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 Mongkutβs 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 Banachβs fixed point theorem in an appropriate weighted space. Examples illustrating the main results are presented.
π
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 [1β3], books [4β7], 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: π
πΌ
π·π‘+π π’ (π‘) = π (π‘, π’ (π‘) , π’ (π (π‘))) , π
1 < πΌπ β€ 2,
Ξπ’ (π‘π ) = πΌπ (π’ (π‘π )) , Ξπ’σΈ (π‘π ) = πΌπβ (π’ (π‘π )) ,
π = 1, 2, . . . , π,
π½
π’ (0) = β π π Jπ‘+π π’ (ππ ) , π=0
π
π’σΈ (0) = 0, π‘π < ππ < π‘π+1 , (1) πΌ
π½
where π π·π‘+π is the Caputo fractional derivative of order πΌπ , Jπ‘+π π π is fractional Riemann-Liouville integral of order π½π > 0, π β πΆ(π½ Γ R Γ R, R), πΌπ , πΌπβ β πΆ(R, R), π β πΆ(π½, π½), π½ = [0, π] (π > 0), 0 = π‘0 < π‘1 < β
β
β
< π‘π < β
β
β
< π‘π < π‘π+1 = π, Ξπ’(π‘π ) = π’(π‘π+ )βπ’(π‘πβ ), and Ξπ’σΈ (π‘π ) = π’σΈ (π‘π+ )βπ’σΈ (π‘πβ ) where π’(π‘π+ ), π’σΈ (π‘π+ ) and π’(π‘πβ ), π’σΈ (π‘πβ ) denote the right and left hand limits of π’(π‘) and π’σΈ (π‘) at π‘ = π‘π (π = 1, 2, . . . , π). 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 [9β 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 πΌ with the lower limit 0 for a function π as π
πΌ
π·0 π (π‘) =
π‘ 1 β« (π‘ β π )πβπΌβ1 π(π) (π ) ππ . Ξ (π β πΌ) 0
(2)
If there are impulse points π‘π such that π‘π β (0, π‘) for some π β N, then the π(π) (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 < πΌπ β€ 1 of the form πΌ
π·π‘ππ π₯ (π‘) = π (π‘, π₯ (π‘)) , Μ (π‘π ) = ππ (π₯ (π‘π )) , Ξπ₯
π‘ β π½, π‘ =ΜΈ π‘π , π = 1, 2, 3, . . . , π,
(3)
π₯ (0) = 0, πΌ
where π·π‘ππ is the Riemann-Liouville fractional derivative of order 0 < πΌπ β€ 1 on intervals π½π , π = 0, 1, 2, . . . , π, 0 = π‘0 < π‘1 < π‘2 < β
β
β
< π‘π < β
β
β
< π‘π < π‘π+1 = π, π : π½ Γ R β R is a Μ π) continuous function, and ππ β πΆ(R, R). The notation Ξπ₯(π‘ is defined by Μ (π‘π ) = πΌπ‘1βπΌπ π₯ (π‘+ ) β πΌπ‘1βπΌπβ1 π₯ (π‘π ) , Ξπ₯ π π πβ1 π = 1, 2, 3, . . . , π,
(4)
1βπΌ
where πΌπ‘π π is the Riemann-Liouville fractional integral of order 1 β πΌπ on interval π½π . It should be noticed that if πΌπ = Μ π ) = Ξπ₯(π‘π ) = π₯(π‘+ ) β π₯(π‘π ) for π = 1 in (4), then Ξπ₯(π‘ π 1, 2, 3, . . . , π. In Section 4, we investigate the initial value problem of impulsive Riemann-Liouville fractional differential equations of the form πΌ
π·π‘ππ π₯ (π‘) = π (π‘, π₯ (π‘)) ,
π‘ β π½, π‘ =ΜΈ π‘π ,
Μ (π‘π ) = ππ (π₯ (π‘π )) , Ξπ₯
π = 1, 2, 3, . . . , π,
Ξβ π₯ (π‘π ) = ππβ (π₯ (π‘π )) ,
π = 1, 2, 3, . . . , π,
(5)
π·πΌ0 β1 π₯ (0) = π½,
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 πΌ > 0 of a continuous function π : (π, π) β R is defined by π·ππΌ π (π‘) =
1 π π π‘ ( ) β« (π‘ β π )πβπΌβ1 π (π ) ππ , Ξ (π β πΌ) ππ‘ π
(7)
π β 1 < πΌ < π, π‘ β (π, π) , where π = [πΌ] + 1, [πΌ] denotes the integer part of a real number πΌ, provided the right-hand side is pointwise defined on (π, π), where Ξ is the gamma function defined by Ξ(πΌ) = β β«0 πβπ π πΌβ1 ππ . Definition 2. For at least π-times differentiable function π : (π, π) β R, the Caputo derivative of fractional order πΌ is defined as π‘ 1 π πΌ β« (π‘ β π )πβπΌβ1 π(π) (π ) ππ , π·π π (π‘) = Ξ (π β πΌ) π (8) π β 1 < πΌ < π, π‘ β (π, π) , where π = [πΌ] + 1. Definition 3. The Riemann-Liouville fractional integral of order π½ > 0 of a continuous function π : (π, π) β R is defined by π‘ 1 β« (π‘ β π )πΌβ1 π (π ) ππ , Ξ (πΌ) π
π‘ β (π, π)
(9)
provided the right-hand side is pointwise defined on (π, π).
where π½ β R, 0 = π‘0 < π‘1 < π‘2 < β
β
β
< π‘π < β
β
β
< π‘π < π‘π+1 = π, π : π½ Γ R β R is a continuous function, ππ , ππβ β πΌ πΆ(R, R) for π = 1, 2, . . . , π, and π·π‘ππ is the Riemann-Liouville fractional derivative of order 1 < πΌπ β€ 2 on intervals π½π for Μ π ) is defined by (4) and π = 0, 1, 2, . . . , π. The notation Ξπ₯(π‘ β Ξ π₯(π‘π ) is defined by 2βπΌπ
2. Preliminaries
πΌππΌ π (π‘) =
π₯ (0) = 0,
Ξβ π₯ (π‘π ) = πΌπ‘π
2βπΌ
where πΌπ‘π π is the Riemann-Liouville fractional integral of order 2 β πΌπ > 0 on π½π . It should be noticed that if πΌπ = 2 Μ π ) = π·π‘ π₯(π‘+ ) β π·π‘ π₯(π‘π ) and Ξβ π₯(π‘π ) = in (6), then Ξπ₯(π‘ π π πβ1 + Ξπ₯(π‘π ) = π₯(π‘π ) β π₯(π‘π ) for π = 1, 2, . . . , π. By using Banachβs 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.
2βπΌ
π₯ (π‘π+ ) β πΌπ‘πβ1 πβ1 π₯ (π‘π ) , π = 1, 2, . . . , π,
(6)
Lemma 4 (see [5]). Let πΌ > 0 and π₯ β πΆ(π, π) β© πΏ(π, π). Then the fractional differential equation π·ππΌ π₯ (π‘) = 0
(10)
has a unique solution π₯ (π‘) = π1 (π‘ β π)πΌβ1 + π2 (π‘ β π)πΌβ2 + β
β
β
+ ππ (π‘ β π)πΌβπ , where ππ β R, π = 1, 2, . . . , π, and π β 1 < π < π.
(11)
Discrete Dynamics in Nature and Society
3
Lemma 5 (see [5]). Let πΌ > 0. Then for π₯ β πΆ(π, π) β© πΏ(π, π) it holds (12)
3. Impulsive Riemann-Liouville Fractional Differential Equations of Orders 0 < πΌπ β€ 1 Let π½ = [0, π], π½0 = [π‘0 , π‘1 ], π½π = (π‘π , π‘π+1 ] for π = 1, 2, 3, . . . , π. Let ππΆ(π½, R) = {π₯ : π½ β R, π₯(π‘) is continuous everywhere except for some π‘π at which π₯(π‘π+ ) and π₯(π‘πβ ) exist, and π₯(π‘πβ ) = π₯(π‘π ), π = 1, 2, 3, . . . , π}. For πΎ β R+ , we introduce the space πΆπΎ,π (π½π , R) = {π₯ : π½π β R : (π‘ β π‘π )πΎ π₯(π‘) β πΆ(π½π , R)} with the norm βπ₯βπΆπΎ,π = supπ‘βπ½π |(π‘ β π‘π )πΎ π₯(π‘)| and ππΆπΎ = {π₯ : π½ β R : for each π‘ β π½π and (π‘ β π‘π )πΎ π₯(π‘) β πΆ(π½π , R), π = 0, 1, 2, . . . , π} with the norm βπ₯βππΆπΎ = max{supπ‘βπ½π |(π‘ β π‘π )πΎ π₯(π‘)| : π = 0, 1, 2, . . . , π}. Clearly ππΆπΎ is a Banach space. In this section we study problem (3). Lemma 6. If π₯ β ππΆ(π½, R) is a solution of (3), then, for any π‘ β π½π , π = 0, 1, 2, . . . , π, πΌ β1
[ β (πΌπ‘1πβ1 π (π‘π , π₯ (π‘π )) 0 1 for π = 0, 1, 2, . . . , π, then the initial value problem (3) has a unique solution on [0, π].
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Proof. In view of Lemma 6, we define the operator K : ππΆ β ππΆ as πΌ β1
Kπ₯ (π‘) =
(π‘ β π‘π ) π Ξ (πΌπ )
[ β (πΌπ‘1πβ1 π (π‘π , π₯ (π‘π )) 0 0, βπ₯, π¦ β R. (46)
πΌ
+ πΌπ‘ππ π (π‘, π₯ (π‘)) , for π‘ β π½π , π = 0, 1, 2, . . . , π with βππ=π = 0 for π < π. It is straightforward to show that Aπ₯ β ππΆπΎ (π½, R); see Theorem 7. Next we will show that Aπ΅π β π΅π , where a ball π΅π is defined by π΅π = {π₯ β ππΆπΎ (π½, R), βπ₯βππΆπΎ β€ π}. Assume that supπ‘βπ½ |π(π‘, 0)| = π, max{|ππ (0)| : π = 1, 2, 3, . . . , π} = π and max{|ππβ (0)| : π = 1, 2, 3, . . . , π} = π. Setting π2β (π (Ξ¦ + π + 1) + π (2π β 1) + ππ Ξ2β σ΅¨ σ΅¨ + σ΅¨σ΅¨σ΅¨π½σ΅¨σ΅¨σ΅¨ (π + 1)) ,
Ξ©2 =
(50)
we choose a constant π such that πβ₯
Ξ©2 . 1 β Ξ©1
Let π₯ β π΅π . For any π‘ β π½π , π = 0, 1, 2, . . . , π, we have
(51)
Discrete Dynamics in Nature and Society
7
πΌ β2
|(Aπ₯) (π‘)| β€
πβ1 (π‘ β π‘π ) π [σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨ σ΅¨ σ΅¨ 1 σ΅¨ σ΅¨σ΅¨π½σ΅¨σ΅¨ π‘π + β ((π‘π β π‘π ) (πΌπ‘πβ1 σ΅¨σ΅¨σ΅¨π (π , π₯ (π ))σ΅¨σ΅¨σ΅¨ (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππ (π₯ (π‘π ))σ΅¨σ΅¨σ΅¨σ΅¨)) Ξ (πΌπ β 1) π=1 [ πΌ β1
π (π‘ β π‘π ) π σ΅¨ σ΅¨ σ΅¨ σ΅¨ + β (πΌπ‘2πβ1 σ΅¨σ΅¨σ΅¨π (π , π₯ (π ))σ΅¨σ΅¨σ΅¨ (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππβ (π₯ (π‘π ))σ΅¨σ΅¨σ΅¨σ΅¨)] + Ξ (πΌπ ) π=1 ]
π [σ΅¨σ΅¨σ΅¨π½σ΅¨σ΅¨σ΅¨ + β (πΌ1 σ΅¨σ΅¨σ΅¨π (π , π₯ (π ))σ΅¨σ΅¨σ΅¨ (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππ (π₯ (π‘π ))σ΅¨σ΅¨σ΅¨σ΅¨)] π‘πβ1 σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ π=1 ] [
πΌ β2
(π‘ β π‘π ) π [σ΅¨σ΅¨ σ΅¨σ΅¨ πΌ σ΅¨ σ΅¨ + πΌπ‘ππ σ΅¨σ΅¨σ΅¨π (π , π₯ (π ))σ΅¨σ΅¨σ΅¨ (π‘) β€ σ΅¨π½σ΅¨ π‘ Ξ (πΌπ β 1) σ΅¨ σ΅¨ π [ πβ1 σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ + β ((π‘π β π‘π ) (πΌπ‘1πβ1 (σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , 0)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π (π , 0)σ΅¨σ΅¨σ΅¨) (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππ (π₯ (π‘π )) β ππ (0)σ΅¨σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨σ΅¨ππ (π₯ (0))σ΅¨σ΅¨σ΅¨σ΅¨)) π=1 π
+ β (πΌπ‘2πβ1 π=1
πΌ β1
(π‘ β π‘π ) π σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ (σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , 0)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π (π , 0)σ΅¨σ΅¨σ΅¨) (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππβ (π₯ (π‘π )) β ππβ (0)σ΅¨σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨σ΅¨ππβ (0)σ΅¨σ΅¨σ΅¨σ΅¨)] + Ξ (πΌπ ) ]
[σ΅¨σ΅¨σ΅¨π½σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨ [
π σ΅¨ σ΅¨ σ΅¨ σ΅¨ πΌ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ σ΅¨ + β (πΌπ‘1πβ1 (σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , 0)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π (π , 0)σ΅¨σ΅¨σ΅¨) (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππ (π₯ (π‘π )) β ππ (0)σ΅¨σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨σ΅¨ππ (0)σ΅¨σ΅¨σ΅¨σ΅¨)] + πΌπ‘ππ (σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , 0)σ΅¨σ΅¨σ΅¨ π=1 ] πΌ β2
πβ1 (π‘ β π‘π ) π [σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨ σ΅¨ + σ΅¨σ΅¨σ΅¨π (π , 0)σ΅¨σ΅¨σ΅¨) (π‘) β€ σ΅¨σ΅¨π½σ΅¨σ΅¨ π‘π + (πΏ 1 π + π) ( β (π‘π β π‘π ) (π‘π β π‘πβ1 )) + (πΏ 2 π + π) (π β 1) Ξ (πΌπ β 1) π=1 [ 2
π
+ (πΏ 1 π + π) (β
(π‘π β π‘πβ1 )
) + (πΏ 3 π + π) π] + ]
2
π=1
πΌ β1
(π‘ β π‘π ) π Ξ (πΌπ )
π
[σ΅¨σ΅¨σ΅¨π½σ΅¨σ΅¨σ΅¨ + (πΏ 1 π + π) ( β (π‘π β π‘πβ1 )) + (πΏ 2 π + π) π] σ΅¨ σ΅¨ π=1 ] [
πΌ
+ (πΏ 1 π + π)
(π‘ β π‘π ) π . Ξ (πΌπ + 1)
Multiplying both sides of the above inequality by (π‘ β π‘π )πΎ for π‘ β π½π , we have πΎ+πΌ β2
(π‘ β π‘π ) π [σ΅¨σ΅¨ σ΅¨σ΅¨ (π‘ β π‘π ) |(Aπ₯) (π‘)| β€ σ΅¨σ΅¨π½σ΅¨σ΅¨ π‘π Ξ (πΌπ β 1) [ πΎ
πβ1
+ (πΏ 1 π + π) ( β (π‘π β π‘π ) (π‘π β π‘πβ1 )) π=1
π
+ (πΏ 1 π + π) ( β (π‘π β π‘πβ1 )) + (πΏ 2 π + π) π] π=1 ] πΎ+πΌ
+ (πΏ 1 π + π)
(π‘ β π‘π ) π π2β [σ΅¨σ΅¨ σ΅¨σ΅¨ β€ β σ΅¨σ΅¨π½σ΅¨σ΅¨ π Ξ2 Ξ (πΌπ + 1) [ πβ1
+ (πΏ 1 π + π) ( β (π‘π β π‘π ) (π‘π β π‘πβ1 )) π=1
+ (πΏ 2 π + π) (π β 1) π
+ (πΏ 1 π + π) ( β
(π‘π β π‘πβ1 )
π=1
πΎ+πΌ β1
+
(π‘ β π‘π ) π Ξ (πΌπ )
+ (πΏ 2 π + π) (π β 1) 2
[σ΅¨σ΅¨σ΅¨π½σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨ [
2
) + (πΏ 3 π + π) π] ]
π
+ (πΏ 1 π + π) (β
π=1 π
(π‘π β π‘πβ1 ) 2
2
σ΅¨ σ΅¨ ) + (πΏ 3 π + π) + σ΅¨σ΅¨σ΅¨π½σ΅¨σ΅¨σ΅¨
+ (πΏ 1 π + π) ( β (π‘π β π‘πβ1 )) + (πΏ 2 π + π) π π=1
(52)
8
Discrete Dynamics in Nature and Society π2β (π (Ξ¦ + π + 1) + π (2π β 1) + ππ Ξ2β σ΅¨ σ΅¨ + σ΅¨σ΅¨σ΅¨π½σ΅¨σ΅¨σ΅¨ (π + 1)) = Ξ©1 π + Ξ©2 .
+
+ (πΏ 1 π + π)] ] = π[
(53)
π2β (πΏ (Ξ¦ + π + 1) + πΏ 2 (2π β 1) + πΏ 3 π)] Ξ2β 1
This implies that Aπ΅π β π΅π . Finally we will show that A is a contraction mapping on π΅π . For π₯, π¦ β ππΆπΎ (π½, R) and for each π‘ β π½ we have
πΌ β2
πβ1 π σ΅¨ (π‘ β π‘π ) σ΅¨σ΅¨ [ β ((π‘π β π‘π ) (πΌ1 σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , π¦ (π ))σ΅¨σ΅¨σ΅¨ (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππ (π₯ (π‘π )) β ππ (π¦ (π‘π ))σ΅¨σ΅¨σ΅¨σ΅¨)) σ΅¨σ΅¨(Aπ₯) (π‘) β (Aπ¦) (π‘)σ΅¨σ΅¨σ΅¨ β€ π‘πβ1 σ΅¨ σ΅¨ σ΅¨ σ΅¨ Ξ (πΌπ β 1) π=1 [ π σ΅¨ σ΅¨ σ΅¨ σ΅¨ + β (πΌπ‘2πβ1 σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , π¦ (π ))σ΅¨σ΅¨σ΅¨ (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππβ (π₯ (π‘π )) β ππβ (π¦ (π‘π ))σ΅¨σ΅¨σ΅¨σ΅¨)] π=1 ] πΌ β1
+
(π‘ β π‘π ) π Ξ (πΌπ )
(54)
π [β (πΌ1 σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , π¦ (π ))σ΅¨σ΅¨σ΅¨ (π‘π ) + σ΅¨σ΅¨σ΅¨σ΅¨ππ (π₯ (π‘π )) β ππ (π¦ (π‘π ))σ΅¨σ΅¨σ΅¨σ΅¨)] + πΌπ‘πΌπ σ΅¨σ΅¨σ΅¨π (π , π₯ (π )) β π (π , π¦ (π ))σ΅¨σ΅¨σ΅¨ π‘πβ1 σ΅¨ σ΅¨ σ΅¨ π σ΅¨ σ΅¨ σ΅¨ ] [π=1
β
(π‘) .
Multiplying both sides of the above inequality by (π‘ β π‘π )πΎ , we have σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨(π‘ β π‘π )πΎ (Aπ₯) (π‘) β (π‘ β π‘π )πΎ (Aπ¦) (π‘)σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨
Example 11. Consider the following impulsive multiorders Riemann-Liouville fractional initial value problem: 2
2
+3π+5)/(π +2π+3) π·π‘(2π π₯= π
πΎ+πΌ β2
β€
(π‘ β π‘π ) π [ σ΅©σ΅© σ΅© πΏ 1 σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© Ξ (πΌπ β 1) [
2π‘ |π₯ (π‘)| cos π‘ (7 + |π₯ (π‘)|)2 + |π₯ (π‘)| sin2 π‘ 1 + , 2 π‘ β [0, tanβ1 (
πβ1
σ΅© σ΅© β
( β (π‘π β π‘π ) (π‘π β π‘πβ1 )) + πΏ 2 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© (π β 1) π=1
2
(π‘π β π‘πβ1 ) σ΅© σ΅© σ΅© σ΅© + πΏ 1 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© ( β ) + πΏ 3 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© π] 2 π=1 ] π
πΎ+πΌ β1
(π‘ β π‘π ) π + Ξ (πΌπ )
Μ (π‘π ) = Ξπ₯
(55)
π
σ΅© σ΅© (π‘ β π‘π ) σ΅© σ΅© + πΏ 2 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© π] + πΏ 1 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© Ξ (πΌπ + 1) ] σ΅© σ΅© β€ Ξ©1 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© .
(57) π ), 3π
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π₯ (π‘π )σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨, 37 (1 β 2 cos ππ) + σ΅¨σ΅¨σ΅¨π₯ (π‘π )σ΅¨σ΅¨σ΅¨
π = 1, 2, . . . , 10, π‘π = tanβ1 (
πΎ+πΌπ
π ), 3π
π₯ (0) = 0 π₯ (0) = π. π·π‘2/3 0
It follows that σ΅© σ΅©σ΅© σ΅© σ΅© σ΅©σ΅©Aπ₯ β Aπ¦σ΅©σ΅©σ΅© β€ Ξ©1 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© .
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π₯ (π‘π )σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨, 9 (π2 + 3π + 5) + σ΅¨σ΅¨σ΅¨π₯ (π‘π )σ΅¨σ΅¨σ΅¨
π = 1, 2, . . . , 10, π‘π = tanβ1 ( Ξβ π₯ (π‘π ) =
[πΏ 1 σ΅©σ΅©σ΅©π₯ β π¦σ΅©σ΅©σ΅© ( β (π‘π β π‘πβ1 )) σ΅© σ΅© π=1 [
11 )] , π‘ =ΜΈ π‘π , 3π
(56)
Since Ξ©1 < 1, A is a contraction mapping on π΅π . Therefore (5) has a unique solution on [0, π].
Here πΌπ = (2π2 + 3π + 5)/(π2 + 2π + 3), π = 0, 1, 2, . . . , 10, π = 10, π = tanβ1 (11/(3π)), π(π‘, π₯) = (2π‘ |π₯|cosπ‘)/((7+|π₯|)2 + |π₯|sin2 π‘)+(1/2), ππ (π₯) = |π₯|/(9(π2 +3π+5)+|π₯|), and ππβ (π₯) = |π₯|/(37(1β2cosππ)+|π₯|). Since |π(π‘, π₯)βπ(π‘, π¦)| β€ (2/49)|π₯β π¦|, |ππ (π₯) β ππ (π¦)| β€ (1/81)|π₯ β π¦|, and |ππβ (π₯) β ππβ (π¦)| β€ (3/37)|π₯βπ¦|, then (π»1 ), (π»2 ), and (π»3 ) are satisfied with πΏ 1 = 2/49, πΏ 2 = 1/81, and πΏ 3 = 3/37, respectively. By choosing
Discrete Dynamics in Nature and Society
9
πΎ = 2, we can find that π2β = 0.781307, Ξ2β = 0.902745, Ξ¦ = 0.332114, and Ξ©1 =
π2β (πΏ (π + π + 1) + πΏ 2 (2π β 1) + πΏ 3 π) Ξ2β 1
(58)
= 0.982275 < 1. Hence, by Theorem 10, the initial value problem (57) has a unique solution on [0, tanβ1 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, βExistence of fractional neutral functional differential equations,β Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1095β1100, 2010. [2] B. Ahmad and J. J. Nieto, βBoundary value problems for a class of sequential integrodifferential equations of fractional order,β Journal of Function Spaces and Applications, vol. 2013, Article ID 149659, 8 pages, 2013. [3] B. Ahmad and S. K. Ntouyas, βFractional differential inclusions with fractional separated boundary conditions,β Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 362β382, 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, βA new impulsive multi-orders fractional differential equation involving multipoint fractional integral boundary conditions,β Abstract and Applied Analysis, vol. 2014, Article ID 932747, 10 pages, 2014. [9] C. Bai, βImpulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative,β Journal of Mathematical Analysis and Applications, vol. 384, no. 2, pp. 211β231, 2011. [10] M. FeΛckan, Y. Zhou, and J. Wang, βOn the concept and existence of solution for impulsive fractional differential equations,β
Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050β3060, 2012. [11] X. Zhang, X. Zhang, and M. Zhang, βOn the concept of general solution for impulsive differential equations of fractional order π β (0, 1),β Applied Mathematics and Computation, vol. 247, pp. 72β89, 2014.
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