- Research Article
- Open access
- Published:
The Existence of Positive Solutions for Third-Order
-Laplacian
-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales
Advances in Difference Equations volume 2009, Article number: 169321 (2009)
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):

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

satisfying the boundary conditions

or

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

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:

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

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

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

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

If , then we define the nabla integral by

Lemma 2.5.
If condition holds, then for
, the boundary value problem (BVP)

has the unique solution

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)

has the unique solution

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

So,

By boundary value condition , we have

By (2.10) and (2.12) we know

This together with Lemma 2.5 implies that

where . The proof is complete.
Lemma 2.7.
Let condition holds If
and
, then the unique solution
of (2.8) satisfies

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

So we only prove By condition
we have

The proof is completed.
Lemma 2.8.
Let condition hold. If
and
, then the unique positive solution
of (BVP) (2.8) satisfies

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

For all , we have from the concavity of
that

that is,

This together with the boundary condition implies that

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

We define two cones by

where ,
is defined in Lemma 2.8 and

Define the operators and
by setting

where ,

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.
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

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:

Remark 2.12.
By we can know that
,
Lemma 2.13.
If satisfies the following condition :

then

Proof.
For , then from (2.30) we have

So that

Therefore,

This implies that for
. Hence by Lemma 2.10(1) it follows that
.
Lemma 2.14.
If satisfies the following condition:

then

Proof.
Let for
. Then
. We claim that

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

So that

For , then

This together with Lemma 2.11(c) implies that

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

By and Lemma 2.14, we have that

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

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

This means that is a fixed point of operator
.
When condition holds, by
and Lemma 2.13, we have that

By and Lemma 2.14, we have that

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

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

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

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,

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

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

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

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

where ,
,
,
,
.
By computing, we can know ,
,
,
. Obviously,
,
.
Let ,
, then
. We define a sign changing nonlinearity as follows:

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.
References
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.
Agarwal RP, O'Regan D: Nonlinear boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2001,44(4):527–535. 10.1016/S0362-546X(99)00290-4
Atici FM, Guseinov GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002,141(1–2):75–99. 10.1016/S0377-0427(01)00437-X
Sun H-R, Li W-T: Positive solutions for nonlinear three-point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,299(2):508–524. 10.1016/j.jmaa.2004.03.079
Bohner M, Peterso A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
Sun HR, Li WT: Positive solutions for nonlinear -point boundary value problems on time scales. Acta Mathematica Sinica 2006,49(2):369–380.
Sun H-R, Li W-T: Existence theory for positive solutions to one-dimensional -Laplacian boundary value problems on time scales. Journal of Differential Equations 2007,240(2):217–248. 10.1016/j.jde.2007.06.004
Su Y-H, Li W-T, Sun H-R: Triple positive pseudo-symmetric solutions of three-point BVPs for -Laplacian dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,68(6):1442–1452.
He Z: Double positive solutions of three-point boundary value problems for -Laplacian dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005,182(2):304–315. 10.1016/j.cam.2004.12.012
He Z, Jiang X: Triple positive solutions of boundary value problems for -Laplacian dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2006,321(2):911–920. 10.1016/j.jmaa.2005.08.090
Xu FY: Positive solutions for third-order nonlinear -Laplacian -point boundary value problems on time scales. Discrete Dynamics in Nature and Society 2008, 2008:-16.
Su YH, Li S, Huang C: Positive solution to a singular -Laplacian BVPs with sign-changing nonlinearity involving derivative on time scales. Advances in Difference Equations 2009, 2009:-21.
Su YH, Li WT: Existence of positive solutions to a singular -Laplacian dynamic equations with sign changing nonlinearity. Acta Mathematica Scientia 2009, 52: 181–196.
Xu FY: Positive solutions for multipoint boundary value problems with one-dimensional -Laplacian operator. Applied Mathematics and Computation 2007,194(2):366–380. 10.1016/j.amc.2007.04.118
Su YH, Li WT: Existence of positive solutions to a singular -Laplacian dynamic equations with sign changing nonlinearity. Acta Mathematica Scientia 2008, 28: 51–60.
Su Y-H: Multiple positive pseudo-symmetric solutions of -Laplacian dynamic equations on time scales. Mathematical and Computer Modelling 2009,49(7–8):1664–1681. 10.1016/j.mcm.2008.10.010
Su Y-H, Li W-T, Sun H-R: Positive solutions of singular -Laplacian BVPs with sign changing nonlinearity on time scales. Mathematical and Computer Modelling 2008,48(5–6):845–858. 10.1016/j.mcm.2007.11.008
Zhou C, Ma D: Existence and iteration of positive solutions for a generalized right-focal boundary value problem with -Laplacian operator. Journal of Mathematical Analysis and Applications 2006,324(1):409–424. 10.1016/j.jmaa.2005.10.086
Anderson DR: Green's function for a third-order generalized right focal problem. Journal of Mathematical Analysis and Applications 2003,288(1):1–14. 10.1016/S0022-247X(03)00132-X
Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. Journal of the London Mathematical Society 2001,63(3):690–704. 10.1112/S002461070100206X
Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.
Acknowledgment
This project was supported by the National Natural Science Foundation of China (10471075, 10771117).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Xu, F., Meng, Z. The Existence of Positive Solutions for Third-Order -Laplacian
-Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales.
Adv Differ Equ 2009, 169321 (2009). https://doi.org/10.1155/2009/169321
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/169321