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Existence Results for Higher-Order Boundary Value Problems on Time Scales
Advances in Difference Equations volume 2009, Article number: 209707 (2009)
Abstract
By using the fixed-point index theorem, we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales ,
;
,
;
,
;
,
, where
,
,
,
,
,
,
, and
is rd-continuous.
1. Introduction
Time scales and time-scale notationare introduced well in the fundamental texts by Bohner and Peterson [1, 2], respectively, as important corollaries. In, the recent years, many authors have paid much attention to the study of boundary value problems on time scales (see, e.g., [3–17]). In particular, we would like to mention some results of Anderson et al. [3, 5, 6, 14, 16], DaCunha et al. [4], and Agarwal and O'Regan [7], which motivate us to consider our problem.
In [3], Anderson and Karaca discussed the dynamic equation on time scales

and the eigenvalue problem

with the same boundary conditions where is a positive parameter. They obtained some results for the existence of positive solutions by using the Krasnoselskii, the Schauder, and the Avery-Henderson fixed-point theorem.
In [4], by using the Gatica-Oliker-Waltman fixed-point theorem, DaCunha, Davis, and Singh proved the existence of a positive solution for the three-point boundary value problem on a time scale given by

where is fixed, and
is singular at
and possibly at
.
Anderson et al. [5] gave a detailed presentation for the following higher-order self-adjoint boundary value problem on time scales:

and got many excellent results.
In related papers, Sun [11] considered the following third-order two-point boundary value problem on time scales:

where and
. Some existence criteria of solution and positive solution are established by using the Leray-Schauder fixed point theorem.
In this paper, we consider the existence of positive solutions for the following higher-order four-point singular boundary value problem (BVP) on time scales


where ,
, and
is rd-continuous. In the rest of the paper, we make the following assumptions:
()
().
In this paper, by constructing one integral equation which is equivalent to the BVP (1.6) and (1.7), we study the existence of positive solutions. Our main tool of this paper is the following fixed-point index theorem.
Theorem 1.1 ([18]).
Suppose is a real Banach space,
is a cone, let
. Let operator
be completely continuous and satisfy
. Then
(i)if , then
(ii)if , then
.
The outline of the paper is as follows. In Section 2, for the convenience of the reader we give some definitions and theorems which can be found in the references, and we present some lemmas in order to prove our main results. Section 3 is developed in order to present and prove our main results. In Section 4 we present some examples to illustrate our results.
2. Preliminaries and Lemmas
For convenience, we list the following definitions which can be found in [1, 2, 9, 14, 17]. 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, and 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
. In this general time-scale setting,
represents the delta (or Hilger) derivative [13, Definition 1.10],

where is the forward jump operator,
is the forward graininess function, and
is abbreviated as
. In particular, if
, then
and
, while if
for any
, then
and

A function is right-dense continuous provided that it is continuous at each right-dense point
(a point where
) and has a left-sided limit at each left-dense point
. The set of right-dense continuous functions on
is denoted by
. It can be shown that any right-dense continuous function
has an antiderivative (a function
with the property
for all
). Then the Cauchy delta integral of
is defined by

where is an antiderivative of
on
. For example, if
, then

and if , then

Throughout we assume that are points in
, and define the time-scale interval
. In this paper, we also need the the following theorem which can be found in [1].
Theorem 2.1.
If and
then

In this paper, let

Then is a Banach space with the norm
. Define a cone
by

Obviously, is a cone in
. Set
. If
on
then we say
is concave on
We can get the following.
Lemma 2.2.
Suppose condition holds. Then there exists a constant
satisfies

Furthermore, the function

is a positive continuous function on , therefore
has minimum on
. Then there exists
such that
.
Lemma 2.3.
Let and
in Lemma 2.2. Then

Proof.
Suppose .
We will discuss it from three perspectives.
(i). It follows from the concavity of
that

then

which means .
(ii). If
, we have

then

If , we have

then

and this means .
(iii). Similarly, we have

then

which means .
From the above, we know . The proof is complete.
Lemma 2.4.
Suppose that conditions hold, then
is a solution of boundary value problem (1.6), (1.7) if and only if
is a solution of the following integral equation:

where

Proof.
Necessity. By the equation of the boundary condition, we see that , then there exists a constant
such that
. Firstly, by delta integrating the equation of the problems (1.6) on
, we have

thus

By and the boundary condition (1.7), let
on (2.23), we have

By the equation of the boundary condition (1.7), we get

then

Secondly, by (2.24) and let on (2.24), we have

Then

Then by delta integrating (2.29) for times on
, we have

Similarly, for , by delta integrating the equation of problems (1.6) on
, we have

Therefore, for any ,
can be expressed as the equation

where is expressed as (2.22).
Sufficiency. Suppose that

then by (2.22), we have

So,

which imply that (1.6) holds. Furthermore, by letting and
on (2.22) and (2.34), we can obtain the boundary value equations of (1.7). The proof is complete.
Now, we define a mapping given by

where is given by (2.22).
Lemma 2.5.
Suppose that conditions hold, the solution
of problem (1.6), (1.7) satisfies

and for in Lemma 2.2, one has

Proof.
If is the solution of (1.6), (1.7), then
is a concave function, and
, thus we have

that is,

By Lemma 2.3, for , we have

then . The proof is complete.
Lemma 2.6.
is completely continuous.
Proof.
Because

is continuous, decreasing on and satisfies
. Then,
for each
and
. This shows that
. Furthermore, it is easy to check that
is completely continuous by Arzela-ascoli Theorem.
For convenience, we set

where is the constant from Lemma 2.2. By Lemma 2.5, we can also set

3. The Existence of Positive Solution
Theorem 3.1.
Suppose that conditions (), (
) hold. Assume that
also satisfies


where
Then, the boundary value problem (1.6), (1.7) has a solution such that
lies between
and
.
Theorem 3.2.
Suppose that conditions (), (
) hold. Assume that
also satisfies


Then, the boundary value problem (1.6), (1.7) has a solution such that
lies between
and
.
Theorem 3.3.
Suppose that conditions (), (
) hold. Assume that
also satisfies
()
()
Then, the boundary value problem (1.6), (1.7) has a solution such that
lies between
and
.
Proof of Theorem 3.1.
Without loss of generality, we suppose that . For any
, by Lemma 2.3, we have

We define two open subsets and
of
:

For any , by (3.1) we have

For and
, we will discuss it from three perspectives.
(i)If , thus for
, by (
) and Lemma 2.4, we have

-
(ii)
If
, thus for
, by (
) and Lemma 2.4, we have

-
(iii)
If
, thus for
, by (
) and Lemma 2.4, we have

Therefore, no matter under which condition, we all have

Then by Theorem 2.1, we have

On the other hand, for , we have
; by (
) we know

thus

Then, by Theorem 2.1, we get

Therefore, by (3.8), (3.11), , we have

Then operator has a fixed point
, and
. Then the proof of Theorem 3.1 is complete .
Proof of Theorem 3.2.
First, by , for
, there exists an adequately small positive number
, as
, we have

Then let , thus by (3.13)

So condition () holds. Next, by condition (
),
, then for
, there exists an appropriately big positive number
, as
, we have

Let , thus by (3.15), condition (
) holds. Therefore by Theorem 3.1 we know that the results of Theorem 3.2 hold. The proof of Theorem 3.2 is complete.
Proof of Theorem 3.3.
Firstly, by condition (),
, then for
, there exists an adequately small positive number
, as
, we have

thus when , we have

Let , so by (3.17), condition (
) holds.
Secondly, by condition (),
, then for
, there exists a suitably big positive number
, as
, we have

If is unbounded, by the continuity of
on
, then there exist a constant
, and a point
such that

Thus, by we know

Choose . Then, we have

If is bounded, we suppose
, there exists an appropriately big positive number
, then choose
, we have

Therefore, condition () holds. Thus, by Theorem 3.1, we know that the result of Theorem 3.3 holds. The proof of Theorem 3.3 is complete.
4. Application
In this section, in order to illustrate our results, we consider the following examples.
Example 4.1.
Consider the following boundary value problem on the specific time scale:

where

and is the constant defined in Lemma 2.2,

Then obviously

By Theorem 2.1, we have

so conditions (),
hold.
By simple calculations, we have

then , that is,
, so condition
holds.
For , it is easy to see that

so condition holds. Then by Theorem 3.2, BVP (4.1) has at least one positive solution.
Example 4.2.
Consider the following boundary value problem on the specific time scale .

where

and is the constant from Lemma 2.2,

Then obviously

By Theorem 2.1, we have

so conditions (),
hold. By simple calculations, we have

then , that is,
, so condition
holds.
For , it is easy to see that

then condition holds. Thus by Theorem 3.3, BVP (4.8) has at least one positive solution.
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Acknowledgment
The authors would like to thank the anonymous referee for his/her valuable suggestions, which have greatly improved this paper.
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Liu, J., Sang, Y. Existence Results for Higher-Order Boundary Value Problems on Time Scales. Adv Differ Equ 2009, 209707 (2009). https://doi.org/10.1155/2009/209707
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DOI: https://doi.org/10.1155/2009/209707
Keywords
- Boundary Value Problem
- Small Positive Number
- Singular Boundary
- Jump Operator
- Nonempty Closed Subset