Multiple Positive Solutions of -Point BVPs for Third-Order -Laplacian Dynamic Equations on Time Scales

Abstract

This paper is concerned with the existence of multiple positive solutions for the third-order -Laplacian dynamic equation with the multipoint boundary conditions , where with . Using the fixed point theorem due to Avery and Peterson, we establish the existence criteria of at least three positive solutions to the problem. As an application, an example is given to illustrate the result. The interesting points are that not only do we consider third-order -Laplacian dynamic equation but also the nonlinear term is involved with the first-order delta derivative of the unknown function.

1. Introduction

The theory of dynamic equations on time scales was introduced by Stefan Hilger in 1988 [1]. This theory has attracted many researchers' attention and interest since it cannot only unify differential and difference equations but also provides accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. In addition, time-scale calculus would allow exploration of a variety of situations in economic, biological, heat transfer, stock market, and epidemic models [2, 3], and so forth.

Recently, there has been much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales; see [4–10] and the references therein. On the one hand, higher-order nonlinear boundary value problems have been studied extensively; see [11–14] and the references therein. On the other hand, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [15–17]. However, very little work has been done to the third-order -Laplacian dynamic equations on time scales [18, 19].

For convenience, throughout this paper, we denote as -Laplacian operator, that is, for with and We also assume that is a closed subset of with ; an interval always means . Other types of intervals are defined similarly.

For example, Sun and Li [16] studied the two-point boundary value problem:

(1.1)

They established the existence theory for positive solutions by using various fixed point theorems [20, 21].

In [15], Su et al. investigated the existence of positive solutions for the following singular -Laplacian -point boundary value problem on time scales:

(1.2)

The main techniques are Schauder fixed point theorem and upper and lower solutions method.

In [19], Han and Kang considered the following third-order -Laplacian dynamic equation on time scales:

(1.3)

By using fixed point theorems in cones, the existence criteria of multiple positive solutions are established.

In [10], Zhao and Sun studied the following second-order nonlinear three-point boundary value problem on time scales:

(1.4)

They gave sufficient condition for the existence of three positive solutions by using a fixed point theorem due to Avery and Peterson [22].

Motivated by [10, 15, 16, 19], in this paper we consider the following third-order -Laplacian dynamic equation on time scales:

(1.5)

subject to the boundary condition

(1.6)

where , for . By using fixed point theorem due to Avery and Peterson [22], we prove that the boundary value problems (1.5) and (1.6) have at least three positive solutions under suitable assumptions. The interesting points are that not only do we consider third-order -Laplacian dynamic equation on time scales but also the nonlinear term is involved with the first-order delta derivative of the unknown function.

Throughout this paper, it is assumed that

• (H1) and , both and do not vanish identically on any closed subinterval of , and there exists such that hold;

• (H2) there exist nonnegative constants and satisfying for

2. Preliminary

To prove the main results in this paper, we will employ several lemmas. And the following lemma is based on the linear BVP:

(2.1)

(2.2)

Lemma 2.1.

If , then the problems (2.1) and (2.2) have the unique nonnegative solution:

(2.3)

Proof.

For any , suppose that is a solution of the BVPs (2.1) and (2.2). By integrating (2.1) from to , and combining the boundary condition, it follows that

(2.4)

Using (2.2), we can easily obtain

(2.5)

So

(2.6)

Then it is easy to see that

(2.7)

So . On the other hand, it is easy to verify that if is as in (2.3), then is a solution of (2.1) and (2.2). Thus in (2.3) is the unique solution of (2.1) and (2.2).

Let = be endowed with the norm

(2.8)

It follows that is a Banach space. Define the cone by

(2.9)

Lemma 2.2.

If , then there exists a constant such that

(2.10)

Proof.

For , implies that

(2.11)

In addition, since

(2.12)

then we have

(2.13)

Therefore, We can choose and the proof is complete.

Lemma 2.3.

If , then for .

Proof.

If , then is decreasing and , and thus and are decreasing. So we have

(2.14)

By the concavity of , for , there is

(2.15)

Then we have

(2.16)

The proof is complete.

Let and be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on . Then for positive real numbers and , we define the following convex sets:

(2.17)

and a closed set

(2.18)

The following fixed point theorem due to Avery and Peterson is fundamental in the proof of our main results.

Lemma 2.4 (see [22]).

Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on satisfying for , such that for some positive numbers and ,

(2.19)

for all . Suppose that is completely continuous and there exist positive numbers , and with such that

• (S1) and for ;

• (S2) for with ;

• (S3) and for with .

Then has at least three fixed points , such that

(2.20)

3. Existence Results

In this section, by using the Avery-Peterson fixed point theorem, we shall give the sufficient conditions for the existence of at least three positive solutions to the BVPs (1.5) and (1.6).

Firstly, we define the nonnegative continuous concave functional , the nonnegative continuous convex functionals , and the nonnegative continuous functional on , respectively, by

(3.1)

For notation convenience, we denote

(3.2)

Now we state and prove our main result.

Theorem 3.1.

Let and suppose that satisfies the following conditions:

• (A1) for

• (A2) for

• (A3) for

Then problems (1.5) and (1.6) have at least three positive solutions , and such that

(3.3)

Proof.

Define an integral operator by

(3.4)

for . It is easy to obtain that is a completely continuous operator and every fixed point of is a solution of (1.5) and (1.6).

Thus we set out to verify that the operator satisfies Avery-Peterson fixed point theorem which will prove the existence of three fixed points of . Now the proof is divided into some steps.

By virtue of , , and Lemma 2.2 we know that there exists a constant such that

(3.5)

We first show that implies that

(3.6)

In fact, for , , by Lemma 2.2, there is It follows from that

(3.7)

Thus (3.6) holds.

Next we show that condition in Lemma 2.4 holds. Let Then it is easy to see that , , and for , so . Also, we have

(3.8)

So . Hence

If , then for It follows from condition that

(3.9)

Therefore we have

(3.10)

That is, condition in Lemma 2.4 is satisfied.

We now prove that in Lemma 2.4 holds. In fact, since then with Lemma 2.3 it follows that

(3.11)

for with . Hence condition in Lemma 2.4 is satisfied.

Finally, we assert that in Lemma 2.4 also holds.

Observe that , so Suppose with Then, by hypothesis we have

(3.12)

Thus condition in Lemma 2.4 holds.

Therefore an application of Lemma 2.4 implies that the BVPs (1.5) and (1.6) have at least three positive solutions , and such that (3.3) holds.

4. Example

In this section, we present an example to explain our result.

Let , , , and , , , , . We consider the following boundary value problem:

(4.1)

where

(4.2)

Choosing , , , , direct calculation shows that

(4.3)

Consequently, satisfies

• (i) for

• (ii) for

• (iii) for

Then all conditions of Theorem 3.1 hold. Thus with Theorem 3.1, the BVP (4.1) has at least three positive solutions , , and such that

(4.4)