# On the Existence of Solutions for Dynamic Boundary Value Problems under Barrier Strips Condition

- Hua Luo
^{1}Email author and - Yulian An
^{2}

**2011**:378686

https://doi.org/10.1155/2011/378686

© Hua Luo and Yulian An. 2011

**Received: **24 November 2010

**Accepted: **20 January 2011

**Published: **14 February 2011

## Abstract

By defining a new terminology, scatter degree, as the supremum of graininess functional value, this paper studies the existence of solutions for a nonlinear two-point dynamic boundary value problem on time scales. We do not need any growth restrictions on nonlinear term of dynamic equation besides a barrier strips condition. The main tool in this paper is the induction principle on time scales.

## Keywords

## 1. Introduction

Calculus on time scales, which unify continuous and discrete analysis, is now still an active area of research. We refer the reader to [1–5] and the references therein for introduction on this theory. In recent years, there has been much attention focused on the existence and multiplicity of solutions or positive solutions for dynamic boundary value problems on time scales. See [6–17] for some of them. Under various growth restrictions on nonlinear term of dynamic equation, many authors have obtained many excellent results for the above problem by using Topological degree theory, fixed-point theorems on cone, bifurcation theory, and so on.

under a barrier strips condition. A barrier strip is defined as follows. There are pairs (two or four) of suitable constants such that nonlinear term does not change its sign on sets of the form , where is a nonnegative constant, and is a closed interval bounded by some pairs of constants, mentioned above.

where is a bounded time scale with , , and . We obtain the existence of at least one solution to problem (1.2) without any growth restrictions on but an existence assumption of barrier strips. Our proof is based upon the well-known Leray-Schauder principle and the induction principle on time scales.

The time scale-related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. Here, in order to make this paper read easily, we recall some necessary definitions here.

Definition 1.1.

for all . The function is called -differentiable on if exists for all .

Definition 1.2.

Lemma 1.3 (see [2, Theorem 1.16 (*SUF*)]).

Lemma 1.4 (see [18, Lemma 3.2]).

Suppose that is -differentiable on , then

(i) is nondecreasing on if and only if ,

(ii) is nonincreasing on if and only if .

Lemma 1.5 (see [4, Theorem 1.4]).

Let be a time scale with . Then the induction principle holds.

Assume that, for a family of statements , , the following conditions are satisfied.

(3)For each with , there is a neighborhood of such that for all , .

(4)For each with , one has for all .

Remark 1.6.

For , we replace with and with , substitute < for >, then the dual version of the above induction principle is also true.

where , , . According to the well-known Leray-Schauder degree theory, we can get the following theorem.

Lemma 1.7.

Then the boundary value problem (1.2) has at least one solution in .

Proof.

The proof is the same as [18, Theorem 4.1].

## 2. Existence Theorem

To state our main result, we introduce the definition of *scatter degree*.

Definition 2.1.

respectively. If , then we call (or ) the scatter degree on .

- (1)

Theorem 2.3.

Let be continuous. Suppose that there are constants , , with , satisfying

Then problem (1.2) has at least one solution in .

Remark 2.4.

Theorem 2.3 extends [19, Theorem 3.2] even in the special case . Moreover, our method to prove Theorem 2.3 is different from that of [19].

Remark 2.5.

where is bounded everywhere and continuous.

It implies that there exist constants
,
, satisfying (*H1*) and (*H2*) in Theorem 2.3. Thus, problem (2.3) has at least one solution in
.

Proof of Theorem 2.3.

We firstly prove that there exists , independent of and , such that .

Let , . We employ the induction principle on time scales (Lemma 1.5) to show that holds step by step.

Case 1.

Case 2.

Case 3.

If , similar to Case 2, then holds.

This contradiction shows that . In the same way, we claim that .

According to (2.22) and Lemma 1.7, the dynamic boundary value problem (1.2) has at least one solution in .

## 3. An Additional Result

Parallel to the definition of delta derivative, the notion of nabla derivative was introduced, and the main relations between the two operations were studied in [7]. Applying to the dual version of the induction principle on time scales (Remark 1.6), we can obtain the following result.

Theorem 3.1.

Let be continuous. Suppose that there are constants , , with , satisfying

has at least one solution.

Remark 3.2.

has at least one solution. Here is bounded everywhere and continuous.

## Declarations

### Acknowledgments

H. Luo was supported by China Postdoctoral Fund (no. 20100481239), the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), and Innovation Method Fund of China (no. 2009IM010400-1-39). Y. An was supported by 11YZ225 and YJ2009-16 (A06/1020K096019).

## Authors’ Affiliations

## References

- Agarwal RP, Bohner M:
**Basic calculus on time scales and some of its applications.***Results in Mathematics*1999,**35**(1-2):3-22.MathSciNetView ArticleMATHGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar - Bohner M, Peterson A (Eds):
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Hilger S:
**Analysis on measure chains—a unified approach to continuous and discrete calculus.***Results in Mathematics*1990,**18**(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar - Kaymakcalan B, Lakshmikantham V, Sivasundaram S:
*Dynamic Systems on Measure Chains, Mathematics and Its Applications*.*Volume 370*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.MATHGoogle Scholar - Agarwal RP, O'Regan D:
**Triple solutions to boundary value problems on time scales.***Applied Mathematics Letters*2000,**13**(4):7-11. 10.1016/S0893-9659(99)00200-1MathSciNetView ArticleMATHGoogle Scholar - Atici FM, Guseinov GS:
**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. Special issue on "Dynamic equations on time scales", edited by R. P. Agarwal, M. Bohner and D. O'Regan 10.1016/S0377-0427(01)00437-XMathSciNetView ArticleMATHGoogle Scholar - Bohner M, Luo H:
**Singular second-order multipoint dynamic boundary value problems with mixed derivatives.***Advances in Difference Equations*2006,**2006:**-15.Google Scholar - Chyan CJ, Henderson J:
**Twin solutions of boundary value problems for differential equations on measure chains.***Journal of Computational and Applied Mathematics*2002,**141**(1-2):123-131. Special issue on "Dynamic equations on time scales", edited by R. P. Agarwal, M. Bohner and D. O'Regan 10.1016/S0377-0427(01)00440-XMathSciNetView ArticleMATHGoogle Scholar - Erbe L, Peterson A, Mathsen R:
**Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain.***Journal of Computational and Applied Mathematics*2000,**113**(1-2):365-380. 10.1016/S0377-0427(99)00267-8MathSciNetView ArticleMATHGoogle Scholar - Gao C, Luo H:
**Positive solutions to nonlinear first-order nonlocal BVPs with parameter on time scales.***Boundary Value Problems*2011,**2011:**-15.Google Scholar - Henderson J:
**Multiple solutions for****order Sturm-Liouville boundary value problems on a measure chain.***Journal of Difference Equations and Applications*2000,**6**(4):417-429. 10.1080/10236190008808238MathSciNetView ArticleMATHGoogle Scholar - Li W-T, Sun H-R:
**Multiple positive solutions for nonlinear dynamical systems on a measure chain.***Journal of Computational and Applied Mathematics*2004,**162**(2):421-430. 10.1016/j.cam.2003.08.032MathSciNetView ArticleMATHGoogle Scholar - Luo H, Ma R:
**Nodal solutions to nonlinear eigenvalue problems on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(4):773-784. 10.1016/j.na.2005.09.043MathSciNetView ArticleMATHGoogle Scholar - Sun H-R:
**Triple positive solutions for****-Laplacian****-point boundary value problem on time scales.***Computers & Mathematics with Applications*2009,**58**(9):1736-1741. 10.1016/j.camwa.2009.07.083MathSciNetView ArticleMATHGoogle Scholar - Sun J-P, Li W-T:
**Existence and nonexistence of positive solutions for second-order time scale systems.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(10):3107-3114. 10.1016/j.na.2007.03.003MathSciNetView ArticleMATHGoogle Scholar - Wang D-B, Sun J-P, Guan W:
**Multiple positive solutions for functional dynamic equations on time scales.***Computers & Mathematics with Applications*2010,**59**(4):1433-1440. 10.1016/j.camwa.2009.12.019MathSciNetView ArticleMATHGoogle Scholar - Ma R, Luo H:
**Existence of solutions for a two-point boundary value problem on time scales.***Applied Mathematics and Computation*2004,**150**(1):139-147. 10.1016/S0096-3003(03)00204-2MathSciNetView ArticleMATHGoogle Scholar - Kelevedjiev P:
**Existence of solutions for two-point boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*1994,**22**(2):217-224. 10.1016/0362-546X(94)90035-3MathSciNetView ArticleMATHGoogle 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.