝐼1 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 𝑦 (𝑠))󵄨󵄨󵄨 (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗 (𝑥 (𝑡𝑗 )) − 𝜑𝑗 (𝑦 (𝑡𝑗 ))󵄨󵄨󵄨󵄨)) 󵄨󵄨(A𝑥) (𝑡) − (A𝑦) (𝑡)󵄨󵄨󵄨 ≤ 𝑡𝑗−1 󵄨 󵄨 󵄨 󵄨 Γ (𝛼𝑘 − 1) 𝑗=1 [ 𝑘 󵄨 󵄨 󵄨 󵄨 + ∑ (𝐼𝑡2𝑗−1 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 𝑦 (𝑠))󵄨󵄨󵄨 (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗∗ (𝑥 (𝑡𝑗 )) − 𝜑𝑗∗ (𝑦 (𝑡𝑗 ))󵄨󵄨󵄨󵄨)] 𝑗=1 ] 𝛼 −1

+

(𝑡 − 𝑡𝑘 ) 𝑘 Γ (𝛼𝑘 )

(54)

𝑘 [∑ (𝐼1 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 𝑦 (𝑠))󵄨󵄨󵄨 (𝑡𝑗 ) + 󵄨󵄨󵄨󵄨𝜑𝑗 (𝑥 (𝑡𝑗 )) − 𝜑𝑗 (𝑦 (𝑡𝑗 ))󵄨󵄨󵄨󵄨)] + 𝐼𝑡𝛼𝑘 󵄨󵄨󵄨𝑓 (𝑠, 𝑥 (𝑠)) − 𝑓 (𝑠, 𝑦 (𝑠))󵄨󵄨󵄨 𝑡𝑗−1 󵄨 󵄨 󵄨 𝑘 󵄨 󵄨 󵄨 ] [𝑗=1

⋅ (𝑡) .

Multiplying both sides of the above inequality by (𝑡 − 𝑡𝑘 )𝛾 , we have 󵄨 󵄨󵄨 󵄨󵄨(𝑡 − 𝑡𝑘 )𝛾 (A𝑥) (𝑡) − (𝑡 − 𝑡𝑘 )𝛾 (A𝑦) (𝑡)󵄨󵄨󵄨 󵄨 󵄨

Example 11. Consider the following impulsive multiorders Riemann-Liouville fractional initial value problem: 2

2

+3𝑘+5)/(𝑘 +2𝑘+3) 𝐷𝑡(2𝑘 𝑥= 𝑘

𝛾+𝛼 −2

≤

(𝑡 − 𝑡𝑘 ) 𝑘 [ 󵄩󵄩 󵄩 𝐿 1 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 Γ (𝛼𝑘 − 1) [

2𝑡 |𝑥 (𝑡)| cos 𝑡 (7 + |𝑥 (𝑡)|)2 + |𝑥 (𝑡)| sin2 𝑡 1 + , 2 𝑡 ∈ [0, tan−1 (

𝑘−1

󵄩 󵄩 ⋅ ( ∑ (𝑡𝑘 − 𝑡𝑗 ) (𝑡𝑗 − 𝑡𝑗−1 )) + 𝐿 2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 (𝑘 − 1) 𝑗=1

2

(𝑡𝑗 − 𝑡𝑗−1 ) 󵄩 󵄩 󵄩 󵄩 + 𝐿 1 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ( ∑ ) + 𝐿 3 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 𝑘] 2 𝑗=1 ] 𝑘

𝛾+𝛼 −1

(𝑡 − 𝑡𝑘 ) 𝑘 + Γ (𝛼𝑘 )

̃ (𝑡𝑘 ) = Δ𝑥

(55)

𝑘

󵄩 󵄩 (𝑡 − 𝑡𝑘 ) 󵄩 󵄩 + 𝐿 2 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 𝑘] + 𝐿 1 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 Γ (𝛼𝑘 + 1) ] 󵄩 󵄩 ≤ Ω1 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .

(57) 𝑘 ), 3𝜋

󵄨 󵄨󵄨 󵄨󵄨𝑥 (𝑡𝑘 )󵄨󵄨󵄨 󵄨 󵄨, 37 (1 − 2 cos 𝑘𝜋) + 󵄨󵄨󵄨𝑥 (𝑡𝑘 )󵄨󵄨󵄨

𝑘 = 1, 2, . . . , 10, 𝑡𝑘 = tan−1 (

𝛾+𝛼𝑘

𝑘 ), 3𝜋

𝑥 (0) = 0 𝑥 (0) = 𝑒. 𝐷𝑡2/3 0

It follows that 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩A𝑥 − A𝑦󵄩󵄩󵄩 ≤ Ω1 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .

󵄨 󵄨󵄨 󵄨󵄨𝑥 (𝑡𝑘 )󵄨󵄨󵄨 󵄨 󵄨, 9 (𝑘2 + 3𝑘 + 5) + 󵄨󵄨󵄨𝑥 (𝑡𝑘 )󵄨󵄨󵄨

𝑘 = 1, 2, . . . , 10, 𝑡𝑘 = tan−1 ( Δ∗ 𝑥 (𝑡𝑘 ) =

[𝐿 1 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ( ∑ (𝑡𝑗 − 𝑡𝑗−1 )) 󵄩 󵄩 𝑗=1 [

11 )] , 𝑡 ≠ 𝑡𝑘 , 3𝜋

(56)

Since Ω1 < 1, A is a contraction mapping on 𝐵𝑟 . Therefore (5) has a unique solution on [0, 𝑇].

Here 𝛼𝑘 = (2𝑘2 + 3𝑘 + 5)/(𝑘2 + 2𝑘 + 3), 𝑘 = 0, 1, 2, . . . , 10, 𝑚 = 10, 𝑇 = tan−1 (11/(3𝜋)), 𝑓(𝑡, 𝑥) = (2𝑡 |𝑥|cos𝑡)/((7+|𝑥|)2 + |𝑥|sin2 𝑡)+(1/2), 𝜑𝑘 (𝑥) = |𝑥|/(9(𝑘2 +3𝑘+5)+|𝑥|), and 𝜑𝑘∗ (𝑥) = |𝑥|/(37(1−2cos𝑘𝜋)+|𝑥|). Since |𝑓(𝑡, 𝑥)−𝑓(𝑡, 𝑦)| ≤ (2/49)|𝑥− 𝑦|, |𝜑𝑘 (𝑥) − 𝜑𝑘 (𝑦)| ≤ (1/81)|𝑥 − 𝑦|, and |𝜑𝑘∗ (𝑥) − 𝜑𝑘∗ (𝑦)| ≤ (3/37)|𝑥−𝑦|, then (𝐻1 ), (𝐻2 ), and (𝐻3 ) are satisfied with 𝐿 1 = 2/49, 𝐿 2 = 1/81, and 𝐿 3 = 3/37, respectively. By choosing

Discrete Dynamics in Nature and Society

9

𝛾 = 2, we can find that 𝑇2∗ = 0.781307, Γ2∗ = 0.902745, Φ = 0.332114, and Ω1 =

𝑇2∗ (𝐿 (𝜙 + 𝑇 + 1) + 𝐿 2 (2𝑚 − 1) + 𝐿 3 𝑚) Γ2∗ 1

(58)

= 0.982275 < 1. Hence, by Theorem 10, the initial value problem (57) has a unique solution on [0, tan−1 11/(3𝜋)].

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

References [1] R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1095–1100, 2010. [2] B. Ahmad and J. J. Nieto, “Boundary value problems for a class of sequential integrodifferential equations of fractional order,” Journal of Function Spaces and Applications, vol. 2013, Article ID 149659, 8 pages, 2013. [3] B. Ahmad and S. K. Ntouyas, “Fractional differential inclusions with fractional separated boundary conditions,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 362–382, 2012. [4] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishers, Boston, Mass, USA, 2012. [5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. [6] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. [7] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. [8] G. Wang, S. Liu, D. Baleanu, and L. Zhang, “A new impulsive multi-orders fractional differential equation involving multipoint fractional integral boundary conditions,” Abstract and Applied Analysis, vol. 2014, Article ID 932747, 10 pages, 2014. [9] C. Bai, “Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 384, no. 2, pp. 211–231, 2011. [10] M. Feˇckan, Y. Zhou, and J. Wang, “On the concept and existence of solution for impulsive fractional differential equations,”

Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012. [11] X. Zhang, X. Zhang, and M. Zhang, “On the concept of general solution for impulsive differential equations of fractional order 𝑞 ∈ (0, 1),” Applied Mathematics and Computation, vol. 247, pp. 72–89, 2014.

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