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Existence Results for HigherOrder Boundary Value Problems on Time Scales
Advances in Difference Equations volumeÂ 2009, ArticleÂ number:Â 209707 (2009)
Abstract
By using the fixedpoint index theorem, we consider the existence of positive solutions for the following nonlinear higherorder fourpoint singular boundary value problem on time scales , ; , ; , ; , , where , , , , , , , and is rdcontinuous.
1. Introduction
Time scales and timescale 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., [3â€“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 AveryHenderson fixedpoint theorem.
In [4], by using the GaticaOlikerWaltman fixedpoint theorem, DaCunha, Davis, and Singh proved the existence of a positive solution for the threepoint 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 higherorder selfadjoint boundary value problem on time scales:
and got many excellent results.
In related papers, Sun [11] considered the following thirdorder twopoint boundary value problem on time scales:
where and . Some existence criteria of solution and positive solution are established by using the LeraySchauder fixed point theorem.
In this paper, we consider the existence of positive solutions for the following higherorder fourpoint singular boundary value problem (BVP) on time scales
where , , and is rdcontinuous. 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 fixedpoint 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 timescale 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 rightdense continuous provided that it is continuous at each rightdense point (a point where ) and has a leftsided limit at each leftdense point . The set of rightdense continuous functions on is denoted by . It can be shown that any rightdense 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 timescale 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 Arzelaascoli 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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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 HigherOrder 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