# Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales

- Liang-Gen Hu
^{1}, - Ti-Jun Xiao
^{2}and - Jin Liang
^{3}Email author

**Received: **21 March 2009

**Accepted: **1 July 2009

**Published: **16 August 2009

## Abstract

We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.

## Keywords

## 1. Introduction

where , , , , , and . The functional is continuous and is continuous. Our nonlinearity may have singularity at and/or and may have singularity at .

To understand the notations used in (1.1), we recall the following definitions which can be found in [1, 2].

(supplemented by and ) are well defined. The point is left-dense, left-scattered, right-dense, right-scattered if , , , , respectively. If has a left-scattered maximum (right-scattered minimum ), define ( ); otherwise, set ( ). By an interval we always mean the intersection of the real interval with the given time scale, that is, . Other types of intervals are defined similarly.

Theoretically, dynamic equations on time scales can build bridges between continuous and discrete mathematics. Practically, dynamic equations have been proposed as models in the study of insect population models, neural networks, and many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical of biological problems [2]. Hence, two-point and multipoint boundary value problems for dynamic equations on time scales have attracted many researchers' attention (see, e.g., [1–19] and references therein). Moreover, singular boundary value problems have also been treated in many papers (see, e.g., [4, 5, 12–14, 18] and references therein).

and obtained the existence of positive solutions by using a fixed point theorem due to Gatica et al. [14], where is decreasing in for every and may have singularity at .

where are fixed integers satisfying , . They obtained some existence theorems of positive solutions by using Krasnosel'skii fixed point theorem.

Recently, Anderson and Karaca [8] studied higher-order three-point boundary value problems on time scales and obtained criteria for the existence of positive solutions.

The purpose of this paper is to investigate further the singular BVP for delay higher-order dynamic equation (1.1). By the use of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to the BVP (1.1) are established.

The paper is organized as follows. In Section 2, we give some lemmas, which will be required in the proof of our main theorem. In Section 3, we prove some theorems on the existence of positive solutions for BVP (1.1). Moreover, we give an example to illustrate our main result.

## 2. Lemmas

Throughout the paper, we assume that .

The following four lemmas can be found in [8].

Lemma 2.1.

Lemma 2.2.

Remark 2.3.

Lemma 2.4.

Remark 2.5.

Remark 2.6.

Lemma 2.7.

Lemma 2.8.

Proof.

So holds. Thus is true by induction.

Lemma 2.9 (see [20]).

Let be a real Banach space and a cone. Assume that is completely continuous operator such that

## 3. Main Results

Assume that

We seek positive solutions , satisfying (1.1). For this end, we transform (1.1) into an integral equation involving the appropriate Green function and seek fixed points of the following integral operator.

Proposition 3.1.

Let (C1), (C2), and (C3) hold, and let , be fixed constants with . Then is completely continuous.

Proof.

We separate the proof into four steps.

Step 1.

Consequently, is bounded and well defined.

Step 2.

Step 3.

Step 4.

From (C1), (C2), and the Lebesgue dominated convergence theorem [10], we see that the right-hand side (3.19) can be sufficiently small for beingbig enough. Hence the sequence of compact operators converges uniformly to on so that operator is compact. Consequently, is completely continuous by using the Arzela-Ascoli theorem [3].

Proposition 3.2.

It holds that is a solution of (1.1) if and only if .

Proof.

From [8, Lemma 3.1], we know that on . So we conclude that is the solution of BVP (1.1).

From condition (C2) and (3.12), we have .

Theorem 3.3.

Assume that there exist positive constants with , and such that

Proof.

If it is false, then there exists some with , that is, which implies that for .

Clearly, (3.31) contradicts (3.29). This means that (3.28) holds.

Suppose on the contrary that there exists some with for all .

yielding a contradiction with for all . This means that (3.32) holds. Therefore, from (3.28), (3.32) and Lemma 2.9, we conclude that the operator has at least one fixed point . From the definition of the cone and (2.18), we see that for all . Thus, Proposition 3.2 implies that is a solution of BVP (1.1). So we obtain the desired result.

Adopting the same argument as in Theorem 3.3, we obtain the following results.

Corollary 3.4.

Theorem 3.5.

Assume that there exist positive constants with , and , such that

Example 3.6.

## Declarations

### Acknowledgments

The authors would like to thank the referees for helpful comments and suggestions. The work was supported partly by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

## Authors’ Affiliations

## References

- 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-XMATHMathSciNetView ArticleGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Application*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Agarwal RP, Bohner M, Rehák P: Half-linear dynamic equations. In
*Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday. Vol. 1, 2*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:1-57.View ArticleGoogle Scholar - Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions of singular Dirichlet problems on time scales via variational methods.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(2):368-381. 10.1016/j.na.2006.05.014MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations.
*Advances in Difference Equations*2008, 2008:-13.Google Scholar - Ahmad B, Nieto JJ:The monotone iterative technique for three-point second-order integrodifferential boundary value problems with
-Laplacian.
*Boundary Value Problems*2007, 2007:-9.Google Scholar - Anderson DR: Solutions to second-order three-point problems on time scales.
*Journal of Difference Equations and Applications*2002, 8(8):673-688. 10.1080/1023619021000000717MATHMathSciNetView ArticleGoogle Scholar - Anderson DR, Karaca IY: Higher-order three-point boundary value problem on time scales.
*Computers & Mathematics with Applications*2008, 56(9):2429-2443. 10.1016/j.camwa.2008.05.018MATHMathSciNetView ArticleGoogle Scholar - Anderson DR, Smyrlis G: Solvability for a third-order three-point BVP on time scales.
*Mathematical and Computer Modelling*2009, 49(9-10):1994-2001. 10.1016/j.mcm.2008.11.009MATHMathSciNetView ArticleGoogle Scholar - Aulbach B, Neidhart L: Integration on measure chains. In Proceedings of the 6th International Conference on Difference Equations, 2004, Boca Raton, Fla, USA. CRC Press; 239-252.Google Scholar
- Boey KL, Wong PJY: Positive solutions of two-point right focal boundary value problems on time scales.
*Computers & Mathematics with Applications*2006, 52(3-4):555-576. 10.1016/j.camwa.2006.08.025MATHMathSciNetView ArticleGoogle Scholar - Cabada A, Cid JÁ: Existence of a solution for a singular differential equation with nonlinear functional boundary conditions.
*Glasgow Mathematical Journal*2007, 49(2):213-224. 10.1017/S0017089507003679MATHMathSciNetView ArticleGoogle Scholar - DaCunha JJ, Davis JM, Singh PK: Existence results for singular three point boundary value problems on time scales.
*Journal of Mathematical Analysis and Applications*2004, 295(2):378-391. 10.1016/j.jmaa.2004.02.049MATHMathSciNetView ArticleGoogle Scholar - Gatica JA, Oliker V, Waltman P: Singular nonlinear boundary value problems for second-order ordinary differential equations.
*Journal of Differential Equations*1989, 79(1):62-78. 10.1016/0022-0396(89)90113-7MATHMathSciNetView ArticleGoogle Scholar - Henderson J, Tisdell CC, Yin WKC: Uniqueness implies existence for three-point boundary value problems for dynamic equations.
*Applied Mathematics Letters*2004, 17(12):1391-1395. 10.1016/j.am1.2003.08.015MATHMathSciNetView ArticleGoogle Scholar - Kaufmann ER, Raffoul YN: Positive solutions for a nonlinear functional dynamic equation on a time scale.
*Nonlinear Analysis: Theory, Methods & Applications*2005, 62(7):1267-1276. 10.1016/j.na.2005.04.031MATHMathSciNetView ArticleGoogle Scholar - Khan RA, Nieto JJ, Otero-Espinar V: Existence and approximation of solution of three-point boundary value problems on time scales.
*Journal of Difference Equations and Applications*2008, 14(7):723-736. 10.1080/10236190701840906MATHMathSciNetView ArticleGoogle Scholar - Liang J, Xiao T-J, Hao Z-C: Positive solutions of singular differential equations on measure chains.
*Computers & Mathematics with Applications*2005, 49(5-6):651-663. 10.1016/j.camwa.2004.12.001MATHMathSciNetView ArticleGoogle Scholar - Yaslan İ: Multiple positive solutions for nonlinear three-point boundary value problems on time scales.
*Computers & Mathematics with Applications*2008, 55(8):1861-1869. 10.1016/j.camwa.2007.07.005MATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar

## Copyright

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.