In this article, we study the following m-point boundary value problem on time scales,

where is a time scale such that , ϕ_p(s) = |s|^p-2s,p > 1,h ∈ C_ld((0, T), (0, +∞)), and f ∈ C([0,+∞), (0,+∞)), . By using several well-known fixed point theorems in a cone, the existence of at least one, two, or three positive solutions are obtained. Examples are also given in this article.

AMS Subject Classification: 34B10; 34B18; 39A10.

The study of dynamic equations on time scales goes back to its founder Hilger [1], and is a new area of still theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Further, the study of time scales has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, epidemic models, etc. [2].

Multipoint boundary value problems of ordinary differential equations (BVPs for short) arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross section and composed of N parts of different densities can be set up as a multi-point boundary value problem [3]. Many problems in the theory of elastic stability can be handled by the method of multi-point problems [4]. Small size bridges are often designed with two supported points, which leads into a standard two-point boundary value condition and large size bridges are sometimes contrived with multi-point supports, which corresponds to a multi-point boundary value condition [5]. The study of multi-point BVPs for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [6]. Since then many authors have studied more general nonlinear multi-point BVPs, and the multi-point BVP on time scales can be seen as a generalization of that in ordinary differential equations.

Recently, the existence and multiplicity of positive solutions for nonlinear differential equations on time scales have been studied by some authors [7-11], and there has been some merging of existence of positive solutions to BVPs with p-Laplacian on time scales [12-19].

He [20] studied

subject to one of the following boundary conditions

where . By using a double fixed-point theorem, the authors get the existence of at least two positive solutions to BVP (1.1) and (1.2).

Anderson [21] studied

subject to one of the following boundary conditions

by using a functional-type cone expansion-compression fixed-point theorem, the author gets the existence of at least one positive solution to BVP (1.3), (1.4) and BVP (1.3), (1.5).

However, to the best of the authors' knowledge, up to now, there are few articles concerned with the existence of m-point boundary value problem with p-Laplacian on time scales. So, in this article, we try to fill this gap. Motivated by the article mentioned above, in this article, we consider the following m-point BVP with one-dimensional p-Laplacian,

where ϕ_p(s) = |s|^p-2s,p > 1,h ∈ C_ld((0,T), (0, +∞)), . δ, β_i > 0, i = 1,..., m - 2.

We will assume throughout

(S1) h ∈ C_ld ((0, T), [0, ∞)) such that ;

(S2) f ∈ C([0, ∞), (0, ∞)), f ≢ 0 on ;

(S3) By ϕ_q we denote the inverse to ϕ_p, where ;

(S4) By t ∈ [a, b] we mean that , where 0 ≤ a ≤ b ≤ T.

In this section, we will give some background materials on time scales.

Definition 2.1. [7,22] For and , define the forward jump operator σ and the backward jump operator ρ, respectively,

for all . If σ(t) > t, t is said to be right scattered, and if ρ(r) < r, r is said to be left scattered. If σ(t) = t, t is said to be right dense, and if ρ(r) = r, r is said to be left dense. If has a right scattered minimum m, define ; Otherwise set . The backward graininess is defined by

If has a left scattered maximum M, define ; Otherwise set . The forward graininess is defined by

Definition 2.2. [7,22] For and , we define the "Δ" derivative of x(t), x^Δ(t), to be the number (when it exists), with the property that, for any ε > 0, there is neighborhood U of t such that

for all s ∈ U. For and , we define the "∇" derivative of x(t),x^Δ (t), to be the number(when it exists), with the property that, for any ε > 0, there is a neighborhood V of t such that

for all s ∈ V.

Definition 2.3. [22] If F^Δ (t) = f(t), then we define the "Δ" integral by

If F^∇ (t) = f(t), then we define the "∇" integral by

Lemma 2.1. [23]The following formulas hold:

(i) ,

(ii) ,

(iii) ,

(iv) .

Lemma 2.2. [7, Theorem 1.75 in p. 28] If f ∈ C_rd and , then

According to [23, Theorem 1.30 in p. 9], we have the following lemma, which can be proved easily. Here, we omit it.

Lemma 2.3. Let and f ∈ C_ld.

(i) If , then

where the integral on the right is the usual Riemann integral from calculus.

(ii) If [a, b] consists of only isolated points, then

(iii) If , where h > 0, then

(iv) If , then

In what follows, we list the fixed point theorems that will be used in this article.

Theorem 2.4. [24]Let E be a Banach space and P ⊂ E be a cone. Suppose Ω₁, Ω₂ ⊂ E open and bounded, . Assume is completely continuous. If one of the following conditions holds

(i) ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω₁ ∩ P, ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω₂ ∩ P;

(ii) ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω₁ ∩ P, ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω₂ ∩ P.

Then, A has a fixed point in .

Theorem 2.5. [25]Let P be a cone in the real Banach space E. Set

If α and γ are increasing, nonnegative continuous functionals on P, let θ be a nonnegative continuous functional on P with θ(0) = 0 such that for some positive constants r, M,

for all . Further, suppose there exists positive numbers a < b < r such that

If is completely continuous operator satisfying

(i) γ(Au) > r for all u ∈ ∂P(γ, r);

(ii) θ(Au) < b for all u ∈ ∂P(θ, r);

(iii) and α(Au) > a for all u ∈ ∂P(α, a).

Then, A has at least two fixed points u₁ and u₂ such that

Let a, b, c be constants, P_r = {u ∈ P : ∥u∥ < r}, P(ψ, b, d) = {u ∈ P : a ≤ ψ(u), ∥u∥ ≤ b}.

Theorem 2.6. [26]Let be a completely continuous map and ψ be a nonnegative continuous concave functional on P such that for , there holds ψ(u) ≤ ∥u∥. Suppose there exist a, b, d with 0 < a < b < d ≤ c such that

(i) and ψ(Au) > b for all u ∈ P(ψ, b, d);

(ii) ∥Au∥ < a for all ;

(iii) ψ(Au) > b for all u ∈ P(ψ, b, d) with ∥Au∥ > d.

Then, A has at least three fixed points u₁,u₂, and u₃ satisfying

Let the Banach space E = C_ld[0, T] be endowed with the norm ∥u∥ = sup_{t ∈ [0,T]} u(t), and cone P ⊂ E is defined as

It is obvious that ∥u∥ = u(T) for u ∈ P. Define A : P → E as

for t ∈ [0, T].

In what follows, we give the main lemmas which are important for getting the main results.

Lemma 2.7. A : P → P is completely continuous.

Proof. First, we try to prove that A : P → P.

Thus, (Au)^Δ (T) = 0 and by Lemma 2.1 we have (Au)^Δ∇ (t) = -h(t)f(t, u(t)) ≤ 0 for t ∈ (0, T). Consequently, A : P → P.

By standard argument we can prove that A is completely continuous. For more details, see [27]. The proof is complete.

Lemma 2.8. For u ∈ P, there holds for t ∈ [0,T].

Proof. For u ∈ P, we have u^Δ∇ (t) ≤ 0, it follows that u^Δ (t) is non-increasing. Therefore, for 0 < t < T,

and

thus

Combining (2.1) and (2.3) we have

as u(0) ≥ 0, it is immediate that

The proof is complete.

First, we give some notations. Set

Theorem 3.1. Assume in addition to (S1) and (S2), the following conditions are satisfied, there exists such that

(H1) ;

(H2) .

Then, BVP (1.6) has at least one positive solution.

Proof. Cone P is defined as above. By Lemma 2.7 we know that A : P → P is completely continuous. Set Ω_r = {u ∈ E, ∥u∥ < r}. In view of (H1), for u ∈ ∂ Ω_r ∩ P,

which means that for u ∈ ∂Ω_r ∩ P, ||Au|| ≤ ||u||.

On the other hand, for u ∈ P, in view of Lemma 2.8, there holds , for t ∈ [ξ₁, T]. Denote Ω_ρ = {u ∈ E, ∥u∥ < ρ}. Then for u ∈ ∂Ω_ρ ∩ P, considering (H2), we have

which implies that for u ∈ ∂Ω_ρ∩P, ∥Au∥ ≥ ∥u∥ Therefore, the immediate result of Theorem 2.4 is that A has at least one fixed point u ∈ (Ω_ρ\Ω_r) ∩ P. Also, it is obvious that the fixed point of A in cone P is equivalent to the positive solution of BVP (1.6), this yields that BVP (1.6) has at least one positive solution u satisfies r ≤ ∥u∥ ≤ ρ. The proof is complete.

Here is an example.

Example 3.2. Let . Consider the following four point BVP on time scale .

where

and h(t) = 1, T = 4, ξ₁ = 2, ξ₂ = 3, δ = 2, β₁ = β₂ = 1,p = q = 2. In what follows, we try to calculate Λ, B. By Lemmas 2.2 and 2.3, we have

where

Thus, . Let . Then, we have

(i)

(ii)

Thus, if all the conditions in Theorem 3.1 satisfied, then BVP (3.1) has at least one positive solution lies between 100 and 1000.

In this section, we will apply fixed point Theorem 2.5 to prove the existence of at least two positive solutions to the nonlinear BVP (1.6).

Fix such that

and define the increasing, nonnegative, continuous functionals γ, θ,α on P by

We can see that, for u ∈ P, there holds

In addition, Lemma 2.8 implies that which means that

We also see that

For convenience, we give some notations,

Theorem 4.1. Assume in addition to (S1), (S2) there exist positive constants such that the following conditions hold

(H3) f(t, u) > ϕ_p(c/M) for t ∈ [ξ₁,T] u ∈ [c,Tc/ξ₁];

(H4) f(t, u) < ϕ_p(b/K) for t ∈ [0,ξ_m-2], u ∈ [b,Tb/ξ_m-2];

(H5) f(t, u) > ϕ_p(a/L) for t ∈ [η,T], u ∈ [a,Ta/η].

Then BVP (1.6) has at least two positive solutions u₁ and u₂ such that

Proof. From Lemma 2.7 we know that A : P(γ, c) → P is completely continuous. In what follows, we will prove the result step by step.

Step one: To verify (i) of theorem 2.5 holds.

We choose u ∈ ∂P(γ,c), then . This implies that u(t) ≥ c for t ∈ [ξ₁,T], considering that , we have

As a consequence of (H3),

Since Au ∈ P, we have

Thus, (i) of Theorem 2.5 is satisfied.

Step two: To verify (ii) of Theorem 2.5 holds.

Let u ∈ ∂P(θ,b), then , this implies that 0 ≤ u(t) ≤ b, t ∈ [0,ξ_m-2] and since u ∈ P, we have ∥u∥ = u(T), note that . So,

From (H4) we know that for t ∈ [0, ξ_m-2] and so

Thus, (ii) of Theorem 2.5 holds.

Step three: To verify (iii) of Theorem 2.5 holds.

Choose , obviously, u₀(t) ∈ P(α, a) and , thus .

Now, let u ∈ ∂P(α, a), then, α(u) = min_t∈[η,T] u(t) = u(η) = a. Recalling that . Thus, we have

From assumption (H5) we know that

and so

Therefore, all the conditions of Theorem 2.5 are satisfied, thus A has at least two fixed points in P(γ,c), which implies that BVP (1.6) has at least two positive solutions u₁,u₂ which satisfies (4.1). The proof is complete.

Example 4.2. Let . Consider the following four point boundary value problem on time scale .

where

and h(t) = t, T = 8, ξ₁ = 1, ξ₂ = 2, δ = 1, β₁ = 1, β₂ = 2,p = 3/2, q = 3. In what follows, we try to calculate K, M, L. By Lemmas 2.2 and 2.3, we have

Let a = 10⁶, b = 10⁸, c = 10⁹, then we have

(i) , for t ∈ [1, 8], u ∈ [10⁹, 8 × 10⁹];

(ii) , for t ∈ [0, 2], u ∈ [10⁸, 4 × 10⁸];

(iii) , for t ∈ [4, 8], u ∈ [10⁶, 2 × 10⁶].

Thus, if all the conditions in Theorem 4.1 are satisfied, then BVP (4.2) has at least two positive solutions satisfying (4.1).

Let , then 0 < ψ(u) ≤ ∥u∥. Denote

In this section, we will use fixed point Theorem 2.6 to get the existence of at least three positive solutions.

Theorem 5.1. Assume that there exists positive number d, ν, g satisfying , such that the following conditions hold.

(H6) f(t, u) < ϕ_p(d/R), t ∈ [0,T],u ∈ [0,d];

(H7) f(t, u) > ϕ_p(ν/D), t ∈ [ξ₁, T], u ∈ [ν, Tυ/ξ₁];

(H8) f(t, u) ≤ ϕ_p(g/R), t ∈ [0,T],u ∈ [0,g],

then BVP (1.6) has at least three positive solutions u₁, u₂, u₃ satisfying

Proof. From Lemma 2.8 we know that A : P → P is completely continuous. Now we only need to show that all the conditions in Theorem 2.6 are satisfied.

For . By (H8), one has

Thus, . Similarly, by (H6), we can prove (ii) of Theorem 2.6 is satisfied.

In what follows, we try to prove that (i) of theorem 2.6 holds. Choose , obviously, ψ(u₁) > ν, thus . For u ∈ P(ψ,ν,Tν/ξ₁),

It remains to prove (iii) of Theorem 2.6 holds. For u ∈ P(ψ, ν, Tυ/ξ₁), with ∥Au∥ > Tν/ξ₁, in view of Lemma 2.8, there holds , which implies that (iii) of Theorem 2.6 holds.

Therefore, all the conditions in Theorem 2.6 are satisfied. Thus, BVP (1.6) has at least three positive solutions satisfying (5.1). The proof is complete.

Example 5.2. Let . Consider the following four point boundary value problem on time scale .

where

and h(t) = e^t, T = 2, ξ₁ = 1/2, ξ₂ = 1, δ = 3, β₁ = 2, β₂ = 3, p = 4, q = 4/3. In what follows, we try to calculate D, R. By Lemmas 2.2 and 2.3, we have

Let d = 40, ν = 50, g = 400, then we have

(i) f(t, u) < 7.027 = (40/20.8832)³ = ϕ_p(d/R), for t ∈ [0, 2], u ∈ [0, 40];

(ii) f(t, u) > 23.2375 = (50/17.5216)³ = ϕ_p(ν/D), for t ∈ [1/2, 2], u ∈ [50, 200];

(iii) f(t, u) < 7027.305 = (400/20.8832)³ = ϕ_p(g/R), for t ∈ [0, 2], u ∈ [0, 400].

Thus, if all the conditions in Theorem 5.1 are satisfied, then BVP (5.2) has at least three positive solutions satisfying (5.1).

The authors declare that they have no competing interests.

WG and HL conceived of the study, and participated in its coordination. JZ drafted the manuscript. All authors read and approved the final manuscript.

The authors were very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The study was supported by Pre-research project and Excellent Teachers project of the Fundamental Research Funds for the Central Universities (2011YYL079, 2011YXL047).

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