báo cáo hóa học:" Research Article Positive Solutions of Singular Complementary Lidstone Boundary Value Problems" ppt

Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 368169, 15 pages doi:10.1155/2010/368169 Research Article Positive Solutions of Singular Complementary Lidstone Boundary Value Problems Ravi P. Agarwal, 1 Donal O’Regan, 2 and Svatoslav Stan ˇek 3 1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA 2 Department of Mathematics, National University of Ireland, Galway, Ireland 3 Department of Mathematical Analysis, Faculty of Science, Palack ´ y University, T ˇ r. 17. listopadu 12, 771 46 Olomouc, Czech Republic Correspondence should be addressed to Ravi P. Agarwal, agarwal@fit.edu Received 7 October 2010; Accepted 21 November 2010 Academic Editor: Irena Rach ˚ unkov ´ a Copyright q 2010 Ravi P. Agarwal 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. We investigate the existence of positive solutions of singular problem −1 m x 2m1  ft,x, , x 2m , x00, x 2i−1 0x 2i−1 T0, 1 ≤ i ≤ m. Here, m ≥ 1 and the Carath ´ eodory function ft, x 0 , ,x 2m  may be singular in all its space variables x 0 , ,x 2m . The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used. 1. Introduction Let T be a positive constant, J 0,T and − −∞, 0,  0, ∞, 0  \{0}.Weconsider the singular complementary Lidstone boundary value problem  −1  m x 2m1  t   f  t, x  t  , ,x 2m  t   ,m≥ 1, 1.1 x  0   0,x 2i−1  0   x 2i−1  T   0, 1 ≤ i ≤ m, 1.2 where f satisfies the local Carath ´ eodory function on J ×Df ∈ CarJ ×D with D  ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2  × 0 × − × 0 ×  ×···×  × 0    4k−1 if m  2k − 1, 2  × 0 × − × 0 ×  ×···× − × 0    4k1 if m  2k. 1.3 2 Boundary Value Problems The function ft, x 0 , ,x 2m  is positive and may be singular at the value zero of all its space variables x 0 , ,x 2m . Let i ∈{0, 1, ,2m}. We say that f is singular at the value zero of its space variable x i if for a.e. t ∈ J and all x j ,0≤ j ≤ 2m, j /  i such that x 0 , ,x i , ,x 2m  ∈D,therelation lim x i →0 f  t, x 0 , ,x i , ,x 2m   ∞ 1.4 holds. Afunctionx ∈ AC 2m Ji.e., x has absolutely continuous 2mth derivative on J is a positive solution of problem 1.1, 1.2 if xt > 0fort ∈ 0,T, x satisfies the boundary conditions 1.2 and 1.1 holds a.e. on J. The regular complementary Lidstone problem  −1  m x 2m1  t   h  t, x  t  , ,x q  t   ,m≥ 1,qfixed, 0 ≤ q ≤ 2m, x  0   α 0 ,x 2i−1  0   α i ,x 2i−1  1   β i , 1 ≤ i ≤ m 1.5 was discussed in 1. Here, h : 0, 1 × q1 → is continuous at least in the interior of the domain of interest. Existence and uniqueness criteria for problem 1.5 are proved by the complementary Lidstone interpolating polynomia l of d egree 2m. No contributions exist, as far as we know, concerning the exi stence of positive solutions of singular complementary Lidstone problems. We observe that differential equations in complementary Lidstone problems as well as derivatives in boundary conditions are odd orders, in contrast to the Lidstone problem  −1  m x 2m  t   p  t, x  t  , ,x r  t   ,m≥ 1,rfixed, 0 ≤ r ≤ 2m − 1, x 2i  0   a i ,x 2i  1   b i , 1 ≤ i ≤ m − 1, 1.6 where the differential equation and derivatives in the boundary conditions are even orders. For a i  b i  0 1 ≤ i ≤ m − 1, regular Lidstone problems were discussed in 2–9, while singular ones in 10–15. The aim of this paper is to give the conditions on the function f in 1.1 which gua- rantee that the singular problem 1.1, 1.2 ha s a solution. The existence results are proved by regularization and sequential techniques, and in limit processes, the Vitali convergence theorem 16, 17 is applied. Throughout the paper, x ∞  max{|xt| : t ∈ J} and x C n   n k0 x k  ∞ , n ≥ 1 stands fo r the norm in C 0 J and C n J, respectively. L 1 J denotes the set of functions Lebesgue integrable on J and meas M the Lebesgue measure of M⊂J. We work with the following conditions on the function f in 1.1. H 1  f ∈ CarJ ×D and there exists a ∈ 0, ∞ such that a ≤ f  t, x 0 , ,x 2m  , 1.7 for a.e. t ∈ J and each x 0 , ,x 2m  ∈D. Boundary Value Problems 3 H 2  For a.e. t ∈ J and for all x 0 , ,x 2m  ∈D, the inequality f  t, x 0 , ,x 2m  ≤ h ⎛ ⎝ t, 2m  j0   x j   ⎞ ⎠  2m  j0 ω j    x j    1.8 is fulfilled, where h ∈ CarJ × 0, ∞ is positive and nondecreasing in the second variable, ω j :  →  is nonincreasing, 0 ≤ j ≤ 2m, lim sup v →∞ 1 v  T 0 h  t, Kv  dt<1,K ⎧ ⎪ ⎨ ⎪ ⎩ T 2m1 − 1 T −1 if T /  1, 2m  1ifT  1,  1 0 ω 2j  s 2  ds<∞,  1 0 ω 2j1  s  ds<∞ if 0 ≤ j ≤ m − 1,  1 0 ω 2m  s  ds<∞. 1.9 The paper is organized as follows. In Section 2, we construct a sequence of auxiliary regular differential equations associated with 1.1.Section3 is devoted to the study of auxiliary regular complementary Lidstone problems. We show that the solvability of these problems is reduced to the existence of a fixed point of an operator H. The existence of a fixed point of H is proved by a fixed point theorem of cone compression type according to Guo- Krasnosel’skii 18, 19. The properties of solutions to auxiliary problems are also investigated here. In Section 4, applying the results of Section 3, the existence of a positive solution of the singular problem 1.1, 1.2 is proved. 2. Regularization Let m be from 1.1.Forn ∈ ,defineχ n ,ϕ n ,τ n,m ∈ C 0  , n ⊂ ,andD n ⊂ 2m1 by the formulas χ n  u   ⎧ ⎪ ⎨ ⎪ ⎩ u for u ≥ 1 n , 1 n for u< 1 n , ϕ n  u   ⎧ ⎪ ⎨ ⎪ ⎩ − 1 n for u>− 1 n , u for u ≤− 1 n , τ n,m  ⎧ ⎨ ⎩ χ n if m  2k − 1, ϕ n if m  2k, n   −∞, − 1 n  ∪  1 n , ∞  , D n  2 × n × × n × ×···× × n    2m1 . 2.1 4 Boundary Value Problems Let f ∈ CarJ ×D.Chosen ∈ and put f ∗ n  t, x 0 ,x 1 ,x 2 ,x 3 ,x 4 , ,x 2m−1 ,x 2m   f  t, χ n  x 0  ,χ n  x 1  ,x 2 ,ϕ n  x 3  ,x 4 , ,τ n,m  x 2m−1  ,x 2m  2.2 for t, x 0 ,x 1 ,x 2 ,x 3 ,x 4 , ,x 2m−1 ,x 2m  ∈ J ×D n . Now, define an auxiliary function f n by means of the following recurrence formulas: f n,0  t, x 0 ,x 1 , ,x 2m   f ∗ n  t, x 0 ,x 1 , ,x 2m  for  t, x 0 ,x 1 , ,x 2m  ∈ J ×D n , f n,i  t, x 0 ,x 1 , ,x 2m   ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ f n,i−1  t, x 0 ,x 1 , ,x 2m  if | x 2i | ≥ 1 n , n 2  f n,i−1  t, x 0 , ,x 2i−1 , 1 n ,x 2i1 , ,x 2m  x 2i  1 n  −f n,i−1  t, x 0 , ,x 2i−1 , − 1 n ,x 2i1 , ,x 2m  x 2i − 1 n  if | x 2i | < 1 n , 2.3 for 1 ≤ i ≤ m,and f n  t, x 0 ,x 1 , ,x 2m   f n,m  t, x 0 ,x 1 , ,x 2m  for  t, x 0 ,x 1 , ,x 2m  ∈ J × 2m1 . 2.4 Then, under condition H 1 , f n ∈ CarJ × 2m1  and a ≤ f n  t, x 0 ,x 1 , ,x 2m  for a.e.t∈ J and all  x 0 ,x 1 , ,x 2m  ∈ 2m1 . 2.5 Condition H 2  gives f n  t, x 0 ,x 1 , ,x 2m  ≤ h ⎛ ⎝ t, 2m  1  2m  j0   x j   ⎞ ⎠  2m  j0  ω j    x j     ω j  1   , for a.e.t∈ J and all  x 0 ,x 1 , ,x 2m  ∈ 2m1 0 , 2.6 f n  t, x 0 ,x 1 , ,x 2m  ≤ h ⎛ ⎝ t, 2m  1  2m  j0   x j   ⎞ ⎠  2m  j0 ω j  1 n  , for a.e.t∈ J and all  x 0 ,x 1 , ,x 2m  ∈ 2m1 . 2.7 We investigate the regular differential equation  −1  m x 2m1  t   f n  t, x  t  , ,x 2m  t   . 2.8 If a function x ∈ AC 2m J satisfies 2.8 for a.e. t ∈ J,thenx is called a solution of 2.8. Boundary Value Problems 5 3. Auxiliary Regular Problems Let j ∈ and denote by G j t, s the Green function of the problem x 2j  t   0,x 2i  0   x 2i  T   0, 0 ≤ i ≤ j − 1. 3.1 Then, G 1  t, s   ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ s T  t −T  for 0 ≤ s ≤ t ≤ T, t T  s −T  for 0 ≤ t ≤ s ≤ T. 3.2 By 2, 3, 20, the Green function G j can be expressed as G j  t, s    T 0 G 1  t, τ  G j−1  τ, s  dτ, j > 1, 3.3 and it is known that see, e.g., 3, 20  −1  j G j  t, s  > 0for  t, s  ∈  0,T  ×  0,T  ,j≥ 1. 3.4 Lemma 3.1 see 10, Lemmas 2.1 and 2.3. For t, s ∈ J × J and j ∈ , the inequalities  −1  j G j  t, s  ≤ T 2j−3 6 j−1 s  T −s  , 3.5  −1  j G j  t, s  ≥ T 2j−5 30 j−1 ts  T −t  T −s  3.6 hold. Let γ ∈ L 1 J and let u ∈ AC 2m−1 J be a solution of the differential equation  −1  m u 2m  t   γ  t  , 3.7 satisfying the Lids tone boundary conditions u 2i  0   u 2i  T   0, 0 ≤ i ≤ m − 1. 3.8 It follows from the definition of the Green function G j that  −1  j u 2j  t    −1  m−j  T 0 G m−j  t, s  γ  s  ds for t ∈ J, 0 ≤ j ≤ m − 1. 3.9 6 Boundary Value Problems It is easy to check that x ∈ AC 2m J is a solution of problem 2.8, 1.2 if and only if x00, and its derivative x  is a solution of a problem involving the functional differential equation  −1  m u 2m  t   f n  t,  t 0 u  s  ds, u  t  , ,u 2m−1  t   3.10 and the Lidstone boundary conditions 3.8.From3.9for j  0,weseethatu ∈ AC 2m−1 J is a solution of problem 3.10, 3.8 exactly if it is a solution of the equation u  t    −1  m  T 0 G m  t, s  f n  s,  s 0 u  τ  dτ, u  s  , ,u 2m−1  s   ds, 3.11 in the set C 2m−1 J.Consequently,x is a solution of problem 2.8, 1.2 if and only if it is a solution of the equation x  t    −1  m  t 0   T 0 G m  s, τ  f n  τ, x  τ, ,x 2m  τ   dτ  ds, 3.12 in the set C 2m J.Itmeansthatx is a solution of problem 2.8, 1.2 if x is a fixed point of the operator H : C 2m J → C 2m J defined as  Hx  t    −1  m  t 0   T 0 G m  s, τ  f n  τ, x  τ, ,x 2m  τ   dτ  ds. 3.13 We prove the existence of a fixed point of H by the following fixed point result of cone compression type according to Guo-Krasnosel’skii see, e.g., 18, 19. Lemma 3.2. Let X be a Banach space, and let P ⊂ X be a cone in X.LetΩ 1 , Ω 2 be bounded open balls of X centered at the origin with Ω 1 ⊂ Ω 2 . Suppose that F : P ∩ Ω 2 \Ω 1  → P is completely continuous operator such that  Fx  ≥  x  for x ∈ P ∩ ∂Ω 1 ,  Fx  ≤  x  for x ∈ P ∩ ∂Ω 2 3.14 holds. Then, F has a fixed point in P ∩ Ω 2 \ Ω 1 . We are now in the position to prove that problem 2.8, 1.2 has a solution. Lemma 3.3. Let (H 1 )and(H 2 ) hold. Then, pr oblem 2.8, 1.2 has a solution. Proof. Let the operator H : C 2m J → C 2m J be given in 3.13,andlet P   x ∈ C 2m  J  : x  t  ≥ 0fort ∈ J  . 3.15 Then, P is a cone in C 2m J and since −1 m G m t, s > 0fort, s ∈ 0,T × 0,T by 3.4 and f n satisfies 2.5,weseethatH : C 2m J → P.ThefactthatH is a completely continuous Boundary Value Problems 7 operat or follows from f n ∈ CarJ × 2m1 , from Lebesgue dominated convergence theorem, and from the Arzel ` a-Ascoli theorem. Choose x ∈ P and put ytHxt for t ∈ J. Then, cf. 2.5  −1  m y 2m1  t   f n  t, x  t  , ,x 2m  t   ≥ a>0fora.e.t∈ J. 3.16 Since y00andy 2i−1 0y 2i−1 T0for1≤ i ≤ m, the equality y j ξ j 0holdswith some ξ j ∈ J for 0 ≤ j ≤ 2m.Wenowusetheequalityy 2m ξ 2m 0 and have    y 2m  t            t ξ 2m y 2m1  s  ds      ≥ a | t − ξ 2m | for t ∈ J. 3.17 Hence, y 2m  ∞ ≥ aT/2, and so  Hx  C 2m > aT 2 . 3.18 Next,wededucefromtherelation    y 2m  t            t ξ 2m f n  s, x  s  , ,x 2m  s   ds      ≤  T 0 f n  s, x  s  , ,x 2m  s   ds 3.19 and from 2.7 that    y 2m  t     ≤  T 0 h  s, 2m  1   x  C 2m  ds  T 2m  j0 ω j  1 n  for t ∈ J. 3.20 Therefore,    y 2m    ∞ ≤  T 0 h  s, 2m  1   x  C 2m  ds  V, 3.21 where V  T  2m j0 ω j 1/n.Sincey j ξ j 0for0≤ j ≤ 2m,wehave    y j    ∞ ≤ T 2m−j    y 2m    ∞ , 0 ≤ j ≤ 2m. 3.22 The last inequality together with 3.21 gives   y   C 2m ≤ K    y 2m    ∞ ≤ K   T 0 h  s, 2m  1   x  C 2m  ds  V  , 3.23 where K is from H 2 .Sincex ∈ P is arbitrary, relations 3.18 and 3.21 imply that for all 8 Boundary Value Problems x ∈ P, inequalities 3.18 and  Hx  C 2m ≤ K   T 0 h  s, 2m  1   x  C 2m  ds  V  3.24 hold. By H 2 ,thereexistsC>0suchthat 1 v   T 0 h  s, 2m  1  Kv  ds  V  ≤ 1 ∀v ≥ C K , 3.25 and therefore, K   T 0 h  s, 2m  1  v  ds  V  ≤ v ∀v ≥ C. 3.26 Let Ω 1   x ∈ C 2m  J  :  x  C 2m < aT 2  , Ω 2   x ∈ C 2m  J  :  x  C 2m <C  . 3.27 Then, it follows from 3.18, 3.24,and3.26 that  Hx  C 2m ≥  x  C 2m for x ∈ P ∩ ∂Ω 1 ,  Hx  C 2m ≤  x  C 2m for x ∈ P ∩ ∂Ω 2 . 3.28 The conclusion now follows from Lemma 3.2 for X  C 2m J and F  H. The properties of solutions to problem 2.8, 1.2 are collected in the following lemma. Lemma 3.4. Let (H 1 )and(H 2 )besatisfied.Letx n be a solution of problem 2.8, 1.2. Then, for all n ∈ , the following assertions hold: i−1 j x 2j1 n t > 0 for t ∈ 0,T, 0 ≤ j ≤ m − 1,and−1 m x 2m1 n t ≥ a for a.e. t ∈ J, ii x n is increasing on J,andfor0 ≤ j ≤ m − 1, −1 j x 2j2 n is decreasing on J,andthereisa unique ξ j,n ∈ 0,T such that x 2j2 n ξ j,n 0, iii there exists a positive constant A such that    x 2m n  t     ≥ A | t − ξ m−1,n | ,    x 2j2 n  t     ≥ A  t − ξ j,n  2 if 0 ≤ j ≤ m − 2,    x 2j1 n  t     ≥ At  T −t  if 0 ≤ j ≤ m − 1, x n  t  ≥ At 2 , 3.29 for t ∈ J, iv the sequence {x n } is bounded in C 2m J. Boundary Value Problems 9 Proof. Let us choose an arbitrary n ∈ .By2.5,  −1  m x 2m1 n  t   f n  t, x n  t  , ,x 2m n  t   ≥ a for a.e.t∈ J, 3.30 and it follows from the definition of the Green function G j that the equality  −1  j x 2j1 n  t    −1  m−j  T 0 G m−j  t, s  f n  s, x n  s  , ,x 2m n  s   ds 3.31 holds for t ∈ J and 0 ≤ j ≤ m − 1. Now, using 1.2, 3.4, 3.30,and3.31,weseethat assertion i is true. Hence, −1 j x 2j2 n is decreasing on J for 0 ≤ j ≤ m−1andx n is increasing on this interval. Due to x 2i−1 n 0x 2i−1 n T0for1≤ i ≤ m, there exists a unique ξ j,n ∈ 0,T such that u 2j2 n ξ j,n 0for0≤ j ≤ m − 1. Consequently, assertion ii holds. Next, in view of 2.5, 3.6,and3.31,    x 2j1 n  t     ≥ T 2m−j−5 a 30 m−j−1 t  T −t   T 0 s  T −s  ds  T 2m−j−2 a 6 ·30 m−j−1 t  T −t  for t ∈ J, 0 ≤ j ≤ m − 1. 3.32 Since x 2j2 n  t    t ξ j,n x 2j3 n  s  ds 3.33 and, by 13, Lemma 6.2,       t ξ j,n s  T −s  ds      ≥ T 6  t − ξ j,n  2 , 3.34 we have    x 2j2 n  t     ≥ T 2m−j−3 a 36 · 30 m−j−2  t − ξ j,n  2 for t ∈ J, 0 ≤ j ≤ m − 2. 3.35 Furthermore,    x 2m n  t            t ξ m−1,n f n  s, x n  s  , ,x 2m n  s   ds      ≥ a | t − ξ m−1,n | ,t∈ J, 3.36 10 Boundary Value Problems and cf. 3.32 for j  0 x n  t    t 0 x  n  s  ds ≥ T 2m−2 a 6 ·30 m−1  t 0 s  T −s  ds  T 2m−2 a 36 · 30 m−1 t 2  3T − 2t  ≥ T 2m−1 a 36 · 30 m−1 t 2 for t ∈ J, 3.37 since x  n > 0on0,T by assertion ii.Let A  a ·min  1,A 1 ,A 2 , T 2m−1 36 · 30 m−1  , 3.38 where A 1  min  T 2m−j−2 6 ·30 m−j−1 :0≤ j ≤ m −1  , A 2  min  T 2m−j−3 36 · 30 m−j−2 :0≤ j ≤ m − 2  . 3.39 Then estimate 3.29 follows from relations 3.32–3.37. It remains to prove the boundedness of the sequence {x n } in C 2m J. We use estimate 3.29, the properties of ω j given in H 2 , and the inequality t  T −t  ≥ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ T 2 t for 0 <t≤ T 2 , T 2  T −t  for T 2 <t<T 3.40 and have  T 0 ω 2m     x 2m n  s      ds ≤  T 0 ω 2m  A | s − ξ m−1,n | ds  1 A   Aξ m−1,n 0 ω 2m  s  ds   AT−ξ m−1,n  0 ω 2m  s  ds  < 2 A  AT 0 ω 2m  s  ds,  T 0 ω 2j2     x 2j2 n  s      ds ≤  T 0 ω 2j2  A  s − ξ j,n  2  ds  1 √ A  √ AT−ξ j,n  − √ Aξ j,n ω 2j2  s 2  ds [...]... method of upper and lower solutions for a Lidstone boundary value problem,” Czechoslovak Mathematical Journal, vol 55 130 , no 3, pp 639–652, 2005 6 Y Ma, “Existence of positive solutions of Lidstone boundary value problems,” Journal of Mathematical Analysis and Applications, vol 314, no 1, pp 97–108, 2006 7 P J Y Wong and R P Agarwal, “Results and estimates on multiple solutions of Lidstone boundary value. .. Equations, vol 5 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2008 14 Z Wei, “Existence of positive solutions for nth-order singular sublinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol 306, no 2, pp 619–636, 2005 15 Z Zhao, “On the existence of positive solutions for n-order singular boundary value problems,”... 86, no 1-2, pp 137–168, 2000 8 Y.-R Yang and S S Cheng, Positive solutions of a Lidstone boundary value problem with variable coefficient function,” Journal of Applied Mathematics and Computing, vol 27, no 1-2, pp 411–419, 2008 9 B Zhang and X Liu, “Existence of multiple symmetric positive solutions of higher order Lidstone problems,” Journal of Mathematical Analysis and Applications, vol 284, no 2,... a positive solution x ∈ AC2m J satisfying inequality 4.1 Acknowledgment This work was supported by the Council of Czech Government MSM no 6198959214 Boundary Value Problems 15 References 1 R P Agarwal, S Pinelas, and P J Y Wong, Complementary Lidstone interpolation and boundary value problems,” Journal of Inequalities and Applications, vol 2009, Article ID 624631, 30 pages, 2009 2 R P Agarwal, Boundary. .. BVPs with a e ˚ singularities in phase variables,” Computers & Mathematics with Applications, vol 46, no 12, pp 1799– 1826, 2003 11 R P Agarwal, D O’Regan, and S Stanˇ k, Singular Lidstone boundary value problem with given e maximal values for solutions, ” Nonlinear Analysis: Theory, Methods & Applications, vol 55, no 7-8, pp 859–881, 2003 a e 12 I Rachunkov´ , S Stanˇ k, and M Tvrdy, “Singularities... Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Teaneck, NJ, USA, 1986 3 R P Agarwal and P J Y Wong, Lidstone polynomials and boundary value problems,” Computers & Mathematics with Applications, vol 17, no 10, pp 1397–1421, 1989 4 J M Davis, J Henderson, and P J Y Wong, “General Lidstone problems: multiplicity and symmetry of solutions, ” Journal of Mathematical... “Singularities and Laplacians in boundary value problems ˚ ´ for nonlinear ordinary differential equations,” in Handbook of Differential Equations: Ordinary Differential Equations Vol III, A Canada, P Dr´ bek, and A Fonda, Eds., Handb Differ Equ., pp a ˜ 607–722, Elsevier/North-Holland, Amsterdam, The Netherlands, 2006 13 I Rachunkov´ , S Stanˇ k, and M Tvrdy, Solvability of Nonlinear Singular Problems for Ordinary... t } for Lemma 3.5 Let (H1 ) and (H2 ) hold Let xn be a solution of problem 2.8 , 1.2 Then, the sequence 2m fn t, xn t , , xn t ⊂ L1 J 3.49 is uniformly integrable on J, that is, for each ε > 0, there exists δ > 0 such that if M ⊂ J and meas M < δ, then 2m M fn t, xn t , , xn t dt < ε for n ∈ Æ 3.50 Boundary Value Problems 13 Proof By Lemma 3.4 iv , there exists S > 0 such that for n ∈ Æ , the... ≤ j ≤ m − 1 Let limn → ∞ xn x and limn → ∞ ξj,n ξj 14 Boundary Value Problems 0 ≤ j ≤ m−1 Then x ∈ C2m J and x satisfies the boundary conditions 1.2 Letting n → ∞ in 3.29 and 4.2 , we get for t ∈ J x 2m t ≥ A|t − ξm−1 |, −1 j x 2j 1 x 2j 2 t ≥ A t − ξj 2 if 0 ≤ j ≤ m − 2 4.3 if 0 ≤ j ≤ m − 1, x t ≥ At2 t ≥ At T − t Keeping in mind the definition of fn , we conclude from 4.3 that 2m lim fn t, xn t ,... exists a positive constant S such that for all v ≥ S the inequality T h t, 2m 1 Kv dt Λ≤v 3.48 0 2m is fulfilled The last inequality together with estimate 3.46 gives xn ∞ < S for n ∈ j 2m−j S for 0 ≤ j ≤ 2n, n ∈ Æ , and assertion iv follows Consequently, xn ∞ < T 2m The following result gives the important property of {fn t, xn t , , xn applying the Vitali convergent theorem in the proof of Theorem . Corporation Boundary Value Problems Volume 2010, Article ID 368169, 15 pages doi:10.1155/2010/368169 Research Article Positive Solutions of Singular Complementary Lidstone Boundary Value Problems Ravi. lower solutions for a Lidstone boundary value problem,” Czechoslovak Mathematical Journal,vol.55130, no. 3, pp. 639–652, 2005. 6 Y. Ma, “Existence of positive solutions of Lidstone boundary value. the exi stence of positive solutions of singular complementary Lidstone problems. We observe that differential equations in complementary Lidstone problems as well as derivatives in boundary conditions