Open Access

Existence of Three Monotone Solutions of Nonhomogeneous Multipoint BVPs for Second-Order Differential Equations

Boundary Value Problems20082008:320603

https://doi.org/10.1155/2008/320603

Received: 9 March 2008

Accepted: 7 July 2008

Published: 27 July 2008

Abstract

This paper is concerned with nonhomogeneous multipoint boundary value problems of second-order differential equations with one-dimensional -Laplacian. Sufficient conditions to guarantee the existence of at least three solutions (may be not positive) of these BVPs are established.

1. Introduction

In recent years, there are several papers concerned with the existence of positive solutions of BVPs for differential equations with nonhomogeneous BCs. Kwong and Wong in [1] studied the following BVP:
(1.1)

where , is a nonnegative and continuous function. Under some assumptions, it was proved that there exists a constant such that

(i)BVP(1.1) has at least two positive solutions if ;

(ii)BVP(1.1) has at least one solution if or ;

(iii)BVP(1.1) has no positive solution if .

Sun et al. in [2] studied the existence of positive solutions for the following three-point boundary value problem:
(1.2)

where , are given. It was proved that there exists such that BVP(1.2) has at least one positive solution if and no positive solution if . To study the existence of positive solutions of above BVPs, the Green's functions of the corresponding problems are established and play an important role in the proofs of the main results.

For the following multipoint boundary value problems
(1.3)

in papers [35], sufficient conditions are found for the existence of solutions of BVP(1.3) based on the existence of lower and upper solutions with certain relations. Using the obtained results, under some other assumptions, the explicit ranges of values of and are presented with which BVP has a solution, has a positive solution, and has no solution, respectively. Furthermore, it is proved that the whole plane for and can be divided into two disjoint connected regions and such that BVP has a solution for and has no solution for .

In a recent paper [6], Liu, by using the Schauder fixed point theorem and imposing growth conditions on , obtained at least one positive solution of the following BVPs:
(1.4)
Motivated by the results obtained in the papers, this paper is concerned with the following BVPs for differential equation with -Laplacian coupled with nonhomogeneous multipoint BCs, that is, the BVPs
(1.5)

where , for all , is continuous and nonnegative, is continuous with , is called -Laplacian, with , its inverse function is denoted by .

Suppose

is continuous with on each subinterval of for all , where ;

;

, satisfy and there exists a constant such that .

The purpose is to establish sufficient conditions for the existence of at least three solutions of BVP(1.5). It is proved that BVP(1.5) has three monotone solutions under the growth conditions imposed on for all . These solutions may not be positive. The proofs of the main results are proved by using fixed point theorem in cones in Banach spaces, Green's functions and the existence of upper and lower solutions are not used in this paper.

The remainder of this paper is organized as follows. The main results are given in Section 2 and an example to show the main results is given in Section 3.

2. Main Results

In this section, we first present some background definitions in Banach spaces and state an important three fixed point theorem. Then the main results are given and proved.

Definition 2.1.

Let be a semi-ordered real Banach space. The nonempty convex closed subset of is called a cone in if for all and and and imply .

Definition 2.2.

A map is a nonnegative continuous concave or convex functional map provided is nonnegative and continuous and satisfies
(2.1)
or
(2.2)

for all and .

Definition 2.3.

An operator is completely continuous if it is continuous and maps bounded sets into relative compact sets.

Definition 2.4.

Let be positive constants, be two nonnegative continuous concave functionals on cone , be three nonnegative continuous convex functionals on cone . Define the convex sets as follows:
(2.3)

Lemma 2.5 (see [7]).

Let be a semi-ordered real Banach space with the norm , let be a cone in , let be two nonnegative continuous concave functionals on cone , let be three nonnegative continuous convex functionals on cone . There exists constant such that
(2.4)

Furthermore, suppose that are constants with . Let be a completely continuous operator. If

and
(2.5)
and
(2.6)

for with ;

for each with then has at least three fixed points , , and such that
(2.7)

Choose . We call for if for all , define the norm for . It is easy to see that is a semi-ordered real Banach space.

Choose . For a cone of the Banach space , define the functionals on by
(2.8)

It is easy to see that are two nonnegative continuous concave functionals on the cone are three nonnegative continuous convex functionals on cone and for all .

Lemma 2.6.

Suppose that for all and is decreasing on . Then
(2.9)

Proof.

Suppose that . If , we get that there exists such that
(2.10)
Then
(2.11)
Similarly we can get that
(2.12)

It follows that for all . The proof is complete.

Consider the following BVP:
(2.13)

Lemma 2.7.

Suppose that is a nonnegative continuous function, and hold. If is a solution of BVP(2.13), then is increasing and positive on .

Proof.

Suppose that satisfies (2.13). It follows from the assumptions that is decreasing on . Then the BCs in (2.13) and imply that
(2.14)
It follows that . We get that for . Then
(2.15)

It follows that . Then for since for all . The proof is complete.

Lemma 2.8.

Suppose that is a nonnegative continuous function, and hold. If is a solution of BVP(2.13), then
(2.16)
and satisfies
(2.17)
and satisfies
(2.18)

Proof.

It follows from (2.13) that
(2.19)
and the BCs in (2.13) imply that
(2.20)
Let
(2.21)
It is easy to see that . On the other hand, it follows from that , one sees that
(2.22)

Hence . Since is increasing for , we get that there exists unique constant such that (2.17) holds. The proof is completed.

Note , and let . Then BVP(1.5) is transformed into the following BVP:
(2.23)
Let
(2.24)

Then is a cone in .

Since
(2.25)
we get that
(2.26)

It is easy to see that there exists a constant such that for all .

Define the nonlinear operator by
(2.27)
where satisfies
(2.28)
and satisfies
(2.29)
Then
(2.30)

Lemma 2.9.

Suppose that , and hold. It is easy to show that

(i) is a solution of the BVP
(2.31)

(ii) for each ;

(iii) is a solution of BVP(1.5) if and only if and is a solution of the operator equation in cone ;

(iv) is completely continuous.

Proof.

The proofs are simple and are omitted.

Theorem 2.10.

Suppose that , and hold and there exist positive constants and , and given by
(2.32)
such that
(2.33)

If

for all ;

for all ;

for all ;

then BVP(1.5) has at least three increasing positive solutions such that
(2.34)

Proof.

To apply Lemma 2.5, we prove that all conditions in Lemma 2.5 are satisfied. By the definitions, it is easy to see that are two nonnegative continuous concave functionals on cone , are three nonnegative continuous convex functionals on cone and for all , there exist constants such that for all . Lemma 2.9 implies that is a positive solution of BVP(1.5) if and only if and is a solution of the operator equation and is completely continuous.

Corresponding to Lemma 2.5,
(2.35)

Now, we prove that all conditions of Lemma 2.5 hold. One sees that . The remainder is divided into four steps.

Step 1.

Prove that .

For , we have . Then for and for all . So implies that
(2.36)
It follows from Lemma 2.9 that . Then Lemma 2.9 implies that
(2.37)
On the other hand, similarly to above discussion, we have from Lemma 2.9 that
(2.38)

It follows that . Then .

Step 2.

Prove that
(2.39)

and for every .

Choose for all . Then and
(2.40)

It follows that .

For , one has that
(2.41)
Then
(2.42)
Thus implies that
(2.43)
Since
(2.44)
we get from Lemma 2.9 that
(2.45)

This completes Step 2.

Step 3.

Prove that and
(2.46)
Choose . Then , and
(2.47)

It follows that .

For , one has that
(2.48)
Hence we get that
(2.49)
Then implies that
(2.50)
So
(2.51)

This completes Step 3.

Step 4.

Prove that for with .

For with , we have that and and . Then
(2.52)

This completes Step 4.

Step 5.

Prove that for each with

For with , we have and and . Then
(2.53)

This completes Step 5.

Then Lemma 2.5 implies that has at least three fixed points , , and in such that
(2.54)
Hence BVP(1.5) has three increasing positive solutions , and such that
(2.55)
Hence
(2.56)

The proof is complete.

3. Examples

Now, we present one example, whose three solutions cannot be obtained by theorems in known papers, to illustrate the main results.

Example 3.1.

Consider the following BVP:
(3.1)

Corresponding to BVP(1.5), one sees that , . It is easy to see that , choose , then .

Choose , then , choose and and are given by
(3.2)
such that
(3.3)
If
(3.4)
let
(3.5)

then

for all ;

for all ;

for all ;

then Theorem 2.10 implies that BVP(3.1) has at least three decreasing and positive solutions such that
(3.6)

Declarations

Acknowledgments

The author is grateful to an anonymous referee for detailed reading and constructive comments which make the presentation of the results readable. This work is supported by Science Foundation of Hunan Educational Committee (08C) and the Natural Science Foundation of Hunan Province, China (no.06JJ5008).

Authors’ Affiliations

(1)
Department of Mathematics, Shaoyang University

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Copyright

© Xingyuan Liu. 2008

This article is published under license to BioMed Central Ltd. 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.