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Existence of Positive Solutions for Semipositone HigherOrder BVPS on Time Scales
Advances in Difference Equations volume 2010, Article number: 235296 (2010)
Abstract
We offer conditions on semipositone function such that the boundary value problem, , , , , , , , has at least one positive solution, where is a time scale and is continuous with for some positive constant .
1. Introduction
Throughout this paper, let be a time scale, for any , the interval defined as . Analogous notations for open and halfopen intervals will also be used in the paper. We also use the notation to denote the real interval . To understand further knowledge about dynamic equations on time scales, the reader may refer to [1–3] for an introduction to the subject.
In this paper, we present results governing the existence of positive solutions to the differential equation on time scales of the form
subject to the twopoint boundary conditions
where and
Throughout, we assume that is continuous with for some positive constant .
Let denote the space of functions
We say that is a positive solution of BVP (1.1) and (1.2), if is a solution of BVP (1.1) and (1.2) and .
Various cases of BVP (1.1) and (1.2) have attracted a lot of attention in the literature. When , BVP (1.1) and (1.2) has been studied by many specialists. For example, Agarwal et al. [4] have established the existence of positive solutions for continuous case of the semipositone SturmLiouville BVPs. Erbe and Peterson [5] andHao et al.[6]dealt with SturmLiouville BVPs on time scale of positone nonlinear term. In addition, Agarwal and O'Regan [7] obtained positive solution of secondorder right focal BVPs on time scale by using nonlinear alternative of LeraySchauder type. In 2005, Chyan and Wong [8] obtained triple solutions of the same BVPs with [7]. Recently,Sun and Li[9, 10] investigated semipositone Dirichlet BVPs on time scale. For higherorder BVPs, continuous case of BVP (1.1) and (1.2) have been investigated by Agarwal and Wong [11], Wong and Agarwal [12] and Wong [13]. The discrete positone case of BVP (1.1) and (1.2) has been tackled by using a fixed point theorem for mappings that are decreasing with respect to a cone in [14]. Especially, timescale case of (1.1) with fourpoint boundary condition has been studied by Liu and Sang [15]. Besides, BVP (1.1) and (1.2) of nonlinear positone term which satisfied Nagumotype conditions have been dealt with in [16]. Motivated by the works mentioned above, the purpose of this paper is to tackle semipositone BVP (1.1) and (1.2). In fact, BVPs appeared in [7–14] can be looked at as special case of BVP (1.1) and (1.2) in this paper. For other related works, we also refer to [17–19].
The paper is outlined as follows. In Section 2, we will present some notations and lemmas which will be used later. In Section 3, by using Krasnoselskii's fixed point theorem in a cone, we offer criteria for the existence of positive solution of BVP (1.1) and (1.2).
2. Preliminary
In this section, we offer some notations and lemmas, which will be used in main results. Throughout this paper, we always use the following notations:
(C_{1}) is the Green's function of the differential equation subject to the boundary conditions (1.2);
(C_{2}) is the Green's function of the differential equation subject to the boundary conditions
(C_{3}) Define , as
where
Lemma 2.1.
For the Green's function the following hold:
Proof.
It is clear that
From the expression of , we can easily obtain
Lemma 2.2.
Let be the solution of BVP
Then
where
and is a positive constant.
Proof.
For ,
For ,
So
By defining as , it is clear that
Then
Further, since , we get
Lemma 2.3 (see [20]).
Let be a Banach space, and let be a cone in . Assume that are open subsets of with , and let be a completely continuous operator such that either

(i)
or
 (ii)
Then, has a fixed point in .
3. Main Results
In this section, by using Lemma 2.3, we offer criteria for the existence of positive solution of BVP (1.1) and (1.2).
Let denote the space of functions
Let be a Banach space with the norm , and let
It is obvious that is a cone in . From , it follows that for all ,
where
Throughout the rest of the section, we assume that the set is such that
exist and satisfy
In addition, we denote that
In order to obtain positive solutions of BVP (1.1) and (1.2), we need to consider the following boundary value problem:
where
and for all ,
Let the operator be defined by
Lemma 3.1.
The operator maps into .
Proof.
From Lemma 2.1, we know that for ,
So for ,
From Lemma 2.1 again, it follows that for ,
Hence, maps into .
Lemma 3.2.
The operator is completely continuous.
Proof.
First we shall prove that the operator is continuous. Let , be such that . From , we have
Then, it is easy to see that as
Hence, we get from Lemma 2.1 that for ,
This shows that is continuous.
Next, to show complete continuity, we will apply ArzelaAscoli theorem. Let be a bounded subset of . Then there exists such that for all ,
where is given in (3.4). Let
Clearly, we have for ,
and for ,
The ArzelaAscoli theorem guarantees that is relatively compact, so is completely continuous.
Theorem 3.3.
Assume that the following conditions hold:

(i)
there exist such that for any ,
(3.22)

(ii)
with such that for any ,
(3.23)where is given in (3.4), are given in (3.5), are given in (3.7), and
(3.24)
Then BVP (1.1) and (1.2) has a positive solution.
Proof.
Without loss of generality, we assume that . Now we seek positive solutions of BVP (3.8). Let
For , it follows from (3.3) that
From , we obtain that for ,
So
Let
For , it follows from Lemma 2.2 and (3.3) that for ,
So
where is given in (3.7) and is given in (3.24).
Combining Lemma 2.1, (3.3), and with (3.31), we obtain that for ,
So
Therefore, it follows from Lemma 2.3 that BVP (3.8) has a solution such that .
Finally, we will prove that is a positive solution of BVP (1.1) and (1.2). Let , then we have from Lemma 2.2 and (3.3) that for ,
In addition,
So, is a positive solution of BVP (1.1) and (1.2). This completes the proof.
Corollary 3.4.
Assume that

(a)
for any ,
(3.36)where is a continuous function which is nondecreasing in for each fixed and is a continuous nonnegative function on ,

(b)
for any ,
(3.37)where is a continuous function which is nondecreasing in for each fixed and is a continuous nonnegative function on ,

(c)
there exists such that
(3.38) 
(d)
there exists with such that
(3.39)
Then BVP (1.1) and (1.2) has a positive solution.
Proof.
From and , we obtain that for ,
So, condition of Theorem 3.3 is satisfied. From and , we obtain that for ,
So, condition ii of Theorem 3.3 is satisfied.
Therefore, from Theorem 3.3, BVP(1.1) and (1.2) has a positive solution.
Corollary 3.5.
Assume that conditions and of Corollary 3.4 and the following condition hold:
where . Then BVP (1.1) and (1.2) has one positive solution.
Proof.
We only need to prove that (3.42) implies condition of Theorem 3.3. From (3.42), we know that there exists ( may be chosen arbitrary large) such that for ,
Hence, for ,
Thus, it follows that
So, condition of Theorem 3.3 is satisfied.
Finally we present an example to illustrate our result.
Example 3.6.
Considerthe following boundary value problem:
where , , , . Obviously,
Let , and . So conditions and in Corollary 3.4 are satisfied. By direct calculation, we obtain that = . Since are nondecreasing, , . In addition, . Take . So
Hence, conditions and in Corollary 3.4 are satisfied. Therefore from Corollary 3.4, (3.46) has at least one positive solution.
Remark 3.7.
In Example 3.6, because nonlinear term may attain negative value, the result in [15] is not applicable.
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Acknowledgment
The authors thank the referee for valuable suggestions which led to improvement of the original manuscript.
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Keywords
 Banach Space
 Nonlinear Term
 Fixed Point Theorem
 Continuous Case
 Real Interval