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Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 362983, 16 pages doi:10.1155/2009/362983 Research Article Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems Minghe Pei 1 and Sung Kag Chang 2 1 Department of Mathematics, Bei Hua University, JiLin 132013, China 2 Department of Mathematics, Yeungnam University, Kyongsan 712-749, South Korea Correspondence should be addressed to Sung Kag Chang, skchang@ynu.ac.kr Received 5 February 2009; Accepted 14 July 2009 Recommended by Kanishka Perera We are concerned with the higher-order nonlinear three-point boundary value problems: x n ft, x, x , ,x n−1 ,n ≥ 3, with the three point boundary conditions gxa,x a, ,x n−1 a 0; x i bμ i ,i 0, 1, ,n− 3; hxc,x c, ,x n−1 c 0, where a<b<c,f: a, c × R n → R −∞, ∞ is continuous, g, h : R n → R are continuous, andμ i ∈ R,i 0, 1, ,n − 3are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result. Copyright q 2009 M. Pei and S. K. Chang. 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 Higher-order boundary value problems were discussed in many papers in recent years; for instance, see 1–22 and references therein. However, most of all the boundary conditions in the above-mentioned references are for two-point boundary conditions 2–11, 14, 17–22,and three-point boundary conditions are rarely seen 1, 12, 13, 16, 18. Furthermore works for nonlinear three point boundary conditions are quite rare in literatures. The purpose of this article is to study the existence and uniqueness of solutions for higher order nonlinear three point boundary value problem x n f t, x, x , ,x n−1 ,n≥ 3, 1.1 2 Boundary Value Problems with nonlinear three point boundary conditions g x a ,x a , ,x n−1 a 0, x i b μ i ,i 0, 1, ,n− 3, h x c ,x c , ,x n−1 c 0, 1.2 where a<b<c, f : a, c × R n → R −∞, ∞ is a continuous function, g,h : R n → R are continuous functions, and μ i ∈ R,i 0, 1, ,n− 3 are arbitrary given constants. The tools we mainly used are the method of upper and lower solutions and Leray-Schauder degree theory. Note that for t he cases of a b or b c in the boundary conditions 1.2, our theorems hold also true. However, for brevity we exclude such cases in this paper. 2. Preliminary In this section, we present some definitions and lemmas that are needed to our main results. Definition 2.1. αt,βt ∈ C n a, c are called lower and upper solutions of BVP 1.1, 1.2, respectively, if α n t ≥ f t, α t ,α t , ,α n−1 t ,t∈ a, c , g α a ,α a , ,α n−1 a ≤ 0, α i b ≤ μ i ,i 0, 1, ,n− 3, h α c ,α c , ,α n−1 c ≤ 0, β n t ≤ f t, β t ,β t , ,β n−1 t ,t∈ a, c , g β a ,β a , ,β n−1 a ≥ 0, β i b ≥ μ i ,i 0, 1, ,n− 3, h β c ,β c , ,β n−1 c ≥ 0. 2.1 Definition 2.2. Let E be a subset of a, c × R n . We say that ft, x 0 ,x 1 , ,x n−1 satisfies the Nagumo condition on E if there exists a continuous function φ : 0, ∞ → 0, ∞ such that f t, x 0 ,x 1 , ,x n−1 ≤ φ | x n−1 | , t, x 0 ,x 1 , ,x n−1 ∈ E, ∞ 0 sds φ s ∞. 2.2 Lemma 2.3 see 10. Let f : a, c × R n → R be a continuous function satisfying the Nagumo condition on E t, x 0 ,x 1 , ,x n−1 ∈ a, c × R n : γ i t ≤ x i ≤ Γ i t ,i 0, 1, ,n− 2 , 2.3 Boundary Value Problems 3 where γ i t, Γ i t : a, c → R are continuous functions such that γ i t ≤ Γ i t ,i 0, 1, ,n− 2,t∈ a, c . 2.4 Then there exists a constant r>0 (depending only on γ n−2 t, Γ n−2 t and φt such that every solution xt of 1.1 with γ i t ≤ x i t ≤ Γ i t ,i 0, 1, ,n− 2,t∈ a, c 2.5 satisfies x n−1 ∞ ≤ r. Lemma 2.4. Let φ : 0, ∞ → 0, ∞ be a continuous function. Then boundary value problem x n x n−2 φ x n−1 ,t∈ a, c , 2.6 x n−2 a x i b x n−2 c 0,i 0, 1, ,n− 3 2.7 has only the trivial solution. Proof. Suppose that x 0 t is a nontrivial solution of BVP 2.6, 2.7. Then there exists t 0 ∈ a, c such that x n−2 0 t 0 > 0orx n−2 0 t 0 < 0. We may assume x n−2 0 t 0 > 0. There exists t 1 ∈ a, c such that max t∈ a,c x n−2 0 t : x n−2 0 t 1 > 0. 2.8 Then x n−1 0 t 1 0, x n 0 t 1 ≤ 0. From 2.6 we have 0 ≥ x n 0 t 1 x n−2 0 t 1 φ x n−1 0 t 1 > 0, 2.9 which is a contradiction. Hence BVP 2.6, 2.7 has only the trivial solution. 3. Main Results We may now formulate and prove our main results on the existence and uniqueness of solutions for nth-order three point boundary value problem 1.1, 1.2. Theorem 3.1. Assume that i there exist lower and upper solutions αt,βt of BVP 1.1, 1.2, respectively, such that −1 n−i α i t ≤ −1 n−i β i t ,t∈ a, b ,i 0, 1, ,n− 2, α i t ≤ β i t ,t∈ b, c ,i 0, 1, ,n− 2; 3.1 4 Boundary Value Problems ii ft, x 0 , ,x n−1 is continuous on a, c×R n , −1 n−i ft, x 0 , ,x n−1 is nonincreasing in x i i 0, 1, ,n− 3 on D b a , and ft, x 0 , ,x n−1 is nonincreasing in x i i 0, 1, ,n− 3 on D c b and satisfies the Nagumo condition on D c a ,where ϕ i t min α i t ,β i t ,ψ i t max α i t ,β i t ,i 0, ,n− 2, D b a t, x 0 , ,x n−1 ∈ a, b × R n : ϕ i t ≤ x i ≤ ψ i t ,i 0, ,n− 2 , D c b t, x 0 , ,x n−1 ∈ b, c × R n : ϕ i t ≤ x i ≤ ψ i t ,i 0, ,n− 2 , D c a t, x 0 , ,x n−1 ∈ a, c × R n : ϕ i t ≤ x i ≤ ψ i t ,i 0, ,n− 2 ; 3.2 iii gx 0 ,x 1 , ,x n−1 is continuous on R n , and −1 n−i gx 0 ,x 1 , ,x n−1 is nonincreasing in x i i 0, 1, ,n− 3 and nondecreasing in x n−1 on n−2 i0 ϕ i a,ψ i a × R; iv hx 0 ,x 1 , ,x n−1 is continuous on R n , and nonincreasing in x i i 0, 1, ,n− 3 and nondecreasing in x n−1 on n−2 i0 ϕ i c,ψ i c × R. Then BVP 1.1, 1.2 has at least one solution xt ∈ C n a, c such that for each i 0, 1, ,n− 2, −1 n−i α i t ≤ −1 n−i x i t ≤ −1 n−i β i t ,t∈ a, b , α i t ≤ x i t ≤ β i t ,t∈ b, c . 3.3 Proof. For each i 0, 1, ,n− 2 define w i t, x ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ψ i t ,x>ψ i t , x, ϕ i t ≤ x ≤ ψ i t , ϕ i t ,x<ϕ i t , 3.4 where ϕ i tmin{α i t,β i t}, ψ i tmax{α i t,β i t}. For λ ∈ 0, 1, we consider the auxiliary equation x n t λf t, w 0 t, x t , ,w n−2 t, x n−2 t ,x n−1 t x n−2 t − λw n−2 t, x n−2 t φ x n−1 t , 3.5 where φ is given by the Nagumo condition, with the boundary conditions x n−2 a λ w n−2 a, x n−2 a − g w 0 a, x a , ,w n−2 a, x n−2 a ,x n−1 a , x i b λμ i ,i 0, 1, ,n− 3, x n−2 c λ w n−2 c, x n−2 c − h w 0 c, x c , ,w n−2 c, x n−2 c ,x n−1 c . 3.6 Boundary Value Problems 5 Then we can choose a constant M n−2 > 0 such that −M n−2 <α n−2 t ≤ β n−2 t <M n−2 ,t∈ a, c , 3.7 f t, α t , ,α n−2 t , 0 − M n−2 α n−2 t φ 0 < 0,t∈ a, c , f t, β t , ,β n−2 t , 0 M n−2 − β n−2 t φ 0 > 0,t∈ a, c , 3.8 α n−2 a − g α a , ,α n−2 a , 0 <M n−2 , β n−2 a − g β a , ,β n−2 a , 0 <M n−2 , 3.9 α n−2 c − h α c , ,α n−2 c , 0 <M n−2 , β n−2 c − h β c , ,β n−2 c , 0 <M n−2 . 3.10 In the following, we will complete the proof in four steps. Step 1. Show that every solution xt of BVP 3.5, 3.6 satisfies x n−2 t <M n−2 ,t∈ a, c , 3.11 independently of λ ∈ 0, 1. Suppose that the estimate |x n−2 t| <M n−2 is not true. Then there exists t 0 ∈ a, c such that x n−2 t 0 ≥ M n−2 or x n−2 t 0 ≤−M n−2 . We may assume x n−2 t 0 ≥ M n−2 . There exists t 1 ∈ a, c such that max t∈ a,c x n−2 t : x n−2 t 1 ≥ M n−2 > 0 . 3.12 There are three cases to consider. Case 1 t 1 ∈ a, c. In this case, x n−1 t 1 0andx n t 1 ≤ 0. For λ ∈ 0, 1,by3.8,weget the following contradiction: 0 ≥ x n t 1 λf t 1 ,w 0 t 1 ,x t 1 , ,w n−2 t 1 ,x n−2 t 1 ,x n−1 t 1 x n−2 t 1 − λw n−2 t 1 ,x n−2 t 1 φ x n−1 t 1 λf t 1 ,w 0 t 1 ,x t 1 , ,w n−3 t 1 ,x n−3 t 1 ,β n−2 t 1 , 0 x n−2 t 1 − λβ n−2 t 1 φ 0 ≥ λ f t 1 ,β t 1 , ,β n−2 t 1 , 0 M n−2 − β n−2 t 1 φ 0 > 0, 3.13 6 Boundary Value Problems and for λ 0, we have the following contradiction: 0 ≥ x n t 1 x n−2 t 1 φ 0 ≥ M n−2 φ 0 > 0. 3.14 Case 2 t 1 a. In this case, max t∈ a,c x n−2 t : x n−2 a ≥ M n−2 > 0 , 3.15 and x n−1 a ≤ 0. For λ 0, by 3.6 we have the following contradiction: 0 <M n−2 ≤ x n−2 a 0. 3.16 For λ ∈ 0, 1,by3.9 and condition iii we can get the following contradiction: M n−2 ≤ x n−2 a , λ w n−2 a, x n−2 a − g w 0 a, x a , ,w n−2 a, x n−2 a ,x n−1 a , ≤ λ β n−2 a − g β a , ,β n−2 a , 0 <M n−2 . 3.17 Case 3 t 1 c. In this case, max t∈ a,c x n−2 t : x n−2 c ≥ M n−2 > 0 , 3.18 and x n−1 c ≥ 0. For λ 0, by 3.6 we have the following contradiction: 0 <M n−2 ≤ x n−2 c 0. 3.19 For λ ∈ 0, 1,by3.10 and condition iv we can get the following contradiction: M n−2 ≤ x n−2 c , λ w n−2 c, x n−2 c − h w 0 c, x c , ,w n−2 c, x n−2 c ,x n−1 c ≤ λ β n−2 c − h β c , ,β n−2 c , 0 <M n−2 . 3.20 By 3.6, the estimates x i t <M i : c − a M i1 μ i ,i 0, 1, ,n− 3,t∈ a, c 3.21 are obtained by integration. Boundary Value Problems 7 Step 2. Show that there exists M n−1 > 0 such that every solution xt of BVP 3.5, 3.6 satisfies x n−1 t <M n−1 ,t∈ a, c , 3.22 independently of λ ∈ 0, 1. Let E { t, x 0 , ,x n−1 ∈ a, c × R n : | x i | ≤ M i ,i 0, 1, ,n− 2 } , 3.23 and define the function F λ : a, c × R n → R as follows: F λ t, x 0 , ,x n−1 λf t, w 0 t, x 0 , ,w n−2 t, x n−2 ,x n−1 x n−2 − λw n−2 t, x n−2 φ | x n−1 | . 3.24 In the following, we show that F λ t, x 0 , ,x n−1 satisfies the Nagumo condition on E, independently of λ ∈ 0, 1. In fact, since f satisfies the Nagumo condition on D c a , we have | F λ t, x 0 , ,x n−1 | λf t, w 0 t, x 0 , ,w n−2 t, x n−2 ,x n−1 x n−2 − λw n−2 t, x n−2 φ | x n−1 | ≤ 1 2M n−2 φ | x n−1 | : φ E | x n−1 | . 3.25 Furthermore, we obtain ∞ 0 s φ E s ds ∞ 0 s 1 2M n−2 φ s ds ∞. 3.26 Thus, F λ satisfies the Nagumo condition on E, independently of λ ∈ 0, 1.Let γ i t −M i , Γ i t M i ,i 0, 1, ,n− 2,t∈ a, c . 3.27 By Step 1 and Lemma 2.3, there exists M n−1 > 0 such that |x n−1 t| <M n−1 for t ∈ a, c. Since M n−2 and φ E do not depend on λ, the estimate |x n−1 t| <M n−1 on a, c is also independent of λ. Step 3. Show that for λ 1, BVP 3.5, 3.6 has at least one solution x 1 t. Define the operators as follows: L : C n a, c ⊂ C n−1 a, c −→ C a, c × R n , 3.28 8 Boundary Value Problems by Lx x n t ,x n−2 a ,x b , ,x n−3 b ,x n−2 c , N λ : C n−1 a, c −→ C a, c × R n , 3.29 by N λ x F λ t, x t , ,x n−1 t ,A λ ,λμ 0 , ,λμ n−3 ,C λ , 3.30 with A λ : λ w n−2 a, x n−2 a − g w 0 a, x a , ,w n−2 a, x n−2 a ,x n−1 a C λ : λ w n−2 c, x n−2 c − h w 0 c, x c , ,w n−2 c, x n−2 c ,x n−1 c . 3.31 Since L −1 is compact, we have the following compact operator: T λ : C n−1 a, c −→ C n−1 a, c , 3.32 defined by T λ x L −1 N λ x . 3.33 Consider the set Ω{x ∈ C n−1 a, c : x i ∞ <M i ,i 0, 1, ,n− 1}. By Steps 1 and 2, the degree degI − T λ , Ω, 0 is well defined for every λ ∈ 0, 1, and by homotopy invariance, we get deg I − T 0 , Ω, 0 deg I − T 1 , Ω, 0 . 3.34 Since the equation x T 0 x has only the trivial solution from Lemma 2.4, by the degree theory we have deg I − T 1 , Ω, 0 deg I − T 0 , Ω, 0 ±1. 3.35 Hence, the equation x T 1 x has at least one solution. That is, the boundary value problem x n t f t, w 0 t, x t , ,w n−2 t, x n−2 t ,x n−1 t x n−2 t − w n−2 t, x n−2 t φ x n−1 t , 3.36 Boundary Value Problems 9 with the boundary conditions x n−2 a w n−2 a, x n−2 a − g w 0 a, x a , ,w n−2 a, x n−2 a ,x n−1 a , x i b μ i ,i 0, 1, ,n− 3, x n−2 c w n−2 c, x n−2 c − h w 0 c, x c , ,w n−2 c, x n−2 c ,x n−1 c , 3.37 has at least one solution x 1 t in Ω. Step 4. Show that x 1 t is a solution of BVP 1.1, 1.2. In fact, the solution x 1 t of BVP 3.36, 3.37 will be a solution of BVP 1.1, 1.2,ifit satisfies ϕ i t ≤ x i 1 t ≤ ψ i t ,i 0, 1, ,n− 2,t∈ a, c . 3.38 By contradiction, suppose that there exists t 0 ∈ a, c such that x n−2 1 t 0 >ψ n−2 t 0 . There exists t 1 ∈ a, c such that max t∈ a,c x n−2 1 t − ψ n−2 t : x n−2 1 t 1 − ψ n−2 t 1 > 0. 3.39 Now there are three cases to consider. Case 1 t 1 ∈ a, c. In this case, since ψ n−2 tβ n−2 t on a, c, we have x n−1 1 t 1 β n−1 t 1 and x n 1 t 1 ≤ β n t 1 . By conditions i and ii, we get the following contradiction: 0 ≥ x n 1 t 1 − β n t 1 ≥ f t 1 ,w 0 t 1 ,x 1 t 1 , ,w n−2 t 1 ,x n−2 1 t 1 ,x n−1 1 t 1 x n−2 1 t 1 − w n−2 t 1 ,x n−2 1 t 1 φ x n−1 1 t 1 − f t 1 ,β t 1 , ,β n−1 t 1 ≥ f t 1 ,β t 1 , ,β n−1 t 1 x n−2 1 t 1 − β n−2 t 1 φ x n−1 1 t 1 − f t 1 ,β t 1 , ··· ,β n−1 t 1 x n−2 1 t 1 − β n−2 t 1 φ x n−1 1 t 1 > 0. 3.40 Case 2 t 1 a. In this case, we have max t∈ a,c x n−2 1 t − ψ n−2 t : x n−2 1 a − β n−2 a > 0, 3.41 10 Boundary Value Problems and x n−1 1 a ≤ β n−1 a.By3.37 and conditions i and iii we can get the following contradiction: β n−2 a <x n−2 1 a , w n−2 a, x n−2 1 a − g w 0 a, x 1 a , ,w n−2 a, x n−2 1 a ,x n−1 1 a ≤ β n−2 a − g β a , ,β n−2 a ,β n−1 a ≤ β n−2 a . 3.42 Case 3 t 1 c. In this case, we have max t∈ a,c x n−2 1 t − ψ n−2 t : x n−2 1 c − β n−2 c > 0, 3.43 and x n−1 1 c ≥ β n−1 c.By3.37 and conditions i and iv we can get the following contradiction: β n−2 c <x n−2 1 c w n−2 c, x n−2 1 c − h w 0 c, x 1 c , ,w n−2 c, x n−2 1 c ,x n−1 1 c ≤ β n−2 c − h β c , ,β n−2 c ,β n−1 c ≤ β n−2 c . 3.44 Similarly, we can show that ϕ n−2 t ≤ x n−2 1 t on a, c. Hence α n−2 t ϕ n−2 t ≤ x n−2 1 t ≤ ψ n−2 t β n−2 t ,t∈ a, c . 3.45 Also, by boundary condition 3.37 and condition i, we have α i b x i 1 b β i b ,i n − 1 − 2j, j 1, 2, , n − 1 2 , α i b ≤ x i 1 b ≤ β i b ,i n − 2 − 2j, j 1, 2, , n − 2 2 . 3.46 Therefore by integration we have for each i 0, 1, ,n− 2, −1 n−i α i t ≤ −1 n−i x i 1 t ≤ −1 n−i β i t ,t∈ a, b , α i t ≤ x i 1 t ≤ β i t ,t∈ b, c , 3.47 [...]... Problems, vol 2007, Article ID 23108, 12 pages, 2007 15 V R G Moorti and J B Garner, Existence- uniqueness theorems for three-point boundary value problems for nth-order nonlinear differential equations,” Journal of Differential Equations, vol 29, no 2, pp 205–213, 1978 16 K N Murty and Y S Rao, “A theory for existence and uniqueness of solutions to three-point boundary value problems,” Journal of Mathematical... Gupta, and J.-M Xu, Existence and uniqueness theorems for three-point boundary value problems,” SIAM Journal on Mathematical Analysis, vol 20, no 3, pp 716–726, 1989 2 R P Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Teaneck, NJ, USA, 1986 3 R P Agarwal and F.-H Wong, Existence of positive solutions for non-positive higher-order BVPs,” Journal of Computational... problems,” Journal of Mathematical Analysis and Applications, vol 167, no 1, pp 43–48, 1992 17 M Pei and S K Chang, Existence and uniqueness of solutions for third-order nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol 327, no 1, pp 23–35, 2007 18 Y Shi and M Pei, “Two-point and three-point boundary value problems for nth-order nonlinear differential equations,”... A Cabada, M R Grossinho, and F Minhos, “Extremal solutions for third-order nonlinear problems ´ with upper and lower solutions in reversed order,” Nonlinear Analysis: Theory, Methods & Applications, vol 62, no 6, pp 1109–1121, 2005 7 Z Du, W Ge, and X Lin, Existence of solutions for a class of third-order nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol 294,... theorems for a third-order threepoint boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol 16, no 11, pp 949–957, 1991 13 J Henderson and R D Taunton, Solutions of boundary value problems by matching methods,” Applicable Analysis, vol 49, no 3-4, pp 235–246, 1993 14 Y Lin and M Pei, “Positive solutions for two-point semipositone right focal eigenvalue problem,” Boundary Value. .. and S Liu, “Solvability of a third-order two-point boundary value problem,” Applied Mathematics Letters, vol 18, no 9, pp 1034–1040, 2005 9 M R Grossinho and F M Minhos, Existence result for some third order separated boundary value ´ problems,” Nonlinear Analysis: Theory, Methods & Applications, vol 47, no 4, pp 2407–2418, 2001 16 Boundary Value Problems 10 M R Grossinho and F M Minhos, “Upper and. .. unique solution x t satisfying 3.3 Proof The existence of a solution for BVP 1.1 , 1.2 satisfying 3.3 follows from Theorem 3.1 Now, we prove the uniqueness of solution for BVP 1.1 , 1.2 To do this, we let x1 t x2 t − x1 t It and x2 t are any two solutions of BVP 1.1 , 1.2 satisfying 3.3 Let z t is easy to show that z t is a solution of the following boundary value problem n−1 zn t di t z i t , 3.53... American Mathematical Society, vol 126, no 8, pp 2389–2397, 1998 21 Q Yao and Y Feng, “The existence of solution for a third-order two-point boundary value problem,” Applied Mathematics Letters, vol 15, no 2, pp 227–232, 2002 22 W L Zhao, Existence and uniqueness of solutions for third order nonlinear boundary value problems,” The Tohoku Mathematical Journal, vol 44, no 4, pp 545–555, 1992 ... Journal of Computational and Applied Mathematics, vol 88, no 1, pp 3–14, 1998 4 R P Agarwal, S R Grace, and D O’Regan, “Semipositone higher-order differential equations,” Applied Mathematics Letters, vol 17, no 2, pp 201–207, 2004 5 A Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol... lower solutions for higher order boundary value ´ problems,” Nonlinear Studies, vol 12, no 2, pp 165–176, 2005 11 M R Grossinho, F M Minhos, and A I Santos, “Solvability of some third-order boundary ´ value problems with asymmetric unbounded nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol 62, no 7, pp 1235–1250, 2005 12 C P Gupta and V Lakshmikantham, Existence and uniqueness . Corporation Boundary Value Problems Volume 2009, Article ID 362983, 16 pages doi:10.1155/2009/362983 Research Article Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value. purpose of this article is to study the existence and uniqueness of solutions for higher order nonlinear three point boundary value problem x n f t, x, x , ,x n−1 ,n≥ 3, 1.1 2 Boundary. eigenvalue problem,” Boundary Value Problems, vol. 2007, Article ID 23108, 12 pages, 2007. 15 V. R. G. Moorti and J. B. Garner, Existence- uniqueness theorems for three-point boundary value problems
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