Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2008, Article ID 143040, 16 pages doi:10.1155/2008/143040 Research Article Positive Solutions for Third-Order Nonlinear p-Laplacian m-Point Boundary Value Problems on Time Scales Fuyi Xu School of Mathematics and Information Science, Shandong University of Technology, Zibo, Shandong 255049, China Correspondence should be addressed to Fuyi Xu, zbxufuyi@163.com Received 7 August 2008; Revised 8 October 2008; Accepted 9 November 2008 Recommended by Leonid Shaikhet We study the following third-order p-Laplacian m-point boundary value problems on time scales: ∇ Δ∇ φp uΔ∇ atft, ut 0, t ∈ 0, T T , βu0 − γuΔ 0 0, uT m−2 0 i1 ai uξi , φp u m−2 p−2 Δ∇ −1 b φ u ξ , where φ s is p-Laplacian operator, that is, φ s |s| s, p > 1, φ i p i p p p φq , i1 1/p 1/q 1, 0 < ξ1 < · · · < ξm−2 < ρT . We obtain the existence of positive solutions by using fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the known results. Copyright q 2008 Fuyi Xu. 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 was initiated by Hilger 1 as a means of unifying and extending theories from differential and difference equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example 2–6. Recently, the boundary value problems with p-Laplacian operator have also been discussed extensively in the literature, for example, see 7–13. A time scale T is a nonempty closed subset of R. We make the blanket assumption that 0, T are points in T. By an interval 0, T, we always mean the intersection of the real interval 0, T with the given time scale; that is 0, T∩T. In 14, Anderson considered the the following third-order nonlinear boundary value problem BVP: x t f t, xt , t1 ≤ t ≤ t3 , x t1 x t2 0, γx t3 δx t3 0. 1.1 2 Discrete Dynamics in Nature and Society Author studied the existence of solutions for the nonlinear boundary value problem by using the Krasnoselskii’s fixed point theorem and Leggett and Williams fixed point theorem, respectively. In 8, 9, He considered the existence of positive solutions of the p-Laplacian dynamic equations on time scales ∇ atf ut 0, φp uΔ t ∈ 0, T T , 1.2 satisfying the boundary conditions u0 − B0 uΔ η 0, uΔ T 0, 1.3 uT − B1 uΔ η 0, 1.4 or uΔ 0 0, where η ∈ 0, ρT . He obtained the existence of at least double and triple positive solutions of the boundary value problems by using a new double fixed point theorem and triple fixed point theorem, respectively. In 13, Zhou and Ma firstly studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with p-Laplacian operator: u0 φp u t qtf t, ut , m αi u ξi , u η 0, 0 ≤ t ≤ 1, u 1 i1 n βi u θi . 1.5 i1 They established a corresponding iterative scheme for the problem by using the monotone iterative technique. However, to the best of our knowledge, little work has been done on the existence of positive solutions for third-order p-Laplacian m-point boundary value problems on time scales. This paper attempts to fill this gap in the literature. In this paper, by using different method, we are concerned with the existence of positive solutions for the following third-order p-Laplacian m-point boundary value problems on time scales: ∇ atf t, ut 0, φp uΔ∇ βu0 − γuΔ 0 0, uT t ∈ 0, T T , m−2 m−2 bi φp uΔ∇ ξi , φp uΔ∇ 0 i1 i1 ai u ξi , 1.6 where φp s is p-Laplacian operator, that is, φp s |s|p−2 s, p > 1, φp−1 φq , 1/p1/q 1, Fuyi Xu 3 and ai , bi , a, f satisfy H1 β, γ ≥ 0, β γ > 0, ai ∈ 0, ∞, i 1, 2, . . . , m − 3, am−2 > 0, 0 < ξ1 < · · · < ξm−2 < m−2 m−2 m−2 ρT , 0 < m−2 i1 bi < 1, 0 < i1 ai ξi < T , d βT − i1 ai ξi γ1 − i1 ai > 0; H2 f : 0, T T × 0, ∞ → R is continuous, a ∈ Cld 0, T T , R and there exists t0 ∈ ξm−2 , T T such that at0 > 0, where R 0, ∞. 2. Preliminaries and lemmas For convenience, we list the following definitions which can be found in 1–5. Definition 2.1. A time scale T is a nonempty closed subset of real numbers R. For t < sup T and r > inf T, define the forward jump operator σ and backward jump operator ρ, respectively, by σt inf{τ ∈ T | τ > t} ∈ T, ρr sup{τ ∈ T | τ < r} ∈ T, 2.1 for all t, r ∈ T. If σt > t, t is said to be right scattered; if ρr < r, r is said to be left scattered; if σt t, t is said to be right dense; if ρr r, r is said to be left dense. If T has a right scattered minimum m, define Tk T − {m}, otherwise set Tk T. If T has a left scattered maximum M, define Tk T − {M}, otherwise set Tk T. Definition 2.2. For f : T → R and t ∈ Tk , the delta derivative of f at the point t is defined to be the number f Δ t, provided it exists, with the property that for each > 0, there is a neighborhood U of t such that f σt − fs − f Δ t σt − s ≤ σt − s, 2.2 for all s ∈ U. For f : T → R and t ∈ Tk , the nabla derivative of f at t, denoted by f ∇ t provided it exists, with the property that for each > 0, there is a neighborhood U of t such that f ρt − fs − f ∇ t ρt − s ≤ ρt − s, 2.3 for all s ∈ U. Definition 2.3. A function f is left-dense continuous i.e., ld-continuous, if f is continuous at each left-dense point in T and its right-sided limit exists at each right-dense point in T. Definition 2.4. If φΔ t ft, then one defines the delta integral by b a ftΔt φb − φa. 2.4 4 Discrete Dynamics in Nature and Society If F ∇ t ft, then one defines the nabla integral by b ft∇t Fb − Fa. 2.5 a Lemma 2.5. If d βT − value problem (BVP) m−2 i1 ai ξi γ1 − m−2 i1 ai > 0, then for h ∈ Cld 0, T T , the boundary uΔ∇ ht 0, t ∈ 0, T T , βu0 − γuΔ 0 0, m−2 ai u ξi uT 2.6 i1 has the unique solution ut − t t − shs∇s 0 βt γ m−2 − ai d i1 ξi βt γ d T T − shs∇s 0 2.7 ξi − s hs∇s. 0 Proof. By direct computation, we can easily get 2.7. So, we omit it. m−2 m−2 m−2 Lemma 2.6. If 0 < m−2 i1 bi < 1, 0 < i1 ai ξi < T , d βT − i1 ai ξi γ1 − i1 ai > 0, then for h ∈ Cld 0, T T , the boundary value problem (BVP) ∇ ht 0, φp uΔ∇ βu0 − γuΔ 0 0, uT t ∈ 0, T T , m−2 m−2 bi φp uΔ∇ ξi φp uΔ∇ 0 i1 i1 ai u ξi , 2.8 has the unique solution t s βt γ T hr∇r B ∇s T − sφq hr∇r B ∇s d 0 0 0 0 ξ s m−2 i βt γ − ai hr∇r B ∇s, ξi − s φq d i1 0 0 ut − where B t − sφq s 2.9 m−2 ξi m−2 i1 bi 0 hr∇r/1 − i1 bi . Proof. Integrating both sides of 1.6 on 0, t, we have φp u Δ∇ t φp u Δ∇ 0 − t hr∇r. 0 2.10 Fuyi Xu 5 So φp u Δ∇ ξi ξi φp uΔ∇ 0 − 2.11 hr∇r. 0 By boundary value condition φp uΔ∇ 0 m−2 i1 bi φp uΔ∇ ξi , we have m−2 ξi i1 bi 0 hr∇r . φp uΔ∇ 0 − 1 − m−2 i1 bi 2.12 By 2.10 and 2.12, we know Δ∇ u t m−2 ξi i1 bi 0 hr∇r t −φq hr∇r . m−2 1 − i1 bi 0 2.13 This together with Lemma 2.5 implies that t s βt γ T ut − t − sφq hr∇r B ∇s T − sφq hr∇r B ∇s d 0 0 0 0 ξi s βt γ m−2 − ai hr∇r B ∇s, ξi − s φq d i1 0 0 where B s 2.14 m−2 ξi m−2 i1 bi 0 hr∇r/1 − i1 bi . The proof is complete. m−2 Lemma 2.7. Let 0 < u of 2.8 satisfies i1 ai ξi < 1, d > 0. If h ∈ Cld 0, T T and ht ≥ 0, then the unique solution ut ≥ 0. 2.15 t ξi m−2 Proof. By uΔ∇ t −φq m−2 i1 bi 0 hr∇r/1 − i1 bi 0 hr∇r ≤ 0, we can know that the graph of ut is concave down on 0, T T . So we only prove u0 ≥ 0, uT ≥ 0. Firstly, we will prove u0 ≥ 0 by the following two perspectives. i If 0 < m−2 i1 ai ≤ 1, we have γ u0 d γ ≥ d γ d T T − sφq s 0 0 T s T − sφq 0 1− hr∇r B ∇s − hr∇r B ∇s − 0 m−2 T i1 0 ai T − sφq s 0 m−2 ξi i1 0 m−2 T i1 0 ai ai ξi − s φq s hr∇r B ∇s 0 T − sφq hr∇r B ∇s ≥ 0. s 0 hr∇r B ∇s 2.16 6 Discrete Dynamics in Nature and Society ii If m−2 i1 γ u0 d γ ≥ d γ d T ai > 1, by 2.8, we have s T − sφq 0 0 T s T − sφq 0 hr∇r B ∇s − hr∇r B ∇s − 0 T T− 0 m−2 ai ξi i1 m−2 m−2 ξi i1 0 m−2 T i1 0 ai ai ai − 1 s φq s ξi − s φq s hr∇r B ∇s 0 ξi − s φq s hr∇r B ∇s 0 hr∇r B ∇s ≥ 0. 0 i1 2.17 On the other hand, we have s βγ T hr∇r B ∇s T − sφq hr∇r B ∇s d 0 0 0 0 ξ s m−2 i βγ − ai hr∇r B ∇s ξi − s φq d i1 0 0 s ξi β m−2 ≥ ai hr∇r B ∇s ξi T − s − T ξi − s φq d i1 0 0 s T m−2 ai ξi T − sφq hr∇r B ∇s uT − T T − sφq i1 s ξi 0 T s s T γ m−2 ai T − sφq hr∇r B ∇s − hr∇r B ∇s ξi − s φq d i1 0 0 0 0 T s s ξi β m−2 ai hr∇r B ∇s ξi T − sφq hr∇r B ∇s T − ξi sφq d i1 0 ξi 0 0 s T γ m−2 T − ξi φq ai hr∇r B ∇s ≥ 0. d i1 0 0 2.18 The proof is completed. Lemma 2.8. Let ai ≥ 0, i 1, . . . , m − 2, 0 < m−2 i1 ai ξi < T, d > 0. If h ∈ Cld 0, T T and ht ≥ 0, then the unique positive solution ut of (BVP) 2.8 satisfies inf t∈ξm−2 ,T T ut ≥ σ||u||, 2.19 where σ min{am−2 T − ξm−2 /T − am−2 ξm−2 , am−2 ξm−2 /T, ξm−2 /T }, ||u|| supt∈0,T T |ut|. Fuyi Xu 7 Proof. Let ut maxt∈0,T T ut ||u||, we shall discuss it from the following two perspectives. Case 1. If 0 < m−2 i1 ai < 1. Firstly, assume t < ξm−2 < T , then mint∈ξm−2 ,T T ut uT . By uT m−2 i1 ai uξi ≥ am−2 uξm−2 , we have uT − u ξm−2 T T 0 − T uT − uT u ξm−2 ut ≤ uT T − ξm−2 T − ξm−2 T − ξm−2 T T T − am−2 ξm−2 ≤ uT 1 − uT . T − ξm−2 am−2 T − ξm−2 am−2 T − ξm−2 2.20 So am−2 T − ξm−2 min ut ≥ ||u||. t∈ξm−2 ,T T T − am−2 ξm−2 2.21 Secondly, assume ξm−2 < t < T , then mint∈ξm−2 ,T T ut uT . Otherwise, we have mint∈ξm−2 ,T T ut uξm−2 , then t ∈ ξm−2 , T T , uξm−2 ≥ uξm−1 ≥ · · · ≥ uξ2 ≥ uξ1 . By 0 < m−2 i1 ai < 1, we have uT m−2 m−2 ai u ξi ≤ ai u ξm−2 < u ξm−2 ≤ uT , i1 2.22 i1 a contradiction. By concave of ut, we get uξm−2 /ξm−2 ≥ ut/t ≥ ut/T . In fact, since uT ≥ am−2 uξm−2 , then uT /am−2 ξm−2 ≥ ut/T , which implies min ut ≥ t∈ξm−2 ,T T am−2 ξm−2 ||u||. T 2.23 Case 2. If m−2 i1 ai > 1. Firstly, assume uξm−2 ≤ uT , then mint∈ξm−2 ,T T ut uξm−2 . By concave of ut, we have t ∈ ξm−2 , tT , which implies uξm−2 /ξm−2 ≥ ut/t ≥ ut/T , then min ut ≥ t∈ξm−2 ,T T ξm−2 ||u||. T 2.24 Secondly, assume uξm−2 > ut, then mint∈ξm−2 ,T T ut uT , and t ∈ ξ1 , T T . If not, t ∈ 0, ξ1 , then uξ1 ≥ · · · ≥ uξm−2 > uT . So, we have uT m−2 m−2 ai u ξi > uT ai ≥ uT , i1 i1 2.25 8 Discrete Dynamics in Nature and Society a contradiction. By m−2 i1 ai > 1, there exists ξ ∈ {ξ1 , ξ2 , . . . , ξm−2 } such that uξ ≤ uT , then uξ1 ≤ uξ2 ≤ · · · ≤ uξm−2 ≤ u1. By concave of ut, we have u1/ξ1 ≥ uξ1 /ξ1 ≥ ut/t ≥ ut/T , then min ut ≥ ξ1 ||u||. 2.26 ut ≥ σ||u||, 2.27 t∈ξm−2 ,T T Therefore, by 2.21–2.26, we have inf t∈ξm−2 ,T T where σ min{am−2 T − ξm−2 /T − am−2 ξm−2 , am−2 ξm−2 /T, ξm−2 /T }. The proof is complete. Let E Cld 0, T T be endowed with the ordering x ≤ y if xt ≤ yt, for all t ∈ 0, T T , and u maxt∈0,T T |ut| is defined as usual by maximum norm. Clearly, it follows that E, u is a Banach space. We define a cone by K u : u ∈ E, ut is concave, nonnegative on 0, T T , inf t∈ξm−2 ,T T ut ≥ σ||u|| . 2.28 Define an operator S : K → E by setting Sut − t t − sφq 0 βt γ d − T 0 s arf r, ur ∇r A ∇s 0 T − sφq s arf r, ur ∇r A ∇s 0 2.29 s ξi βt γ m−2 ai arf r, ur ∇r A ∇s, ξi − s φq d i1 0 0 ξi m−2 where A m−2 i1 bi 0 arfr, ur∇r/1 − i1 bi . Obviously, u is a solution of boundary value problem 1.6 if and only if u is a fixed point of operator S. Lemma 2.9. S : K → K is completely continuous. Proof. By H2 and Lemmas 2.7-2.8, we easily get SK ⊂ K. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove S is completely continuous. Lemma 2.10 see 15. Let K be a cone in a Banach space X. Let D be an open bounded subset of X ∅ and DK / Ax K. Assume that A : DK → K is a compact map such that x / with DK D ∩ K / for x ∈ ∂DK . Then the following results hold. Fuyi Xu 9 1 If Ax ≤ x , x ∈ ∂DK , then iK A, DK 1. Ax λx0, for all x ∈ ∂DK and all λ > 0, then 2 If there exists x0 ∈ K \ {0} such that x / iK A, DK 0. 3 Let U be open in X such that U ⊂ DK . If iK A, DK 1 and iK A, UK 0, then A has a fixed point in DK \ UK . The same result holds if iK A, DK 0 and iK A, UK 1, where iK A, DK denotes fixed point index. One defines Kρ ut ∈ K : u < ρ , Ωρ ut ∈ K : min xt < σρ . ξm−2 ≤t≤T 2.30 Lemma 2.11 see 15. Ωρ defined above has the following properties: a Kσρ ⊂ Ωρ ⊂ Kρ ; b Ωρ is open relative to K; c x ∈ ∂Ωρ if and only if minξm−2 ≤t≤T xt σρ; d if x ∈ ∂Ωρ , then σρ ≤ xt ≤ ρ, for t ∈ ξm−2 , T T . For the convenience, we introduce the following notations: ρ fσρ ft, u : u ∈ σρ, ρ , min min ξm−2 ≤t≤T φp ρ u→α 1 1 M d T ξm−2 ft, u : u ∈ 0, ρ , max max 0≤t≤T φp ρ ft, u α : ∞ or 0 , φp u m−2 ξi T βT γ T 1 i1 bi 0 ar∇r T − s∇sφq ar∇r , m d 1 − m−2 0 0 i1 bi s m−2 m−2 T −sφq ar∇r ∇s min βξm−2 γ, β max ai ξ1 , am−2 ξm−2 γ ai . f α lim sup max 0≤t≤T ft, u , φp u ρ f0 fα lim inf max u→α ξm−2 ≤t≤T ξm−2 i1 i1 2.31 Lemma 2.12. If f satisfies the following condition: ρ f0 ≤ φp m, u/ Su, u ∈ ∂Kρ , 2.32 then iK S, Kρ 1. 2.33 10 Discrete Dynamics in Nature and Society Proof. For u ∈ ∂Kρ , then from 2.32, we have m−2 ξi i1 bi 0 arfr, ur∇r arf r, ur ∇r A arf r, ur ∇r 1 − m−2 0 0 i1 bi m−2 ξi T i1 bi 0 arfr, ur∇r ≤ arf r, ur ∇r 1 − m−2 0 i1 bi ξi T m−2 i1 bi 0 ar∇r ≤ φp mρ ar∇r . 1 − m−2 0 i1 bi s s 2.34 So that s φq arf r, ur ∇r A ≤ mρφq m−2 ξi i1 bi 0 ar∇r ar∇r . m−2 1 − i1 bi 0 T 0 2.35 Therefore, Sut ≤ βt γ d T T − sφq 0 s arf r, ur ∇r A ∇s 0 βT γmρ ≤ d T T − s∇sφq 0 m−2 ξi i1 bi 0 ar∇r ar∇r 1 − m−2 0 i1 bi T 2.36 ρ. This implies that ||Su|| ≤ ||u|| for u ∈ ∂Kρ . Hence, by Lemma 2.101 it follows that iK S, Kρ 1. Lemma 2.13. If f satisfies the following condition: ρ fσρ ≥ φp Mσ, u/ Su, u ∈ ∂Ωρ , 2.37 then iK S, Ωρ 0. 2.38 Proof. Let et ≡ 1 for t ∈ 0, T T . Then e ∈ ∂K1 . We claim that u/ Su λe, u ∈ ∂Ωρ , λ > 0. 2.39 Fuyi Xu 11 ρ In fact, if not, there exist u0 ∈ ∂Ωρ and λ0 > 0 such that u0 Su0 λ0 e. By fσρ ≥ φp Mσ, we have m−2 ξi i1 bi 0 arfr, u0 r∇r arf r, u0 r ∇r A arf r, u0 r ∇r 1 − m−2 0 0 i1 bi s ≥ arf r, ur ∇r s s ξm−2 ≥ φp Mσρ s 2.40 ar∇r . ξm−2 So that s s arf r, ur ∇r A ≥ Mσρφq φq 0 ar∇r . 2.41 ξm−2 By 16, Theorem 2.2iv, for t > 0, we have t 0 t − sφq s 0 Δ arfr, u0 r∇r A ∇s t t 0 sφq s 0 arfr, u0 r∇r A ∇s tσt ≥ 0. 2.42 So, for i 1, 2, . . . , m − 2, we have ξm−2 0 s ξm−2 − s φq 0 arfr, u0 r∇r A ∇s ξm−2 s ξi − s φq 0 arfr, u0 r∇r A ∇s ξi 0 ≥ ξi 2.43 Therefore, Su0 ξm−2 T s β ξm−2 t − sφq arf r, u0 r ∇r A ∇s ≥ d 0 0 ξm−2 s arf r, u0 r ∇r A ∇s −T ξm−2 − s φq γ d T 0 t − sφq 0 − s 0 arf r, u0 r ∇r A ∇s 0 ξm−2 0 ξm−2 − s φq s 0 . arf r, u0 r ∇r A ∇s 12 Discrete Dynamics in Nature and Society ≥ ≥ βξm−2 γ d T T − sφq s ξm−2 arf r, u0 r ∇r A ∇s 0 βξm−2 γMσρ d T T − sφq s ξm−2 ar∇r ∇s, ξm−2 2.44 s ξi β m−2 Su0 T ≥ ai arf r, u0 r ∇r A ∇s t − ξi sφq d i1 0 0 ξi T T − sφq s ξi arf r, u0 r ∇r A ∇s 0 s T γ m−2 t − sφq arf r, u0 r ∇r A ∇s d i1 0 0 − ξi T − sφq s 0 0 ≥ β m−2 ai ξi d i1 arf r, u0 r ∇r A ∇s T t − sφq s ξi arf r, u0 r ∇r A ∇s 0 s T γ m−2 t − sφq arf r, u0 r ∇r A ∇s d i1 ξi 0 s m−2 m−2 T Mσρ β max ≥ ai ξ1 , am−2 ξm−2 γ T − sφq ar∇r ∇s. d ξm−2 ξm−2 i1 i1 2.45 Obviously, we can know min Su0 t min Su0 ξm−2 , Su0 T t∈ξm−2 ,T T Mσρ ≥ d T T − sφq s ξm−2 ar∇r ∇s ξm−2 × min βξm−2 γ, β max m−2 ai ξ1 , am−2 ξm−2 i1 γ m−2 2.46 i1 ≥ σρ. For t ∈ ξm−2 , T T , then u0 t Su0 t λ0 et ≥ min Su0 t λ0 t∈ξm−2 ,T min Su0 ξm−2 , Su0 T λ0 ≥ σρ λ0 . 2.47 Fuyi Xu 13 This together with Lemma 2.11c implies that σρ ≥ σρ λ0 , 2.48 a contradiction. Hence, by Lemma 2.102, it follows that iK S, Ωρ 0. 3. Main results We now give our results on the existence of positive solutions of BVP 1.6. Theorem 3.1. Suppose conditions H1 and H2 hold, and assume that one of the following conditions holds. ρ ρ H3 There exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < σρ2 such that f0 1 ≤ φp m, fσρ2 2 ≥ φp Mσ. ρ ρ H4 There exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < ρ2 such that f0 2 ≤ φp m, fσρ1 1 ≥ φp Mσ. Then, the boundary value problem 1.6 has at least one positive solution. ρ Proof. Assume that H3 holds, we show that S has a fixed point u1 in Ωρ2 \K ρ1 . By f0 1 ≤ φp m and Lemma 2.12, we have that iK S, Kρ1 1. 3.1 ρ By fσρ2 2 ≥ φp Mσ and Lemma 2.13, we have that iK S, Kρ2 0. 3.2 By Lemma 2.11a and ρ1 < σρ2 , we have K ρ1 ⊂ Kσρ2 ⊂ Ωρ2 . It follows from Lemma 2.103 that S has a fixed point u1 in Ωρ2 \ K ρ1 . When condition H4 holds, the proof is similar to the above, so we omit it here. As a special case of Theorem 3.1, we obtain the following result. Corollary 3.2. Suppose conditions H1 and H2 hold, and assume that one of the following conditions holds. H5 0 ≤ f 0 < φp m and φp M < f∞ ≤ ∞. H6 0 ≤ f ∞ < φp m and φp M < f0 ≤ ∞. Then, the boundary value problem 1.6 has at least one positive solution. Theorem 3.3. Assume conditions H1 and H2 hold, and suppose that one of the following conditions holds. H7 There exist ρ1 , ρ2 , and ρ3 ∈ 0, ∞ with ρ1 < σρ2 and ρ2 < ρ3 such that ρ f0 1 ≤ φp m, ρ fσρ2 2 ≥ φp Mσ, u/ Su, ∀ u ∈ ∂Ωρ2 , ρ f0 3 ≤ φp m. 3.3 14 Discrete Dynamics in Nature and Society H8 There exist ρ1 , ρ2 , and ρ3 ∈ 0, ∞ with ρ1 < ρ2 < σρ3 such that ρ f0 2 ≤ φp m, ρ fσρ1 1 ≥ φp Mσ, u/ Su, ∀ u ∈ ∂Kρ2 , ρ fσρ3 3 ≥ φp Mσ. 3.4 ρ Then, the boundary value problem 1.6 has at least two positive solutions. Moreover, if in H7 f0 1 ≤ ρ φp m is replaced by f0 1 < φp m, then the BVP 1.6 has a third positive solution u3 ∈ Kρ1 . Proof. Assume that condition H7 holds, we show that either S has a fixed point u1 in ∂Kρ1 or Ωρ2 \ K ρ1 . If u / Su for u ∈ ∂Kρ1 ∪ ∂Kρ3 . By Lemma 2.12 and Lemma 2.13, we have that iK S, Kρ1 1, iK S, Kρ3 1, iK S, Kρ2 0. 3.5 By Lemma 2.11a and ρ1 < σρ2 , we have K ρ1 ⊂ Kσρ2 ⊂ Ωρ2 . It follows from Lemma 2.103 that S has a fixed point u1 in Ωρ2 \ K ρ1 . Similarly, S has a fixed point u2 in Kρ3 \ Ωρ2 . When condition H8 holds, the proof is similar to the above, so we omit it here. As a special case of Theorem 3.3, we obtain the following result. Corollary 3.4. Assume conditions H1 and H2 hold, if there exists ρ > 0 such that one of the following conditions holds. ρ Su, ∀u ∈ ∂Ωρ and 0 ≤ f ∞ < φp m. H9 0 ≤ f 0 < φp m, fσρ ≥ φp Mσ, u / ρ H10 φp M < f0 ≤ ∞, f0 ≤ φp m, u / Su, ∀u ∈ ∂Kρ and φp M < f∞ ≤ ∞. Then, the boundary value problem 1.6 has at least two positive solutions. 4. Some examples In this section, we present some simple examples to explain our results. We only study the case T R, 0, T T 0, 1. Example 4.1. Consider the following three-point boundary value problem with p-Laplacian: u 0 0, φp u atft, u 0, 0 < t < 1, 1 1 1 1 , φp u 0 φp u , u1 u 2 3 4 3 4.1 where β 0, γ 1, a1 1/2, b1 1/4, ξ1 1/3, at 1, p q 2. By computing, we can know σ 1/6, M 819/16, m 9/10. Let ρ1 1, ρ2 208, then σρ1 < ρ1 < σρ2 < ρ2 . We Fuyi Xu 15 define a nonlinearity f as follows: ⎧ 3 ⎪ 9t3 1 ⎪ ⎪ −u , ⎪ ⎪ ⎪ 10 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 9t3 6π 1π ⎪ ⎪ sin u − , ⎪ ⎪ ⎪ 10 52 52 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 9t3 208 6 819 6 6 ft, u − u u− , ⎪ 10 202 202 96 202 202 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 819 t3 u − 208 , ⎪ ⎪ ⎪ 6 ⎪ 96 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 819 ⎪ 208 2 ⎪ 3 ⎪ 1 u − 208 , t 208 − ⎩ 96 6 1 0 < t < 1, u ∈ 0, , 6 1 0 < t < 1, u ∈ ,1 , 6 208 0 < t < 1, u ∈ 1, , 6 4.2 208 , 208 , 0 < t < 1, u ∈ 6 0 < t < 1, u ∈ 208, ∞ . Then, by the definition of f, we have ρ i f0 1 ≤ φp m 9/10; ρ ii fσρ2 2 ≥ φp Mσ 819/19968. So condition H3 holds, by Theorem 3.1, boundary value problem 4.1 has at least one positive solution. Acknowledgment Project supported by the National Natural Science Foundation of China 10471075. References 1 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. 2 R. P. Agarwal and D. O’Regan, “Nonlinear boundary value problems on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 44, no. 4, pp. 527–535, 2001. 3 F. M. Atici and G. Sh. Guseinov, “On Green’s functions and positive solutions for boundary value problems on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 75–99, 2002. 4 H.-R. Sun and W.-T. Li, “Positive solutions for nonlinear three-point boundary value problems on time scales,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 508–524, 2004. 5 M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. 6 H. R. Sun and W. T. Li, “Positive solutions for nonlinear m-point boundary value problems on time scales,” Acta Mathematica Sinica, vol. 49, no. 2, pp. 369–380, 2006 Chinese. 7 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. 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 Z. He and X. Jiang, “Triple positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 911–920, 2006. 16 Discrete Dynamics in Nature and Society 10 D.-X. Ma, Z.-J. Du, and W.-G. Ge, “Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator,” Computers & Mathematics with Applications, vol. 50, no. 5-6, pp. 729–739, 2005. 11 Y. Wang and W. Ge, “Positive solutions for multipoint boundary value problems with a onedimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 6, pp. 1246– 1256, 2007. 12 Y. Wang and C. Hou, “Existence of multiple positive solutions for one-dimensional p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 144–153, 2006. 13 C. Zhou and D. Ma, “Existence and iteration of positive solutions for a generalized right-focal boundary value problem with p-Laplacian operator,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 409–424, 2006. 14 D. R. Anderson, “Green’s function for a third-order generalized right focal problem,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 1–14, 2003. 15 K. Q. Lan, “Multiple positive solutions of semilinear differential equations with singularities,” Journal of the London Mathematical Society, vol. 63, no. 3, pp. 690–704, 2001. 16 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkhäuser, Boston, Mass, USA, 2001.