Impulsive Multiorders Riemann-Liouville Fractional Differential ...

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Apr 6, 2015 - is the Caputo fractional derivative of order , J. . +. is fractional Riemann-Liouville integral of order > 0, ∈.
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 Β© 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

𝐷𝑑+π‘˜ 𝑒 (𝑑) = 𝑓 (𝑑, 𝑒 (𝑑) , 𝑒 (πœƒ (𝑑))) , π‘˜

1 < π›Όπ‘˜ ≀ 2,

Δ𝑒 (π‘‘π‘˜ ) = πΌπ‘˜ (𝑒 (π‘‘π‘˜ )) , Δ𝑒󸀠 (π‘‘π‘˜ ) = πΌπ‘˜βˆ— (𝑒 (π‘‘π‘˜ )) ,

π‘˜ = 1, 2, . . . , 𝑝,

𝛽

𝑒 (0) = βˆ‘ πœ† π‘˜ J𝑑+π‘˜ 𝑒 (πœ‚π‘˜ ) , π‘˜=0

π‘˜

𝑒󸀠 (0) = 0, π‘‘π‘˜ < πœ‚π‘˜ < π‘‘π‘˜+1 , (1) 𝛼

𝛽

where 𝑐 𝐷𝑑+π‘˜ is the Caputo fractional derivative of order π›Όπ‘˜ , J𝑑+π‘˜ π‘˜ π‘˜ is fractional Riemann-Liouville integral of order π›½π‘˜ > 0, 𝑓 ∈ 𝐢(𝐽 Γ— R Γ— R, R), πΌπ‘˜ , πΌπ‘˜βˆ— ∈ 𝐢(R, R), πœƒ ∈ 𝐢(𝐽, 𝐽), 𝐽 = [0, 𝑇] (𝑇 > 0), 0 = 𝑑0 < 𝑑1 < β‹… β‹… β‹… < π‘‘π‘˜ < β‹… β‹… β‹… < 𝑑𝑝 < 𝑑𝑝+1 = 𝑇, Δ𝑒(π‘‘π‘˜ ) = 𝑒(π‘‘π‘˜+ )βˆ’π‘’(π‘‘π‘˜βˆ’ ), and Δ𝑒󸀠 (π‘‘π‘˜ ) = 𝑒󸀠 (π‘‘π‘˜+ )βˆ’π‘’σΈ€  (π‘‘π‘˜βˆ’ ) where 𝑒(π‘‘π‘˜+ ), 𝑒󸀠 (π‘‘π‘˜+ ) and 𝑒(π‘‘π‘˜βˆ’ ), 𝑒󸀠 (π‘‘π‘˜βˆ’ ) denote the right and left hand limits of 𝑒(𝑑) and 𝑒󸀠 (𝑑) at 𝑑 = π‘‘π‘˜ (π‘˜ = 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 (𝑑) =

𝑑 1 ∫ (𝑑 βˆ’ 𝑠)π‘›βˆ’π›Όβˆ’1 𝑓(𝑛) (𝑠) 𝑑𝑠. Ξ“ (𝑛 βˆ’ 𝛼) 0

(2)

If there are impulse points π‘‘π‘˜ such that π‘‘π‘˜ ∈ (0, 𝑑) for some π‘˜ ∈ N, then the 𝑓(𝑛) (π‘‘π‘˜ ) 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 (π‘‘π‘˜ , π‘‘π‘˜+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 < π›Όπ‘˜ ≀ 1 of the form 𝛼

π·π‘‘π‘˜π‘˜ π‘₯ (𝑑) = 𝑓 (𝑑, π‘₯ (𝑑)) , Μƒ (π‘‘π‘˜ ) = πœ‘π‘˜ (π‘₯ (π‘‘π‘˜ )) , Ξ”π‘₯

𝑑 ∈ 𝐽, 𝑑 =ΜΈ π‘‘π‘˜ , π‘˜ = 1, 2, 3, . . . , π‘š,

(3)

π‘₯ (0) = 0, 𝛼

where π·π‘‘π‘˜π‘˜ is the Riemann-Liouville fractional derivative of order 0 < π›Όπ‘˜ ≀ 1 on intervals π½π‘˜ , π‘˜ = 0, 1, 2, . . . , π‘š, 0 = 𝑑0 < 𝑑1 < 𝑑2 < β‹… β‹… β‹… < π‘‘π‘˜ < β‹… β‹… β‹… < π‘‘π‘š < π‘‘π‘š+1 = 𝑇, 𝑓 : 𝐽 Γ— R β†’ R is a Μƒ π‘˜) continuous function, and πœ‘π‘˜ ∈ 𝐢(R, R). The notation Ξ”π‘₯(𝑑 is defined by Μƒ (π‘‘π‘˜ ) = 𝐼𝑑1βˆ’π›Όπ‘˜ π‘₯ (𝑑+ ) βˆ’ 𝐼𝑑1βˆ’π›Όπ‘˜βˆ’1 π‘₯ (π‘‘π‘˜ ) , Ξ”π‘₯ π‘˜ π‘˜ π‘˜βˆ’1 π‘˜ = 1, 2, 3, . . . , π‘š,

(4)

1βˆ’π›Ό

where πΌπ‘‘π‘˜ π‘˜ is the Riemann-Liouville fractional integral of order 1 βˆ’ π›Όπ‘˜ on interval π½π‘˜ . It should be noticed that if π›Όπ‘˜ = Μƒ π‘˜ ) = Ξ”π‘₯(π‘‘π‘˜ ) = π‘₯(𝑑+ ) βˆ’ π‘₯(π‘‘π‘˜ ) for π‘˜ = 1 in (4), then Ξ”π‘₯(𝑑 π‘˜ 1, 2, 3, . . . , π‘š. In Section 4, we investigate the initial value problem of impulsive Riemann-Liouville fractional differential equations of the form 𝛼

π·π‘‘π‘˜π‘˜ π‘₯ (𝑑) = 𝑓 (𝑑, π‘₯ (𝑑)) ,

𝑑 ∈ 𝐽, 𝑑 =ΜΈ π‘‘π‘˜ ,

Μƒ (π‘‘π‘˜ ) = πœ‘π‘˜ (π‘₯ (π‘‘π‘˜ )) , Ξ”π‘₯

π‘˜ = 1, 2, 3, . . . , π‘š,

Ξ”βˆ— π‘₯ (π‘‘π‘˜ ) = πœ‘π‘˜βˆ— (π‘₯ (π‘‘π‘˜ )) ,

π‘˜ = 1, 2, 3, . . . , π‘š,

(5)

𝐷𝛼0 βˆ’1 π‘₯ (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 : (π‘Ž, 𝑏) β†’ R is defined by π·π‘Žπ›Ό 𝑓 (𝑑) =

1 𝑑 𝑛 𝑑 ( ) ∫ (𝑑 βˆ’ 𝑠)π‘›βˆ’π›Όβˆ’1 𝑓 (𝑠) 𝑑𝑠, Ξ“ (𝑛 βˆ’ 𝛼) 𝑑𝑑 π‘Ž

(7)

𝑛 βˆ’ 1 < 𝛼 < 𝑛, 𝑑 ∈ (π‘Ž, 𝑏) , where 𝑛 = [𝛼] + 1, [𝛼] denotes the integer part of a real number 𝛼, provided the right-hand side is pointwise defined on (π‘Ž, 𝑏), where Ξ“ is the gamma function defined by Ξ“(𝛼) = ∞ ∫0 π‘’βˆ’π‘  π‘ π›Όβˆ’1 𝑑𝑠. Definition 2. For at least 𝑛-times differentiable function 𝑓 : (π‘Ž, 𝑏) β†’ R, the Caputo derivative of fractional order 𝛼 is defined as 𝑑 1 𝑐 𝛼 ∫ (𝑑 βˆ’ 𝑠)π‘›βˆ’π›Όβˆ’1 𝑓(𝑛) (𝑠) 𝑑𝑠, π·π‘Ž 𝑓 (𝑑) = Ξ“ (𝑛 βˆ’ 𝛼) π‘Ž (8) 𝑛 βˆ’ 1 < 𝛼 < 𝑛, 𝑑 ∈ (π‘Ž, 𝑏) , where 𝑛 = [𝛼] + 1. Definition 3. The Riemann-Liouville fractional integral of order 𝛽 > 0 of a continuous function 𝑓 : (π‘Ž, 𝑏) β†’ R is defined by 𝑑 1 ∫ (𝑑 βˆ’ 𝑠)π›Όβˆ’1 𝑓 (𝑠) 𝑑𝑠, Ξ“ (𝛼) π‘Ž

𝑑 ∈ (π‘Ž, 𝑏)

(9)

provided the right-hand side is pointwise defined on (π‘Ž, 𝑏).

where 𝛽 ∈ R, 0 = 𝑑0 < 𝑑1 < 𝑑2 < β‹… β‹… β‹… < π‘‘π‘˜ < β‹… β‹… β‹… < π‘‘π‘š < π‘‘π‘š+1 = 𝑇, 𝑓 : 𝐽 Γ— R β†’ R is a continuous function, πœ‘π‘˜ , πœ‘π‘˜βˆ— ∈ 𝛼 𝐢(R, R) for π‘˜ = 1, 2, . . . , π‘š, and π·π‘‘π‘˜π‘˜ is the Riemann-Liouville fractional derivative of order 1 < π›Όπ‘˜ ≀ 2 on intervals π½π‘˜ for Μƒ π‘˜ ) is defined by (4) and π‘˜ = 0, 1, 2, . . . , π‘š. The notation Ξ”π‘₯(𝑑 βˆ— Ξ” π‘₯(π‘‘π‘˜ ) is defined by 2βˆ’π›Όπ‘˜

2. Preliminaries

πΌπ‘Žπ›Ό 𝑓 (𝑑) =

π‘₯ (0) = 0,

Ξ”βˆ— π‘₯ (π‘‘π‘˜ ) = πΌπ‘‘π‘˜

2βˆ’π›Ό

where πΌπ‘‘π‘˜ π‘˜ is the Riemann-Liouville fractional integral of order 2 βˆ’ π›Όπ‘˜ > 0 on π½π‘˜ . It should be noticed that if π›Όπ‘˜ = 2 Μƒ π‘˜ ) = 𝐷𝑑 π‘₯(𝑑+ ) βˆ’ 𝐷𝑑 π‘₯(π‘‘π‘˜ ) and Ξ”βˆ— π‘₯(π‘‘π‘˜ ) = in (6), then Ξ”π‘₯(𝑑 π‘˜ π‘˜ π‘˜βˆ’1 + Ξ”π‘₯(π‘‘π‘˜ ) = π‘₯(π‘‘π‘˜ ) βˆ’ π‘₯(π‘‘π‘˜ ) for π‘˜ = 1, 2, . . . , π‘š. 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.

2βˆ’π›Ό

π‘₯ (π‘‘π‘˜+ ) βˆ’ πΌπ‘‘π‘˜βˆ’1 π‘˜βˆ’1 π‘₯ (π‘‘π‘˜ ) , π‘˜ = 1, 2, . . . , π‘š,

(6)

Lemma 4 (see [5]). Let 𝛼 > 0 and π‘₯ ∈ 𝐢(π‘Ž, 𝑏) ∩ 𝐿(π‘Ž, 𝑏). Then the fractional differential equation π·π‘Žπ›Ό π‘₯ (𝑑) = 0

(10)

has a unique solution π‘₯ (𝑑) = π‘˜1 (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 + π‘˜2 (𝑑 βˆ’ π‘Ž)π›Όβˆ’2 + β‹… β‹… β‹… + π‘˜π‘› (𝑑 βˆ’ π‘Ž)π›Όβˆ’π‘› , where π‘˜π‘– ∈ R, 𝑖 = 1, 2, . . . , 𝑛, and 𝑛 βˆ’ 1 < π‘ž < 𝑛.

(11)

Discrete Dynamics in Nature and Society

3

Lemma 5 (see [5]). Let 𝛼 > 0. Then for π‘₯ ∈ 𝐢(π‘Ž, 𝑏) ∩ 𝐿(π‘Ž, 𝑏) it holds (12)

3. Impulsive Riemann-Liouville Fractional Differential Equations of Orders 0 < π›Όπ‘˜ ≀ 1 Let 𝐽 = [0, 𝑇], 𝐽0 = [𝑑0 , 𝑑1 ], π½π‘˜ = (π‘‘π‘˜ , π‘‘π‘˜+1 ] for π‘˜ = 1, 2, 3, . . . , π‘š. Let 𝑃𝐢(𝐽, R) = {π‘₯ : 𝐽 β†’ R, π‘₯(𝑑) is continuous everywhere except for some π‘‘π‘˜ at which π‘₯(π‘‘π‘˜+ ) and π‘₯(π‘‘π‘˜βˆ’ ) exist, and π‘₯(π‘‘π‘˜βˆ’ ) = π‘₯(π‘‘π‘˜ ), π‘˜ = 1, 2, 3, . . . , π‘š}. For 𝛾 ∈ R+ , we introduce the space 𝐢𝛾,π‘˜ (π½π‘˜ , R) = {π‘₯ : π½π‘˜ β†’ R : (𝑑 βˆ’ π‘‘π‘˜ )𝛾 π‘₯(𝑑) ∈ 𝐢(π½π‘˜ , R)} with the norm β€–π‘₯‖𝐢𝛾,π‘˜ = supπ‘‘βˆˆπ½π‘˜ |(𝑑 βˆ’ π‘‘π‘˜ )𝛾 π‘₯(𝑑)| and 𝑃𝐢𝛾 = {π‘₯ : 𝐽 β†’ R : for each 𝑑 ∈ π½π‘˜ and (𝑑 βˆ’ π‘‘π‘˜ )𝛾 π‘₯(𝑑) ∈ 𝐢(π½π‘˜ , R), π‘˜ = 0, 1, 2, . . . , π‘š} with the norm β€–π‘₯‖𝑃𝐢𝛾 = max{supπ‘‘βˆˆπ½π‘˜ |(𝑑 βˆ’ π‘‘π‘˜ )𝛾 π‘₯(𝑑)| : π‘˜ = 0, 1, 2, . . . , π‘š}. Clearly 𝑃𝐢𝛾 is a Banach space. In this section we study problem (3). Lemma 6. If π‘₯ ∈ 𝑃𝐢(𝐽, R) is a solution of (3), then, for any 𝑑 ∈ π½π‘˜ , π‘˜ = 0, 1, 2, . . . , π‘š, 𝛼 βˆ’1

[ βˆ‘ (𝐼𝑑1π‘˜βˆ’1 𝑓 (π‘‘π‘˜ , π‘₯ (π‘‘π‘˜ )) 0 1 for π‘˜ = 0, 1, 2, . . . , π‘š, 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 : 𝑃𝐢 β†’ 𝑃𝐢 as 𝛼 βˆ’1

Kπ‘₯ (𝑑) =

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ Ξ“ (π›Όπ‘˜ )

[ βˆ‘ (𝐼𝑑1π‘˜βˆ’1 𝑓 (π‘‘π‘˜ , π‘₯ (π‘‘π‘˜ )) 0 0, βˆ€π‘₯, 𝑦 ∈ R. (46)

𝛼

+ πΌπ‘‘π‘˜π‘˜ 𝑓 (𝑑, π‘₯ (𝑑)) , for 𝑑 ∈ π½π‘˜ , π‘˜ = 0, 1, 2, . . . , π‘š with βˆ‘π‘π‘—=π‘Ž = 0 for 𝑏 < π‘Ž. It is straightforward to show that Aπ‘₯ ∈ 𝑃𝐢𝛾 (𝐽, R); see Theorem 7. Next we will show that Aπ΅π‘Ÿ βŠ‚ π΅π‘Ÿ , where a ball π΅π‘Ÿ is defined by π΅π‘Ÿ = {π‘₯ ∈ 𝑃𝐢𝛾 (𝐽, R), β€–π‘₯‖𝑃𝐢𝛾 ≀ π‘Ÿ}. Assume that supπ‘‘βˆˆπ½ |𝑓(𝑑, 0)| = 𝑀, max{|πœ‘π‘˜ (0)| : π‘˜ = 1, 2, 3, . . . , π‘š} = 𝑁 and max{|πœ‘π‘˜βˆ— (0)| : π‘˜ = 1, 2, 3, . . . , π‘š} = 𝑃. Setting 𝑇2βˆ— (𝑀 (Ξ¦ + 𝑇 + 1) + 𝑁 (2π‘š βˆ’ 1) + π‘ƒπ‘š Ξ“2βˆ— 󡄨 󡄨 + 󡄨󡄨󡄨𝛽󡄨󡄨󡄨 (𝑇 + 1)) ,

Ξ©2 =

(50)

we choose a constant π‘Ÿ such that π‘Ÿβ‰₯

Ξ©2 . 1 βˆ’ Ξ©1

Let π‘₯ ∈ π΅π‘Ÿ . For any 𝑑 ∈ π½π‘˜ , π‘˜ = 0, 1, 2, . . . , π‘š, we have

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𝛼 βˆ’2

|(Aπ‘₯) (𝑑)| ≀

π‘˜βˆ’1 (𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ [󡄨󡄨 󡄨󡄨 󡄨 󡄨 󡄨 1 󡄨 󡄨󡄨𝛽󡄨󡄨 π‘‘π‘˜ + βˆ‘ ((π‘‘π‘˜ βˆ’ 𝑑𝑗 ) (πΌπ‘‘π‘—βˆ’1 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠))󡄨󡄨󡄨 (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (π‘₯ (𝑑𝑗 ))󡄨󡄨󡄨󡄨)) Ξ“ (π›Όπ‘˜ βˆ’ 1) 𝑗=1 [ 𝛼 βˆ’1

π‘˜ (𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ 󡄨 󡄨 󡄨 󡄨 + βˆ‘ (𝐼𝑑2π‘—βˆ’1 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠))󡄨󡄨󡄨 (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘—βˆ— (π‘₯ (𝑑𝑗 ))󡄨󡄨󡄨󡄨)] + Ξ“ (π›Όπ‘˜ ) 𝑗=1 ]

π‘˜ [󡄨󡄨󡄨𝛽󡄨󡄨󡄨 + βˆ‘ (𝐼1 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠))󡄨󡄨󡄨 (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (π‘₯ (𝑑𝑗 ))󡄨󡄨󡄨󡄨)] π‘‘π‘—βˆ’1 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 𝑗=1 ] [

𝛼 βˆ’2

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ [󡄨󡄨 󡄨󡄨 𝛼 󡄨 󡄨 + πΌπ‘‘π‘˜π‘˜ 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠))󡄨󡄨󡄨 (𝑑) ≀ 󡄨𝛽󡄨 𝑑 Ξ“ (π›Όπ‘˜ βˆ’ 1) 󡄨 󡄨 π‘˜ [ π‘˜βˆ’1 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 + βˆ‘ ((π‘‘π‘˜ βˆ’ 𝑑𝑗 ) (𝐼𝑑1π‘—βˆ’1 (󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 0)󡄨󡄨󡄨 + 󡄨󡄨󡄨𝑓 (𝑠, 0)󡄨󡄨󡄨) (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (π‘₯ (𝑑𝑗 )) βˆ’ πœ‘π‘— (0)󡄨󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (π‘₯ (0))󡄨󡄨󡄨󡄨)) 𝑗=1 π‘˜

+ βˆ‘ (𝐼𝑑2π‘—βˆ’1 𝑗=1

𝛼 βˆ’1

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 (󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 0)󡄨󡄨󡄨 + 󡄨󡄨󡄨𝑓 (𝑠, 0)󡄨󡄨󡄨) (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘—βˆ— (π‘₯ (𝑑𝑗 )) βˆ’ πœ‘π‘—βˆ— (0)󡄨󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘—βˆ— (0)󡄨󡄨󡄨󡄨)] + Ξ“ (π›Όπ‘˜ ) ]

[󡄨󡄨󡄨𝛽󡄨󡄨󡄨 󡄨 󡄨 [

π‘˜ 󡄨 󡄨 󡄨 󡄨 𝛼 󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 + βˆ‘ (𝐼𝑑1π‘—βˆ’1 (󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 0)󡄨󡄨󡄨 + 󡄨󡄨󡄨𝑓 (𝑠, 0)󡄨󡄨󡄨) (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (π‘₯ (𝑑𝑗 )) βˆ’ πœ‘π‘— (0)󡄨󡄨󡄨󡄨 + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (0)󡄨󡄨󡄨󡄨)] + πΌπ‘‘π‘˜π‘˜ (󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 0)󡄨󡄨󡄨 𝑗=1 ] 𝛼 βˆ’2

π‘˜βˆ’1 (𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ [󡄨󡄨 󡄨󡄨 󡄨 󡄨 + 󡄨󡄨󡄨𝑓 (𝑠, 0)󡄨󡄨󡄨) (𝑑) ≀ 󡄨󡄨𝛽󡄨󡄨 π‘‘π‘˜ + (𝐿 1 π‘Ÿ + 𝑀) ( βˆ‘ (π‘‘π‘˜ βˆ’ 𝑑𝑗 ) (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) + (𝐿 2 π‘Ÿ + 𝑁) (π‘˜ βˆ’ 1) Ξ“ (π›Όπ‘˜ βˆ’ 1) 𝑗=1 [ 2

π‘˜

+ (𝐿 1 π‘Ÿ + 𝑀) (βˆ‘

(𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )

) + (𝐿 3 π‘Ÿ + 𝑃) π‘˜] + ]

2

𝑗=1

𝛼 βˆ’1

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ Ξ“ (π›Όπ‘˜ )

π‘˜

[󡄨󡄨󡄨𝛽󡄨󡄨󡄨 + (𝐿 1 π‘Ÿ + 𝑀) ( βˆ‘ (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) + (𝐿 2 π‘Ÿ + 𝑁) π‘˜] 󡄨 󡄨 𝑗=1 ] [

𝛼

+ (𝐿 1 π‘Ÿ + 𝑀)

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ . Ξ“ (π›Όπ‘˜ + 1)

Multiplying both sides of the above inequality by (𝑑 βˆ’ π‘‘π‘˜ )𝛾 for 𝑑 ∈ π½π‘˜ , we have 𝛾+𝛼 βˆ’2

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ [󡄨󡄨 󡄨󡄨 (𝑑 βˆ’ π‘‘π‘˜ ) |(Aπ‘₯) (𝑑)| ≀ 󡄨󡄨𝛽󡄨󡄨 π‘‘π‘˜ Ξ“ (π›Όπ‘˜ βˆ’ 1) [ 𝛾

π‘˜βˆ’1

+ (𝐿 1 π‘Ÿ + 𝑀) ( βˆ‘ (π‘‘π‘˜ βˆ’ 𝑑𝑗 ) (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) 𝑗=1

π‘˜

+ (𝐿 1 π‘Ÿ + 𝑀) ( βˆ‘ (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) + (𝐿 2 π‘Ÿ + 𝑁) π‘˜] 𝑗=1 ] 𝛾+𝛼

+ (𝐿 1 π‘Ÿ + 𝑀)

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ 𝑇2βˆ— [󡄨󡄨 󡄨󡄨 ≀ βˆ— 󡄨󡄨𝛽󡄨󡄨 𝑇 Ξ“2 Ξ“ (π›Όπ‘˜ + 1) [ π‘šβˆ’1

+ (𝐿 1 π‘Ÿ + 𝑀) ( βˆ‘ (π‘‘π‘š βˆ’ 𝑑𝑗 ) (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) 𝑗=1

+ (𝐿 2 π‘Ÿ + 𝑁) (π‘˜ βˆ’ 1) π‘˜

+ (𝐿 1 π‘Ÿ + 𝑀) ( βˆ‘

(𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )

𝑗=1

𝛾+𝛼 βˆ’1

+

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ Ξ“ (π›Όπ‘˜ )

+ (𝐿 2 π‘Ÿ + 𝑁) (π‘š βˆ’ 1) 2

[󡄨󡄨󡄨𝛽󡄨󡄨󡄨 󡄨 󡄨 [

2

) + (𝐿 3 π‘Ÿ + 𝑃) π‘˜] ]

π‘š

+ (𝐿 1 π‘Ÿ + 𝑀) (βˆ‘

𝑗=1 π‘š

(𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 ) 2

2

󡄨 󡄨 ) + (𝐿 3 π‘Ÿ + 𝑃) + 󡄨󡄨󡄨𝛽󡄨󡄨󡄨

+ (𝐿 1 π‘Ÿ + 𝑀) ( βˆ‘ (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) + (𝐿 2 π‘Ÿ + 𝑁) π‘š 𝑗=1

(52)

8

Discrete Dynamics in Nature and Society 𝑇2βˆ— (𝑀 (Ξ¦ + 𝑇 + 1) + 𝑁 (2π‘š βˆ’ 1) + π‘ƒπ‘š Ξ“2βˆ— 󡄨 󡄨 + 󡄨󡄨󡄨𝛽󡄨󡄨󡄨 (𝑇 + 1)) = Ξ©1 π‘Ÿ + Ξ©2 .

+

+ (𝐿 1 π‘Ÿ + 𝑀)] ] = π‘Ÿ[

(53)

𝑇2βˆ— (𝐿 (Ξ¦ + 𝑇 + 1) + 𝐿 2 (2π‘š βˆ’ 1) + 𝐿 3 π‘š)] Ξ“2βˆ— 1

This implies that Aπ΅π‘Ÿ βŠ‚ π΅π‘Ÿ . Finally we will show that A is a contraction mapping on π΅π‘Ÿ . For π‘₯, 𝑦 ∈ 𝑃𝐢𝛾 (𝐽, R) and for each 𝑑 ∈ 𝐽 we have

𝛼 βˆ’2

π‘˜βˆ’1 π‘˜ 󡄨 (𝑑 βˆ’ π‘‘π‘˜ ) 󡄨󡄨 [ βˆ‘ ((π‘‘π‘˜ βˆ’ 𝑑𝑗 ) (𝐼1 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 𝑦 (𝑠))󡄨󡄨󡄨 (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (π‘₯ (𝑑𝑗 )) βˆ’ πœ‘π‘— (𝑦 (𝑑𝑗 ))󡄨󡄨󡄨󡄨)) 󡄨󡄨(Aπ‘₯) (𝑑) βˆ’ (A𝑦) (𝑑)󡄨󡄨󡄨 ≀ π‘‘π‘—βˆ’1 󡄨 󡄨 󡄨 󡄨 Ξ“ (π›Όπ‘˜ βˆ’ 1) 𝑗=1 [ π‘˜ 󡄨 󡄨 󡄨 󡄨 + βˆ‘ (𝐼𝑑2π‘—βˆ’1 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 𝑦 (𝑠))󡄨󡄨󡄨 (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘—βˆ— (π‘₯ (𝑑𝑗 )) βˆ’ πœ‘π‘—βˆ— (𝑦 (𝑑𝑗 ))󡄨󡄨󡄨󡄨)] 𝑗=1 ] 𝛼 βˆ’1

+

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ Ξ“ (π›Όπ‘˜ )

(54)

π‘˜ [βˆ‘ (𝐼1 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 𝑦 (𝑠))󡄨󡄨󡄨 (𝑑𝑗 ) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‘π‘— (π‘₯ (𝑑𝑗 )) βˆ’ πœ‘π‘— (𝑦 (𝑑𝑗 ))󡄨󡄨󡄨󡄨)] + πΌπ‘‘π›Όπ‘˜ 󡄨󡄨󡄨𝑓 (𝑠, π‘₯ (𝑠)) βˆ’ 𝑓 (𝑠, 𝑦 (𝑠))󡄨󡄨󡄨 π‘‘π‘—βˆ’1 󡄨 󡄨 󡄨 π‘˜ 󡄨 󡄨 󡄨 ] [𝑗=1

β‹… (𝑑) .

Multiplying both sides of the above inequality by (𝑑 βˆ’ π‘‘π‘˜ )𝛾 , we have 󡄨 󡄨󡄨 󡄨󡄨(𝑑 βˆ’ π‘‘π‘˜ )𝛾 (Aπ‘₯) (𝑑) βˆ’ (𝑑 βˆ’ π‘‘π‘˜ )𝛾 (A𝑦) (𝑑)󡄨󡄨󡄨 󡄨 󡄨

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

2

+3π‘˜+5)/(π‘˜ +2π‘˜+3) 𝐷𝑑(2π‘˜ π‘₯= π‘˜

𝛾+𝛼 βˆ’2

≀

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ [ σ΅„©σ΅„© σ΅„© 𝐿 1 σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 Ξ“ (π›Όπ‘˜ βˆ’ 1) [

2𝑑 |π‘₯ (𝑑)| cos 𝑑 (7 + |π‘₯ (𝑑)|)2 + |π‘₯ (𝑑)| sin2 𝑑 1 + , 2 𝑑 ∈ [0, tanβˆ’1 (

π‘˜βˆ’1

σ΅„© σ΅„© β‹… ( βˆ‘ (π‘‘π‘˜ βˆ’ 𝑑𝑗 ) (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) + 𝐿 2 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 (π‘˜ βˆ’ 1) 𝑗=1

2

(𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 ) σ΅„© σ΅„© σ΅„© σ΅„© + 𝐿 1 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 ( βˆ‘ ) + 𝐿 3 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 π‘˜] 2 𝑗=1 ] π‘˜

𝛾+𝛼 βˆ’1

(𝑑 βˆ’ π‘‘π‘˜ ) π‘˜ + Ξ“ (π›Όπ‘˜ )

Μƒ (π‘‘π‘˜ ) = Ξ”π‘₯

(55)

π‘˜

σ΅„© σ΅„© (𝑑 βˆ’ π‘‘π‘˜ ) σ΅„© σ΅„© + 𝐿 2 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 π‘˜] + 𝐿 1 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 Ξ“ (π›Όπ‘˜ + 1) ] σ΅„© σ΅„© ≀ Ξ©1 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 .

(57) π‘˜ ), 3πœ‹

󡄨 󡄨󡄨 󡄨󡄨π‘₯ (π‘‘π‘˜ )󡄨󡄨󡄨 󡄨 󡄨, 37 (1 βˆ’ 2 cos π‘˜πœ‹) + 󡄨󡄨󡄨π‘₯ (π‘‘π‘˜ )󡄨󡄨󡄨

π‘˜ = 1, 2, . . . , 10, π‘‘π‘˜ = tanβˆ’1 (

𝛾+π›Όπ‘˜

π‘˜ ), 3πœ‹

π‘₯ (0) = 0 π‘₯ (0) = 𝑒. 𝐷𝑑2/3 0

It follows that σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©σ΅„©Aπ‘₯ βˆ’ A𝑦󡄩󡄩󡄩 ≀ Ξ©1 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 .

󡄨 󡄨󡄨 󡄨󡄨π‘₯ (π‘‘π‘˜ )󡄨󡄨󡄨 󡄨 󡄨, 9 (π‘˜2 + 3π‘˜ + 5) + 󡄨󡄨󡄨π‘₯ (π‘‘π‘˜ )󡄨󡄨󡄨

π‘˜ = 1, 2, . . . , 10, π‘‘π‘˜ = tanβˆ’1 ( Ξ”βˆ— π‘₯ (π‘‘π‘˜ ) =

[𝐿 1 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 ( βˆ‘ (𝑑𝑗 βˆ’ π‘‘π‘—βˆ’1 )) σ΅„© σ΅„© 𝑗=1 [

11 )] , 𝑑 =ΜΈ π‘‘π‘˜ , 3πœ‹

(56)

Since Ξ©1 < 1, A is a contraction mapping on π΅π‘Ÿ . Therefore (5) has a unique solution on [0, 𝑇].

Here π›Όπ‘˜ = (2π‘˜2 + 3π‘˜ + 5)/(π‘˜2 + 2π‘˜ + 3), π‘˜ = 0, 1, 2, . . . , 10, π‘š = 10, 𝑇 = tanβˆ’1 (11/(3πœ‹)), 𝑓(𝑑, π‘₯) = (2𝑑 |π‘₯|cos𝑑)/((7+|π‘₯|)2 + |π‘₯|sin2 𝑑)+(1/2), πœ‘π‘˜ (π‘₯) = |π‘₯|/(9(π‘˜2 +3π‘˜+5)+|π‘₯|), and πœ‘π‘˜βˆ— (π‘₯) = |π‘₯|/(37(1βˆ’2cosπ‘˜πœ‹)+|π‘₯|). Since |𝑓(𝑑, π‘₯)βˆ’π‘“(𝑑, 𝑦)| ≀ (2/49)|π‘₯βˆ’ 𝑦|, |πœ‘π‘˜ (π‘₯) βˆ’ πœ‘π‘˜ (𝑦)| ≀ (1/81)|π‘₯ βˆ’ 𝑦|, and |πœ‘π‘˜βˆ— (π‘₯) βˆ’ πœ‘π‘˜βˆ— (𝑦)| ≀ (3/37)|π‘₯βˆ’π‘¦|, 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 𝑇2βˆ— = 0.781307, Ξ“2βˆ— = 0.902745, Ξ¦ = 0.332114, and Ξ©1 =

𝑇2βˆ— (𝐿 (πœ™ + 𝑇 + 1) + 𝐿 2 (2π‘š βˆ’ 1) + 𝐿 3 π‘š) Ξ“2βˆ— 1

(58)

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

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.

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

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Volume 2014

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Discrete Dynamics in Nature and Society

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International Journal of

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Volume 2014