Research Article Twin Positive Solutions of a Nonlinear m-Point Boundary Value Problem for Third-Order p-Laplacian Dynamic Equations on Time Scales

Several existence theorems of twin positive solutions are established for a nonlinear 𝑚-point boundary value problem of third-order 𝑝-Laplacian dynamic equations on time scales by using a fixed point theorem. We present two theorems and four corollaries which generalize the results of related literature. As an application, an example to demonstrate our results is given. The obtained conditions are different from some known results.


Introduction
A time scale T is a nonempty closed subset of R. We make the blanket assumption that 0 and 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 this paper, we will be concerned with the existence of positive solutions of the p-Laplacian dynamic equations on time scales: where φ p s is p-Laplacian operator; that is, φ p s |s| p−2 s, p > 1, φ −1 p φ q , 1/p 1/q 1, 0 < ξ 1 < • • • < ξ m−2 < ρ T , and H 1 a i , b i ∈ 0, ∞ , i 1, 2, . . ., satisfy 0 < m−2 i 1 a i < 1 and m−2 i 1 b i < 1; H 2 a t ∈ C ld 0, T , 0, ∞ and there exists t 0 ∈ ξ m−2 , T such that a t 0 > 0; We point out that the Δ-derivative and the ∇-derivative in 1.2 and the C ld space in H 2 are defined in Section 2.
Recently, there has been much attention paid to the existence of positive solutions for third-order nonlinear boundary value problems of differential equations.For example, see 1-10 and the listed references.Anderson 2 considered the following third-order nonlinear problem: 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.

1.3
He used the Krasnoselskii and the Leggett and Williams fixed-point theorems to prove the existence of solutions to the nonlinear problem 1.3 .Li 6 considered the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem: Under various assumptions on a and f, they established intervals of the parameter λ which yield the existence of at least two and infinitely many positive solutions of the boundary value problem by using Krasnoselski's fixed-point theorem of cone expansion-compression type.Liu et al. 7 discussed the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third-order differential equations: x t λα t f t, x t 0, a < t < b, x a x a x b 0. 1.5 Green's function and the fixed-point theorem of cone expansion-compression type are utilized in their paper.In 8 , Sun considered the following nonlinear singular third-order three-point boundary value problem: He obtained various results on the existence of single and multiple positive solutions to the boundary value problem 1.6 by using a fixed-point theorem of cone expansion-compression type due to Krasnosel'skii.In 10 , Zhou and Ma 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 1.7 by using the monotone iterative technique.
On the other hand, the existence of positive solutions for third-order nonlinear boundary value problems of difference equations is also extensively studied by a number of authors see 1, 3, 5, 9 and the listed references .The present work is motivated by a recent paper 4 .In 4 , Henderson and Yin considered the existence of solutions for a third-order boundary value problem on a time-scale equation of the form which is uniform for the third-order difference equation and the third-order differential equation.

Preliminaries and lemmas
For convenience, we list the following definitions which can be found in 4, 11-15 .
Definition 2.1.Let T be a time scale.For t < sup T and r > inf T, define the forward jump operator σ and the backward jump operator ρ, respectively, by for all t, r ∈ T. If σ t > t, t is said to be right-scattered, and if ρ r < r, r is said to be leftscattered; if σ t t, t is said to be right-dense, and if ρ r r, r is said to be left-dense.If T has a right-scattered minimum m, define T k T − {m}; otherwise set T k T. If T has a left-scattered maximum M, define T k T − {M}; otherwise set T k T. 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 that 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 is denoted by f ∇ t provided that it exists , with the property that for each > 0 there is a neighborhood U of t such that

2.5
To prove the main results in this paper, we will employ several lemmas.These lemmas are based on the linear BVP where

2.9
Proof.i Let u be a solution, then we will show that 2.8 holds.By taking the nabla integral of problem 2.6 on 0, t , we have By taking the nabla integral of 2.11 on 0, t , we can get

2.12
By taking the delta integral of 2.12 on 0, t , we can get Similarly, let t 0 on 2.10 , then we have φ p u Δ∇ 0 A; let t ξ i on 2.10 , then we have Let t 0 on 2.12 , then we have

2.15
Let t T on 2.13 , then we have Similarly, let t ξ i on 2.13 , then we have By the boundary condition 2.7 , we can get B 0, 2.18

2.20
By the boundary condition 2.7 , we can obtain

2.21
Substituting 2.20 in the above expression, one has ii We show that the function u given in 2.8 is a solution.
Let u be as in 2.8 .By 12, Theorem 2.10 iii and taking the delta derivative of 2.8 , we have 23 moreover, we get

2.24
Taking the nabla derivative of this expression yields φ p u Δ∇ ∇ −h t .Also, routine calculation verifies that u satisfies the boundary value conditions in 2.7 so that u given in 2.8 is a solution of 2.6 and 2.7 .The proof is complete.Lemma 2.6.Assume H 1 holds.For h ∈ C ld 0, T and h ≥ 0, the unique solution u of 2.6 and 2.7 satisfies u t ≥ 0 for t ∈ 0, T .

2.25
Proof.Let W. Han and G. Zhang 7 According to Lemma 2.5, we get

2.28
If t ∈ 0, T , we have
Lemma 2.7.Assume H 1 holds.If h ∈ C ld 0, T and h ≥ 0, then the unique solution u of 2.6 and 2.7 satisfies

2.31
Proof.It is easy to check that u Δ t − t 0 ϕ 0 s ∇s ≤ 0; this implies that u u 0 , min t∈ 0,T u t u T .

2.32
It is easy to see that u Δ t 2 ≤ u Δ t 1 for any t 1 , t 2 ∈ 0, T with t 1 ≤ t 2 .Hence, u Δ t is a decreasing function on 0, T .This means that the graph of u t is concave down on 0, T .
Let the norm on C ld 0, T be the maximum norm.Then, the C ld 0, T is a Banach space.It is easy to see that BVP 1.1 -1.2 has a solution u u t if and only if u is a fixed point of the operator where where γ is the same as in Lemma 2.7.It is obvious that K is a cone in C ld 0, T .By Lemma 2.7, A K ⊂ K.So by applying Arzela-Ascoli theorem on time scales 16 , we can obtain that A K is relatively compact.In view of Lebesgue's dominated convergence theorem on time scales 13 , it is easy to prove that A is continuous.Hence, A : K→K is completely continuous.
Proof.First, we show that A maps bounded set into bounded set.Assume c > 0 is a constant and u ∈ K c {u ∈ K : u ≤ c}.Note that the continuity of f guarantees that there is c > 0 such that f t, u t ≤ φ p c for t ∈ 0, T .So

2.40
That is, AK c is uniformly bounded.
In addition, notice that for any t 1 , t 2 ∈ 0, T , we have

2.41
So, by applying Arzela-Ascoli theorem on time scales 16 , we obtain that AK c is relatively compact.
Finally, we prove that A : K c →K is continuous.Suppose that {u n } ∞ n 1 ⊂ K c and u n t converges to u * t uniformly on 0, T .Hence, {Au n t } ∞ n 1 is uniformly bounded and equicontinuous on 0, T .The Arzela-Ascoli theorem on time scales 16 tells us that there exists uniformly convergent subsequence in {Au n t } ∞ n 1 .Let {Au n m t } ∞ m 1 be a subsequence which converges to v t uniformly on 0, T .In addition,

2.42
Observe the expression of {Au n m t }, and then letting m→∞, we obtain where

2.44
Here, we have used the Lebesgue dominated convergence theorem on time scales 13 .From the definition of A, we know that v t Au * t on 0, T .This shows that each subsequence of {Au n t } ∞ n 1 uniformly converges to Au * t .Therefore, the sequence {Au n t } ∞ n 1 uniformly converges to Au * t .This means that A is continuous at u * ∈ K c .So, A is continuous on K c since u * is arbitrary.Thus, A is completely continuous.This proof is complete.Lemma 2.9.Let ϕ s φ q s 0 a τ f τ, u τ ∇τ − A .

2.45
For Then, F has a fixed point in K ∩ Ω 2 \ Ω 1 .

Main results
Theorem 3.1.Assume H 1 , H 2 , and H 3 hold, and assume that the following conditions hold:

3.3
Then, problem 1.1 -1.2 has at least two positive solutions Theorem 3.2.Assume H 1 , H 2 , and H 3 hold, and assume that the following conditions hold:
Proof of Theorem 3.

3.11
It follows that
On the other hand, since f : 0, T × 0, ∞ → 0, ∞ is continuous, by condition A 4 , there exist In the same way, we can prove that if u ∈ ∂Ω b 2 , then Au ≥ u .Now, we consider the operator A on Ω b 1 \Ω a 1 and Ω a 2 \Ω b 2 , respectively.By Lemma 2.10, we assert that the operator A has two fixed points u

3.18
It follows that

3.19
Noticing λ 3 − p 1 > 0, we get By condition B 4 , we can see that there exist b

Further discussion
If the conditions of Theorems 3.1 and 3.2 are weakened, we will get the existence of single positive solution of problem 1.1 -1.2 .
Corollary 4.1.Assume H 1 , H 2 , and H 3 hold, and assume that the following conditions hold: Then, problem 1.1 -1.2 has at least one positive solution.
Corollary 4.2.Assume H 1 , H 2 , and H 3 hold, and assume that the following conditions hold:

4.4
Then, problem 1.1 -1.2 has at least one positive solution.
Corollary 4.3.Assume H 1 , H 2 , and H 3 hold, and assume that the following conditions hold:

4.8
Then, problem 1.1 -1.2 has at least one positive solution.
The proof of the above results is similar to those of Theorems 3.1 and 3.2; thus we omit it.

Some examples
In this section, we present a simple example to explain our results.We only study the case T R, 0, T 0, 1 .Let f t, 0 ≡ 0. Consider the following BVP: √ u 3 , t, u ∈ 0, 1 × 3, ∞ .

5.3
We have

5.5
It follows that f satisfies the conditions A 1 -A 4 of Theorem 3.1; then problem 5.1 has at least two positive solutions.
Let E be a Banach space, and let K ⊂ E be a cone.Assume Ω 1 , Ω 2 are open bounded subsets of E with 0 ∈ Ω 1 , Ω 1 ⊂ Ω 2 , and let 4 , then Au ≤ u .Now, we study the operator A on Ω a 3 \ Ω b 3 and Ω b 4 \ Ω a 4 , respectively.By Lemma 2.10, we assert that the operator A has two fixed points u * 3 , u * 4 ∈ K such that b 3 ≤ u * 3 ≤ a 3 and a 4 ≤ u * 4 ≤ b 4 .Therefore, u * i , i 3, 4, are positive solutions of problem 1.1 -1.2 .