Let T be a time scale. We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with 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 is obtained.

1. Introduction

The study of dynamic equations on time scales goes back to its founder Hilger [1] 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 [2, 3]). 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 [4–14] and the references therein).

In [7], Sun and Li considered the existence of positive solution of the following dynamic equations on time scales:

(1.1)

(1.2)

where β, γ ≥ 0, β + γ > 0, η ∈ (0, ρ(T)), 0 < α < T/η. They obtained the existence of single and multiple positive solutions of the problem (1.1) and (1.2) by using fixed-point theorem and Leggett-Williams fixed-point theorem (see [15]), respectively.

In [4], Anderson discussed the following dynamic equation on time scales:

(1.3)

He obtained some results for the existence of one positive solution of the problem (1.3) based on the limits

and f_∞ = lim _u→∞ f(u)/u.

In [5], Kaufmann studied the problem (1.3) and obtained the existence results of at least two positives solutions.

In [14], Wang et al. discussed the following dynamic equation on time scales by using Avery-Peterson fixed theorem (see [14]):

(1.4)

(1.5)

(1.5')

They obtained some results for the existence three positive solutions of the problem (1.4), (1.5) and (1.4), and (1.5′), respectively.

In [15], Lee and Sim discussed the following equation:

(1.6)

By applying the global bifurcation theorem and figuring the shape of unbounded subcontinua of solutions, they obtain many different types of global existence results of positive solutions.

However, there are not many concerning the p-Laplacian problems on time scales. Especially, for the singular multipoint boundary value problems for 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.

Recently, in [16], we have studied the existence of positive solutions for the following nonlinear two-point singular boundary value problem with p-Laplacian operator:

(1.7)

By using the fixed-point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for nonlinear singular boundary value problem (1.7) with p-Laplacian operator is obtained.

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):

(1.8)

(1.9)

(1.10)

where ϕ_p(s) is p-Laplacian operator, that is, ϕ_p(s) = |s|^p−2s, p > 1,

, 1/p + 1/q = 1. ξ, η ∈ (0, T), τ ∈ [0, T] is prescribed and ξ < η, g : (0, T)→[0, ∞), α > 0, β ≥ 0, γ > 0, δ ≥ 0 and B₀, B₁ are both nondecreasing continuous odd functions defined on (−∞, +∞).

In this paper, by constructing one integral equation which is equivalent to the problem (1.8), (1.9) and (1.8), and (1.10), we research the existence of positive solutions for nonlinear singular boundary value problem (1.8), (1.9) and (1.8), and (1.10) when g and f satisfy some suitable conditions.

Our main tool of this paper is the following fixed point index theory.

Theorem 1.1 (see [17], [18].)Suppose that 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 Tx ≠ x, ∀ x ∈ ∂Ω_r. Then

(i)
if ∥Tx∥ ≤ ∥x∥, ∀ x ∈ ∂Ω_r, then i(T, Ω_r, K) = 1;
(ii)
if ∥Tx∥ ≥ ∥x∥, ∀ x ∈ ∂Ω_r, then i(T, Ω_r, K) = 0.

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 single solution of the systems (1.8) and (1.9). In Section 4, we study the existence of at least two solutions of the systems (1.8) and (1.9). In Section 5, we discuss the existence of single and many solutions of the systems (1.8) and (1.10). In Section 6, we give two examples as the application.

2. Preliminaries and Lemmas

For convenience, we can found some basic definitions in [1, 19, 20].

In the rest of this paper, T is closed subset of R with 0 ∈ T_k, T ∈ T^k. And let B = {u ∈ C_ld[−τ, T]}, then B is a Banach space with the norm ∥u∥ = max _{t∈[−τ,T]} |u(t)|. And let

(2.1)

Obviously, K is a cone in B. Set K_r = {u ∈ K : ∥u∥ ≤ r}.

Definition 2.1. u(t) is called a solution of SBVP (1.8) and (1.9) if it satisfies the following:

(1)
u ∈ C[−τ, 0]∩C_ld(0, T),
(2)
u(t) > 0 for all t ∈ (0, T) and satisfies conditions (1.9),
(3)
holds for t ∈ (0, T).

In the rest of the paper, we also make the following assumptions:

(H₁) f ∈ C_ld([0, +∞) ², [0, +∞)),
(H₂) g(t) ∈ C_ld((0, T), [0, +∞)) and there exists t₀ ∈ (0, T), such that
(2.2)
(H₃) ζ(t) ∈ C([−τ, 0], ζ(t) > 0 on [−τ, 0) and ζ(0) = 0,
(H₄) B₀, B₁ are both increasing, continuous, odd functions defined on (−∞, +∞), and at least one of them satisfies the condition that there exists one b > 0 such that
(2.3)

It is easy to check that condition (H₂) implies that

(2.4)

We can easily get the following Lemmas.

Lemma 2.2. Suppose that condition (H₂) holds. Then there exists a constant θ ∈ (0, 1/2) that satisfies

(2.5)

Furthermore, the function

(2.6)

is positive continuous functions on [θ, T − θ]; therefore, A(t) has minimum on [θ, T − θ]. Hence, we suppose that there exists L > 0 such that A(t) ≥ L, t ∈ [θ, T − θ].

Proof. At first, it is easily seen that A(t) is continuous on [θ, T − θ]. Nest, let

(2.7)

Then, from condition (H₂), we have the function A₁(t) is strictly monotone nondecreasing on [θ, T − θ] and A₁(θ) = 0, the function A₂(t) is strictly monotone nonincreasing on [θ, T − θ] and A₂(T − θ) = 0, which implies L = min _{t∈[θ,T−θ]} A(t) > 0. The proof is complete.

Lemma 2.3 (see [16].)Let u ∈ K and θ of Lemma 2.2, then

(2.8)

Lemma 2.4. Suppose that conditions (H₁), (H₂), (H₃), and (H₄) hold, u(t) ∈ B∩C_ld(0,1) is a solution of the following boundary value problems:

(2.9)

(2.10)

(2.10')

where

(2.11)

Then,

is a positive solution to the SBVP (1.8) and (1.9) or (1.8) and (1.10).

Proof. It is easy to check that satisfies (1.8) and (1.9) or (1.8) and (1.10).

So in the rest section of this paper, we focus on SBVP (2.9), (2.10), and (2.9), (2.10′).

Lemma 2.5. Suppose that conditions (H₁), (H₂), (H₃), or (H₁), (H₂), (H₃), (H₄), hold, u(t) ∈ B∩C_ld(0,1) is a solution of boundary value problems (2.9), (2.10) or (2.9), (2.10′), respectively, if and only if u(t) ∈ B is a solution of the following integral equation, respectively:

(2.12)

where

(2.13)

(2.13')

Here σ, ϱ is unique solution of the equation, respectively,

(2.14)

where

(2.15)

Equation g₁(t) = g₂(t),

has unique solution in (0, T). Because g₁(t),

is strictly monotone increasing on [0, T), and g₁(0) = 0,

is strictly monotone decreasing on (0, T], and g₂(T) = 0,

Proof. We only proof the first section of the results.

Necessity. Obviously, for t ∈ (−τ, 0), we have u(t) = ζ(t).

If t ∈ (0, T), by the equation of the boundary condition and we have u^Δ(ξ) ≥ 0, u^Δ(η) ≤ 0, then there exist is a constant σ ∈ [ξ, η]⊂(0, T) such that u^Δ(σ) = 0.

Firstly, by integrating the equation of the problems (2.9) on (σ, t), we have

(2.16)

then

(2.17)

thus

(2.18)

By u^Δ(σ) = 0 and condition (2.16), t = η on (2.16), we have

(2.19)

By the equation of the boundary condition (2.10), we have

(2.20)

then

(2.21)

Then, by (2.18) and let t = T on (2.18), we have

(2.22)

Then

(2.23)

Similarly, for t ∈ (0, σ), by integrating the equation of problems (2.9) on (0, σ), we have

(2.24)

Therefore, for any t ∈ [0, T], u(t) can be expressed as equation

(2.25)

where w(t) is expressed as (2.13).

Sufficiency. Suppose that . Then by (2.13), we have

(2.26)

So,

. These imply that (2.9) holds. Furthermore, by letting t = 0 and t = T on (2.13) and (2.26), we can obtain the boundary value equations of (2.10). The proof is complete.

Now, we define an operetor equation T given by

(2.27)

where

is given by (2.13) and (2.13′).

From the definition of and above discussion, we deduce that for each . Moreover, we have the following Lemma.

Lemma 2.6. is completely continuous.

Proof. We only proof the completely continuous of T.

Because

(2.28)

is continuous, decreasing on [0, T], and satisfies that (Tu)^Δ(σ) = 0, then, Tu ∈ K for each u ∈ K and (Tu)(σ) = max _t∈[0,T] (Tu)(t). This shows that TK ⊂ K. Furthermore, it is easy to check by Arzela-ascoli Theorem that T : K → K is completely continuous.

Lemma 2.7. Suppose that conditions (H₁), (H₂), and (H₃) hold, the solution u(t) ∈ K of problem (2.9) and (2.10) satisfy

(2.29)

Proof. Firstly, we can have

(2.30)

The proof is complete.

For convenience, we set

(2.31)

where L is the constant from Lemma 2.2. By Lemma 2.5, we can also set

(2.32)

3. The Existence of Single Positive Solution to (1.8) and (1.9)

In this section, we present our main results.

Theorem 3.1. Suppose that condition (H₁), (H₂), and (H₃) hold. Assume that f also satisfies

(A₁): f(u₁, u₂) ≥ (mr)^p−1, for θr ≤ u₂ ≤ r, 0 ≤ u₁ ≤ H,
(A₂): f(u₁, u₂) ≤ (MR)^p−1, for 0 ≤ u₂ ≤ R, 0 ≤ u₁ ≤ H,

where m ∈ (θ^*, ∞), M ∈ (0, θ_*).

Then, the SBVP (2.9), (2.10) has a solution u such that ∥u∥ lies between r and R. Furthermore by Lemma 2.4, is a positive solution to the SBVP (1.8) and (1.9).

Proof of Theorem 3.1. Without loss of generality, we suppose that r < R. For any u ∈ K, by Lemma 2.3, we have

(3.1)

We define two open subset Ω₁ and Ω₂ of E:

(3.2)

For any u ∈ ∂Ω₁, by (3.1), we have

(3.3)

For t ∈ [θ, T − θ] and u ∈ ∂Ω₁, we shall discuss it from three perspectives.

(i)
If σ ∈ [θ, T − θ], thus for u ∈ ∂Ω₁, by (A₁) and Lemma 2.4, we have

(3.4)

(ii)
If σ ∈ (T − θ, T], thus for u ∈ ∂Ω₁, by (A₁) and Lemma 2.4, we have

(3.5)

(iii)
If σ ∈ (0, θ), thus for u ∈ ∂Ω₁, by (A₁) and Lemma 2.4, we have

(3.6)

Therefore, no matter under which condition, we all have

(3.7)

Then by Theorem 1.1, we have

(3.8)

On the other hand, for u ∈ ∂Ω₂, we have u(t)≤∥u∥ = R, and by (A₂), we know that

(3.9)

thus

(3.10)

Then, by Theorem 1.1, we have

(3.11)

Therefore, by (3.8), and (3.11), r < R, we have

(3.12)

Then operator T has a fixed point

, and r ≤ ∥u∥≤R. This completes the proof of Theorem 3.1.

Theorem 3.2. Suppose that condition (H₁), (H₂), and (H₃) hold. Assume that f also satisfies

(A₃): ,
(A₄): .

Then, the SBVP (2.9), (2.10) has a solution u which is bounded in ∥·∥. Furthermore, by Lemma 2.4,

is a positive solution to the SBVP (1.8), (1.9).

Proof of Theorem 3.2. First, by , for , there exists an adequately small positive number ρ, as 0 ≤ u₂ ≤ ρ, u₂ ≠ 0, u₁ ≤ H, we have

(3.13)

Then let R = ρ, M = θ_*/4 ∈ (0, θ_*), thus by (3.13),

(3.14)

So condition (A₂) holds.

Next, by condition (A₄), , then for , there exists an appropriately big positive number r ≠ R, as u₂ ≥ θr, u₁ ≤ H, we have

(3.15)

Let m = 2θ^* > θ^*, thus by (3.15), condition (A₁) holds. Therefore, by Theorem 3.1, we know that the results of Theorem 3.2 hold. The proof of Theorem 3.2 is complete.

Theorem 3.3. Suppose that conditions (H₁), (H₂), (H₃) hold. Assume that f also satisfies

(A₅): ,
(A₆): .

Then, the SBVP (2.9), (2.10) has a solution u which is bounded in ∥·∥. Furthermore by Lemma 2.4,

is a positive solution to the SBVP (1.8), (1.9).

Proof of Theorem 3.3. First, by condition (A₆), , then for , there exists an adequately small positive number r, as 0 ≤ u₂ ≤ r, u₂ ≠ 0, u₁ ≤ H, we have

(3.16)

thus when θr ≤ u₂ ≤ r, u₁ ≤ H, we have

(3.17)

Let m = 2θ^* > θ^*, so by (3.17), condition (A₁) holds.

Next, by condition (A₅): , then for , there exists an suitably big positive number ρ ≠ r, as u₂ ≥ ρ, u₁ ≤ H, we have

(3.18)

If f is unbounded, by the continuity of f on [0, ∞) ², then exists constant R (≠r) ≥ ρ, and a point (u₀₁, u₀₂)∈[0, ∞) ² such that

(3.19)

Thus, by ρ ≤ u₀₂ ≤ R, u₁ ≤ H, we know

(3.20)

Choose M = θ_*/4 ∈ (0, θ_*). Then, we have

(3.21)

If f is bounded, we suppose

, there exists an appropriately big positive number

, then choose M = θ_*/4 ∈ (0, θ_*), we have

(3.22)

Therefore, condition (A₂) holds. Therefore, by Theorem 3.1, we know that the results of Theorem 3.3 holds. The proof of Theorem 3.3 is complete.

4. The Existence of Many Positive Solutions to (1.8) and (1.9)

Next, we will discuss the existence of many positive solutions.

Theorem 4.1. Suppose that conditions (H₁), (H₂), (H₃), and (A₂) in Theorem 3.1 hold. Assume that f also satisfies

(A₇): f₀ = +∞,
(A₈): f_∞ = +∞.

Then, the SBVP (2.9), (2.10) has at last two solutions u₁, u₂ such that

(4.1)

Furthermore, by Lemma 2.4,

, −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.9).

Proof of Theorem 4.1. First, by condition (A₇), for any N > 2/θL, there exists a constant ρ_* ∈ (0, R) such that

(4.2)

Set

, for any

, by (4.2) and Lemma 2.3, similar to the previous proof of Theorem 3.1, we can have from three perspectives

(4.3)

Then by Theorem 1.1, we have

(4.4)

Next, by condition (A₈), for any

, there exists a constant ρ₀ > 0 such that

(4.5)

We choose a constant ρ^* > max {R, ρ₀/θ}, obviously ρ_* < R < ρ^*. Set

. For any

, by Lemma 2.3, we have

(4.6)

Then by (4.5) and also similar to the previous proof of Theorem 3.1, we can also have from three perspectives

(4.7)

Then by Theorem 1.1, we have

(4.8)

Finally, set Ω_R = {u ∈ K : ∥u∥<R}, For any u ∈ ∂Ω_R, by (A₂), Lemma 2.3 and also similar to the latter proof of Theorem 3.1, we can also have

(4.9)

Then by Theorem 1.1, we have

(4.10)

Therefore, by (4.4), (4.8), (4.10), ρ_* < R < ρ^*, we have

(4.11)

Then T has fixed-point

, and fixed-point

. Obviously, u₁, u₂ are all positive solutions of problem (2.9), (2.10) and ρ_* < ∥u₁∥<R < ∥u₂∥<ρ^*. The proof of Theorem 4.1 is complete.

Theorem 4.2. Suppose that conditions (H₁), (H₂), (H₃), and (A₁) in Theorem 3.1 hold. Assume that f also satisfies

(A₉): f⁰ = 0,
(A₁₀): f^∞ = 0.

Then, the SBVP (2.9), (2.10) has at last two solutions u₁, u₂ such that 0 < ∥u₁∥<r < ∥u₂∥. Furthermore, by Lemma 2.4,

, −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Proof of Theorem 4.2. First, by f⁰ = 0, for ϵ₁ ∈ (0, θ_*), there exists a constant ρ_* ∈ (0, r) such that

(4.12)

Set

, for any

, by (4.12), we have

(4.13)

that is

(4.14)

Then by Theorem 1.1, we have

(4.15)

Next, let , and note that f^*(x) is monotone increasing with respect to x ≥ 0. Then from f^∞ = 0, it is easy to see that

(4.16)

Therefore, for any ϵ₂ ∈ (0, θ_*), there exists a constant ρ^* > r such that

(4.17)

Set

, for any

, by (4.17), we have

(4.18)

that is

(4.19)

Then by Theorem 1.1, we have

(4.20)

Finally, set Ω_r = {u ∈ K : ∥u∥<r}. For any u ∈ ∂Ω_r, by (A₁), Lemma 2.3 and also similar to the previous proof of Theorem 3.1, we can also have

(4.21)

Then by Theorem 1.1, we have

(4.22)

Therefore, by (4.15), (4.20), (4.22), ρ_* < r < ρ^*, we have

(4.23)

Then T have fixed point

, and fixed point

. Obviously, u₁, u₂ are all positive solutions of problem (1.8), (1.9) and ρ_* < ∥u₁∥<r < ∥u₂∥<ρ^*. The proof of Theorem 4.2 is complete.

Similar to Theorem 3.1, we also obtain the following Theorems.

Theorem 4.3. Suppose that conditions (H₁), (H₂), (H₃) and (A₂) in Theorem 3.1, (A₄) in Theorem 3.2 and (A₆) in Theorem 3.3 hold. Then, the SBVP (2.9), (2.10) has at last two solutions u₁, u₂ such that 0 < ∥u₁∥<R < ∥u₂∥. Furthermore by Lemma 2.4, , −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Theorem 4.4. Suppose that conditions (H₁), (H₂), (H₃) and (A₁) in Theorem 3.1, (A₃) in Theorem 3.2 and (A₅) in Theorem 3.3 hold. Then, the SBVP (2.9), (2.10) have at last two solutions u₁, u₂ such that 0 < ∥u₁∥<r < ∥u₂∥. Furthermore by Lemma 2.4, , , −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

5. The Existence of Many Positive Solutions to (1.8) and (1.10)

In the following, we will deal with problem (1.8), (1.10), the method is similar to that in Sections 3 and 4, so we omit many proof in this section.

Theorem 5.1. Suppose that condition (H₁), (H₂), (H₃), (H₄) hold. Assume that f also satisfies

f(u₁, u₂) ≥ (mr)^p−1, for θr ≤ u₂ ≤ r, 0 ≤ u₁ ≤ H,
f(u₁, u₂) ≤ (MR)^p−1, for 0 ≤ u₂ ≤ R, 0 ≤ u₁ ≤ H,

where m ∈ (θ^*, ∞), M ∈ (0, θ_**). Then, the SBVP (2.9), (2.13) has a solution u such that ∥u∥ lies between r and R. Furthermore by Lemma 2.4,

, −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Theorem 5.2. Suppose that condition (H₁), (H₂), (H₃), (H₄) hold. Assume that f also satisfies

Then, the SBVP (2.9), (2.13) has a solution u which is bounded in ∥·∥. Furthermore by Lemma 2.4,

, −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Theorem 5.3. Suppose that condition (H₁), (H₂), (H₃), (H₄) hold. Assume that f also satisfies

Then, the SBVP (2.9), (2.13) has a solution u which is bounded in ∥·∥. Furthermore by Lemma 2.4,

, −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Theorem 5.4. Suppose that conditions (H₁), (H₂), (H₃), (H₄) and in Theorem 5.1 hold. Assume that f also satisfies

f₀ = +∞,
f_∞ = +∞.

Then, the SBVP (2.9), (2.13) has at least two solutions u₁, u₂ such that

(5.1)

Furthermore by Lemma 2.4,

, −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Theorem 5.5. Suppose that conditions (H₁), (H₂), (H₃), (H₄) and in Theorem 5.1 hold. Assume that f also satisfies

f⁰ = 0,
f^∞ = 0.

Then, the SBVP (2.9), (2.13) has at least two solutions u₁, u₂ such that 0 < ∥u₁∥<r < ∥u₂∥. Furthermore by Lemma 2.4,

, −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Theorem 5.6. Suppose that conditions (H₁), (H₂), (H₃), (H₄) and (A₂) in Theorem 5.1, (A₄) in Theorem 5.2 and in Theorem 3.3 hold. Then, the SBVP (2.9), (2.13) has at least two solutions u₁, u₂ such that 0 < ∥u₁∥<R < ∥u₂∥. Furthermore by Lemma 2.4, , −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

Theorem 5.7. Suppose that conditions (H₁), (H₂), (H₃), (H₄) and in Theorem 5.1, in Theorem 5.2 and in Theorem 3.3 hold. Then, the SBVP (2.9), (2.13) has at least two solutions u₁, u₂ such that 0 < ∥u₁∥<r < ∥u₂∥. Furthermore by Lemma 2.4, , −τ ≤ t ≤ T is a positive solution to the SBVP (1.8), (1.10).

6. Application

In the section, we present two simple examples to explain our result.

Example 6.1. Let T = {1 − (1/2) ^N}∪{1}, where N denotes the set of all nonnegative integers. Consider the following 3-order singular boundary value problem (SBVP) with p-Laplacian

(6.1)

where

(6.2)

So, by Lemma 2.4, we discuss the following SBVP:

(6.3)

where

(6.4)

Then obviously,

(6.5)

so conditions (H₁), (H₂), (H₃) hold.

Next,

(6.6)

then

, that is,

, so condition (A₃) holds.

For θ = 1/4, it is easy to see by calculating that

(6.7)

Because of

(6.8)

then

(6.9)

so condition (A₄) holds. Then by Theorem 3.2, SBVP (6.3) has at least a positive solution u(t). So,

is the positive solution of SBVP (6.1).

Example 6.2. Consider the following 3-order singular boundary value problem (SBVP) with p-Laplacian:

(6.10)

where

(6.11)

So, by Lemma 2.4, we discuss the following SBVP:

(6.12)

where

(6.13)

Then obviously,

(6.14)

so conditions (H₁), (H₂), (H₃), (A₇), (A₈) hold.

Next,

(6.15)

we choose R = 3, M = 2, and for θ = 1/4, because of the monotone increasing of f(u₁, u₂, u₃) on [0, ∞) ³, then

(6.16)

Therefore, by

(6.17)

we know that

(6.18)

so condition (A₂) holds. Then by Theorem 4.1, SBVP (6.12) has at least two positive solutions v₁, v₂ and 0 < ∥v₁∥ < 3 < ∥v₂∥. Then, by Lemma 2.4,

are the positive solutions of the SBVP (6.10).

Acknowledgments

The first the second authors were supported financially by Shandong Province Natural Science Foundation (ZR2009AQ004), NSFC (11026108, 11071141), and the third author was supported by Shandong Province planning Foundation of Social Science (09BJGJ14), and Shandong Province Natural Science Foundation (Z2007A04).

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Solutions for p-Laplacian Dynamic Delay Differential Equations on Time Scales

Abstract

1. Introduction

2. Preliminaries and Lemmas

3. The Existence of Single Positive Solution to (1.8) and (1.9)

4. The Existence of Many Positive Solutions to (1.8) and (1.9)

5. The Existence of Many Positive Solutions to (1.8) and (1.10)

6. Application

Acknowledgments

References

References

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Solutions for p-Laplacian Dynamic Delay Differential Equations on Time Scales

Abstract

1. Introduction

2. Preliminaries and Lemmas

3. The Existence of Single Positive Solution to (1.8) and (1.9)

4. The Existence of Many Positive Solutions to (1.8) and (1.9)

5. The Existence of Many Positive Solutions to (1.8) and (1.10)

6. Application

Acknowledgments

References

References

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