Báo cáo hóa học: " Research Article Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order" ppt

Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 404917, 16 pages doi:10.1155/2011/404917 Research Article Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order Liu Yang1, and Haibo Chen1 Department of Mathematics, Central South University, Changsha, Hunan 410075, China Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan 421008, China Correspondence should be addressed to Liu Yang, yangliu19731974@yahoo.com.cn Received 18 September 2010; Accepted January 2011 Academic Editor: Mouffak Benchohra Copyright q 2011 L Yang and H Chen 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 We study a nonlocal boundary value problem of impulsive fractional differential equations By means of a fixed point theorem due to O’Regan, we establish sufficient conditions for the existence of at least one solution of the problem For the illustration of the main result, an example is given Introduction Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, aerodynamics, chemistry, and engineering and biological sciences, involves derivatives of fractional order Fractional differential equations also provide an excellent tool for the description of memory and hereditary properties of many materials and processes In consequence, fractional differential equations have emerged as a significant development in recent years, see 1–3 As one of the important topics in the research differential equations, the boundary value problem has attained a great deal of attention from many researchers, see 4–11 and the references therein As pointed out in 12 , the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena There are three noteworthy papers dealing with the nonlocal boundary value problem of fractional differential equations Benchohra et al 12 investigated the following nonlocal boundary value problem c Dα u t f t, u t u0 g u, 0, < t < T, < α ≤ 2, u T where c Dα denotes the Caputo’s fractional derivative uT , 1.1 Advances in Difference Equations Zhong and Lin 13 studied the following nonlocal and multiple-point boundary value problem c Dα u t f t, u t < t < 1, < α ≤ 2, 0, 1.2 m−2 u0 g u, u0 u bi u ξi u1 i Ahmad and Sivasundaram 14 studied a class of four-point nonlocal boundary value problem of nonlinear integrodifferential equations of fractional order by applying some fixed point theorems On the other hand, impulsive differential equations of fractional order play an important role in theory and applications, see the references 15–21 and references therein However, as pointed out in 15, 16 , the theory of boundary value problems for nonlinear impulsive fractional differential equations is still in the initial stages Ahmad and Sivasundaram 15, 16 studied the following impulsive hybrid boundary value problems for fractional differential equations, respectively, c Dq u t Δu tk f t, u t Ik u t − k < q ≤ 2, t ∈ J1 0, Dq u t Δu tk Δu tk u0 c , u f t, u t Ik u t − k 0, tk ∈ 0, , k , u1 u 1 < q ≤ 2, t ∈ J1 0, Δu tk , J k u t− k 0, \ t1 , t2 , , , J k u t− k , 0, \ t1 , t2 , , , tk ∈ 0, , k 1, 2, , p, 1.4 q1 u s ds, βu 1.3 0, αu 1, 2, , p, αu q2 u s ds βu 0 Motivated by the facts mentioned above, in this paper, we consider the following problem: c Dq u t Δu tk f t, u t , u t , Ik u t − k αu , Δu tk βu < q ≤ 2, t ∈ J1 J k u t− k g1 u , , αu 0, \ t1 , t2 , , , tk ∈ 0, , k βu 1, 2, , p, 1.5 g2 u , where J 0, , f : J × Ê × Ê → Ê is a continuous function, and Ik , Jk : Ê → Ê are continuous functions, Δu tk u tk − u t− with u tk limh → u tk h , u t− k k 1, 2, , p, t0 < t1 < t2 < · · · < < 1, α > 0, β ≥ 0, and limh → 0− u tk h , k g1 , g2 : PC J, Ê → Ê are two continuous functions We will define PC J, Ê in Section To the best of our knowledge, this is the first time in the literatures that a nonlocal boundary value problem of impulsive differential equations of fractional order is considered Advances in Difference Equations In addition, the nonlinear term f t, u t , u t involves u t Evidently, problem 1.5 not only includes boundary value problems mentioned above but also extends them to a much wider case Our main tools are the fixed point theorem of O’Regan Some recent results in the literatures are generalized and significantly improved see Remark 3.6 The organization of this paper is as follows In Section 2, we will give some lemmas which are essential to prove our main results In Section 3, main results are given, and an example is presented to illustrate our main results Preliminaries At first, we present here the necessary definitions for fractional calculus theory These definitions and properties can be found in recent literature Definition 2.1 see 1–3 The Riemann-Liouville fractional integral of order α > of a function y : 0, ∞ → Ê is given by t Γα α I0 y t t−s α−1 y s ds, 2.1 where the right side is pointwise defined on 0, ∞ Definition 2.2 see 1–3 The Caputo fractional derivative of order α > of a function y : 0, ∞ → Ê is given by c Dα u t t Γ n−α t−s n−α−1 n y s ds, 2.2 where n α 1, α denotes the integer part of the number α, and the right side is pointwise defined on 0, ∞ Lemma 2.3 see 1–3 Let α > 0, then the fractional differential equation solutions ut where ci ∈ Ê, i 0, 1, , n − 1, n c0 c1 t q c2 t2 ··· cn−1 tn−1 , c Dq u t has 2.3 Lemma 2.4 see 1–3 Let α > 0, then one has α I0 c D α u t where ci ∈ Ê, i 0, 1, , n − 1, n u t q c0 c1 t c2 t2 ··· cn−1 tn−1 , 2.4 Second, we define {x : J → Ê; x ∈ C tk , tk PC J, Ê x tk , k 1, , p} with x t− k ,Ê , k 0, 1, , p and x tk , x t− exist k Advances in Difference Equations {x ∈ PC J, Ê ; x t ∈ C tk , tk , Ê , k 0, 1, , p 1, x tk , PC1 J, Ê x t− exist, and x is left continuous at tk , k 1, , p} Let C PC1 J, Ê ; it k is a Banach space with the norm x supt∈J { x t PC , x t PC }, where x PC supt∈J |x t | Like Definition 2.1 in 16 , we give the following definition Definition 2.5 A function u ∈ C with its Caputo derivative of order q existing on J1 is a solution of 1.5 if it satisfies 1.5 To deal with problem 1.5 , we first consider the associated linear problem and give its solution Lemma 2.6 Assume that ⎧ ⎨ t0 , t1 , Ji ⎩ ti , ti i i , 0, 1, 2, , p, 2.5 ⎧ ⎨0, ⎩ 1, Xt t ∈ J0 , t ∈ J0 For any σ ∈ C 0, , the solution of the problem c Δu tk Dq u t < q ≤ 2, t ∈ J1 σ t, Ik u t − k αu , Δu tk βu J k u t− k g1 u , 0, \ t1 , t2 , , , , αu tk ∈ 0, , k βu 1, 2, , p, g2 u is given by t ut ti t − s q−1 σ s ds Γ q β −t α − s q−1 σ s ds Γ q tk tk−1 0