Báo cáo hóa học: " Research Article Existence of Positive Solutions for Multipoint Boundary Value Problem with p-Laplacian on Time Scales" ppt

Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 312058, 18 pages doi:10.1155/2009/312058 Research Article Existence of Positive Solutions for Multipoint Boundary Value Problem with p-Laplacian on Time Scales Meng Zhang, 1 Shurong Sun, 1 and Zhenlai Han 1, 2 1 School of Science, University of Jinan, Jinan, Shandong 250022, China 2 School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China Correspondence should be addressed to Shurong Sun, sshrong@163.com Received 11 March 2009; Accepted 8 May 2009 Recommended by Victoria Otero-Espinar We consider the existence of positive solutions for a class of second-order multi-point boundary value problem with p-Laplacian on time scales. By using the well-known Krasnosel’ski’s fixed- point theorem, some new existence criteria for positive solutions of the boundary value problem are presented. As an application, an example is given to illustrate the main results. Copyright q 2009 Meng Zhang 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. 1. Introduction The theory of time scales has become a new important mathematical branch since it was introduced by Hilger 1. Theoretically, the time scales approach not only unifies calculus of differential and difference equations, but also solves other problems that are a mix of stop start and continuous behavior. Practically, the time scales calculus has a tremendous potential for application, for example, Thomas believes that time scales calculus is the best way to understand Thomas models populations of mosquitoes that carry West Nile virus 2. In addition, Spedding have used this theory to model how students suffering from the eating disorder bulimia are influenced by their college friends; with the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks 2. By using the theory on time scales we can also study insect population, biology, heat transfer, stock market, epidemic models 2–6, and so forth. At the same time, motivated by the wide application of boundary value problems in physical and applied mathematics, boundary value problems for dynamic equations with p-Laplacian on time scales have received lots of interest 7–16. 2 Advances in Difference Equations In 7, Anderson et al. considered the following three-point boundary value problem with p-Laplacian on time scales:  ϕ p u Δ t  ∇  c  t  f  u  t   0,t∈  a, b  , u  a  − B 0  u Δ  v    0,u Δ  b   0, 1.1 where v ∈ a, b,f ∈ C ld 0, ∞, 0, ∞,c ∈ C ld a, b, 0, ∞,andK m x ≤ B 0 x ≤ K M x for some positive constants K m ,K M . They established the existence results for at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type. For the same boundary value problem, He in 8 using a new fixed point theorem due to Avery and Henderson obtained the existence results for at least two positive solutions. In 9, Sun and Li studied the following one-dimensional p-Laplacian boundary value problem on time scales:  ϕ p u Δ t  Δ  h  t  f  u σ  t   0,t∈  a, b  , u  a  − B 0  u Δ  a    0,u Δ  σ  b   0, 1.2 where ht is a nonnegative rd-continuous function defined in a, b and satisfies that there exists t 0 ∈ a, b such that ht 0  > 0,fu is a nonnegative continuous function defined on 0, ∞,B 1 x ≤ B 0 x ≤ B 2 x for some positive constants B 1 ,B 2 . They established the existence results for at least single, twin, or triple positive solutions of the above problem by using Krasnosel’skii’s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett-Williams fixed point theorem. For the Sturm-Liouville-like boundary value problem, in 17 Ji and Ge investigated a class of Sturm-Liouville-like four-point boundary value problem with p-Laplacian:  ϕ p  u   t     f  t, u  t   0,t∈  0, 1  , u  0  − αu   ξ   0,u  1   βu   η   0, 1.3 where ξ<η,f∈ C0, 1 × 0, ∞, 0, ∞. By using fixed-point theorem for operators on a cone, they obtained some existence of at least three positive solutions for the above problem. However, to the best of our knowledge, there has not any results concerning the similar problems on time scales. Motivated by the above works, in this paper we consider the following multi-point boundary value problem on time scales:  ϕ p  u Δ  t   Δ  h  t  f  u  t   0,t∈ a, b T , αu  a  − βu Δ  ξ   0,γu  σ 2  b    δu Δ  η   0,u Δ  θ   0, 1.4 Advances in Difference Equations 3 where T is a time scale,ϕ p u|u| p−2 u, p > 1,α>0,β≥ 0,γ>0,δ≥ 0,a<ξ<θ<η<b, and we denote ϕ p  −1  ϕ q with 1/p  1/q  1. In the following, we denote a, b :a, b T a, b ∩ T for convenience. And we list the following hypotheses: C 1  fu is a nonnegative continuous function defined on 0, ∞; C 2  h : a, σ 2 b → 0, ∞ is rd-continuous with h · f / ≡ 0. 2. Preliminaries In this section, we provide some background material to facilitate analysis of problem 1.4. Let the Banach space E  {u : a, σ 2 b → R is rd-continuous} be endowed with the norm u  sup t∈a,σ 2 b |ut| and choose the cone P ⊂ E defined by P   u ∈ E : u  t  ≥ 0,t∈  a, σ 2  b   ,u ΔΔ  t  ≤ 0,t∈  a, b   . 2.1 It is easy to see that t he solution of BVP 1.4 can be expressed as u  t   ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ β α ϕ q   θ ξ h  r  f  u  r  Δr    t a ϕ q   θ s h  r  f  u  r  Δr  Δs, a ≤ t ≤ θ, δ γ ϕ q   η θ h  r  f  u  r  Δr    σ 2  b  t ϕ q   s θ h  r  f  u  r  Δr  Δs, θ ≤ t ≤ σ 2  b  . 2.2 If V 1  V 2 , where V 1  β α ϕ q   θ ξ h  r  f  u  r  Δr    θ a ϕ q   θ s h  r  f  u  r  Δr  Δs, V 2  δ γ ϕ q   η θ h  r  f  u  r  Δr    σ 2 b θ ϕ q   s θ h  r  f  u  r  Δr  Δs, 2.3 we define the operator A : P → E by Au  t   ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ β α ϕ q   θ ξ h  r  f  u  r  Δr    t a ϕ q   θ s h  r  f  u  r  Δr  Δs, a ≤ t ≤ θ, δ γ ϕ q   η θ h  r  f  u  r  Δr    σ 2 b t ϕ q   s θ h  r  f  u  r  Δr  Δs, θ ≤ t ≤ σ 2  b  . 2.4 4 Advances in Difference Equations It is easy to see u  uθ, Aut ≥ 0fort ∈ a, σ 2 b, and if Autut, then ut is the positive solution of BVP 1.4. From the definition of A, for each u ∈ P, we have Au ∈ P, and Au  Auθ. In fact, Au Δ  t   ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ q   θ t h  r  f  u  r  Δr  ≥ 0,a≤ t ≤ θ, −ϕ q   t θ h  r  f  u  r  Δr  ≤ 0,θ≤ t ≤ σ 2  b  2.5 is continuous and nonincreasing in a, σ 2 b. Moreover, ϕ q x is a monotone increasing continuously differentiable function,   θ t hsfusΔs  Δ   −  t θ hsfusΔs  Δ  −h  t  f  u  t  ≤ 0, 2.6 then by the chain rule on time scales, we obtain Au ΔΔ  t  ≤ 0, 2.7 so, A : P → P. For the notational convenience, we denote L 1   β α  θ − a  ϕ q   θ a h  r  Δr  , L 2   δ γ  σ 2  b  − θ  ϕ q   σ 2 b θ h  r  Δr  , M 1  β α ϕ q   θ ξ h  r  Δr    θ ξ ϕ q   θ s h  r  Δr  Δs, M 2  δ γ ϕ q   η θ h  r  Δr    η θ ϕ q   s θ h  r  Δr  Δs, M 3  min  ξ − a θ − a , σ 2  b  − η σ 2  b  − θ  , M 4  max  θ − a ξ − a , σ 2  b  − θ σ 2  b  − η  . 2.8 Advances in Difference Equations 5 Lemma 2.1. A : P → P is completely continuous. Proof. First, we show that A maps bounded set into bounded set. Assume that c>0 is a constant and u ∈ P c . Note that the continuity of f guarantees that there exists K>0 such that fu ≤ ϕ p K.So Au  Au  θ   β α ϕ q   θ ξ h  r  f  u  r  Δr    θ a ϕ q   θ s h  r  f  u  r  Δr  Δs ≤ β α ϕ q   θ a h  r  ϕ p  K  Δr    θ a ϕ q   θ a h  r  ϕ p  K  Δr  Δs  K  β α  θ − a  ϕ q   θ a h  r  Δr   KL 1 , Au  Au  θ   δ γ ϕ q   η θ h  r  f  u  r  Δr    σ 2 b θ ϕ q   s θ h  r  f  u  r  Δr  Δs ≤ δ γ ϕ q   σ 2 b ξ h  r  ϕ p  K  Δr    σ 2 b θ ϕ q   σ 2 b θ h  r  ϕ p  K  Δr  Δs  K  δ γ  σ 2  b  − θ  ϕ q   σ 2 b θ h  r  Δr   KL 2 . 2.9 That is, A P c is uniformly bounded. In addition, it is easy to see | Au  t 1  − Au  t 2  | ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C | t 1 − t 2 | ϕ q   θ a h  r  Δr  ,t 1 ,t 2 ∈  a, θ  , C | t 1 − t 2 | ϕ q   σ 2 b a h  r  Δr  ,t 1 ∈  a, θ  ,t 2 ∈  θ, σ 2  b   or t 2 ∈  a, θ  ,t 1 ∈  θ, σ 2  b   , C | t 1 − t 2 | ϕ q   σ 2  b  θ h  r  Δr  ,t 1 ,t 2 ∈  a, θ  . 2.10 6 Advances in Difference Equations So, by applying Arzela-Ascoli Theorem on time scales, we obtain that A P c is relatively compact. Second, we will show that A : P c → P is continuous. Suppose that {u n } ∞ n1 ⊂ P c and u n t converges to u 0 t uniformly on a, σ 2 b. Hence, {Au n t} ∞ n1 is uniformly bounded and equicontinuous on a, σ 2 b. The Arzela-Ascoli Theorem on time scales tells us that there exists uniformly convergent subsequence in {Au n t} ∞ n1 .Let{Au n l t} ∞ l1 be a subsequence which converges to vt uniformly on a, σ 2 b. In addition, 0 ≤ Au n  t  ≤ min { KL 1 ,KL 2 } . 2.11 Observe that Au n  t   ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ β α ϕ q   θ ξ h  r  f  u n  r  Δr    t a ϕ q   θ s h  r  f  u n  r  Δr  Δs, a ≤ t ≤ θ, δ γ ϕ q   η θ h  r  f  u n  r  Δr    σ 2 b t ϕ q   s θ h  r  f  u n  r  Δr  Δs, θ ≤ t ≤ σ 2  b  . 2.12 Inserting u n l into the above and then letting l →∞,weobtain v  t   ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ β α ϕ q   θ ξ h  r  f  u 0  r  Δr    t a ϕ q   θ s h  r  f  u 0  r  Δr  Δs, a ≤ t ≤ θ, δ γ ϕ q   η θ h  r  f  u 0  r  Δr    σ 2 b t ϕ q   s θ h  r  f  u 0  r  Δr  Δs, θ ≤ t ≤ σ 2  b  , 2.13 here we have used the Lebesgues dominated convergence theorem on time scales. From the definition of A, we know that vtAu 0 t on a, σ 2 b. This shows that each subsequence of {Au n t} ∞ n1 uniformly converges to Au 0 t. Therefore, the sequence {Au n t} ∞ n1 uniformly converges to Au 0 t. This means that A is continuous at u 0 ∈ P c .So,A is continuous on P c since u 0 is arbitrary. Thus, A is completely continuous. The proof is complete. Lemma 2.2. Let u ∈ P, then ut ≥ t − a/θ − au for t ∈ a, θ, and ut ≥ σ 2 b − t/σ 2 b − θu for t ∈ θ, σ 2 b. Proof. Since u ΔΔ t ≤ 0, it follows that u Δ t is nonincreasing. Hence, for a<t<θ, u  t  − u  a    t a u Δ  s  Δs ≥  t − a  u Δ  t  , u  θ  − u  t    θ t u Δ  s  Δs ≤  θ − t  u Δ  t  , 2.14 Advances in Difference Equations 7 from which we have u  t  ≥ u  a  θ − t    t − a  u  θ  θ − a ≥ t − a θ − a u  θ   t − a θ − a u. 2.15 For θ ≤ t ≤ σ 2 b, u  σ 2  b   − u  t    σ 2 b t u Δ  s  Δs ≤  σ 2  b  − t  u Δ  t  , u  t  − u  θ    t θ u Δ  s  Δs ≥  t − θ  u Δ  t  , 2.16 we know u  t  ≥  σ 2  b  − t  u  θ    t − θ  u  σ 2  b   σ 2  b  − θ ≥ σ 2  b  − t σ 2  b  − θ u  θ   σ 2  b  − t σ 2  b  − θ u. 2.17 The proof is complete. Lemma 2.3 18. Let P be a cone in a Banach space E. Assum that Ω 1 , Ω 2 are open subsets of E with 0 ∈ Ω 1 , Ω 1 ⊂ Ω 2 . If A : P ∩  Ω 2 \ Ω 1  −→ P 2.18 is a completely continuous operator such that either i Ax≤x, ∀x ∈ P ∩ ∂Ω 1 and Ax≥x, ∀x ∈ P ∩ ∂Ω 2 , or ii Ax≥x, ∀x ∈ P ∩ ∂Ω 1 and Ax≤x, ∀x ∈ P ∩ ∂Ω 2 . Then A has a fixed point in P ∩  Ω 2 \ Ω 1 . 3. Main Results In this section, we present our main results with respect to BVP 1.4. For the sake of convenience, we define f 0  lim u → 0  fu/ϕ p u,f ∞  lim u →∞ fu/ϕ p u,i 0  number of zeros in the set {f 0 ,f ∞ },andi ∞  number of ∞ in the set {f 0 ,f ∞ }. Clearly, i 0 ,i ∞  0, 1, or 2 and there are six possible cases: i i 0  0andi ∞  0; ii i 0  0andi ∞  1; iii i 0  0andi ∞  2; 8 Advances in Difference Equations iv i 0  1andi ∞  0; v i 0  1andi ∞  1; vi i 0  2andi ∞  0. Theorem 3.1. BVP 1.4 has at least one positive solution in the case i 0  1 and i ∞  1. Proof. First, we consider the case f 0  0andf ∞  ∞. Since f 0  0, then there exists H 1 > 0 such that fu ≤ ϕ p εϕ p uϕ p εu, for 0 <u≤ H 1 , where ε satisfies max { εL 1 ,εL 2 } ≤ 1. 3.1 If u ∈ P, with u  H 1 , then Au  Au  θ   β α ϕ q   θ ξ h  r  f  u  r  Δr    θ a ϕ q   θ s h  r  f  u  r  Δr  Δs ≤ β α ϕ q   θ a h  r  f  u  r  Δr    θ a ϕ q   θ a h  r  f  u  r  Δr  Δs ≤ β α ϕ q   θ a h  r  ϕ p  εu  Δr    θ a ϕ q   θ a h  r  ϕ p  εu  Δr  Δs  uεL 1 ≤u, Au  Au  θ   δ γ ϕ q   η θ h  r  f  u  r  Δr    σ 2 b θ ϕ q   s θ h  r  f  u  r  Δr  Δs ≤ δ γ ϕ q   σ 2 b θ h  r  f  u  r  Δr    σ 2 b θ ϕ q   σ 2 b θ h  r  f  u  r  Δr  Δs ≤ δ γ ϕ q   σ 2 b θ h  r  ϕ p  εu  Δr    σ 2 b θ ϕ q   σ 2 b θ h  r  ϕ p  εu  Δr  Δs  uεL 2 ≤u. 3.2 It follows that if Ω H 1  {u ∈ E : u <H 1 }, then Au≤u for u ∈ P ∩ ∂Ω H 1 . Advances in Difference Equations 9 Since f ∞  ∞, then there exists H  2 > 0 such that fu ≥ ϕ p kϕ p uϕ p ku, for u ≥ H  2 , where k>0 is chosen such that min  k ξ − a θ − a M 1 ,k σ 2  b  − η σ 2  b  − θ M 2  ≥ 1. 3.3 Set H 2  max{2H 1 , θ − a/ξ − aH  2 , σ 2 b − θ/σ 2 b − ηH  2 }, and Ω H 2  {u ∈ E : u <H 2 }. If u ∈ P with u  H 2 , then min t∈ξ,θ u  t   u  ξ  ≥ ξ − a θ − a u≥H  2 , min t∈θ,η u  t   u  η  ≥ σ 2  b  − η σ 2  b  − θ u≥H  2 . 3.4 So that Au  Au  θ   β α ϕ q   θ ξ h  r  f  u  r  Δr    θ a ϕ q   θ s h  r  f  u  r  Δr  Δs ≥ β α ϕ q   θ ξ h  r  ϕ p  ku  Δr    θ ξ ϕ q   θ s h  r  ϕ p  ku  Δr  Δs ≥ β α ϕ q   θ ξ h  r  ϕ p  k ξ − a θ − a u  Δr    θ ξ ϕ q   θ s h  r  ϕ p  k ξ − a θ − a u  Δr  Δs  uk ξ − a θ − a M 1 ≥u, Au  Au  θ   δ γ ϕ q   η θ h  r  f  u  r  Δr    σ 2 b θ ϕ q   s θ h  r  f  u  r  Δr  Δs ≥ δ γ ϕ q   η θ h  r  ϕ p  k σ 2  b  − η σ 2  b  − θ u  Δr    η θ ϕ q   s θ h  r  ϕ p  k σ 2  b  − η σ 2  b  − θ u  Δr  Δs  uk σ 2  b  − η σ 2  b  − θ M 2 ≥u. 3.5 10 Advances in Difference Equations In other words, if u ∈ P ∩ ∂Ω H 2 , then Au≥u. Thus by i of Lemma 2.3, it follows that A has a fixed point in P ∩  Ω H 2 \ Ω H 1  with H 1 ≤u≤H 2 . Now we consider the case f 0  ∞ and f ∞  0. Since f 0  ∞, there exists H 3 > 0, such that fu ≥ ϕ p mϕ p uϕ p mu for 0 <u≤ H 3 , where m is such that min  mM 1 ξ − a θ − a ,mM 2 σ 2  b  − η σ 2  b  − θ  ≥ 1. 3.6 If u ∈ P with u  H 3 , then we have Au  Au  θ   β α ϕ q   θ ξ h  r  f  u  r  Δr    θ a ϕ q   θ s h  r  f  u  r  Δr  Δs ≥ β α ϕ q   θ ξ h  r  ϕ p  m ξ − a θ − a u  Δr    θ ξ ϕ q   θ s h  r  ϕ p  m ξ − a θ − a u  Δr  Δs  um ξ − a θ − a M 1 ≥u, Au  Au  θ   δ γ ϕ q   η θ h  r  f  u  r  Δr    σ 2 b θ ϕ q   s θ h  r  f  u  r  Δr  Δs ≥ δ γ ϕ q   η θ h  r  ϕ p  m σ 2  b  −η σ 2  b  −θ u  Δr    η θ ϕ q   s θ h  r  ϕ p  m σ 2  b  − η σ 2  b  − θ u  Δr  Δs  um σ 2  b  − η σ 2  b  − θ M 2 ≥u. 3.7 Thus, we let Ω H 3  {u ∈ E : u <H 3 }, so that Au≥u for u ∈ P ∩ ∂Ω H 3 . Next consider f ∞  0. By definition, there exists H  4 > 0 such that fu ≤ ϕ p εϕ p u ϕ p εu for u ≥ H  4 , where ε>0satisfies max { εL 1 ,εL 2 } ≤ 1. 3.8 [...]... Henderson, Existence of solutions for a one dimensional p-Laplacian on time- scales,” Journal of Difference Equations and Applications, vol 10, no 10, pp 889–896, 2004 8 Z He, “Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol 182, no 2, pp 304–315, 2005 9 H.-R Sun and W.-T Li, Existence. .. theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales,” Journal of Differential Equations, vol 240, no 2, pp 217– 248, 2007 18 Advances in Difference Equations 10 W.-T Li and H R Sun, Positive solutions for second-order m-point boundary value problems on time scales,” Acta Mathematica Sinica, vol 22, no 6, pp 1797–1804, 2006 11 H R Sun, Boundary value problems... Ge, Existence of positive solutions for a class of m-point boundary value problems,” Advances in Difference Equations, vol 2008, Article ID 845121, 9 pages, 2008 15 M Feng, X Li, and W Ge, “Triple positive solutions of fourth-order four-point boundary value problems for p-Laplacian dynamic equations on time scales,” Advances in Difference Equations, vol 2008, Article ID 496078, 9 pages, 2008 16 C Song,... problems for dynamic equations on measure chains, Ph D thesis, Lanzhou University, Lanzhou, China, 2004 12 D R Anderson, Existence of solutions for a first-order p-Laplacian BVP on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol 69, no 12, pp 4521–4525, 2008 13 Y.-H Su, W.-T Li, and H.-R Sun, Positive solutions of singular p-Laplacian BVPs with sign changing nonlinearity on time. .. 9 pages, 2008 16 C Song, “Eigenvalue problems for p-Laplacian functional dynamic equations on time scales,” Advances in Difference Equations, vol 2008, Article ID 879140, 9 pages, 2008 17 D Ji and W Ge, Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with p-Laplacian, ” Nonlinear Analysis: Theory, Methods & Applications, vol 68, no 9, pp 2638–2646,... evolution model for phytoremediation of metals,” Discrete and Continuous Dynamical Systems Series B, vol 5, no 2, pp 411–422, 2005 5 M Bohner and A Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨ user, Boston, Mass, USA, 2001 a 6 M Bohner and A Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkh¨ user, Boston, a Mass, USA, 2003 7 D Anderson, R Avery,... the proof of Theorem 3.2, we obtain Au ≤ u , u ∈ P ∩ ∂Ωp 3.29 The result is obtained, and the proof is complete Theorem 3.8 Suppose that f∞ ∈ 0, ϕp A1 positive solution and f0 ∞ hold Then BVP 1.4 has at least one ∞, similar to the second part of Theorem 3.1, we have Au ≥ u for Proof Since f0 u ∈ P ∩ ∂ΩH3 By f∞ ∈ 0, ϕp A1 , similar to the second part of proof of Theorem 3.4, we have Au ≤ u for u ∈... least one positive solution The proof is complete From Theorems 3.7 and 3.8, we can get the following corollaries Corollary 3.9 Suppose that f∞ has at least one positive solution ∞ and the condition C3 in Theorem 3.2 hold Then BVP 1.4 Corollary 3.10 Suppose that f0 has at least one positive solution ∞ and the condition C3 in Theorem 3.2 hold Then BVP 1.4 Theorem 3.11 Suppose that i0 0, i∞ 2, and the condition... 0, i∞ 2, and the condition C3 of Theorem 3.2 hold Then BVP 1.4 has at least two positive solutions u1 , u2 ∈ P such that 0 < u1 < p < u2 Proof By using the method of proving Theorems 3.1 and 3.2, we can deduce the conclusion easily, so we omit it here Theorem 3.12 Suppose that i0 2, i∞ 0, and the condition C4 of Theorem 3.2 hold Then BVP 1.4 has at least two positive solutions u1 , u2 ∈ P such that... the BVP 4.1 has at least one positive solution Acknowledgments This research is supported by the Natural Science Foundation of China 60774004 , China Postdoctoral Science Foundation Funded Project 20080441126 , Shandong Postdoctoral Funded Project 200802018 , the Natural Science Foundation of Shandong Y2007A27, Y2008A28 , and the Fund of Doctoral Program Research of University of Jinan B0621, XBS0843 . Corporation Advances in Difference Equations Volume 2009, Article ID 312058, 18 pages doi:10.1155/2009/312058 Research Article Existence of Positive Solutions for Multipoint Boundary Value Problem with p-Laplacian. 2009 Recommended by Victoria Otero-Espinar We consider the existence of positive solutions for a class of second-order multi-point boundary value problem with p-Laplacian on time scales. By using the well-known. USA, 2003. 7 D. Anderson, R. Avery, and J. Henderson, Existence of solutions for a one dimensional p-Laplacian on time- scales,” Journal of Difference Equations and Applications, vol. 10, no. 10,