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Hindawi Publishing Corporation BoundaryValue Problems Volume 2009, Article ID 273063, 18 pages doi:10.1155/2009/273063 ResearchArticlePositiveSolutionsforaClassofCoupledSystemofSingularThree-PointBoundaryValue Problems Naseer Ahmad Asif and Rahmat Ali Khan Centre for Advanced Mathematics and Physics, Campus of College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Peshawar Road, Rawalpindi 46000, Pakistan Correspondence should be addressed to Rahmat Ali Khan, rahmat alipk@yahoo.com Received 27 February 2009; Accepted 15 May 2009 Recommended by Juan J. Nieto Existence ofpositivesolutionsforacoupledsystemof nonlinear three-pointboundaryvalue problems of the type −x tft, xt,yt, t ∈ 0, 1, −y tgt, xt,yt, t ∈ 0, 1, x0y00, x1αxη, y1αyη, is established. The nonlinearities f, g : 0, 1 × 0, ∞ × 0, ∞ → 0, ∞ are continuous and may be singular at t 0,t 1,x 0, and/or y 0, while the parameters η, α satisfy η ∈ 0, 1 , 0 <α<1/η. An example is also included to show the applicability of our result. Copyright q 2009 N. A. Asif and R. A. Khan. 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. 1. Introduction Multipoint boundaryvalue problems BVPs arise in different areas of applied mathematics and physics. For example, the vibration ofa guy wire composed of N parts with a uniform cross-section and different densities in different parts can be modeled as a Multipoint boundaryvalue problem 1. Many problems in the theory of elastic stability can also be modeled as Multipoint boundaryvalue problem 2. The study of Multipoint boundaryvalue problems for linear second order ordinary differential equations was initiated by Il’in and Moiseev, 3, 4, and extended to nonlocal linear elliptic boundaryvalue problems by Bitsadze et al. 5, 6. Existence theory for nonlinear three-pointboundaryvalue problems was initiated by Gupta 7. Since then the study of nonlinear three-point BVPs has attracted much attention of many researchers, see 8–11 and references therein forboundaryvalue problems with ordinary differential equations and also 12 forboundaryvalue problems on time scales. Recently, the study ofsingular BVPs has attracted the attention of many authors, see for example, 13–18 and the recent monograph by Agarwal et al. 19. 2 BoundaryValue Problems The study ofsystemof BVPs has also fascinated many authors. Systemof BVPs with continuous nonlinearity can be seen in 20–22 and the case ofsingular nonlinearity can be seen in 8, 21, 23–26.Wei25, developed the upper and lower solutions method for the existence ofpositivesolutionsof the following coupledsystemof BVPs: −x t f t, x t ,y t ,t∈ 0, 1 , −y t g t, x t ,y t ,t∈ 0, 1 , x 0 0,x 1 0, y 0 0,y 1 0, 1.1 where f, g ∈ C0, 1×0, ∞×0, ∞, 0, ∞, and may be singular at t 0, t 1, x 0and/or y 0. By using fixed point theorem in cone, Yuan et al. 26 studied the following coupledsystemof nonlinear singularboundaryvalue problem: x 4 t f t, x t ,y t ,t∈ 0, 1 , −y t g t, x t ,y t ,t∈ 0, 1 , x 0 x 1 x 0 x 1 0, y 0 y 1 0, 1.2 f, g are allowed to be superlinear and are singular at t 0and/ort 1. Similarly, results are studied in 8, 21, 23. In this paper, we generalize the results studied in 25, 26 to the following more general singularsystemforthree-point nonlocal BVPs: −x t f t, x t ,y t ,t∈ 0, 1 , −y t g t, x t ,y t ,t∈ 0, 1 , x 0 0,x 1 αx η , y 0 0,y 1 αy η , 1.3 where η ∈ 0, 1,0<α<1/η, f, g ∈ C0, 1 × 0, ∞ × 0, ∞, 0, ∞. We allow f and g to be singular at t 0, t 1, and also x 0and/ory 0. We study the sufficient conditions for existence ofpositive solution for the singularsystem 1.3 under weaker hypothesis on f and g as compared to the previously studied results. We do not require the system 1.3 to have lower and upper solutions. Moreover, the cone we consider is more general than the cones considered in 20, 21, 26. By singularity, we mean the functions ft, x, y and gt, x, y are allowed to be unbounded at t 0, t 1, x 0, and/or y 0. To the best of our knowledge, existence ofpositivesolutionsforasystem 1.3 with singularity with respect to dependent variables has not been studied previously. Moreover, our conditions and results are different from those BoundaryValue Problems 3 studied in 21, 24–26. Throughout this paper, we assume that f, g : 0, 1 × 0, ∞ × 0, ∞ → 0, ∞ are continuous and may be singular at t 0, t 1, x 0, and/or y 0. We also assume that the following conditions hold: A 1 f·, 1, 1,g·, 1, 1 ∈ C0, 1, 0, ∞ and satisfy a : 1 0 t 1 − t f t, 1, 1 dt < ∞,b: 1 0 t 1 − t g t, 1, 1 dt < ∞. 1.4 A 2 There exist real constants α i ,β i such that α i ≤ 0 ≤ β i < 1, i 1, 2, β 1 β 2 < 1andfor all t ∈ 0, 1, x,y ∈ 0, ∞, c β 1 f t, x, y ≤ f t, cx, y ≤ c α 1 f t, x, y , if 0 <c≤ 1, c α 1 f t, x, y ≤ f t, cx, y ≤ c β 1 f t, x, y , if c ≥ 1, c β 2 f t, x, y ≤ f t, x, cy ≤ c α 2 f t, x, y , if 0 <c≤ 1, c α 2 f t, x, y ≤ f t, x, cy ≤ c β 2 f t, x, y , if c ≥ 1. 1.5 A 3 There exist real constants γ i ,ρ i such that γ i ≤ 0 ≤ ρ i < 1, i 1, 2, ρ 1 ρ 2 < 1andfor all t ∈ 0, 1, x,y ∈ 0, ∞, c ρ 1 g t, x, y ≤ g t, cx, y ≤ c γ 1 g t, x, y , if 0 <c≤ 1, c γ 1 g t, x, y ≤ g t, cx, y ≤ c ρ 1 g t, x, y , if c ≥ 1, c ρ 2 g t, x, y ≤ g t, x, cy ≤ c γ 2 g t, x, y , if 0 <c≤ 1, c γ 2 g t, x, y ≤ g t, x, cy ≤ c ρ 2 g t, x, y , if c ≥ 1, 1.6 for example, a function that satisfies the assumptions A 2 and A 3 is h t, x, y m i1 n j1 p ij t x r i y s j , 1.7 where p ij ∈ C0, 1, 0, ∞, r i ,s j < 1, i 1, 2, ,m; j 1, 2, ,nsuch that max 1≤i≤m r i max 1≤j≤n s j < 1. 1.8 The main result of this paper is as follows. Theorem 1.1. Assume that A 1 –A 3 hold. Then the system 1.3 has at least one positive solution. 4 BoundaryValue Problems 2. Preliminaries For each u ∈ E : C0, 1, we write u max{ut : t ∈ 0, 1}.LetP {u ∈ E : ut ≥ 0,t ∈ 0, 1}. Clearly, E, · is a Banach space and P is a cone. Similarly, for each x, y ∈ E × E, we write x, y 1 x y. Clearly, E × E, · 1 is a Banach space and P × P is a cone in E × E. For any real constant r>0, define Ω r {x, y ∈ E × E : x, y 1 <r}. By apositive solution of 1.3, we mean a vector x, y ∈ C0, 1∩C 2 0, 1×C0, 1∩ C 2 0, 1 such that x, y satisfies 1.3 and x>0, y>0on0, 1. The proofs of our main result Theorem 1.1 is based on the Guo’s fixed-point theorem. Lemma 2.1 Guo’s Fixed-Point Theorem 27. Let K be a cone ofa real Banach space E, Ω 1 , Ω 2 be bounded open subsets of E and θ ∈ Ω 1 ⊂ Ω 2 . Suppose that T : K ∩ Ω 2 \ Ω 1 → K is completely continuous such that one of the following condition hold: i Tx≤x for x ∈ ∂Ω 1 ∩ K and Tx≥x for x ∈ ∂Ω 2 ∩ K; ii Tx≤x for x ∈ ∂Ω 2 ∩ K and Tx≥x for x ∈ ∂Ω 1 ∩ K. Then, T has a fixed point in K ∩ Ω 2 \ Ω 1 . The following result can be easily verified. Result 1. Let t 1 ,t 2 ∈ R such that t 1 <t 2 .Letx ∈ Ct 1 ,t 2 , x ≥ 0 and concave on t 1 ,t 2 . Then, xt ≥ min{t − t 1 ,t 2 − t}max s∈t 1 ,t 2 xs for all t ∈ t 1 ,t 2 . Choose n 0 ∈{3, 4, 5, } such that n 0 > max{1/η, 1/1 − η, 2 − α/1 − αη}. For fixed n ∈{n 0 ,n 0 1,n 0 2, } and z ∈ C0, 1, the linear three-point BVP −u t z t ,t∈ 1 n , 1 − 1 n , u 1 n 0,u 1 − 1 n αu η , 2.1 has a unique solution u t 1−1/n 1/n H n t, s z s ds, 2.2 where H n : 1/n, 1 − 1/n × 1/n, 1 − 1/n → 0, ∞ is the Green’s function and is given by H n t, s ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t−1/n 1− 1/n−s 1 − 2/n α/n − αη − α t − 1/n η − s 1 − 2/n α/n − αη − t − s , 1 n ≤ s ≤ t ≤ 1 − 1 n ,s≤ η, t − 1/n 1 − 1/n − s 1 − 2/n α/n − αη − α t − 1/n η − s 1 − 2/n α/n − αη , 1 n ≤ t ≤ s ≤ 1 − 1 n ,s≤ η, t − 1/n 1 − 1/n − s 1 − 2/n α/n − αη , 1 n ≤ t ≤ s ≤ 1 − 1 n ,s≥ η, t − 1/n 1 − 1/n − s 1 − 2/n α/n − αη − t − s , 1 n ≤ s ≤ t ≤ 1 − 1 n ,s≥ η. 2.3 BoundaryValue Problems 5 We note that H n t, s → Ht, s as n →∞, where H t, s ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t 1 − s 1 − αη − αt η − s 1 − αη − t − s , 0 ≤ s ≤ t ≤ 1,s≤ η, t 1 − s 1 − αη − αt η − s 1 − αη , 0 ≤ t ≤ s ≤ 1,s≤ η, t 1 − s 1 − αη , 0 ≤ t ≤ s ≤ 1,s≥ η, t 1 − s 1 − αη − t − s , 0 ≤ s ≤ t ≤ 1,s≥ η, 2.4 is the Green’s function corresponding the boundaryvalue problem −u t z t ,t∈ 0, 1 , u 0 0,u 1 αu η 2.5 whose integral representation is given by u t 1 0 H t, s z s ds. 2.6 Lemma 2.2 see 9. Let 0 <α<1/η.Ifz ∈ C0, 1 and z ≥ 0, then then unique solution u of the problem 2.5 satisfies min t∈η,1 u t ≥ γu, 2.7 where γ min{αη, α1 − η/1 − αη,η}. We need the following properties of the Green’s function H n in the sequel. Lemma 2.3 see 11. The function H n can be written as H n t, s G n t, s α t − 1/n 1 − 2/n α/n − αη G n η, s , 2.8 where G n t, s n n − 2 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ s − 1 n 1 − 1 n − t , 1 n ≤ s ≤ t ≤ 1 − 1 n , t − 1 n 1 − 1 n − s , 1 n ≤ t ≤ s ≤ 1 − 1 n . 2.9 6 BoundaryValue Problems Following the idea in 10, we calculate upper bound for the Green’s function H n in the following lemma. Lemma 2.4. The function H n satisfies H n t, s ≤ μ n s − 1 n 1 − 1 n − s , t, s ∈ 1 n , 1 − 1 n × 1 n , 1 − 1 n , 2.10 where μ n max{1,α}/1 − 2/n α/n − αη. Proof. For t, s ∈ 1/n, 1 − 1/n × 1/n, 1 − 1/n, we discuss various cases. Case 1. s ≤ η, s ≤ t;using2.3,weobtain H n t, s s − 1 n α − 1 t − 1/n s − 1/n 1 − 2/n α/n − αη . 2.11 If α>1, the maximum occurs at t 1 − 1/n, hence H n t, s ≤ H n 1 − 1 n ,s α s − 1/n 1 − 1/n − η 1 − 2/n α/n − αη ≤ α s − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ μ n s − 1 n 1 − 1 n − s , 2.12 and if α ≤ 1, the maximum occurs at t s, hence H n t, s ≤ H n s, s s − 1/n 1 − 1/n − s α s − η 1 − 2/n α/n − αη ≤ s − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ μ n s − 1 n 1 − 1 n − s . 2.13 Case 2. s ≤ η, s ≥ t;using2.3, we have H n t, s t − 1/n 1 − 1/n − s 1 − 2/n α/n − αη − α t − 1/n η − s 1 − 2/n α/n − αη ≤ t − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ s − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ μ n s − 1 n 1 − 1 n − s . 2.14 Case 3. s ≥ η, t ≤ s;using2.3, we have H n t, s t − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ s − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ μ n s − 1 n 1 − 1 n − s . 2.15 BoundaryValue Problems 7 Case 4. s ≥ η, t ≥ s;using2.3, we have H n t, s s − 1 n t − 1 n α η − 1/n − s − 1/n 1 − 2/n α/n − αη . 2.16 For αη − 1/n >s− 1/n, the maximum occurs at t 1 − 1/n, hence H n t, s ≤ H n 1 − 1 n ,s α η − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ α s − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ μ n s − 1 n 1 − 1 n − s . 2.17 For αη − 1/n ≤ s − 1/n, the maximum occurs at t s,so H n t, s ≤ H n s, s s − 1/n 1 − 1/n − s 1 − 2/n α/n − αη ≤ μ n s − 1 n 1 − 1 n − s . 2.18 Now, we consider the nonlinear nonsingular systemof BVPs −x t f t, max x t 1 n , 1 n , max y t 1 n , 1 n ,t∈ 1 n , 1 − 1 n , −y t g t, max x t 1 n , 1 n , max y t 1 n , 1 n ,t∈ 1 n , 1 − 1 n , x 1 n 0,x 1 − 1 n αx η , y 1 n 0,y 1 − 1 n αy η . 2.19 We write 2.19 as an equivalent systemof integral equations x t 1−1/n 1/n H n t, s f s, max x s 1 n , 1 n , max y s 1 n , 1 n ds, y t 1−1/n 1/n H n t, s g s, max x s 1 n , 1 n , max y s 1 n , 1 n ds. 2.20 By a solution of the system 2.19, we mean a solution of the corresponding systemof integral equations 2.20. Define a retraction σ n : 0, 1 → 1/n, 1 −1/n by σ n tmax{1/n, min{t, 1− 1/n}} and an operator T n : E × E → P × P by T n x, y A n x, y ,B n x, y , 2.21 where operators A n ,B n : E × E → P are defined by 8 BoundaryValue Problems A n x, y t 1−1/n 1/n H n σ n t ,s f s, max x s 1 n , 1 n , max y s 1 n , 1 n ds, B n x, y t 1−1/n 1/n H n σ n t ,s g s, max x s 1 n , 1 n , max y s 1 n , 1 n ds. 2.22 Clearly, if x n ,y n ∈ E × E is a fixed point of T n , then x n ,y n is a solution of the system 2.19. Lemma 2.5. Assume that A 1 –A 3 holds. Then T n : P × P → P × P is completely continuous. Proof. Clearly, for any x, y ∈ P × P, A n x, y,B n x, y ∈ P. We show that the operator A n : P × P → P is uniformly bounded. Let d>0 be fixed and consider D x, y ∈ P × P : x, y 1 ≤ d . 2.23 Choose a constant c ∈ 0, 1 such that cx 1/3 ≤ 1, cy 1/3 ≤ 1, x, y ∈ D. Then, for every x, y ∈ D,using2.22, Lemma 2.4, A 1 and A 2 , we have A n x, y t 1−1/n 1/n H n σ n t ,s f s, x s 1 n ,y s 1 n ds 1−1/n 1/n H n σ n t ,s f s, c x s 1/n c ,c y s 1/n c ds ≤ 1 c β 1 1−1/n 1/n H n σ n t ,s f s, c x s 1 n ,c y s 1/n c ds ≤ 1 c β 1 1 c β 2 1−1/n 1/n H n σ n t ,s f s, c x s 1 n ,c y s 1 n ds ≤ c α 1 −β 1 −β 2 1−1/n 1/n H n σ n t ,s xs 1 n α 1 f s, 1,c y s 1 n ds ≤ c α 1 −β 1 α 2 −β 2 1−1/n 1/n H n σ n t ,s xs 1 n α 1 ys 1 n α 2 f s, 1, 1 ds ≤ c α 1 −β 1 α 2 −β 2 1−1/n 1/n H n σ n t ,s 1 n α 1 1 n α 2 f s, 1, 1 ds ≤ μ n c α 1 −β 1 α 2 −β 2 n −α 1 −α2 1−1/n 1/n s − 1 n 1 − 1 n − s f s, 1, 1 ds ≤ μ n c α 1 −β 1 α 2 −β 2 n −α 1 −α2 1−1/n 1/n s 1 − s f s, 1, 1 ds ≤ μ n c α 1 −β 1 α 2 −β 2 n −α 1 −α2 1 0 s 1 − s f s, 1, 1 ds aμ n c α 1 −β 1 α 2 −β 2 n −α 1 −α2 , 2.24 BoundaryValue Problems 9 which implies that A n x, y ≤aμ n c α 1 −β 1 α 2 −β 2 n −α 1 −α2 , 2.25 that is, A n D is uniformly bounded. Similarly, using 2.22, Lemma 2.4 , A 1 and A 3 ,we can show that B n D is also uniformly bounded. Thus, T n D is uniformly bounded. Now we show that A n D is equicontinuous. Define ω max max t,x,y∈1/n,1−1/n×0,d×0,d f t, x 1 n ,y 1 n , max t,x,y∈1/n,1−1/n×0,d×0,d g t, x 1 n ,y 1 n . 2.26 Let t 1 ,t 2 ∈ 0, 1 such that t 1 ≤ t 2 . Since H n t, s is uniformly continuous on 1/n, 1 − 1/n × 1/n, 1 − 1/n, for any ε>0, there exist δ δε > 0 such that |t 1 − t 2 | <δimplies | H n σ n t 1 ,s − H n σ n t 2 ,s | < ε ω 1 − 2/n for s ∈ 1 n , 1 − 1 n . 2.27 For x, y ∈ D,using2.22–2.27, we have A n x, y t 1 − A n x, y t 2 1−1/n 1/n H n σ n t 1 ,s − H n σ n t 2 ,s f s, x s 1 n ,y s 1 n ds ≤ 1−1/n 1/n | H n σ n t 1 ,s − H n σ n t 2 ,s | f s, x s 1 n ,y s 1 n ds ≤ ω 1−1/n 1/n | H n σ n t 1 ,s − H n σ n t 2 ,s | ds <ω ε ω 1 − 2/n 1−1/n 1/n ds ε 1 − 2/n 1 − 2 n ε. 2.28 Hence, A n x, y t 1 − A n x, y t 2 <ε, ∀ x, y ∈ D, | t 1 − t 2 | <δ, 2.29 which implies that A n D is equicontinuous. Similarly, using 2.22–2.27, we can show that B n D is also equicontinuous. Thus, T n D is equicontinuous. By Arzel ` a-Ascoli theorem, T n D is relatively compact. Hence, T n is a compact operator. 10 BoundaryValue Problems NowweshowthatT n is continuous. Let x m ,y m , x, y ∈ P × P such that x m ,y m − x, y 1 → 0asm → ∞. Then by using 2.22 and Lemma 2.4, we have A n x m ,y m t − A n x, y t 1−1/n 1/n H n σ n t ,s f s, x m s 1 n ,y m s 1 n − f s, x s 1 n ,y s 1 n ds ≤ 1−1/n 1/n H n σ n t ,s f s, x m s 1 n ,y m s 1 n − f s, x s 1 n ,y s 1 n ds ≤ μ n 1−1/n 1/n s− 1 n 1− 1 n −s f s, x m s 1 n ,y m s 1 n −f s, x s 1 n ,y s 1 n ds. 2.30 Consequently, A n x m ,y m − A n x, y ≤ μ n 1−1/n 1/n s − 1 n 1 − 1 n − s × f s, x m s 1 n ,y m s 1 n − f s, x s 1 n ,y s 1 n ds. 2.31 By Lebesgue dominated convergence theorem, it follows that A n x m ,y m − A n x, y −→ 0asm −→ ∞. 2.32 Similarly, by using 2.22 and Lemma 2.4, we have B n x m ,y m − B n x, y −→ 0asm −→ ∞. 2.33 From 2.32 and 2.33, it follows that T n x m ,y m − T n x, y 1 −→ 0asm −→ ∞, 2.34 that is, T n : P × P → P × P is continuous. Hence, T n : P × P → P × P is completely continuous. 3. Main Results Proof of Theorem 1.1. Let M max{μ n 0 , max{1,α}/1 − αη}. 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Recently, the study of singular BVPs has attracted the attention of many