Hindawi Publishing Corporation International Journal of Differential Equations Volume 2011, Article ID 808175, 12 pages doi:10.1155/2011/808175 Research Article The Existence of Positive Solutions for Singular Impulse Periodic Boundary Value Problem Zhaocai Hao and Tanggui Chen Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China Correspondence should be addressed to Zhaocai Hao, zchjal@163.com Received 15 May 2011; Accepted 1 July 2011 Academic Editor: Jian-Ping Sun Copyright q 2011 Z. Hao and T. 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 obtain new result of the existence of positive solutions of a class of singular impulse periodic boundary value problem via a nonlinear alternative principle of Leray-Schauder. We do not require the monotonicity of functions in paper Zhang and Wang, 2003. An example is also given to illustrate our result. 1. Introduction Because of wide interests in physics and engineering, periodic boundary value problems have been investigated by many authors see 1–19. In most real problems, only the positive solution is significant. In this paper, we consider the following periodic boundary value problem PBVP in short with impulse effects: Δu|ttk −u t Mut ft, ut, t ∈ J , Ik utk , −Δu ttk Jk utk , k 1, 2, . . . l, u0 u2π, 1.1 u 0 u 2π. Here, J 0, 2π, 0 < t1 < t2 < · · · < tl < 2π, J J \{t1 , t2 , . . . , tl }, M > 0, f ∈ CJ ×R , R , Ik ∈ CR , R, Jk ∈√CR , R , R 0, ∞, R 0, ∞ with −1/mJk u < Ik u < 1/mJk u, u ∈ R , m M. Δu|ttk utk − ut−k , Δu |ttk u tk − u t−k , where ui tk and ui t−k , i 0, 1, respectively, denote the right and left limit of ui t at t tk . 2 International Journal of Differential Equations In 7, Liu applied Krasnoselskii’s and Leggett-Williams fixed-point theorem to establish the existence of at least one, two, or three positive solutions to the first-order periodic boundary value problems x t atxt ft, xt, a.e. t ∈ 0, T \ t1 , . . . , tp , Δx|ttk Ik xtk , k 1, . . . , p, 1.2 x0 xT . Jiang 5 has applied Krasnoselskii’s fixed point theorem to establish the existence of positive solutions of problem x t Mxt ft, xt, x0 x2π, t ∈ 0, 2π, x 0 x 2π. 1.3 The work 5 proved that periodic boundary value problem PBVP in short 1.3 without singularity have at least one positive solutions provided ft, x is superlinear or sublinear at x 0 and x ∞. In 14, Tian et al. researched PBVP 1.1 without singularity. They obtained the existence of multiple positive solutions of PBVP 1.1 by replacing the suplinear condition or sublinear condition of 4 with the following limit inequality condition: A1 l 2πf0 J0 i σ > 2πM, l 2πf∞ J∞ i σ > 2πM, i1 1.4 i1 A2 2πf 0 l i1 J i σ < 2πM, 0 2πf ∞ l ∞ J i σ < 2πM. 1.5 i1 Nieto 10 introduced the concept of a weak solution for a damped linear equation with Dirichlet boundary conditions and impulses. These results will allow us in the future to deal with the corresponding nonlinear problems and look for solutions as critical points of weakly lower semicontinuous functionals. We note that the function f involved in above papers 5, 7, 10, 14 does not have singularity. Xiao et al. 16 investigate the multiple positive solutions of singular boundary value problem for second-order impulsive singular differential equations on the halfline, where the function ft, u is singular only at t 0 and/or t 1. Reference 19 studied PBVP 1.3, where the function f has singularity at x 0. The authors present the existence of multiple positive solutions via the Krasnoselskii’s fixed point theorem under the following conditions. International Journal of Differential Equations 3 A1 There exist nonnegative valued ξx, ηx ∈ C0, ∞ and P t, Qt ∈ L1 0, 2π such that 0 ≤ ft, x ≤ P tξx Qtηx, a.e. t, x ∈ 0, 2π × 0, ∞, ⎧ ⎫ ⎨ ⎬ x > Bj , sup 2π 2π x∈0,∞⎩ P tdtξx/ηx 0 Qtdt η δj t ⎭ 0 1.6 where ηx is nonincreasing and ξx/ηx is nondecreasing on 0, ∞, A2 min lim inf 2π 0 fx, wdx : δj t ≤ w ≤ t t t → 0 > 1 , Aj 1.7 > 1 . Aj 1.8 A3 min lim inf 2π 0 fx, wdx : δj t ≤ w ≤ t t → ∞ t Here, δj , Aj , Bj are some constants. In this paper, the nonlinear term ft, u is singular at u 0, and positive solution of PBVP 1.1 is obtained by a nonlinear alternative principle of Leray-Schauder type in cone. We do not require the monotonicity of functions η, ξ/η used in 19. An example is also given to illustrate our result. This paper is organized as follows. In Section 1, we give a brief overview of recent results on impulsive and periodic boundary value problems. In Section 2, we present some preliminaries such as definitions and lemmas. In Section 3, the existence of one positive solution for PBVP 1.1 will be established by using a nonlinear alternative principle of LeraySchauder type in cone. An example is given in Section 4. 2. Preliminaries Consider the space P CJ, R {u : u is a map from J into R such that ut is continuous at t/ tk , left continuous at t tk , and utk exists, for k 1, 2, . . . l.}. It is easy to say that P CJ, R is a Banach space with the norm upc supt∈J |ut|. Let P C1 J, R {u ∈ P CJ, R : u t tk , and u tk , u t−k exist and u t is left continuous at exists at t / tk and is continuous at t / t tk , for k 1, 2, . . . l.} with the norm upc1 max{upc , u pc }. Then, P C1 J, R is also a Banach space. 4 International Journal of Differential Equations Lemma 2.1 see 15. u ∈ P C1 J, R ∩ C2 J , R is a solution of PBVP 1.1 if and only if u ∈ P CJ is a fixed point of the following operator T : T ut 2π Gt, sfs, usds 0 l k1 l ∂Gt, s Gt, tk Jk utk ∂s k1 Ik utk , 2.1 stk where Gt, s is the Green’s function to the following periodic boundary value problem: −u Mu 0, u0 u2π, Gt, s : u 0 u 2π, 2.2 ⎧⎧ ⎪⎪emt−s em2π−ts , 1 ⎨⎨ 0 ≤ s ≤ t ≤ 2π, Γ⎪ ⎩⎪ ⎩ems−t em2π−st , 0 ≤ t ≤ s ≤ 2π, here, Γ 2me2mπ − 1. It is clear that 2emπ e2mπ 1 Gπ ≤ Gt, s ≤ G0 . Γ Γ 2.3 K u ∈ P CJ, R : ut ≥ σupc , t ∈ J , 2.4 Define where σ 1 e2mπ . 2.5 The following nonlinear alternative principle of Leray-Schauder type in cone is very important for us. Lemma 2.2 see 4. Assume that Ω is a relatively open subset of a convex set K in a Banach space P CJ, R. Let T : Ω → K be a compact map with 0 ∈ Ω. Then, either i T has a fixed point in Ω, or, ii there is a u ∈ ∂Ω and a λ < 1 such that u λT u. 3. Main Results In this section, we establish the existence of positive solutions of PBVP 1.1. International Journal of Differential Equations 5 Theorem 3.1. Assume that the following three hypothesis hold: H1 there exists nonnegative functions ξu, ηu, γu ∈ C0, ∞ and pt, qt ∈ L1 0, 2π such that ft, u ≤ ptξu qtηu, max Jk u ≤ γu, t, u ∈ 0, 2π × 0, ∞, 3.1 t, u ∈ 0, 2π × 0, ∞, 1≤k≤l 3.2 H2 there exists a positive number r > 0 such that A 2 2π max ξx x∈σr,r psds max ηx 2π x∈σr,r 0 qsds Alγr < r, 3.3 0 H3 for the constant r in (H2 ), there exists a function Φr > 0 such that ft, u > Φr t, 2π t, u ∈ 0, 2π × 0, r, Φr sds > 0. 3.4 0 Then PBVP 1.1 has at least one positive periodic solution with 0 < u < r, where √ e2mπ 1 e2π M 1 A . √ √ me2mπ − 1 M e2π M − 1 3.5 Proof. The existence of positive solutions is proved by using the Leray-Schauder alternative principle given in Lemma 2.2. We divide the rather long proof into six steps. Step 1. From 3.3, we may choose n0 ∈ {1, 2, . . .} such that A 2 2π max ξx x∈σr,r psds max ηx x∈σr,r 0 2π qsds Alγr 0 1 < r. n0 3.6 Let N0 {n0 , n0 1, . . .}. For n ∈ N0 . We consider the family of equations −u t Mut λfn t, ut Δu|ttk Ik utk , M , n t ∈ J , − Δu ttk Jk utk , u0 u2π, u 0 u 2π, k 1, 2, . . . l, 3.7 6 International Journal of Differential Equations where λ ∈ 0, 1 and 1 , fn t, u f t, max u, n t, u ∈ J × 0, ∞. 3.8 For every λ and n ∈ N0 , define an operator as follows: 2π l Gt, sfn s, usds Gt, tk Jk utk Tλ,n ut λ 0 l ∂Gt, s ∂s k1 k1 Ik utk , 3.9 u ∈ K. stk Then, we may verify that Tλ,n : K −→ K is completely continuous. 3.10 To find a positive solution of 3.7 is equivalent to solve the following fixed point problem in P CJ, R: u Tλ,n u 1 . n 3.11 Let Ω {x ∈ K : x < r}, 3.12 then Ω is a relatively open subset of the convex set K. Step 2. We claim that any fixed point u of 3.11 for any λ ∈ 0, 1 must satisfies u / r. Otherwise, we assume that u is a solution of 3.11 for some λ ∈ 0, 1 such that u r. Note that fn t, u ≥ 0. ut ≥ 1/n for all t ∈ J and r ≥ ut ≥ 1/n σu − 1/n. By the choice of n0 , 1/n ≤ 1/n0 < r. Hence, for all t ∈ J, we get 1 1 1 1 1 1 r ≥ ut ≥ σ u − ≥ σ u − ≥ σ r − > σr. n n n n n n 3.13 From 3.2, we have Jk utk ≤ max Jk utk ≤ γutk ≤ γr. 3.14 1≤k≤l Consequently, for any fixed point u of 3.11, by 3.8, 3.13, and 3.14, we have ut λ 2π 0 ≤ 2π 0 l l ∂Gt, s 1 Gt, sfn s, usds k 1 Gt, tk Jk utk Ik utk n ∂s stk k1 k1 Gt, sfs, usds l k1 l ∂Gt, s Gt, tk Jk utk ∂s k1 Ik utk stk 1 n International Journal of Differential Equations ≤ 7 2π Gt, sfs, usds 0 1 mt−tk e em2π−ttk Jk utk Γ tk ≤t emtk −t em2π−tk t Jk utk −emt−tk em2π−ttk mIk utk tk ≤t tk >t tk >t 1 emtk −t − em2π−tk t mIk utk n 2π Gt, sfs, usds 0 1 mt−tk e Jk utk − mIk utk Γ tk ≤t em2π−ttk Jk utk mIk utk emtk −t Jk utk mIk utk tk ≤t tk >t 1 em2π−tk t Jk utk − mIk utk . n tk >t 3.15 It follows from −1/mJk u < Ik u < 1/mJk u that Jk utk − mIk utk > 0, Jk utk mIk utk > 0. 3.16 So, we get from 3.1, 3.2, and 3.3 that ut ≤ 2π 0 l 2 e2mπ 1 1 Gt, sfs, usds Jk utk Γ n k1 2π ! 1 Gt, s psξus qsηus ds Alγr n 0 0 2π 2π 1 A ≤ psds max ξx qsds max ξx Alγr . 2 0 x∈σr,r x∈σr,r n 0 0 ≤ 3.17 Therefore, A r u ≤ 2 2π 0 psds max ξx x∈σr,r 2π qsds max ξx Alγr 0 This is a contraction, and so the claim is proved. x∈σr,r 1 < r. n0 3.18 8 International Journal of Differential Equations Step 3. From the above claim and the Leray-Schauder alternative principle, we know that operator 3.9 with λ 1 has a fixed point denoted by un in Ω. So, 3.7 with λ 1 has a positive solution un with un t ≥ un < r, 1 , n t ∈ J. 3.19 Step 4. We show that {un } have a uniform positive lower bound; that is, there exists a constant δ > 0, independent of n ∈ N0 , such that min{un t} ≥ δ. 3.20 t In fact, from 3.4, 3.8, 3.16, and 3.19, we get un t 2π Gt, sfn s, un sds 0 k1 Ik un tk stk Gt, sfs, un sds 0 k1 ≥ 1 n l Gt, tk Jk un tk k1 l ∂Gt, s ∂s 2π Gt, tk Jk un tk k1 l ∂Gt, s ∂s 2π l Ik un tk stk 1 n Gt, sΦr sds 0 1 mt−tk e Jk un tk − mIk un tk Γ tk ≤t em2π−ttk Jk un tk mIk un tk tk ≤t emtk −t Jk un tk mIk un tk tk >t 1 m2π−tk t e Jk un tk − mIk un tk n tk >t ≥ 2π Gt, sΦr sds 0 ≥ 2emπ Γ 2π Φr sds : δ > 0. 0 3.21 International Journal of Differential Equations 9 Step 5. We prove that un < H, n ≥ n0 3.22 for some constant H > 0. Equations 3.19 and 3.20 tell us that δ ≤ un t ≤ r, so we may let M1 M2 maxGt t, s, max ft, u, t, s∈J t∈J, u∈δ,r M3 max u∈δ,r l Jk u. 3.23 k1 Then, un supun t t∈J l 2π sup Gt t, sfs, un sds Gt t, tk Jk un tk t∈J 0 k1 l ∂ ∂t k1 " # ∂Gt, s I u t k n k ∂s stk 2π sup Gt t, sfs, un sds 0 t∈J m mt−tk e Jk un tk − mIk un tk Γ tk ≤t − em2π−ttk Jk un tk mIk un tk tk ≤t − emtk −t Jk un tk mIk un tk tk >t m2π−tk t e Jk un tk − mIk un tk t >t k l 2m e2mπ 1 ≤ sup Gt t, s fs, un sds Jk un tk Γ 0 t∈J k1 2m e2mπ 1 ≤ 2πM1 M2 M3 : H. Γ 2π 3.24 Step 6. Now, we pass the solution un of the truncation equation 3.7 with λ 1 to that of the original equation 1.1. The fact that un < r and 3.22 show that {un }n∈N0 is a bounded and equi-continuous family on 0, 2π. Then, the Arzela-Ascoli Theorem guarantees that {un }n∈N0 has a subsequence {unj }j∈N , converging uniformly on 0, 2π. From the fact un < r and 10 International Journal of Differential Equations 3.20, u satisfies δ ≤ ut ≤ r for all t ∈ J. Moreover, unj also satisfies the following integral equation: unj t 2π 0 Gt, sf s, unj s ds l l 1 ∂Gt, s Gt, tk Jk unj tk Ik unj tk n . ∂s j stk k1 k1 3.25 Let j → ∞, and we get ut 2π Gt, sfs, usds 0 l k1 l ∂Gt, s Gt, tk Jk utk ∂s k1 3.26 Ik utk , stk where the uniform continuity of ft, u on J × δ, r is used. Therefore, u is a positive solution of PBVP 1.1. This ends the proof. 4. An Example Consider the following impulsive PBVP: |sin u| −u t Mut t 1 3/2 t1 |cos u|, u Δu|ttk 2 min{c1 , c2 , . . . , cl } utk , √ 2 M u0 u2π, −Δu ttk ck utk , t ∈ J , k 1, 2, . . . , l, 4.1 u 0 u 2π, where ck > 0 are constants. Then, PBVP 4.1 has at least one positive solution u with 0 < u < 1. To see this, we will apply Theorem 3.1. Let |sin u| t1 |cos u|, ft, u t2 1 3/2 u 4.2 then ft, u has a repulsive singularity at u 0 lim ft, u ∞, u → 0 uniformaly in t. 4.3 International Journal of Differential Equations 11 Denote pt t2 , qt t, ξu 1 |sin u| , u3/2 ηu 1 |cos u|, 4.4 γu max{c1 , c2 , . . . , cl }u, r 1, Φr t t t2 . Then, it is easy to say that 3.1, 3.2, and 3.3 hold. From 3.5, we know lim A lim √ M → ∞ M → ∞ e2π M √ M 1 √ e2π M −1 0. 4.5 So, we may choose M large enough to guarantee that 3.3 holds. Then, the result follows from Theorem 3.1. Remark 4.1. Functions ξ, η in example 4.1 do not have the monotonicity required as in 19. So, the results of 19 cannot be applied to PBVP 4.1. Acknowledgments The authors are grateful to the anonymous referees for their helpful suggestions and comments. Zhaocai Hao acknowledges support from NSFC 10771117, Ph.D. Programs Foundation of Ministry of Education of China 20093705120002, NSF of Shandong Province of China Y2008A24, China Postdoctoral Science Foundation 20090451290, ShanDong Province Postdoctoral Foundation 200801001, and Foundation of Qufu Normal University BSQD07026. References 1 M. Benchohra, J. Henderson, and S. Ntouyas, “Impulsive differential equations and inclusions,” Contemporary Mathematics and Its Applications, vol. 2, pp. 1–380, 2006. 2 J. Chu and Z. Zhou, “Positive solutions for singular non-linear third-order periodic boundary value problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 64, no. 7, pp. 1528–1542, 2006. 3 W. Ding and M. Han, “Periodic boundary value problem for the second order impulsive functional differential equations,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 709–726, 2004. 4 D. J. Guo, J. Sun, and Z. Liu, Nonlinear Ordinary Differential Equations Functional Technologies, ShanDong Science Technology, 1995. 5 D. Jiang, “On the existence of positive solutions to second order periodic BVPs,” Acta Mathematica Scientia, vol. 18, pp. 31–35, 1998. 6 Y. H. Lee and X. Liu, “Study of singular boundary value problems for second order impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 159–176, 2007. 7 Y. Liu, “Positive solutions of periodic boundary value problems for nonlinear first-order impulsive differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 5, pp. 2106– 2122, 2009. 8 Y. Liu, “Multiple solutions of periodic boundary value problems for first order differential equations,” Computers and Mathematics with Applications, vol. 54, no. 1, pp. 1–8, 2007. 12 International Journal of Differential Equations 9 J. J. Nieto and R. Rodrı́guez-López, “Boundary value problems for a class of impulsive functional equations,” Computers and Mathematics with Applications, vol. 55, no. 12, pp. 2715–2731, 2008. 10 J. J. Nieto, “Variational formulation of a damped Dirichlet impulsive problem,” Applied Mathematics Letters, vol. 23, pp. 940–942, 2010. 11 J. J. Nieto and D. O’Regan, “Singular boundary value problems for ordinary differential equations,” Boundary Value Problems, vol. 2009, Article ID 895290, 2 pages, 2009. 12 S. Peng, “Positive solutions for first order periodic boundary value problem,” Applied Mathematics and Computation, vol. 158, no. 2, pp. 345–351, 2004. 13 A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. 14 Y. Tian, D. Jiang, and W. Ge, “Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 123–132, 2008. 15 Z. L. Wei, “Periodic boundary value problem for second order impulsive integro differential equations of mixed type in Banach space,” Journal of Mathematical Analysis and Applications, vol. 195, pp. 214–229, 1995. 16 J. Xiao, J. J. Nieto, and Z. Luo, “Multiple positive solutions of the singular boundary value problem for second-order impulsive differential equations on the half-line,” Boundary Value Problems, vol. 2010, Article ID 281908, 13 pages, 2010. 17 X. Yang and J. Shen, “Periodic boundary value problems for second-order impulsive integrodifferential equations,” Journal of Computational and Applied Mathematics, vol. 209, no. 2, pp. 176–186, 2007. 18 Q. Yao, “Positive solutions of nonlinear second-order periodic boundary value problems,” Applied Mathematics Letters, vol. 20, no. 5, pp. 583–590, 2007. 19 Z. Zhang and J. Wang, “On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 99–107, 2003. Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Algebra Hindawi Publishing Corporation http://www.hindawi.com Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Journal of Hindawi Publishing Corporation http://www.hindawi.com Stochastic Analysis Abstract and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com International Journal of Mathematics Volume 2014 Volume 2014 Discrete Dynamics in Nature and Society Volume 2014 Volume 2014 Journal of Journal of Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Applied Mathematics Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014