Higher-order Dynamic Delay Differential Equations on Time Scales

Copyright q 2012 Hua Su 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 study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-order p-Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained.


Introduction
The study of dynamic equations on time scales goes back to its founder Stefan Hilger [12], and is a new area of still fairly theoretical exploration in mathematics.Boundary value problems for delay differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory (see [5,4]).In recent years, many authors have begun to pay attention to the study of boundary-value problems or with p-Laplacian equations or with p-Laplacian dynamic equations on time scales (see [1][2][3][6][7][8][9][10][11][12][13][14] and the references therein).
In [14], Wang discussed the following dynamic equation on time scales by using Avery-Peterson fixed theorem (see [11]): (φ p (u )) + q(t)f (t, u(t), u(t − 1), u (t)) = 0, t ∈ (0, 1), (1.3) and They obtained some results for the existence three positive solutions of the problem (1.3), (1.4) and (1.3), (1.4 ), respectively.However, there are not many concerning the p-Laplacian problems on time scales.Especially, for the singular multi-point boundary value problems for higher-order p-Laplacian dynamic delay differential equations on time scales, with the author's acknowledge, no one has studied the existence of positive solutions in this case.Now, motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following nonlinear four-point singular boundary value problem with higher-order p-Laplacian dynamic delay differential equations operator on time scales (SBVP): where Our main tool of this paper is the following fixed point index theory.Theorem 1.1 [1] .Suppose E is a real Banach space, K ⊂ E is a cone, let Ω r = {u ∈ K : u ≤ r}.Let operator T : Ω r −→ K be completely continuous and satisfy This paper is organized as follows.In section 2, we present some preliminaries and lemmas that will be used to prove our main results.In section 3, we discuss the existence of solution of the systems (1.5), (1.6).In section 4, we give a examples as the application.

Preliminary Notes
For convenience, we list here the following definitions which are needed later.We begin by presenting some basic definitions which can be found in [1,2].
A time scale T is an arbitrary nonempty closed subset of real numbers R. The operators σ and ρ from T to T , are called the forward jump operator and the backward jump operator, respectively.
Let f : T → R and t ∈ T k (assume t is not left-scattered if t = sup T), then 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 Similarly, for t ∈ T (assume t is not right-scattered if t = inf T), the nabla derivative of f at the point t is defined in [1] to be the number f ∇ (t) (provided it exists) with the property that for each > 0 there is a neighborhood U of t such that is the forward difference operator while x ∇ (t) = x(t) − x(t − 1) is the backward difference operator.
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. It is well-known that if f is ld-continuous, then there is a function Define f Δ n (t) to be the delta derivative of In the rest of this article, T is closed subset of R with 0 ∈ T k , T ∈ T k .And let Then B is a Banach space with the norm u = max In the rest of the paper, we also make the following assumptions: ) and there exists t 0 ∈ (0, T ), such that We can easily get the following Lemmas.Lemma 2.1 Suppose condition (H 2 ) holds.Then there exists a constant Furthermore, the function The proof of the above two lemmas are similar to the proof of in [6, Lemma 2.1 and Lemma 2.2], so we omit it.

Lemma 2.3 Suppose that conditions (H
) is a solution of the following boundary value problems for 0 < t < T , where where (2.3)
From the definition of T and above discussion, we deduce that for each u ∈ K, T u ∈ K.Moreover, we have the following Lemma.Lemma 2.5 T : is continuous, decreasing on [0, T ] and satisfies (T u) . This shows that T K ⊂ K. Furthermore, it is easy to check by Arzela-ascoli Theorem that T : K → K is completely continuous.Lemma 2.6 Suppose that conditions (H 1 ), (H 2 ), (H 3 ) hold, the solution u(t) of problem (2.1), (2.2) satisfy: and for θ ∈ (0, 1 2 ) in Lemma 2.1, we have Proof.Firstly, we can have max Next, if u(t) is the solution of problem (2.1), (2.2), then u Δ n−2 (t) is concave function, and The proof is complete.For convenience, we set where L is the constant from Lemma 2.1.By Lemma 2.4, we can also set

Main Results
In this section, we present our main results.Theorem 3.1 Suppose that condition (H 1 ), (H 2 ), (H 3 ) hold.Assume that f also satisfy (A 1 ): Then, the (SBVP) (2.1), (2.2) has a solution u such that u lies between r and R. The proof of Theorem 3.1 Without loss of generality, we suppose that r < R. For any u ∈ K, by Lemma 2.2, we have We define two open subset Ω 1 and Ω 2 of E: For any u ∈ ∂Ω 1 , by (3.1) we have For t ∈ [θ, T − θ] and u ∈ ∂Ω 1 , we shall discuss it from three perspectives.(i) If σ ∈ [θ, T − θ], thus for u ∈ ∂Ω 1 , by (A 1 ) and Lemma 2.3, we have ) and Lemma 2.3, we have (iii) If σ ∈ (0, θ), thus for u ∈ ∂Ω 1 , by (A 1 ) and Lemma 2.3, we have Therefore, no matter under which condition, we all have Then by Theorem 1.1, we have On the other hand, for u ∈ ∂Ω 2 , we have u(t) ≤ u = R, by (A 2 ) we know Then, by Theorem 1.1, we have Therefore, by (3.2), (3.3), r < R, we have Then operator T has a fixed point u ∈ (Ω 1 \ Ω 2 ), and r ≤ u ≤ R.This completes the proof of Theorem 3.1.
Theorem 3.2 Suppose that condition (H 1 ), (H 2 ), (H 3 ) hold.Assume that f also satisfy Then, the (SBVP) (2.1), (2.2) has a solution u such that u lies between r and R. The proof of Theorem 3.2 First, by − ϕ, there exists an adequately small positive number ρ, as 0 ≤ u n ≤ ρ, u n = 0, we have So condition (A 2 ) holds.

Next, by condition (
, there exists an appropriately big positive number r = R, as u n ≥ θr, we have Let m = 2θ * > θ * , thus by (3.5), condition (A 1 ) holds.Therefor by Theorem 3.1 we know that the results of Theorem 3.2 holds.The proof of Theorem 3.2 is complete.
Theorem 3.3 Suppose that condition (H 1 ), (H 2 ), (H 3 ) hold.Assume that f also satisfy Then, the (SBVP) (2.1), (2.2) has a solution u such that u lies between r and R. The proof of Theorem 3.3.

First, by condition (
, there exists an adequately small positive number r, as 0 ≤ u n ≤ r, u n = 0, we have thus when θr ≤ u n ≤ r, we have Let m = 2θ * > θ * , so by (3.6), condition (A 1 ) holds.
Next, by condition (A 5 ): λ, there exists an suitably big positive number ρ = r, as u n ≥ ρ, we have Thus, by ρ ≤ u 0n ≤ R, we know there exists an appropriately big positive number R > 4 θ * M , then choose Therefore, condition (A 2 ) holds.Therefore, by Theorem 3.1, we know that the results of Theorem 3.3 holds.The proof of 3.3 is complete.