Báo cáo hóa học: " Research Article Existence and Uniqueness of Solutions for Singular Higher Order Continuous and Discrete Boundary Value Problems" pptx

Hindawi Publishing Corporation Boundary Value Problems Volume 2008, Article ID 123823, 11 pages doi:10.1155/2008/123823 Research Article Existence and Uniqueness of Solutions for Singular Higher Order Continuous and Discrete Boundary Value Problems Chengjun Yuan, 1, 2 Daqing Jiang, 1 and You Zhang 1 1 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China 2 School of Mathematics and Computer, Harbin University, Harbin 150086, Heilongjiang, China Correspondence should be addressed to Chengjun Yuan, ycj7102@163.com Received 4 July 2007; Accepted 31 December 2007 Recommended by Raul Manasevich By mixed monotone method, the existence and uniqueness are established for singular higher-order continuous and discrete boundary value problems. The theorems obtained are very general and complement previous known results. Copyright q 2008 Chengjun Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which p ermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In recent years, the study of higher-order continuous and discrete boundary value problems has been studied extensively in the literature see 1–17 and their references. Most of the results told us that the equations had at least single and multiple positive solutions. Recently, some authors have dealt with the uniqueness of solutions for singular higher- order continuous boundary value problems by using mixed monotone method, for example, see 6, 14, 15. However, there are few works on the uniqueness of solutions for singular dis- crete boundary value problems. In this paper, we state a unique fixed point theorem for a class of mixed monotone op- erators, see 6, 14, 18. In virtue of the theorem, we consider the existence and uniqueness of solutions for the following singular higher-order continuous and discrete boundary value problems 1.1 and 1.2 by using mixed monotone method. We first discuss the existence and uniqueness of solutions for the following singular higher-order continuous boundary value problem y n tλqtgyhy0, 0<t<1,λ>0, y i 0y n−2 10, 0 ≤ i ≤ n − 2, 1.1 2 Boundary Value Problems where n ≥ 2, qt ∈ C0, 1, 0, ∞, g : 0, ∞ → 0, ∞ is continuous and nondecreasing; h : 0, ∞ → 0, ∞ is continuous and nonincreasing, and h may be singular at y  0. Next, we consider the existence and uniqueness of solutions for the following singular higher-order discrete boundary value problem Δ n yiλqin−1gyin−1 hyin−1  0,i∈ N  {0, 1, 2, ,T − 1},λ>0, Δ k y0Δ n−2 yT  10, 0 ≤ k ≤ n − 2, 1.2 where n ≥ 2, N   {0, 1, 2, ,Tn}, qi ∈ CN  , 0, ∞, g : 0, ∞ → 0, ∞ is continuous and nondecreasing; h : 0, ∞ → 0, ∞ is continuous and nonincreasing, and h may be singular at y  0. Throughout this paper, the topology on N  will be the discrete topology. 2. Preliminaries Let P be a normal c one of a Banach space E,ande ∈ P with e≤1,e /  θ. Define Q e  {x ∈ P | x /  θ, there exist constants m, M > 0 such that me ≤ x ≤ Me}. 2.1 Now we give a definition see 18. Definition 2.1 see 18. Assume A : Q e × Q e → Q e . A is said to be mixed monotone if Ax, y is nondecreasing in x and nonincreasing in y,thatis,ifx 1 ≤ x 2 x 1 ,x 2 ∈ Q e  implies Ax 1 ,y ≤ Ax 2 ,y for any y ∈ Q e ,andy 1 ≤ y 2 y 1 ,y 2 ∈ Q e  implies Ax, y 1  ≥ Ax, y 2  for any x ∈ Q e . x ∗ ∈ Q e is said to be a fixed point of A if Ax ∗ ,x ∗ x ∗ . Theorem 2.2 see 6, 14. Suppose that A: Q e × Q e → Q e is a mixed monotone operator and ∃ a constant α, 0 ≤ α<1, such that A  tx, 1 t y  ≥ t α Ax, y,forx,y∈ Q e , 0 <t<1. 2.2 Then A has a unique fixed point x ∗ ∈ Q e . Moreover, for any x 0 ,y 0  ∈ Q e × Q e , x n  A  x n−1 ,y n−1  ,y n  A  y n−1 ,x n−1  ,n 1, 2, , 2.3 satisfy x n −→ x ∗ ,y n −→ x ∗ , 2.4 where   x n − x ∗    o  1 − r α n  ,   y n − x ∗    o  1 − r α n  , 2.5 0 <r<1, r is a constant from x 0 ,y 0 . Theorem 2.3 see 6, 14, 18. Suppose that A: Q e × Q e → Q e is a mixed monotone operator and ∃ a constant α ∈ 0, 1 such that 2.2 holds. If x ∗ λ is a unique solution of equation Ax, xλx, λ>02.6 in Q e ,thenx ∗ λ − x ∗ λ 0 →0,λ→ λ 0 . If 0 <α<1/2, then 0 <λ 1 <λ 2 implies x ∗ λ 1 ≥ x ∗ λ 2 , x ∗ λ 1 /  x ∗ λ 2 ,and lim λ→∞   x ∗ λ    0, lim λ→0    x ∗ λ   ∞. 2.7 Chengjun Yuan et al. 3 3. Uniqueness positive solution of differential equations 1.1 This section discusses singular higher-order boundary value problem 1.1. Throughout this section, we let Gt, s be the Green’s function to −y   0,y0y10, we note that Gt, s ⎧ ⎨ ⎩ t1 − s, 0 ≤ t ≤ s ≤ 1, s1 − t, 0 ≤ s ≤ t ≤ 1, 3.1 and one can show that Gt, tGs, s ≤ Gt, s ≤ Gt, t, for Gt, s ≤ Gs, s, t, s ∈ 0, 1 × 0, 1.  3.2 Suppose that y is a positive solution of 1.1.Let xty n−2 t, 3.3 from y i 0y n−2 10, 0 ≤ i ≤ n−2, and Taylor Formula, we define operator T : C 2 0, 1 → C n 0, 1,by ytTxt  t 0 t − s n−3 n − 3! xsds, for 3 ≤ n, ytTxtxt, for n  2 3.4 Then we have x 2 tλft, Txt  0, 0 <t<1,λ>0, x0x10. 3.5 Then from 3.4, we have the next lemma. Lemma 3.1. If xt is a solution of 3.5,thenyt is a solution of 1.1. Further, if yt is a solution of 1.1,implythatxt is a solution of 3.5. Let P  {x ∈ C0, 1 | xt ≥ 0, for all t ∈ 0, 1 }. Obviously, P is a normal cone of Banach space C0, 1. Theorem 3.2. Suppose that there exists α ∈ 0, 1 such that gtx ≥ t α gx, 3.6 h  t −1 x  ≥ t α hx, 3.7 for any t ∈ 0, 1 and x>0,andq ∈ C0, 1, 0, ∞ satisfies  1 0  s n−1 n − 2s  −α qsds < ∞. 3.8 Then 1.1 has a unique positive solution y ∗ λ t. And moreover, 0 <λ 1 <λ 2 implies y ∗ λ 1 ≤ y ∗ λ 2 ,y ∗ λ 1 /  y ∗ λ 2 . If α ∈ 0, 1/2,then lim λ→0    y ∗ λ    0, lim λ→∞   y ∗ λ   ∞. 3.9 4 Boundary Value Problems Proof. Since 3.7  holds, let t −1 x  y, one has hy ≥ t α hty. 3.10 Then hty ≤ 1 t α hy, for t ∈ 0, 1,y>0. 3.11 Let y  1. The above inequality is ht ≤ 1 t α h1, for t ∈ 0, 1. 3.12 From 3.7, 3.11,and3.12, one has h  t −1 x  ≥ t α hx,h  1 t  ≥ t α h1,htx ≤ 1 t α hx,ht ≤ 1 t α h1, for t ∈ 0, 1,x>0. 3.13 Similarly, from 3.6, one has gtx ≥ t α gx,gt ≥ t α g1, for t ∈ 0, 1,x>0. 3.14 Let t  1/x, x > 1, one has gx ≤ x α g1, for x ≥ 1. 3.15 Let etGt, tt1 − t, and we define Q e   x ∈ C0, 1 | 1 M Gt, t ≤ xt ≤ MGt, t,t∈ 0, 1  , 3.16 where M>1 is chosen such that M>max   λg1  1 0 qsds  λh1  1 0  s n−1 n − 2s n!  −α qsds  1/1−α ,  λg1  1 0 Gs, s  s n−1 n − 2s n!  α qsds  λh1  1 0 Gs, sqsds  −1/1−α  . 3.17 First, from 3.4 and 3.16, for any x ∈ Q e ,wehavethefollowing. When 3 ≤ n, 1 M t n−1 n − 2t n! ≤  t 0 1 M Gs, s t − s n−3 n − 3! ds ≤ Txt ≤  t 0 MGs, s t − s n−3 n − 3! ds ≤ M t n−1 n − 2t n! ≤ M, for t ∈ 0, 1, 3.18 Chengjun Yuan et al. 5 when n  2, 1 M t n−1 n − 2t n! ≤ Txtxt ≤ M tn − 2t n! ≤ M, for t ∈ 0, 1, 3.19 then 1 M t n−1 n − 2t n! ≤ Txt ≤ M t n−1 n − 2t n! ≤ M, for t ∈ 0, 1. 3.20 For any x, y ∈ Q e , we define A λ x, ytλ  1 0 Gt, sqsgTxs  hTysds, for t ∈ 0, 1. 3.21 First, we show that A λ : Q e × Q e → Q e . Let x, y ∈ Q e , from 3.14, 3.15,and3.20,wehave gTxt ≤ gM ≤ M α g1, for t ∈ 0, 1, 3.22 and from 3.13,wehave hTyt ≤ h  1 M t n−1 n − 2t n!  ≤  t n−1 n − 2t n!  −α h  1 M  ≤ M α  t n−1 n − 2t n!  −α h1, for t ∈ 0, 1. 3.23 Then, from 3.2, 3.21, 3.22 and 3.23,wehave On the other hand, for any x, y ∈ Q e ,from3.13 and 3.14,wehave gTxt ≥ g  1 M t n−1 n − 2t n!  ≥  t n−1 n − 2t n!  α g  1 M  ≥  t n−1 n − 2t n!  α 1 M α g1, hTyt ≥ hMh  1 1/M  ≥ 1 M α h1, for t ∈ 0, 1. 3.24 Thus, from 3.2, 3.21 and 3.24,wehave A λ x, yt ≥ λGt, t   1 0 Gs, sqsM −α  s n−1 n − 2s n!  α g1ds   1 0 Gs, sqsM −α h1ds ≥ 1 M Gt, t, for t ∈ 0, 1. 3.25 So, A λ is well defined and A λ Q e × Q e  ⊂ Q e . 6 Boundary Value Problems Next, for any l ∈ 0, 1, one has A λ  lx, l −1 y  tλ  1 0 Gt, sqs  glTxs  h  l −1 Tys  ds ≥ λ  1 0 Gt, sqs  l α gTxs  l α hTys  ds  l α A λ x, yt, for t ∈ 0, 1. 3.26 So the conditions of Theorems 2.2 and 2.3 hold. Therefore, there exists a unique x ∗ λ ∈ Q e such that A λ x ∗ ,x ∗ x ∗ λ . It is easy to check that x ∗ λ is a unique positive solution of 3.5 for given λ>0. Moreover, Theorem 2.3 means that if 0 <λ 1 <λ 2 , then x ∗ λ 1 t ≤ x ∗ λ 2 t, x ∗ λ 1 t /  x ∗ λ 2 t and if α ∈ 0, 1/2,then lim λ→0    x ∗ λ    0, lim λ→∞   x ∗ λ   ∞. 3.27 Next, from Lemma 3.1 and 3.4,wegetthaty ∗ λ  Tx ∗ λ is a unique positive solution of 1.1 for given λ>0. Moreover, if 0 <λ 1 <λ 2 , then y ∗ λ 1 t ≤ y ∗ λ 2 t, y ∗ λ 1 t /  y ∗ λ 2 t and if α ∈ 0, 1/2,then lim λ→0    y ∗ λ    0, lim λ→∞   y ∗ λ   ∞. 3.28 This completes the proof. Example 3.3. Consider the following singular boundary value problem: y n tλ  μy a ty −b t   0,t∈ 0, 1, y i 0y n−2 10, 0 ≤ i ≤ n − 2, 3.29 where λ, a, b > 0, μ ≥ 0, max{a, b} < 1/n − 1. Applying Theorem 3.2,letα  max{a, b} < 1/n − 1, qt1, gyμy a , hyy −b , then gty ≥ t α gy,h  t −1  ≥ t α hy,  1 0  s n−1 n − 2s  −α ds < ∞. 3.30 Thus all conditions in Theorem 3.2 are satisfied. We can find 3.29 has a unique positive so- lution y ∗ λ t. In addition, 0 <λ 1 <λ 2 implies y ∗ λ 1 ≤ y ∗ λ 2 ,y ∗ λ 1 /  y ∗ λ 2 .Ifα  max{a, b}∈0, 1/2, then lim λ→0    y ∗ λ    0, lim λ→∞   y ∗ λ   ∞. 3.31 Chengjun Yuan et al. 7 4. Uniqueness positive solution of difference equations 1.2 This section discusses singular higher-order boundary value problem 1.2. Throughout this section, we let Ki, j be Green’s function to −Δ 2 yiui  10, i ∈ N, y0yT  10, we note that Ki, j ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ jT  1 − i T  1 , 0 ≤ j ≤ i − 1, iT  1 − j T  1 ,i≤ j ≤ T  1, 4.1 and one can show that Ki, i ≥ Ki, j,Kj, j ≥ Ki, j,Ki, j ≥ Ki, i T  1 , for 0 ≤ i ≤ T  1, 1 ≤ j ≤ T. 4.2 Suppose that y is a positive solution of 1.2.Let xiΔ n−2 yi, for 0 ≤ i ≤ T  1. 4.3 From Δ i y0Δ n−2 yT  10, 0 ≤ i ≤ n − 2, and Δ m yi − 1Δ m−1 yi − Δ m−1 yi − 1,sowe define operator T,by Txiyi  n − 1 i1  l1 C n−2 i−ln−1 xl, for 0 ≤ i ≤ T. 4.4 Then Δ 2 xiλFi  n − 1,Txi  0, 0 ≤ i ≤ T − 1,λ>0, x0xT  10. 4.5 Lemma 4.1. If xi is a solution of 4.5,thenyi is a solutionn of 1.2. Proof. Since we remark that xi is a solution of 4.5,ifandonlyif xi T  j1 Ki, jλFj  n − 1,Txj, for 0 ≤ i ≤ T  1. 4.6 Let Txiyi  n − 1, for 0 ≤ i ≤ T. 4.7 From 4.4 we find Δ i y0Δ n−2 yT  10, 0 ≤ i ≤ n − 2, and xiΔ n−2 yi,sothatyi is asolutionof1.2. Further, if yi is a solution o f 1.2,implythatxi is a solution of 4.5. Let P  {x ∈ CN  , 0, ∞ | xi ≥ 0, for all i ∈ N  }. Obviously, P is a normal cone of Banach space CN  , 0, ∞. 8 Boundary Value Problems Theorem 4.2. Suppose that there exists α ∈ 0, 1 such that gtx ≥ t α gx, h  t −1 x  ≥ t α gx, 4.8 for any t ∈ 0, 1 and x>0,andq ∈ CN  , 0, ∞. Then 1.2 has a unique positive solution y ∗ λ i. And moreover, 0 <λ 1 <λ 2 implies y ∗ λ 1 ≤ y ∗ λ 2 , y ∗ λ 1 /  y ∗ λ 2 .Ifα ∈ 0, 1/2,then lim λ→0    y ∗ λ    0, lim λ→∞   y ∗ λ   ∞. 4.9 Proof. The proof is the same as that of Theorem 3.2,from4.12 and 4.13, one has h  t −1 x  ≥ t α hx,h  1 t  ≥ t α h1,htx ≤ 1 t α hx,ht ≤ 1 t α h1, for t ∈ 0, 1,x>0; gtx ≥ t α gx,gt ≥ t α g1, for t ∈ 0, 1,x>0. 4.10 gtx ≥ t α gx,gt ≥ t α g1, for t ∈ 0, 1,x>0. 4.11 Let t  1/x, x > 1, one has gx ≤ x α g1, for x ≥ 1. 4.12 Let eiKi, i/T  1, and we define Q e   x ∈ P | 1 M ei ≤ xi ≤ Mei, for 0 ≤ i ≤ T  1  , 4.13 where M>1 is chosen such that M>max   λT  1g1 T  j1 qj  n − 1  j1  l1 C n−2 j−ln−1  α  λT  1 1α h1 T  j1 K −α j, jqj  n − 1  1/1−α ;  λg1 T  j1 qj  n − 1  Kj, j T  1  α  λh1 T  j1 qj  n − 1  j1  l1 C n−2 j−ln−1  −α  −1/1−α  . 4.14 From 4.4 and 4.13, for any x ∈ Q e ,wehave 1 M ej ≤ Txj j1  l1 C n−2 j−ln−1 xl ≤ Mej j1  l1 C n−2 j−ln−1 , for 0 ≤ j ≤ T. 4.15 Chengjun Yuan et al. 9 For any x, y ∈ Q e , we define A λ x, yiλ T  j1 Ki, jqj  n − 1gTxj  hTyj, for 0 ≤ i ≤ T  1. 4.16 First we show that A λ : Q e × Q e → Q e . Let x, y ∈ Q e , from 4.11 and 4.12,wehave gTxj ≤ g  Mej j1  l1 C n−2 j−ln−1  ≤g  M j1  l1 C n−2 j−ln−1  ≤M α  j1  l1 C n−2 j−ln−1  α g1, for 1≤j ≤ T , 4.17 and from 4.10,wehave hTyj ≤ h  1 M ej  ≤ e −α jh  1 M  ≤ M α e −α jh1, for 1 ≤ j ≤ T. 4.18 Then, from 4.2 and the above, we have A λ x, yi ≤ λKi, i T  j1 qj  n − 1gTxj  T  1hTyj ≤ eiM α λT  1  g1 T  j1 qj  n − 1  j1  l1 C n−2 j−ln−1  α  h1 T  j1 e −α jqj  n − 1  ≤ eiM α λT  1  g1 T  j1 qj  n − 1  j1  l1 C n−2 j−ln−1  α  h1 T  j1  Kj, j T  1  −α qj  n − 1  ≤ Mei, for 0 ≤ i ≤ T  1. 4.19 On the other hand, for any x, y ∈ Q e ,from4.10 and 4.12,wehave gxj ≥ g  1 M ej  ≥ e α j 1 M α g1, for 1 ≤ j ≤ T, 4.20 hyj ≥ h  Mej j1  l1 C n−2 j−ln−1  ≥ M −α  j1  l1 C n−2 j−ln−1  −α h1, for 1 ≤ j ≤ T. 4.21 Thus, from 4.2 and 4.16,wehave A λ x, yi ≥ λei  T  j0 qj  n − 1gTxj  T  j0 qj  n − 1hTyj  ≥ λeiM −α  g1 T  j0 qj  Ki, i T  1  α  h1 T  j0 qj  n − 1  j1  l1 C n−2 j−ln−1  −α  ≥ 1 M ei, for 0 ≤ i ≤ T  1. 4.22 10 Boundary Value Problems So, A λ is well defined and A λ Q e × Q e  ⊂ Q e . Next, for any l ∈ 0, 1, one has A λ  lx, l −1 y  iλ T  j1 Ki, jqj  n − 1  gTlxj  h  T  l −1  yj   λ T  j1 Ki, jqj  n − 1  glTxj  h  l −1 Tyj  ≥ λ T  j1 Ki, jqj  n − 1  l α gTxj  l α hTyj  ds  l α A λ x, yi, for 0 ≤ i ≤ T  1. 4.23 So the conditions of Theorems 2.2 and 2.3 hold. Therefore, there exists a unique x ∗ λ ∈ Q e such that A λ x ∗ ,x ∗ x ∗ λ . It is easy to check that x ∗ λ is a unique positive solution of 4.5 for given λ>0. Moreover, Theorem 2.3 means that if 0 <λ 1 <λ 2 , then x ∗ λ 1 t ≤ x ∗ λ 2 t, x ∗ λ 1 t /  x ∗ λ 2 t and if α ∈ 0, 1/2,then lim λ→0    x ∗ λ    0, lim λ→∞   x ∗ λ   ∞. 4.24 Next, on using Lemma 3.1,from4.5,wegetthaty ∗ λ  Tx ∗ λ is a unique positive solution of 1.2 for given λ>0. Moreover, if 0 <λ 1 <λ 2 , then y ∗ λ 1 t ≤ y ∗ λ 2 t, y ∗ λ 1 t /  y ∗ λ 2 t and if α ∈ 0, 1/2,then lim λ→0    y ∗ λ    0, lim λ→∞   y ∗ λ   ∞. 4.25 This completes the proof. Example 4.3. Consider the following singular boundary value problem: Δ n yi − 1λ  μy a iy −b i   0,i∈ N, Δ i y0Δ n−2 y10, 0 ≤ i ≤ n − 2, 4.26 where λ, a, b > 0, μ ≥ 0, max{a, b} < 1. Let qi1, gyμy a , hyy −b , α  max{a, b} < 1, then gty ≥ t α gy,h  t −1 y  ≥ t α hy, 4.27 thus all conditions in Theorem 4.2 are satisfied. We can find 4.26 has a unique positive so- lution y ∗ λ t. In addition, 0 <λ 1 <λ 2 implies y ∗ λ 1 ≤ y ∗ λ 2 ,y ∗ λ 1 /  y ∗ λ 2 .Ifα  max{a, b}∈0, 1/2, then lim λ→0    y ∗ λ    0, lim λ→∞   y ∗ λ   ∞. 4.28 [...]... Existence of solutions for singular boundary problems for higher order differential equations,” Rendiconti del Seminario Matem` tico e Fisico di Milano, vol 65, pp 249– a 264, 1995 13 R P Agarwal and F.-H Wong, Existence of positive solutions for higher order difference equations,” Applied Mathematics Letters, vol 10, no 5, pp 67–74, 1997 14 X Lin, D Jiang, and X Li, Existence and uniqueness of solutions. .. Agarwal and D O’Regan, Existence theory for single and multiple solutions to singular positone boundary value problems,” Journal of Differential Equations, vol 175, no 2, pp 393–414, 2001 8 P R Agarwal and D O’Regan, Singular discrete boundary value problems,” Applied Mathematics Letters, vol 12, no 4, pp 127–131, 1999 9 R P Agarwal and F.-H Wong, Existence of positive solutions for higher order difference... Xue, and W Ge, “Triple solutions for a higher- order difference equation,” Journal of Inequalities in Pure and Applied Mathematics, vol 6, no 1, Article 10, pp 1–11, 2005 11 P J Y Wong and R P Agarwal, “On the existence of solutions of singular boundary value problems for higher order difference equations,” Nonlinear Analysis Theory, Methods & Applications, vol 28, no 2, pp 277–287, 1997 12 R P Agarwal and. .. solutions for singular fourth -order boundary value problems,” Journal of Computational and Applied Mathematics, vol 196, no 1, pp 155–161, 2006 15 Z Zengqin, Uniqueness of positive solutions for singular nonlinear second -order boundary- value problems,” Nonlinear Analysis Theory, Methods & Applications, vol 23, no 6, pp 755–765, 1994 16 R P Agarwal and I Kiguradze, “Two-point boundary value problems for higher- order. .. order differential equations,” Journal of Differential Equations, vol 3, pp 1–8, 1995 5 C J Chyan and J Henderson, “Positive solutions for singular higher order nonlinear equations,” Differential Equations and Dynamical Systems, vol 2, no 2, pp 153–160, 1994 6 X Lin, D Jiang, and X Li, Existence and uniqueness of solutions for singular k, n − k conjugate boundary value problems,” Computers & Mathematics... for higher- order linear differential equations with strong singularities,” Boundary Value Problems, vol 2006, Article ID 83910, 32 pages, 2006 17 X Hao, L Liu, and Y Wu, “Positive solutions for nonlinear nth -order singular nonlocal boundary value problems,” Boundary Value Problems, vol 2007, Article ID 74517, 10 pages, 2007 18 D Guo, The Order Methods in Nonlinear Analysis, Shandong Technical and Science... boundary value problems for superlinear higherorder ODEs,” Computers and Mathematics with Applications, vol 40, no 2-3, pp 249–259, 2000 3 P W Eloe and J Henderson, Singular nonlinear boundary value problems for higher order ordinary differential equations,” Nonlinear Analysis Theory, Methods & Applications, vol 17, no 1, pp 1–10, 1991 4 P W Eloe and J Henderson, “Postive solutions for higher order differential... of China Grants no 10571021 and 10701020 The work was supported by Subject Foundation of Harbin University Grant no HXK200714 References 1 Y Guo and J Tian, “Positive solutions of m-point boundary value problems for higher order ordinary differential equations,” Nonlinear Analysis Theory, Methods & Applications, vol 66, no 7, pp 1573–1586, 2007 2 D Jiang, “Multiple positive solutions to singular boundary . Corporation Boundary Value Problems Volume 2008, Article ID 123823, 11 pages doi:10.1155/2008/123823 Research Article Existence and Uniqueness of Solutions for Singular Higher Order Continuous and Discrete Boundary. continuous and discrete boundary value problems 1.1 and 1.2 by using mixed monotone method. We first discuss the existence and uniqueness of solutions for the following singular higher- order continuous. theorem for a class of mixed monotone op- erators, see 6, 14, 18. In virtue of the theorem, we consider the existence and uniqueness of solutions for the following singular higher- order continuous