The Existence of Positive Solutions for Third-Order -Laplacian -Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales

Abstract

We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.

1. Introduction

The theory of time scales was initiated by Hilger [1] as a mean of unifying and extending theories from differential and difference equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example [2–6]. Recently, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [7–18]. However, to the best of our knowledge, there are not many results concerning the higher-order -Laplacian mutilpoint boundary value problem on time scales.

A time scale is a nonempty closed subset of . We make the blanket assumption that are points in . By an interval , we always mean the intersection of the real interval with the given time scale; that is .

In [19], Anderson considered the following third-order nonlinear boundary value problem (BVP):

(1.1)

author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii's fixed point theorem and Leggett and Williams fixed point theorem, respectively.

In [9, 10], He considered the existence of positive solutions of the -Laplacian dynamic equations on time scales

(1.2)

satisfying the boundary conditions

(1.3)

or

(1.4)

where . He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In [18], Zhou and Ma firstly studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with -Laplacian operator

(1.5)

They established a corresponding iterative scheme for the problem by using the monotone iterative technique.

All the above works were done under the assumption that the nonlinear term is nonnegative. The key conditions used in the above papers ensure that positive solution is concave down. If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down. As a result, it is difficult to find positive solutions of the -Laplacian equation when the nonlinearity changes sign. In particular, little work has been done on the existence of positive solutions for higher order -Laplacian -point boundary value problems with nonlinearity being nonnegative on time scales. Therefore, it is a natural problem to consider the existence of positive solution for higher order -Laplacian equations with sign changing nonlinearity on time scales. This paper attempts to fill this gap in literature.

In this paper, by using different method, we are concerned with the existence of positive solutions for the following third-order -Laplacian -point boundary value problems on time scales:

(1.6)

where is -Laplacian operator, that is, , , and , , , satisfy

, , , ;

is continuous, , and there exists such that .

2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in [1–5].

Definition 2.1.

A time scale is a nonempty closed subset of real numbers . For and , define the forward jump operator and backward jump operator , respectively, by

(2.1)

for all . If , is said to be right scattered, if , is said to be left scattered; if , is said to be right dense, and if , is said to be left dense. If has a right scattered minimum , define ; otherwise set . If has a left scattered maximum , define ; otherwise set .

Definition 2.2.

For and , the delta derivative of at the point is defined to be the number (provided that it exists), with the property that for each , there is a neighborhood of such that

(2.2)

for all .

For and , the nabla derivative of at , denoted by (provided it exists) with the property that for each , there is a neighborhood of such that

(2.3)

for all .

Definition 2.3.

A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in .

Definition 2.4.

If , then we define the delta integral by

(2.4)

If , then we define the nabla integral by

(2.5)

Lemma 2.5.

If condition holds, then for , the boundary value problem (BVP)

(2.6)

has the unique solution

(2.7)

Proof.

By caculating, we can easily get (2.7). So we omit it.

Lemma 2.6.

If condition holds, then for , the boundary value problem (BVP)

(2.8)

has the unique solution

(2.9)

where .

Proof.

Integrating both sides of equation in (2.8) on , we have

(2.10)

So,

(2.11)

By boundary value condition , we have

(2.12)

By (2.10) and (2.12) we know

(2.13)

This together with Lemma 2.5 implies that

(2.14)

where . The proof is complete.

Lemma 2.7.

Let condition holds If and , then the unique solution of (2.8) satisfies

(2.15)

Proof.

By , we can know that the graph of is concave down on , and is nonincreasing on . This together with the assumption that the boundary condition implies that for . This implies that

(2.16)

So we only prove By condition we have

(2.17)

The proof is completed.

Lemma 2.8.

Let condition hold. If and , then the unique positive solution of (BVP) (2.8) satisfies

(2.18)

where , .

Proof.

By , we can know that the graph of is concave down on , and is nonincreasing on . This together with the assumption that the boundary condition implies that for . This implies that

(2.19)

For all , we have from the concavity of that

(2.20)

that is,

(2.21)

This together with the boundary condition implies that

(2.22)

This completes the proof.

Let be endowed with the ordering if for all and is defined as usual by maximum norm. Clearly, it follows that is a Banach space.

For the convenience, let

(2.23)

We define two cones by

(2.24)

where , is defined in Lemma 2.8 and

(2.25)

Define the operators and by setting

(2.26)

where ,

(2.27)

where , , and . Obviously, is a solution of the BVP(1.6) if and only if is a fixed point of operator .

Lemma 2.9.

is completely continuous.

Proof.

It is easy to see that by and Lemma 2.8. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operator is completely continuous.

Lemma 2.10 (see [20, 21]).

Let be a cone in a Banach space . Let be an open bounded subset of with and . Assume that is a compact map such that for . Then the following results hold.

(1)If , , then .

(2)If there exists such that for all and all , then .

(3)Let be open in such that . If and , then has a fixed point in . The same result holds if and , where denotes fixed point index.

We define

(2.28)

Lemma 2.11 (see [20]).

defined above has the following properties:

(a)

(b) is open relative to K;

(c) if and only if

(d)if , then for .

For the convenience, we introduce the following notations:

(2.29)

Remark 2.12.

By we can know that ,

Lemma 2.13.

If satisfies the following condition :

(2.30)

then

(2.31)

Proof.

For , then from (2.30) we have

(2.32)

So that

(2.33)

Therefore,

(2.34)

This implies that for . Hence by Lemma 2.10(1) it follows that .

Lemma 2.14.

If satisfies the following condition:

(2.35)

then

(2.36)

Proof.

Let for . Then . We claim that

(2.37)

In fact, if not, there exist and such that . By , we have

(2.38)

So that

(2.39)

For , then

(2.40)

This together with Lemma 2.11(c) implies that

(2.41)

a contradiction. Hence by Lemma 2.10(2) it follows that .

3. Main Results

We now give our results on the existence of positive solutions of BVP (1.6).

Theorem 3.1.

Suppose that conditions and hold, and assume that one of the following conditions holds.

There exist with such that

(i), ;

(ii), , moreover , .

There exist with such that

(i), ;

(ii), .

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

Proof.

Assume that holds, we show that has a fixed point in . By and Lemma 2.13, we have that

(3.1)

By and Lemma 2.14, we have that

(3.2)

By Lemma 2.11(a) and , we have . It follows from Lemma 2.10(3) that has a fixed point in . Clearly,

(3.3)

which implies that , . By condition (ii), we have , , that is, . Hence,

(3.4)

This means that is a fixed point of operator .

When condition holds, by and Lemma 2.13, we have that

(3.5)

By and Lemma 2.14, we have that

(3.6)

By Lemma 2.11(a) and , we have . It follows from Lemma 2.10(3) that has a fixed point in . Obviously,

(3.7)

which implies that , . By condition (ii), we have , , that is, . Hence,

(3.8)

This means that is a fixed point of operator . Therefore, the BVP (1.6) has at least one positive solution.

Theorem 3.2.

Assume that conditions and hold, and suppose that one of the following conditions holds.

There exist , and with , and such that

(i), ;

(ii), , moreover , , ;

(iii), .

There exist , and with such that

(i), ;

(ii), , ,;

(iii), , moreover, , .

Then, the BVP (1.6) has at least two positive solutions.

Proof.

Assume that condition holds, we show that has a fixed point either in or in . If for . by Lemmas 2.13 and 2.14, we have that

(3.9)

By Lemma 2.11(a) and , we have . It follows from Lemma 2.10(3) that has a fixed point in . Similarly, has a fixed point in . Clearly,

(3.10)

which implies that , . By condition (ii), we have , , that is, . Hence,

(3.11)

This means that is a fixed point of operator . On the other hand, from and Lemma 2.11(a), we have . Clearly,

(3.12)

which implies that . By and condition (ii), we have , that is, . Hence,

(3.13)

This means that is a fixed point of operator . Then, the BVP (1.6) has at least two positive solutions.

When condition holds, the proof is similar to the above, and so we omit it here.

4. An Example

In the section, we present some simple examples to explain our results.

Example 4.1.

Let , . Consider the following three-point boundary value problem with -Laplacian

(4.1)

where , , , , .

By computing, we can know , , , . Obviously, , .

Let , , then . We define a sign changing nonlinearity as follows:

(4.2)

Then, by the definition of we have

(i), ;

(ii), , moreover , .

So condition holds, and by Theorem 3.1, BVP (4.1) has at least one positive solution.