Research Article Positive Solutions for Third-Order Nonlinear p-Laplacian m-Point Boundary Value Problems

We study the following third-order p-Laplacian m-point boundary value problems on time scales: φp uΔ∇ ∇ a t f t, u t 0, t ∈ 0, T T, βu 0 − γuΔ 0 0, u T ∑m−2 i 1 aiu ξi , φp u Δ∇ 0 ∑m−2 i 1 biφp u Δ∇ ξi , where φp s is p-Laplacian operator, that is, φp s |s|p−2s, p > 1, φ−1 p φq, 1/p 1/q 1, 0 < ξ1 < · · · < ξm−2 < ρ T . 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.


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, x t , t 1 ≤ t ≤ t 3 , x t 1 x t 2 0, γx t 3 δx t 3 0. 1.1 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 φ p u Δ ∇ a t f u t 0, t ∈ 0, T T , 1.2 satisfying the boundary conditions where η ∈ 0, ρ T .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: 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: where R and there exists t 0 ∈ ξ m−2 , T T such that a t 0 > 0, where R 0, ∞ .

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 Definition 2.2.For f : T → R and t ∈ T k , 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 for all s ∈ U.
For f : T → R and t ∈ T k , 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 has the unique solution

2.7
Proof.By direct computation, we can easily get 2.7 .So, we omit it.

the boundary value problem (BVP)
has the unique solution Proof.Integrating both sides of 1.6 on 0, t , we have

2.11
By boundary value condition By 2.10 and 2.12 , we know This together with Lemma 2.5 implies that r ∇r ≤ 0, we can know that the graph of u t is concave down on 0, T T .So we only prove u 0 ≥ 0, u T ≥ 0.
Firstly, we will prove u 0 ≥ 0 by the following two perspectives. 2.17 On the other hand, we have

2.18
The proof is completed. where Proof.Let u t max t∈ 0,T T u t ||u||, we shall discuss it from the following two perspectives.

2.20
So

2.24
Secondly, assume u ξ m−2 > u t , then min t∈ ξ m−2 ,T T u t u T , and where The proof is complete.
Let E C ld 0, T T be endowed with the ordering x ≤ y if x t ≤ y t , for all t ∈ 0, T T , and u max t∈ 0,T T |u t | is defined as usual by maximum norm.Clearly, it follows that E, u is a Banach space.
We define a cone by

2.29
where 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 H 2 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 with D K D ∩ K / ∅ and D K / K. Assume that A : D K → K is a compact map such that x / Ax for x ∈ ∂D K .Then the following results hold.
2 If there exists x 0 ∈ K \ {0} such that x / Ax λx 0, for all x ∈ ∂D K and all λ > 0, then i K A, D K 0.
, where i K A, D K denotes fixed point index.

One defines
x t < σρ .

2.30
Lemma 2.11 see 15 .Ω ρ defined above has the following properties: For the convenience, we introduce the following notations:

2.33
Proof.For u ∈ ∂K ρ , then from 2.32 , we have

2.34
So that

2.40
So that By 16, Theorem 2.2 iv , for t > 0, we have

2.44
a r ∇r ∇s.

Main results
We now give our results on the existence of positive solutions of BVP 1.6 .
Theorem 3.1.Suppose conditions H 1 and H 2 hold, and assume that one of the following conditions holds.
Then, the boundary value problem 1.6 has at least one positive solution.
Proof.Assume that H 3 holds, we show that S has a fixed point u 1 in Ω ρ 2 \K ρ 1 .By f ρ 1 0 ≤ φ p m and Lemma 2.12, we have that Then, the boundary value problem 1.6 has at least one positive solution.Then, the boundary value problem 1.6 has at least two positive solutions.

Example 4 . 1 .
Consider the following three-point boundary value problem with p-Laplacian: By Lemma 2.11 a and ρ 1 < σρ 2 , we haveK ρ 1 ⊂ K σρ 2 ⊂ Ω ρ 2 .It follows from Lemma 2.10 3 that S has a fixed point u 1 in Ω ρ 2 \ K ρ 1 .When condition H 4 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 H 1 and H 2 hold, and assume that one of the following conditions holds.
So condition H 3 holds, by Theorem 3.1, boundary value problem 4.1 has at least one positive solution.