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Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 456426, 19 pages doi:10.1155/2011/456426 Research Article Positive Solutions of nth-Order Nonlinear Impulsive Differential Equation with Nonlocal Boundary Conditions Meiqiang Feng,1 Xuemei Zhang,2 and Xiaozhong Yang2 School of Science, Beijing Information Science & Technology University, Beijing 100192, China Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China Correspondence should be addressed to Meiqiang Feng, meiqiangfeng@sina.com Received 25 March 2010; Accepted May 2010 Academic Editor: Feliz Manuel Minhos ´ Copyright q 2011 Meiqiang Feng 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 This paper is devoted to study the existence, nonexistence, and multiplicity of positive solutions for the nth-order nonlocal boundary value problem with impulse effects The arguments are based upon fixed point theorems in a cone An example is worked out to demonstrate the main results Introduction The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al ; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations; see, for instance, 3–14 and their references At the same time, a class of boundary value problems with integral boundary conditions arise naturally in thermal conduction problems 15 , semiconductor problems 16 , hydrodynamic problems 17 Such problems include two, three, and multipoint boundary value problems as special cases and attract much attention; see, for instance, 7, 8, 11, 18–44 and references cited therein In particular, we would like to mention some results of Eloe and Ahmad 19 and Pang et al 22 In 19 , by applying the fixed point Boundary Value Problems theorem in cones due to the work of Krasnosel’kii and Guo, Eloe and Ahmad established the existence of positive solutions of the following nth boundary value problem: xn t x a t f t, x t ··· x x t ∈ 0, , 0, x n−2 1.1 0, αx η In 22 , Pang et al considered the expression and properties of Green’s function for the nth-order m-point boundary value problem xn t x a t f x t ··· x 0, < t < 1, x n−2 0, 1.2 m−2 βi x ξi , x i where < ξ1 < ξ2 < · · · < ξm−2 < 1, βi > 0, m−2 βi ξim−1 < Furthermore, they obtained the i existence of positive solutions by means of fixed point index theory Recently, Yang and Wei 23 and the author of 24 improved and generalized the results of Pang et al 22 by using different methods, respectively On the other hand, it is well known that fixed point theorem of cone expansion and compression of norm type has been applied to various boundary value problems to show the existence of positive solutions; for example, see 7, 8, 11, 19, 23, 24 However, there are few papers investigating the existence of positive solutions of nth impulsive differential equations by using the fixed point theorem of cone expansion and compression The objective of the present paper is to fill this gap Being directly inspired by 19, 22 , using of the fixed point theorem of cone expansion and compression, this paper is devoted to study a class of nonlocal BVPs for nth-order impulsive differential equations with fixed moments Consider the following nth-order impulsive differential equations with integral boundary conditions: xn t −Δx n−1 |t x x ··· f t, x t tk 0, I k x tk , x n−2 t ∈ J, t / tk , k 1, 2, , m, 1.3 0, h t x t dt x 0, ∞ , tk k 1, 2, , m Here J 0, , f ∈ C J × R , R , Ik ∈ C R , R , and R where m is fixed positive integer are fixed points with < t1 < t2 < · · · < tk < · · · < tm < 1, Δx n−1 |t tk x n−1 tk − x n−1 t− , where x n−1 tk and x n−1 t− represent the right-hand k k limit and left-hand limit of x n−1 t at t tk , respectively, h ∈ L1 0, is nonnegative For the case of h ≡ 0, problem 1.3 reduces to the problem studied by Samo˘lenko and ı Perestyuk in By using the fixed point index theory in cones, the authors obtained some Boundary Value Problems sufficient conditions for the existence of at least one or two positive solutions to the two-point BVPs Motivated by the work above, in this paper we will extend the results of 4, 19, 22– 24 to problem 1.3 On the other hand, it is also interesting and important to discuss the 1, 2, , m, , n ≥ 2, and existence of positive solutions for problem 1.3 when Ik / k h / Many difficulties occur when we deal with them; for example, the construction of ≡ cone and operator So we need to introduce some new tools and methods to investigate the existence of positive solutions for problem 1.3 Our argument is based on fixed point theory in cones 45 To obtain positive solutions of 1.3 , the following fixed point theorem in cones is fundamental which can be found in 45, page 93 Lemma 1.1 Let Ω1 and Ω2 be two bounded open sets in Banach space E, such that ∈ Ω1 and Ω1 ⊂ Ω2 Let P be a cone in E and let operator A : P ∩ Ω2 \ Ω1 → P be completely continuous Suppose that one of the following two conditions is satisfied: ≤ i Ax/x, ∀x ∈ P ∩ ∂Ω1 ; Ax/x, ∀x ∈ P ∩ ∂Ω2 ; ≥ ≥ ii Ax/x, ∀x ∈ P ∩ ∂Ω1 ; Ax/x, ∀x ∈ P ∩ ∂Ω2 ≤ Then, A has at least one fixed point in P ∩ Ω2 \ Ω1 Preliminaries In order to define the solution of problem 1.3 , we will consider the following space Let J J \ {t1 , t2 , , tn }, and P Cn−1 0, x ∈ C 0, : x n−1 | tk ,tk ∈ C t k , tk , 2.1 x n−1 t− k x n−1 tk , ∃ x n−1 tk , k 1, 2, , m Then P Cn−1 0, is a real Banach space with norm x pcn−1 max x ∞, x ∞ , x ∞ , , x n−1 ∞ , 2.2 where x n−1 ∞ supt∈J |x n−1 t |, n 1, 2, A function x ∈ P Cn−1 0, ∩ Cn J is called a solution of problem 1.3 if it satisfies 1.3 To establish the existence of multiple positive solutions in P Cn−1 0, ∩ Cn J of problem 1.3 , let us list the following assumptions: H1 f ∈ C J × R , R , Ik ∈ C R , R ; H2 μ ∈ 0, , where μ h t tn−1 dt 4 Boundary Value Problems Lemma 2.1 Assume that H1 and H2 hold Then x ∈ P Cn−1 0, ∩ Cn J is a solution of problem 1.3 if and only if x is a solution of the following impulsive integral equation: m H t, s f s, x s ds x t H t, tk Ik x tk , 2.3 k where H t, s G1 t, s G2 t, s , ⎧ n−1 − s n−1 − t − s n−1 , ≤ s ≤ t ≤ 1, ⎨t n − ! ⎩tn−1 − s n−1 , ≤ t ≤ s ≤ 1, G1 t, s tn−1 G2 t, s 1− 0 Proof First suppose that x ∈ P Cn−1 0, ∩ Cn J see by integration of 1.3 that x n−1 t f s, x s ds x t − 2.6 is a solution of problem 1.3 It is easy to n−1 h t G1 t, s dt t x n−1 − 2.5 h t tn−1 dt 2.4 0 τ1 pcn−1 3.19 So, we have m t ∈ J ⇒ x00 t ≥ T x00 t ≥ H t, tk Ik x00 tk t∈ tm ,1 ≥ γ ∗ H0 k m Mk x00 tk 3.20 k ≥ γ ∗ H0 M∗ m x00 tk k From 3.20 , we obtain that x00 t1 ≥ γ ∗ H0 M∗ m x00 tk , k x00 t2 ≥ γ ∗ H0 M∗ m x00 tk , k 3.21 x00 tk ≥ γ ∗ H0 M∗ m x00 tk k So, we have m k x00 tk ≥ mγ ∗ H0 M∗ m x00 tk k 3.22 14 Boundary Value Problems From the definition of M∗ , we can find that m m x00 tk > m k x00 tk , x00 ∈ K, x00 pcn−1 R 3.23 k Similar to the proof in case , we can show that m x00 tk > Then, from 3.23 , k we have m < 1, which is a contraction Hence, 3.10 holds Applying i of Lemma 1.1 to 3.2 and 3.10 yields that T has a fixed point x ∈ K r,R {x : r ≤ x pcn−1 ≤ R} Thus, it follows that BVP 1.3 has at least one positive solution, and the theorem is proved Theorem 3.2 Assume that H1 and H2 hold In addition, letting f and Ik satisfy the following conditions: H5 f ∞ H6 f0 and I ∞ k ∞ or I0 k 0, k ∞, k 1, 2, , m; 1, 2, , m, BVP 1.3 has at least one positive solution Proof Considering H5 , there exists r > such that f t, x ≤ εr, Ik x ≤ εk r, and k m 1, 2, , m, for x ≥ r, t ∈ J, where ε, εk > satisfy max{H0 , G0 } ε k εk < Similar to the proof of 3.2 , we can show that T x/x, ≥ x ∈ K, x r pc1 3.24 Next, turning to H6 Under condition H6 , similar to the proof of 3.10 , we can also show that T x/x, ≤ x ∈ K, x R pc1 3.25 Applying i of Lemma 1.1 to 3.24 and 3.25 yields that T has a fixed point x ∈ K r,R {x : r ≤ x pcn−1 ≤ R} Thus, it follows that BVP 1.3 has one positive solution, and the theorem is proved Theorem 3.3 Assume that H1 , H2 , H3 , and H5 hold In addition, letting f and Ik satisfy the following condition: H7 there is a ς > such that γ ∗ ς ≤ x ≤ ς and t ∈ J implies f t, x ≥ τς, Ik x ≥ τk ς, k 1, 2, , 3.26 m m where τ, τk ≥ satisfy τ k τk > 0, τ tm H 1/2, s ds k τk H 1/2, tk > 1, BVP 1.3 has at least two positive solutions x∗ and x∗∗ with < x∗ pcn−1 < ς < x∗∗ pcn−1 Proof We choose ρ, ξ with < ρ < ς < ξ If H3 holds, similar to the proof of 3.2 , we can prove that T x/x, ≥ x ∈ K, x pc1 ρ 3.27 Boundary Value Problems 15 If H5 holds, similar to the proof of 3.24 , we have T x/x, ≥ x ∈ K, x pcn−1 ξ 3.28 T x/x, ≤ x ∈ K, x pcn−1 ς 3.29 Finally, we show that In fact, if there exists x2 ∈ K with x2 ς, then by 2.23 , we have pcn−1 x2 t ≥ γ ∗ x2 γ ∗ ς, pcn−1 3.30 and it follows from H7 that x2 t ≥ 1 , s f s, x2 s ds H tm ≥ς τ H tm >ς x2 , s ds m H k m τk H k 1 , tk , tk Ik x2 tk 3.31 pcn−1 , that is, x2 pcn−1 > x2 pcn−1 , which is a contraction Hence, 3.29 holds Applying Lemma 1.1 to 3.27 , 3.28 , and 3.29 yields that T has two fixed points x∗ , x∗∗ with x∗ ∈ K ρ,ς , x∗∗ ∈ K ς,ξ Thus it follows that BVP 1.3 has two positive solutions x∗ , x∗∗ with < x∗ pcn−1 < ς < x∗∗ pcn−1 The proof is complete Our last results corresponds to the case when problem 1.3 has no positive solution Write Δ H0 m 3.32 Theorem 3.4 Assume H1 , H2 , f t, x < Δ−1 x, t ∈ J, x > 0, and Ik x < Δ−1 x, ∀x > 0, then problem 1.3 has no positive solution 16 Boundary Value Problems Proof Assume to the contrary that problem 1.3 has a positive solution, that is, T has a fixed point y Then y ∈ K, y > for t ∈ 0, , and y ∞ ≤ m H s f s, y s ds H tk I k y tk < k H s Δ−1 y s ds H tk Δ−1 y k ≤ H0 Δ y H0 Δ−1 ∞ ∞ −1 H0 Δ y k m ∞ 3.33 m −1 y m y ∞ ∞ , which is a contradiction, and this completes the proof To illustrate how our main results can be used in practice we present an example Example 3.5 Consider the following boundary value problem: −x t t5 t ∈ J, t / , 1x5 x, −Δx |t1 1/2 x3 , 3.34 x x x 0, tx t dt x Conclusion BVP 3.34 has at least one positive solution Proof BVP 3.34 can be regarded as a BVP of the form 1.3 , where h t t, μ t · t3 dt G1 t, s 1 , t1 , f t, x t5 1x5 x, ⎧ 3 ⎨t − s − t − s , ≤ s ≤ t ≤ 1, ⎩t − s , ≤ t ≤ s ≤ 1, G2 t, s 3 t s − 2s2 24 3 s − s I1 x x3 , 3.35 Boundary Value Problems 17 It is not difficult to see that conditions H1 and H2 hold In addition, f0 lim 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W Ge, “Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces,” Journal of Computational... 1915–1926, 2010 X Zhang, X Yang, and W Ge, ? ?Positive solutions of nth-order impulsive boundary value problems with integral boundary conditions in Banach spaces,” Nonlinear Analysis Theory, Methods &... results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol 2009, Article ID 708576, p 11, 2009 Boundary