Open Access

Existence of Positive Solutions for Semipositone Higher-Order BVPS on Time Scales

Advances in Difference Equations20102010:235296

https://doi.org/10.1155/2010/235296

Received: 4 December 2009

Accepted: 29 March 2010

Published: 10 May 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 half-open 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 [13] 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
(1.1)
subject to the two-point boundary conditions
(1.2)
where and
(1.3)

Throughout, we assume that is continuous with for some positive constant .

Let denote the space of functions
(1.4)

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 Sturm-Liouville BVPs. Erbe and Peterson [5] andHao et al.[6]dealt with Sturm-Liouville BVPs on time scale of positone nonlinear term. In addition, Agarwal and O'Regan [7] obtained positive solution of second-order right focal BVPs on time scale by using nonlinear alternative of Leray-Schauder 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 higher-order 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, time-scale case of (1.1) with four-point boundary condition has been studied by Liu and Sang [15]. Besides, BVP (1.1) and (1.2) of nonlinear positone term which satisfied Nagumo-type 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 [714] 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 [1719].

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:

(C1) is the Green's function of the differential equation subject to the boundary conditions (1.2);

(C2) is the Green's function of the differential equation subject to the boundary conditions
(2.1)
(C3) Define , as
(2.2)
where
(2.3)

Lemma 2.1.

For the Green's function the following hold:
(2.4)

Proof.

It is clear that
(2.5)
From the expression of , we can easily obtain
(2.6)

Lemma 2.2.

Let be the solution of BVP
(2.7)
Then
(2.8)
where
(2.9)

and is a positive constant.

Proof.

For ,
(2.10)
For ,
(2.11)
So
(2.12)
By defining as , it is clear that
(2.13)
Then
(2.14)
Further, since , we get
(2.15)

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
  1. (i)

    or

     
  2. (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
(3.1)
Let be a Banach space with the norm , and let
(3.2)
It is obvious that is a cone in . From , it follows that for all ,
(3.3)
where
(3.4)
Throughout the rest of the section, we assume that the set is such that
(3.5)
exist and satisfy
(3.6)
In addition, we denote that
(3.7)
In order to obtain positive solutions of BVP (1.1) and (1.2), we need to consider the following boundary value problem:
(3.8)
where
(3.9)
and for all ,
(3.10)
Let the operator be defined by
(3.11)

Lemma 3.1.

The operator maps into .

Proof.

From Lemma 2.1, we know that for ,
(3.12)
So for ,
(3.13)
From Lemma 2.1 again, it follows that for ,
(3.14)

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
(3.15)
Then, it is easy to see that as
(3.16)
Hence, we get from Lemma 2.1 that for ,
(3.17)

This shows that is continuous.

Next, to show complete continuity, we will apply Arzela-Ascoli theorem. Let be a bounded subset of . Then there exists such that for all ,
(3.18)
where is given in (3.4). Let
(3.19)
Clearly, we have for ,
(3.20)
and for ,
(3.21)

The Arzela-Ascoli theorem guarantees that is relatively compact, so is completely continuous.

Theorem 3.3.

Assume that the following conditions hold:
  1. (i)
    there exist such that for any ,
    (3.22)
     
  1. (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
(3.25)
For , it follows from (3.3) that
(3.26)
From , we obtain that for ,
(3.27)
So
(3.28)
Let
(3.29)
For , it follows from Lemma 2.2 and (3.3) that for ,
(3.30)
So
(3.31)

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 ,
(3.32)
So
(3.33)

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 ,
(3.34)
In addition,
(3.35)

So, is a positive solution of BVP (1.1) and (1.2). This completes the proof.

Corollary 3.4.

Assume that
  1. (a)
    for any ,
    (3.36)

    where is a continuous function which is nondecreasing in for each fixed and is a continuous nonnegative function on ,

     
  2. (b)
    for any ,
    (3.37)

    where is a continuous function which is nondecreasing in for each fixed and is a continuous nonnegative function on ,

     
  3. (c)
    there exists such that
    (3.38)
     
  4. (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 ,
(3.40)
So, condition of Theorem 3.3 is satisfied. From and , we obtain that for ,
(3.41)

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:
(3.42)

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 ,
(3.43)
Hence, for ,
(3.44)
Thus, it follows that
(3.45)

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:
(3.46)
where , , , . Obviously,
(3.47)
Let , and . So conditions and in Corollary 3.4 are satisfied. By direct calculation, we obtain that = . Since are nondecreasing, , . In addition, . Take . So
(3.48)

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.

Declarations

Acknowledgment

The authors thank the referee for valuable suggestions which led to improvement of the original manuscript.

Authors’ Affiliations

(1)
Department of Mathematics, Beihua University

References

  1. Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics. Resultate der Mathematik 1999,35(1-2):3-22.MATHMathSciNetView ArticleGoogle Scholar
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar
  3. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHView ArticleGoogle Scholar
  4. Agarwal RP, Hong H-L, Yeh C-C: The existence of positive solutions for the Sturm-Liouville boundary value problems. Computers & Mathematics with Applications 1998,35(9):89-96. 10.1016/S0898-1221(98)00060-1MATHMathSciNetView ArticleGoogle Scholar
  5. Erbe L, Peterson A: Positive solutions for a nonlinear differential equation on a measure chain. Mathematical and Computer Modelling 2000,32(5-6):571-585. 10.1016/S0895-7177(00)00154-0MATHMathSciNetView ArticleGoogle Scholar
  6. Hao Z-C, Xiao T-J, Liang J: Existence of positive solutions for singular boundary value problem on time scales. Journal of Mathematical Analysis and Applications 2007,325(1):517-528. 10.1016/j.jmaa.2006.01.083MATHMathSciNetView ArticleGoogle Scholar
  7. 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)00290-4MATHMathSciNetView ArticleGoogle Scholar
  8. Chyan CJ, Wong PJY: Triple solutions of focal boundary value problems on time scale. Computers & Mathematics with Applications 2005,49(7-8):963-979. 10.1016/j.camwa.2005.01.015MATHMathSciNetView ArticleGoogle Scholar
  9. Sun J-P, Li W-T: Existence of positive solutions to semipositone Dirichlet BVPs on time scales. Dynamic Systems and Applications 2007,16(3):571-578.MATHMathSciNetGoogle Scholar
  10. Sun J-P, Li W-T: Solutions and positive solutions to semipositone Dirichlet BVPs on time scales. Dynamic Systems and Applications 2008,17(2):303-311.MATHMathSciNetGoogle Scholar
  11. Agarwal RP, Wong F-H: Existence of positive solutions for non-positive higher-order BVPs. Journal of Computational and Applied Mathematics 1998,88(1):3-14. 10.1016/S0377-0427(97)00211-2MATHMathSciNetView ArticleGoogle Scholar
  12. Wong PJY, Agarwal RP: On eigenvalue intervals and twin eigenfunctions of higher-order boundary value problems. Journal of Computational and Applied Mathematics 1998,88(1):15-43. 10.1016/S0377-0427(97)00202-1MATHMathSciNetView ArticleGoogle Scholar
  13. Wong F-H: An application of Schauder's fixed point theorem with respect to higher order BVPs. Proceedings of the American Mathematical Society 1998,126(8):2389-2397. 10.1090/S0002-9939-98-04709-1MATHMathSciNetView ArticleGoogle Scholar
  14. Wong PJY, Agarwal RP: On the existence of solutions of singular boundary value problems for higher order difference equations. Nonlinear Analysis: Theory, Methods &Applications 1997,28(2):277-287. 10.1016/0362-546X(95)00151-KMATHMathSciNetView ArticleGoogle Scholar
  15. Liu J, Sang Y: Existence results for higher-order boundary value problems on time scales. Advances in Difference Equations 2009, 2009:-18.Google Scholar
  16. Grossinho MR, Minhós FM: Upper and lower solutions for higher order boundary value problems. Nonlinear Studies 2005,12(2):165-176.MATHMathSciNetGoogle Scholar
  17. Agarwal RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore; 1986:xii+307.MATHView ArticleGoogle Scholar
  18. Lin Y, Pei M: Positive solutions for two-point semipositone right focal eigenvalue problem. Boundary Value Problems 2007, 2007:-12.Google Scholar
  19. Lin Y, Pei M: Positive solutions of two-point right focal eigenvalue problems on time scales. Advances in Difference Equations 2007, 2007:-15.Google Scholar
  20. Guo D, 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.Google Scholar

Copyright

© Y. Lin and M. Pei. 2010

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.